Subgroups — C₆₇×#35149 #30714400

Order 17152.

5314 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#536×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#535×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#525×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2849	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2613	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2432	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2431	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2407	C₂	1	8	yes	maximal, complemented
10	C₆₇×#2310	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2309	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2280	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	—	2	8	no	—
14	C₂²×#41×C₆₇	—	2	8	no	—
15	C₂²×#41×C₆₇	—	2	8	no	—
16	C₂²×#41×C₆₇	C₂²	1	4	yes	—
17	C₂²×#41×C₆₇	C₂²	1	4	yes	—
18	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
21	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
23	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
24	#536×C₆₇	—	2	8	no	—
25	#535×C₆₇	—	2	8	no	—
26	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
27	C₂×#127×C₆₇	C₂²	1	2	yes	complemented
28	C₂×#124×C₆₇	C₂²	1	2	yes	complemented
29	C₂×#124×C₆₇	C₂²	1	2	yes	complemented
30	#525×C₆₇	—	2	8	no	—
31	C₂×#123×C₆₇	C₂²	1	2	yes	—
32	C₂×#123×C₆₇	C₂²	1	4	yes	—
33	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
34	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
35	C₂×#121×C₆₇	C₂²	1	4	yes	complemented
36	C₂×#121×C₆₇	C₂²	1	2	yes	complemented
37	C₂×#120×C₆₇	C₂²	1	4	yes	—
38	C₂×#120×C₆₇	C₂²	1	2	yes	—
39	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
40	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
43	#394×C₆₇	C₂²	1	16	yes	complemented
44	#392×C₆₇	C₂²	1	16	yes	complemented
45	#391×C₆₇	C₂²	1	4	yes	complemented
46	#391×C₆₇	C₂²	1	4	yes	complemented
47	#390×C₆₇	C₂²	1	4	yes	complemented
48	#390×C₆₇	C₂²	1	4	yes	complemented
49	#386×C₆₇	C₂²	1	16	yes	complemented
50	#385×C₆₇	C₂²	1	16	yes	complemented
51	#381×C₆₇	C₂²	1	2	yes	—
52	#381×C₆₇	C₂²	1	4	yes	—
53	#380×C₆₇	C₂²	1	4	yes	complemented
54	#379×C₆₇	C₂²	1	2	yes	—
55	#379×C₆₇	C₂²	1	4	yes	—
56	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
57	C₂⁵×C₆₇	C₂³	1	2	yes	—
58	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×#41×C₆₇	—	2	128	no	—
60	C₂×#41×C₆₇	—	2	16	no	—
61	C₂×#41×C₆₇	—	2	16	no	—
62	C₂×#41×C₆₇	—	2	16	no	—
63	C₂×#41×C₆₇	—	2	16	no	—
64	C₂×#41×C₆₇	—	2	16	no	—
65	C₂×#41×C₆₇	—	2	16	no	—
66	C₂×#41×C₆₇	—	2	4	no	—
67	C₂×#41×C₆₇	—	2	4	no	—
68	C₂×#41×C₆₇	—	2	4	no	—
69	C₂²×Q₈×C₆₇	—	2	4	no	—
70	C₂²×Q₈×C₆₇	—	2	4	no	—
71	C₂²×Q₈×C₆₇	D₄	1	8	yes	complemented
72	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
74	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
75	C₂²×D₄×C₆₇	—	2	8	no	—
76	C₂²×D₄×C₆₇	—	2	8	no	—
77	C₂²×D₄×C₆₇	—	2	8	no	—
78	C₂²×D₄×C₆₇	—	2	4	no	—
79	C₂²×D₄×C₆₇	—	2	4	no	—
80	C₂²×D₄×C₆₇	D₄	1	8	yes	complemented
81	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
82	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
84	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
85	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
