Subgroups — C₆₇×#35152 #30714403

Order 17152.

2950 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#535×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2849	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2622	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2447	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2445	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2422	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2420	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2317	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2311	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2309	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2297	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#2296	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
17	#535×C₆₇	—	2	8	no	—
18	C₂×#127×C₆₇	C₂²	1	2	yes	—
19	C₂×#124×C₆₇	C₂²	1	2	yes	—
20	C₂×#123×C₆₇	C₂²	1	8	yes	complemented
21	C₂×#121×C₆₇	C₂²	1	2	yes	—
22	C₂×#121×C₆₇	C₂²	1	2	yes	—
23	C₂×#120×C₆₇	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₆₇	—	2	8	no	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
27	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
32	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
33	#397×C₆₇	C₂²	1	8	yes	complemented
34	#397×C₆₇	C₂²	1	8	yes	complemented
35	#394×C₆₇	C₂²	1	8	yes	complemented
36	#394×C₆₇	C₂²	1	8	yes	complemented
37	#391×C₆₇	C₂²	1	2	yes	complemented
38	#391×C₆₇	C₂²	1	2	yes	complemented
39	#390×C₆₇	C₂²	1	4	yes	—
40	#390×C₆₇	C₂²	1	4	yes	—
41	#390×C₆₇	C₂²	1	2	yes	complemented
42	#390×C₆₇	C₂²	1	2	yes	complemented
43	#389×C₆₇	C₂²	1	2	yes	complemented
44	#389×C₆₇	C₂²	1	2	yes	complemented
45	#389×C₆₇	C₂²	1	4	yes	complemented
46	#386×C₆₇	C₂²	1	2	yes	—
47	#386×C₆₇	C₂²	1	2	yes	—
48	#386×C₆₇	C₂²	1	4	yes	—
49	#386×C₆₇	C₂²	1	8	yes	complemented
50	#385×C₆₇	C₂²	1	2	yes	—
51	#385×C₆₇	C₂²	1	2	yes	—
52	#385×C₆₇	C₂²	1	4	yes	—
53	#385×C₆₇	C₂²	1	8	yes	complemented
54	#384×C₆₇	C₂²	1	2	yes	complemented
55	#382×C₆₇	C₂²	1	2	yes	complemented
56	#381×C₆₇	C₂²	1	8	yes	complemented
57	#380×C₆₇	C₂²	1	4	yes	complemented
58	#379×C₆₇	C₂²	1	4	yes	—
59	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
60	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
61	C₂⁵×C₆₇	C₂³	1	2	yes	—
62	C₂×#41×C₆₇	—	2	4	no	—
63	C₂×#41×C₆₇	—	2	4	no	—
64	C₂×#41×C₆₇	—	2	4	no	—
65	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₆₇	—	2	4	no	—
67	C₂²×D₄×C₆₇	—	2	4	no	—
68	C₂²×D₄×C₆₇	D₄	1	8	yes	—
69	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
70	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
72	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
73	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
74	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
75	C₂³×C₄×C₆₇	—	2	8	no	—
76	C₂³×C₄×C₆₇	—	4	8	no	—
77	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
79	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
80	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	#127×C₆₇	—	2	8	no	—
84	#124×C₆₇	—	2	8	no	—
85	#123×C₆₇	—	4	16	no	—
86	#121×C₆₇	—	2	8	no	—
87	#121×C₆₇	—	2	8	no	—
88	#120×C₆₇	—	2	16	no	—
89	C₄×D₄×C₆₇	—	2	16	no	—
90	C₄×D₄×C₆₇	—	2	16	no	—
91	#117×C₆₇	—	2	4	no	—
92	#117×C₆₇	—	2	4	no	—
93	#117×C₆₇	—	2	4	no	—
94	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
116	C₂×C₄²×C₆₇	—	2	4	no	—
117	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
118	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
119	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
120	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
121	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
122	#95×C₆₇	C₂³	1	4	yes	—
123	#95×C₆₇	C₂³	1	4	yes	—
124	#95×C₆₇	C₂³	1	4	yes	—
125	#95×C₆₇	C₂³	1	4	yes	—
126	#95×C₆₇	C₂³	1	8	yes	—
127	#95×C₆₇	C₂³	1	8	yes	—
128	C₂⁴×C₆₇	—	2	4	no	—
129	C₂⁴×C₆₇	—	2	4	no	—
130	C₂⁴×C₆₇	—	4	4	no	—