86	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
87	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
88	C₂³×C₄×C₆₇	—	2	8	no	—
89	C₂³×C₄×C₆₇	—	2	8	no	—
90	C₂³×C₄×C₆₇	—	2	8	no	—
91	C₂³×C₄×C₆₇	—	2	4	no	—
92	C₂³×C₄×C₆₇	—	2	4	no	—
93	C₂³×C₄×C₆₇	D₄	1	8	yes	complemented
94	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
95	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
96	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
97	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
98	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
99	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
100	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
101	C₄⋊Q₈×C₆₇	—	2	8	no	—
102	#127×C₆₇	—	2	8	no	—
103	#124×C₆₇	—	2	8	no	—
104	#124×C₆₇	—	2	8	no	—
105	#123×C₆₇	—	2	8	no	—
106	#123×C₆₇	—	2	16	no	—
107	C₂²⋊Q₈×C₆₇	—	2	16	no	—
108	C₂²⋊Q₈×C₆₇	—	2	8	no	—
109	#121×C₆₇	—	2	16	no	—
110	#121×C₆₇	—	2	8	no	—
111	#120×C₆₇	—	2	8	no	—
112	#120×C₆₇	—	2	16	no	—
113	C₄×D₄×C₆₇	—	2	16	no	—
114	#117×C₆₇	—	2	4	no	—
115	#117×C₆₇	—	2	4	no	—
116	#117×C₆₇	—	2	4	no	—
117	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
118	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
123	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
125	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
126	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
127	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
128	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
129	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
130	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
132	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
133	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
134	#95×C₆₇	C₂³	1	8	yes	—
135	#95×C₆₇	C₂³	1	16	yes	—
136	C₂⁴×C₆₇	—	2	2	no	—
137	C₂⁴×C₆₇	—	2	4	no	—
138	C₂⁴×C₆₇	—	2	4	no	—
139	C₂⁴×C₆₇	—	2	2	no	—
140	C₂⁴×C₆₇	—	4	4	no	—
141	C₂⁴×C₆₇	—	4	8	no	—
142	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
143	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
144	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
145	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
146	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
147	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
148	#41×C₆₇	—	2	128	no	—
149	#41×C₆₇	—	2	128	no	—
150	#41×C₆₇	—	4	16	no	—
151	C₂×Q₈×C₆₇	—	2	32	no	—
152	C₂×Q₈×C₆₇	—	2	4	no	—
153	C₂×Q₈×C₆₇	—	2	4	no	—
154	C₂×Q₈×C₆₇	—	2	4	no	—
155	C₂×Q₈×C₆₇	—	2	4	no	—
156	C₂×Q₈×C₆₇	—	2	2	no	—
157	C₂×Q₈×C₆₇	—	2	2	no	—
158	C₂×Q₈×C₆₇	—	2	2	no	—
159	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
160	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
161	C₂×D₄×C₆₇	—	2	64	no	—
162	C₂×D₄×C₆₇	—	2	16	no	—
163	C₂×D₄×C₆₇	—	2	16	no	—
164	C₂×D₄×C₆₇	—	2	8	no	—
165	C₂×D₄×C₆₇	—	2	4	no	—
166	C₂×D₄×C₆₇	—	2	4	no	—
167	C₂×D₄×C₆₇	—	2	4	no	—
168	C₂×D₄×C₆₇	—	2	16	no	—
169	C₂×D₄×C₆₇	—	2	32	no	—
170	C₂×D₄×C₆₇	—	2	16	no	—
171	C₂×D₄×C₆₇	—	2	4	no	—
172	C₂×D₄×C₆₇	—	2	4	no	—
173	C₂×D₄×C₆₇	—	2	4	no	—
174	C₂×D₄×C₆₇	—	2	4	no	—
175	C₂×D₄×C₆₇	—	2	2	no	—
176	C₂×D₄×C₆₇	—	2	2	no	—
177	C₂×D₄×C₆₇	—	2	2	no	—
178	C₂×D₄×C₆₇	—	4	8	no	—
179	C₂×D₄×C₆₇	—	4	16	no	—
180	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
181	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
182	C₂²×C₄×C₆₇	—	2	64	no	—
183	C₂²×C₄×C₆₇	—	2	2	no	—
184	C₂²×C₄×C₆₇	—	2	2	no	—
185	C₂²×C₄×C₆₇	—	2	4	no	—
186	C₂²×C₄×C₆₇	—	2	4	no	—
187	C₂²×C₄×C₆₇	—	2	16	no	—
188	C₂²×C₄×C₆₇	—	2	32	no	—
189	C₂²×C₄×C₆₇	—	2	16	no	—
190	C₂²×C₄×C₆₇	—	2	16	no	—
191	C₂²×C₄×C₆₇	—	2	4	no	—
192	C₂²×C₄×C₆₇	—	2	4	no	—
193	C₂²×C₄×C₆₇	—	2	4	no	—
194	C₂²×C₄×C₆₇	—	2	4	no	—
195	C₂²×C₄×C₆₇	—	2	4	no	—
196	C₂²×C₄×C₆₇	—	2	4	no	—
197	C₂²×C₄×C₆₇	—	2	4	no	—
198	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
199	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
200	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
201	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
202	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
203	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
204	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
205	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
206	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
207	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
208	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
210	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
211	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
212	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
213	C₄⋊C₄×C₆₇	—	2	4	no	—
214	C₄⋊C₄×C₆₇	—	2	4	no	—
215	C₄⋊C₄×C₆₇	—	2	4	no	—
216	C₄⋊C₄×C₆₇	—	2	8	no	—
217	C₄⋊C₄×C₆₇	—	2	16	no	—
218	C₂²⋊C₄×C₆₇	—	2	4	no	—
219	C₂²⋊C₄×C₆₇	—	2	4	no	—
220	C₂²⋊C₄×C₆₇	—	2	4	no	—
221	C₂²⋊C₄×C₆₇	—	2	8	no	—
222	C₂²⋊C₄×C₆₇	—	2	16	no	—
223	C₂²⋊C₄×C₆₇	—	2	8	no	—
224	C₂²⋊C₄×C₆₇	—	2	16	no	—
225	C₂²⋊C₄×C₆₇	—	4	16	no	—
226	C₂²⋊C₄×C₆₇	—	4	8	no	—
227	C₄²×C₆₇	—	2	4	no	—
228	C₄²×C₆₇	—	2	4	no	—
229	C₄²×C₆₇	—	2	4	no	—
230	C₂³×C₆₇	—	2	4	no	—
231	C₂³×C₆₇	—	2	16	no	—
232	C₂³×C₆₇	—	2	8	no	—
233	C₂³×C₆₇	—	2	4	no	—
234	C₂³×C₆₇	—	2	2	no	—
235	C₂³×C₆₇	—	2	4	no	—
236	C₂³×C₆₇	—	2	2	no	—
237	C₂³×C₆₇	—	2	4	no	—
238	C₂³×C₆₇	—	2	16	no	—
239	C₂³×C₆₇	—	2	4	no	—
240	C₂³×C₆₇	—	2	2	no	—
241	C₂³×C₆₇	—	2	2	no	—
242	C₂³×C₆₇	—	2	2	no	—
243	C₂³×C₆₇	—	2	8	no	—
244	C₂³×C₆₇	—	4	8	no	—
245	C₂³×C₆₇	—	4	16	no	—
246	C₂³×C₆₇	—	4	16	no	—
247	C₂³×C₆₇	—	4	8	no	—
248	C₂³×C₆₇	—	4	8	no	—
249	C₂³×C₆₇	—	4	16	no	—
250	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
251	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
252	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
253	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
254	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
255	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
256	Q₈×C₆₇	—	2	16	no	—
257	Q₈×C₆₇	—	2	16	no	—
258	Q₈×C₆₇	—	4	4	no	—
259	D₄×C₆₇	—	2	64	no	—
260	D₄×C₆₇	—	2	16	no	—
261	D₄×C₆₇	—	2	16	no	—
262	D₄×C₆₇	—	4	16	no	—
263	D₄×C₆₇	—	4	8	no	—
264	D₄×C₆₇	—	4	4	no	—
265	D₄×C₆₇	—	8	16	no	—
266	C₂×C₄×C₆₇	—	2	64	no	—
267	C₂×C₄×C₆₇	—	2	8	no	—
268	C₂×C₄×C₆₇	—	2	16	no	—
269	C₂×C₄×C₆₇	—	2	2	no	—
270	C₂×C₄×C₆₇	—	2	2	no	—
271	C₂×C₄×C₆₇	—	2	4	no	—
272	C₂×C₄×C₆₇	—	2	4	no	—
273	C₂×C₄×C₆₇	—	2	16	no	—
274	C₂×C₄×C₆₇	—	2	16	no	—
275	C₂×C₄×C₆₇	—	2	4	no	—
276	C₂×C₄×C₆₇	—	2	4	no	—
277	C₂×C₄×C₆₇	—	2	4	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	2	2	no	—
280	C₂×C₄×C₆₇	—	2	2	no	—
281	C₂×C₄×C₆₇	—	2	8	no	—
282	C₂×C₄×C₆₇	—	2	8	no	—
283	C₂×C₄×C₆₇	—	2	8	no	—
284	C₂×C₄×C₆₇	—	4	8	no	—
285	C₂×C₄×C₆₇	#120	1	16	yes	—
286	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
287	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
288	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
289	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
290	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
291	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
292	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
293	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
294	C₂²×C₆₇	—	2	4	no	—
295	C₂²×C₆₇	—	2	4	no	—
296	C₂²×C₆₇	—	2	4	no	—
297	C₂²×C₆₇	—	2	8	no	—
298	C₂²×C₆₇	—	2	4	no	—
299	C₂²×C₆₇	—	2	4	no	—
300	C₂²×C₆₇	—	2	4	no	—
301	C₂²×C₆₇	—	2	16	no	—
302	C₂²×C₆₇	—	4	16	no	—
303	C₂²×C₆₇	—	4	32	no	—
304	C₂²×C₆₇	—	4	8	no	—
305	C₂²×C₆₇	—	4	4	no	—
306	C₂²×C₆₇	—	4	4	no	—
307	C₂²×C₆₇	—	4	8	no	—
308	C₂²×C₆₇	—	4	2	no	—
309	C₂²×C₆₇	—	4	4	no	—
310	C₂²×C₆₇	—	4	4	no	—
311	C₂²×C₆₇	—	4	4	no	—
312	C₂²×C₆₇	C₂×#120	1	4	yes	—
313	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
314	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
315	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
316	C₂²×C₆₇	#585	1	4	yes	—
317	C₄×C₆₇	—	2	4	no	—
318	C₄×C₆₇	—	2	8	no	—
319	C₄×C₆₇	—	4	2	no	—
320	C₄×C₆₇	—	4	4	no	—
321	C₄×C₆₇	—	4	4	no	—
322	C₄×C₆₇	—	4	4	no	—
323	C₄×C₆₇	C₂×#120	1	4	yes	—
324	C₄×C₆₇	C₂×#120	1	4	yes	—
325	#35149	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
326	C₂×C₆₇	—	2	4	no	—
327	C₂×C₆₇	—	4	4	no	—
328	C₂×C₆₇	—	4	8	no	—
329	C₂×C₆₇	—	8	2	no	—
330	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
331	C₂×C₆₇	#3449	1	2	yes	—
332	C₂×C₆₇	#3485	1	2	yes	—
333	C₂×C₆₇	#3487	1	2	yes	—
334	C₂³×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
335	C₂×#536	C₂×C₆₇	1	1	yes	characteristic, complemented
336	C₂×#535	C₂×C₆₇	1	1	yes	characteristic, complemented
337	C₂×#525	C₂×C₆₇	1	1	yes	characteristic, complemented
338	#2849	C₂×C₆₇	1	4	yes	complemented
339	#2613	C₂×C₆₇	1	4	yes	complemented
340	#2432	C₂×C₆₇	1	4	yes	complemented
341	#2431	C₂×C₆₇	1	4	yes	complemented
342	#2407	C₂×C₆₇	1	8	yes	complemented
343	#2310	C₂×C₆₇	1	1	yes	characteristic, complemented
344	#2309	C₂×C₆₇	1	1	yes	characteristic, complemented
345	#2280	C₂×C₆₇	1	1	yes	characteristic, complemented
346	C₆₇	#35149	1	1	yes	characteristic, direct product factor, Sylow
347	C₂²×#41	—	2	8	no	—
348	C₂²×#41	—	2	8	no	—
349	C₂²×#41	—	2	8	no	—
350	C₂²×#41	C₂²×C₆₇	1	4	yes	—
351	C₂²×#41	C₂²×C₆₇	1	4	yes	—
352	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
353	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
354	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
355	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
356	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
357	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
358	#536	—	2	8	no	—
359	#535	—	2	8	no	—
360	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
361	C₂×#127	C₂²×C₆₇	1	2	yes	complemented
362	C₂×#124	C₂²×C₆₇	1	2	yes	complemented
363	C₂×#124	C₂²×C₆₇	1	2	yes	complemented
364	#525	—	2	8	no	—
365	C₂×#123	C₂²×C₆₇	1	2	yes	—
366	C₂×#123	C₂²×C₆₇	1	4	yes	—
367	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
368	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
369	C₂×#121	C₂²×C₆₇	1	4	yes	complemented
370	C₂×#121	C₂²×C₆₇	1	2	yes	complemented
371	C₂×#120	C₂²×C₆₇	1	4	yes	—
372	C₂×#120	C₂²×C₆₇	1	2	yes	—
373	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
374	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
375	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
376	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
377	#394	C₂²×C₆₇	1	16	yes	complemented
378	#392	C₂²×C₆₇	1	16	yes	complemented
379	#391	C₂²×C₆₇	1	4	yes	complemented
380	#391	C₂²×C₆₇	1	4	yes	complemented
381	#390	C₂²×C₆₇	1	4	yes	complemented
382	#390	C₂²×C₆₇	1	4	yes	complemented
383	#386	C₂²×C₆₇	1	16	yes	complemented
384	#385	C₂²×C₆₇	1	16	yes	complemented
385	#381	C₂²×C₆₇	1	4	yes	—
386	#381	C₂²×C₆₇	1	2	yes	—
387	#380	C₂²×C₆₇	1	4	yes	complemented
388	#379	C₂²×C₆₇	1	4	yes	—
389	#379	C₂²×C₆₇	1	2	yes	—
390	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
391	C₂⁵	C₂³×C₆₇	1	2	yes	—
392	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
393	C₂×#41	—	2	128	no	—
394	C₂×#41	—	2	16	no	—
395	C₂×#41	—	2	16	no	—
396	C₂×#41	—	2	16	no	—
397	C₂×#41	—	2	16	no	—
398	C₂×#41	—	2	16	no	—
399	C₂×#41	—	2	16	no	—
400	C₂×#41	—	2	4	no	—
401	C₂×#41	—	2	4	no	—
402	C₂×#41	—	2	4	no	—
403	C₂²×Q₈	—	2	4	no	—
404	C₂²×Q₈	—	2	4	no	—
405	C₂²×Q₈	D₄×C₆₇	1	8	yes	complemented
406	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
407	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
408	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
409	C₂²×D₄	—	2	8	no	—
410	C₂²×D₄	—	2	8	no	—
411	C₂²×D₄	—	2	8	no	—
412	C₂²×D₄	—	2	4	no	—
413	C₂²×D₄	—	2	4	no	—
414	C₂²×D₄	D₄×C₆₇	1	8	yes	complemented
415	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
416	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
417	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
418	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
419	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
420	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
421	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
422	C₂³×C₄	—	2	8	no	—
423	C₂³×C₄	—	2	8	no	—
424	C₂³×C₄	—	2	8	no	—
425	C₂³×C₄	—	2	4	no	—
426	C₂³×C₄	—	2	4	no	—
427	C₂³×C₄	D₄×C₆₇	1	8	yes	complemented
428	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
429	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
430	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
431	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
432	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
433	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
434	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
435	C₄⋊Q₈	—	2	8	no	—
436	#127	—	2	8	no	—
437	#124	—	2	8	no	—
438	#124	—	2	8	no	—
439	#123	—	2	8	no	—
440	#123	—	2	16	no	—
441	C₂²⋊Q₈	—	2	16	no	—
442	C₂²⋊Q₈	—	2	8	no	—
443	#121	—	2	16	no	—
444	#121	—	2	8	no	—
445	#120	—	2	8	no	—
446	#120	—	2	16	no	—
447	C₄×D₄	—	2	16	no	—
448	#117	—	2	4	no	—
449	#117	—	2	4	no	—
450	#117	—	2	4	no	—
451	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
452	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
453	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
454	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
455	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
456	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
457	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
458	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
459	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
460	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
461	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
462	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
463	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
464	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
465	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
466	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
467	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
468	#95	C₂³×C₆₇	1	8	yes	—
469	#95	C₂³×C₆₇	1	16	yes	—
470	C₂⁴	—	2	2	no	—
471	C₂⁴	—	2	4	no	—
472	C₂⁴	—	2	4	no	—
473	C₂⁴	—	2	2	no	—
474	C₂⁴	—	4	4	no	—
475	C₂⁴	—	4	8	no	—
476	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
477	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
478	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
479	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
480	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
481	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
482	#41	—	2	128	no	—
483	#41	—	2	128	no	—
484	#41	—	4	16	no	—
485	C₂×Q₈	—	2	32	no	—
486	C₂×Q₈	—	2	4	no	—
487	C₂×Q₈	—	2	4	no	—
488	C₂×Q₈	—	2	4	no	—
489	C₂×Q₈	—	2	4	no	—
490	C₂×Q₈	—	2	2	no	—
491	C₂×Q₈	—	2	2	no	—
492	C₂×Q₈	—	2	2	no	—
493	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
494	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
495	C₂×D₄	—	2	64	no	—
496	C₂×D₄	—	2	16	no	—
497	C₂×D₄	—	2	16	no	—
498	C₂×D₄	—	2	8	no	—
499	C₂×D₄	—	2	4	no	—
500	C₂×D₄	—	2	4	no	—
501	C₂×D₄	—	2	4	no	—
502	C₂×D₄	—	2	16	no	—
503	C₂×D₄	—	2	32	no	—
504	C₂×D₄	—	2	16	no	—
505	C₂×D₄	—	2	4	no	—
506	C₂×D₄	—	2	4	no	—
507	C₂×D₄	—	2	4	no	—
508	C₂×D₄	—	2	4	no	—
509	C₂×D₄	—	2	2	no	—
510	C₂×D₄	—	2	2	no	—
511	C₂×D₄	—	2	2	no	—
512	C₂×D₄	—	4	8	no	—
513	C₂×D₄	—	4	16	no	—
514	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
515	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
516	C₂²×C₄	—	2	64	no	—
517	C₂²×C₄	—	2	2	no	—
518	C₂²×C₄	—	2	2	no	—
519	C₂²×C₄	—	2	4	no	—
520	C₂²×C₄	—	2	4	no	—
521	C₂²×C₄	—	2	16	no	—
522	C₂²×C₄	—	2	32	no	—
523	C₂²×C₄	—	2	16	no	—
524	C₂²×C₄	—	2	16	no	—
525	C₂²×C₄	—	2	4	no	—
526	C₂²×C₄	—	2	4	no	—
527	C₂²×C₄	—	2	4	no	—
528	C₂²×C₄	—	2	4	no	—
529	C₂²×C₄	—	2	4	no	—
530	C₂²×C₄	—	2	4	no	—
531	C₂²×C₄	—	2	4	no	—
532	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
533	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
534	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
535	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
536	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
537	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
538	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
539	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
540	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
541	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
542	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
543	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
544	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
545	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
546	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	4	no	—
549	C₄⋊C₄	—	2	4	no	—
550	C₄⋊C₄	—	2	8	no	—
551	C₄⋊C₄	—	2	16	no	—
552	C₂²⋊C₄	—	2	4	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	2	8	no	—
556	C₂²⋊C₄	—	2	16	no	—
557	C₂²⋊C₄	—	2	8	no	—
558	C₂²⋊C₄	—	2	16	no	—
559	C₂²⋊C₄	—	4	16	no	—
560	C₂²⋊C₄	—	4	8	no	—
561	C₄²	—	2	4	no	—
562	C₄²	—	2	4	no	—
563	C₄²	—	2	4	no	—
564	C₂³	—	2	4	no	—
565	C₂³	—	2	16	no	—
566	C₂³	—	2	8	no	—
567	C₂³	—	2	4	no	—
568	C₂³	—	2	2	no	—
569	C₂³	—	2	4	no	—
570	C₂³	—	2	2	no	—
571	C₂³	—	2	4	no	—
572	C₂³	—	2	16	no	—
573	C₂³	—	2	4	no	—
574	C₂³	—	2	2	no	—
575	C₂³	—	2	2	no	—
576	C₂³	—	2	2	no	—
577	C₂³	—	2	8	no	—
578	C₂³	—	4	8	no	—
579	C₂³	—	4	16	no	—
580	C₂³	—	4	16	no	—
581	C₂³	—	4	8	no	—
582	C₂³	—	4	8	no	—
583	C₂³	—	4	16	no	—
584	C₂³	C₂²×D₄×C₆₇	1	8	yes	—
585	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
586	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
587	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
588	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
589	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
590	Q₈	—	2	16	no	—
591	Q₈	—	2	16	no	—
592	Q₈	—	4	4	no	—
593	D₄	—	2	64	no	—
594	D₄	—	2	16	no	—
595	D₄	—	2	16	no	—
596	D₄	—	4	16	no	—
597	D₄	—	4	8	no	—
598	D₄	—	4	4	no	—
599	D₄	—	8	16	no	—
600	C₂×C₄	—	2	64	no	—
601	C₂×C₄	—	2	8	no	—
602	C₂×C₄	—	2	16	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	4	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	16	no	—
608	C₂×C₄	—	2	16	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	4	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	8	no	—
616	C₂×C₄	—	2	8	no	—
617	C₂×C₄	—	2	8	no	—
618	C₂×C₄	—	4	8	no	—
619	C₂×C₄	#120×C₆₇	1	16	yes	—
620	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
621	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
622	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
623	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
624	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
625	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
626	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
627	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
628	C₂²	—	2	4	no	—
629	C₂²	—	2	4	no	—
630	C₂²	—	2	4	no	—
631	C₂²	—	2	8	no	—
632	C₂²	—	2	4	no	—
633	C₂²	—	2	4	no	—
634	C₂²	—	2	4	no	—
635	C₂²	—	2	16	no	—
636	C₂²	—	4	16	no	—
637	C₂²	—	4	32	no	—
638	C₂²	—	4	8	no	—
639	C₂²	—	4	4	no	—
640	C₂²	—	4	4	no	—
641	C₂²	—	4	8	no	—
642	C₂²	—	4	2	no	—
643	C₂²	—	4	4	no	—
644	C₂²	—	4	4	no	—
645	C₂²	—	4	4	no	—
646	C₂²	C₂×#120×C₆₇	1	4	yes	—
647	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
648	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
649	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
650	C₂²	#585×C₆₇	1	4	yes	—
651	C₄	—	2	4	no	—
652	C₄	—	2	8	no	—
653	C₄	—	4	2	no	—
654	C₄	—	4	4	no	—
655	C₄	—	4	4	no	—
656	C₄	—	4	4	no	—
657	C₄	C₂×#120×C₆₇	1	4	yes	—
658	C₄	C₂×#120×C₆₇	1	4	yes	—
659	C₂	—	2	4	no	—
660	C₂	—	4	4	no	—
661	C₂	—	4	8	no	—
662	C₂	—	8	2	no	—
663	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
664	C₂	C₆₇×#3449	1	2	yes	—
665	C₂	C₆₇×#3485	1	2	yes	—
666	C₂	C₆₇×#3487	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×#41×C₆₇	C₂	semidirect
2. C₂×#536×C₆₇	C₂	semidirect
3. C₂×#535×C₆₇	C₂	semidirect
4. C₂×#525×C₆₇	C₂	semidirect
5. C₆₇×#2849	C₂	semidirect
6. C₆₇×#2613	C₂	semidirect
7. C₆₇×#2432	C₂	semidirect
8. C₆₇×#2431	C₂	semidirect
9. C₆₇×#2407	C₂	semidirect
10. C₆₇×#2310	C₂	semidirect
11. C₆₇×#2309	C₂	semidirect
12. C₆₇×#2280	C₂	semidirect
26. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
27. C₂×#127×C₆₇	C₂²	semidirect
28. C₂×#124×C₆₇	C₂²	semidirect
29. C₂×#124×C₆₇	C₂²	semidirect
33. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
34. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
35. C₂×#121×C₆₇	C₂²	semidirect
36. C₂×#121×C₆₇	C₂²	semidirect
39. C₂×C₄×D₄×C₆₇	C₂²	semidirect
43. #394×C₆₇	C₂²	semidirect
44. #392×C₆₇	C₂²	semidirect
45. #391×C₆₇	C₂²	semidirect
46. #391×C₆₇	C₂²	semidirect
47. #390×C₆₇	C₂²	semidirect
48. #390×C₆₇	C₂²	semidirect
49. #386×C₆₇	C₂²	semidirect
50. #385×C₆₇	C₂²	semidirect
53. #380×C₆₇	C₂²	semidirect
56. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
71. C₂²×Q₈×C₆₇	D₄	semidirect
80. C₂²×D₄×C₆₇	D₄	semidirect
93. C₂³×C₄×C₆₇	D₄	semidirect
325. #35149	C₆₇	direct (N × H)
334. C₂³×#41	C₂×C₆₇	semidirect
335. C₂×#536	C₂×C₆₇	semidirect
336. C₂×#535	C₂×C₆₇	semidirect
337. C₂×#525	C₂×C₆₇	semidirect
338. #2849	C₂×C₆₇	semidirect
339. #2613	C₂×C₆₇	semidirect
340. #2432	C₂×C₆₇	semidirect
341. #2431	C₂×C₆₇	semidirect
342. #2407	C₂×C₆₇	semidirect
343. #2310	C₂×C₆₇	semidirect
344. #2309	C₂×C₆₇	semidirect
345. #2280	C₂×C₆₇	semidirect
346. C₆₇	#35149	direct (N × H)
360. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
361. C₂×#127	C₂²×C₆₇	semidirect
362. C₂×#124	C₂²×C₆₇	semidirect
363. C₂×#124	C₂²×C₆₇	semidirect
367. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
368. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
369. C₂×#121	C₂²×C₆₇	semidirect
370. C₂×#121	C₂²×C₆₇	semidirect
373. C₂×C₄×D₄	C₂²×C₆₇	semidirect
377. #394	C₂²×C₆₇	semidirect
378. #392	C₂²×C₆₇	semidirect
379. #391	C₂²×C₆₇	semidirect
380. #391	C₂²×C₆₇	semidirect
381. #390	C₂²×C₆₇	semidirect
382. #390	C₂²×C₆₇	semidirect
383. #386	C₂²×C₆₇	semidirect
384. #385	C₂²×C₆₇	semidirect
387. #380	C₂²×C₆₇	semidirect
390. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
405. C₂²×Q₈	D₄×C₆₇	semidirect
414. C₂²×D₄	D₄×C₆₇	semidirect
427. C₂³×C₄	D₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#41×C₆₇	2
C₂×#536×C₆₇	2
C₂×#535×C₆₇	2
C₂×#525×C₆₇	2
C₆₇×#2849	2
C₆₇×#2613	2
C₆₇×#2432	2
C₆₇×#2431	2
C₆₇×#2407	2
C₆₇×#2310	2
C₆₇×#2309	2
C₆₇×#2280	2
#35149	67

Subgroups — C67×#35149 #30714400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35149 #30714400