131	C₂⁴×C₆₇	—	4	4	no	—
132	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
133	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
134	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
135	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
136	#41×C₆₇	—	4	16	no	—
137	C₂×Q₈×C₆₇	—	2	2	no	—
138	C₂×Q₈×C₆₇	—	2	2	no	—
139	C₂×Q₈×C₆₇	—	2	2	no	—
140	C₂×D₄×C₆₇	—	2	4	no	—
141	C₂×D₄×C₆₇	—	2	4	no	—
142	C₂×D₄×C₆₇	—	2	4	no	—
143	C₂×D₄×C₆₇	—	2	8	no	—
144	C₂×D₄×C₆₇	—	2	8	no	—
145	C₂×D₄×C₆₇	—	2	32	no	—
146	C₂×D₄×C₆₇	—	2	2	no	—
147	C₂×D₄×C₆₇	—	2	4	no	—
148	C₂×D₄×C₆₇	—	2	4	no	—
149	C₂×D₄×C₆₇	—	2	4	no	—
150	C₂×D₄×C₆₇	—	2	4	no	—
151	C₂×D₄×C₆₇	—	2	2	no	—
152	C₂×D₄×C₆₇	—	2	2	no	—
153	C₂×D₄×C₆₇	—	4	8	no	—
154	C₂×D₄×C₆₇	—	4	8	no	—
155	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
156	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	2	16	no	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	4	no	—
164	C₂²×C₄×C₆₇	—	2	4	no	—
165	C₂²×C₄×C₆₇	—	2	16	no	—
166	C₂²×C₄×C₆₇	—	4	8	no	—
167	C₂²×C₄×C₆₇	—	4	4	no	—
168	C₂²×C₄×C₆₇	—	4	2	no	—
169	C₂²×C₄×C₆₇	—	4	32	no	—
170	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
171	C₂²×C₄×C₆₇	#41	1	4	yes	—
172	C₂²×C₄×C₆₇	#41	1	4	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₄⋊C₄×C₆₇	—	2	4	no	—
184	C₄⋊C₄×C₆₇	—	2	4	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	4	no	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	2	8	no	—
190	C₄⋊C₄×C₆₇	—	4	4	no	—
191	C₄⋊C₄×C₆₇	—	4	8	no	—
192	C₂²⋊C₄×C₆₇	—	2	16	no	—
193	C₂²⋊C₄×C₆₇	—	2	4	no	—
194	C₂²⋊C₄×C₆₇	—	2	8	no	—
195	C₂²⋊C₄×C₆₇	—	2	8	no	—
196	C₂²⋊C₄×C₆₇	—	2	4	no	—
197	C₂²⋊C₄×C₆₇	—	2	4	no	—
198	C₂²⋊C₄×C₆₇	—	2	8	no	—
199	C₂²⋊C₄×C₆₇	—	4	8	no	—
200	C₂²⋊C₄×C₆₇	—	4	4	no	—
201	C₂²⋊C₄×C₆₇	—	4	8	no	—
202	C₂²⋊C₄×C₆₇	—	4	8	no	—
203	C₄²×C₆₇	—	2	4	no	—
204	C₄²×C₆₇	—	2	4	no	—
205	C₄²×C₆₇	—	2	4	no	—
206	C₄²×C₆₇	—	2	4	no	—
207	C₄²×C₆₇	—	2	4	no	—
208	C₂³×C₆₇	—	2	2	no	—
209	C₂³×C₆₇	—	2	2	no	—
210	C₂³×C₆₇	—	2	16	no	—
211	C₂³×C₆₇	—	2	4	no	—
212	C₂³×C₆₇	—	2	4	no	—
213	C₂³×C₆₇	—	2	8	no	—
214	C₂³×C₆₇	—	2	8	no	—
215	C₂³×C₆₇	—	2	2	no	—
216	C₂³×C₆₇	—	2	2	no	—
217	C₂³×C₆₇	—	2	2	no	—
218	C₂³×C₆₇	—	2	4	no	—
219	C₂³×C₆₇	—	2	4	no	—
220	C₂³×C₆₇	—	4	8	no	—
221	C₂³×C₆₇	—	4	8	no	—
222	C₂³×C₆₇	—	4	16	no	—
223	C₂³×C₆₇	—	4	16	no	—
224	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
225	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
226	C₂³×C₆₇	C₂×#41	1	2	yes	—
227	C₂³×C₆₇	C₂×#41	1	2	yes	—
228	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
229	Q₈×C₆₇	—	4	4	no	—
230	D₄×C₆₇	—	2	16	no	—
231	D₄×C₆₇	—	2	16	no	—
232	D₄×C₆₇	—	4	8	no	—
233	D₄×C₆₇	—	4	4	no	—
234	D₄×C₆₇	—	8	8	no	—
235	D₄×C₆₇	—	8	8	no	—
236	C₂×C₄×C₆₇	—	2	8	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	2	8	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	4	32	no	—
258	C₂×C₄×C₆₇	—	4	4	no	—
259	C₂×C₄×C₆₇	—	4	8	no	—
260	C₂×C₄×C₆₇	—	4	8	no	—
261	C₂×C₄×C₆₇	#123	1	16	yes	complemented
262	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
265	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
266	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
267	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
268	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
269	C₂²×C₆₇	—	2	4	no	—
270	C₂²×C₆₇	—	2	2	no	—
271	C₂²×C₆₇	—	2	2	no	—
272	C₂²×C₆₇	—	2	4	no	—
273	C₂²×C₆₇	—	2	8	no	—
274	C₂²×C₆₇	—	2	8	no	—
275	C₂²×C₆₇	—	2	8	no	—
276	C₂²×C₆₇	—	4	32	no	—
277	C₂²×C₆₇	—	4	8	no	—
278	C₂²×C₆₇	—	4	8	no	—
279	C₂²×C₆₇	—	4	2	no	—
280	C₂²×C₆₇	—	4	4	no	—
281	C₂²×C₆₇	—	4	2	no	—
282	C₂²×C₆₇	—	4	2	no	—
283	C₂²×C₆₇	—	4	4	no	—
284	C₂²×C₆₇	C₂×#123	1	4	yes	—
285	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
286	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
287	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
288	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
289	C₂²×C₆₇	#585	1	2	yes	—
290	C₄×C₆₇	—	2	4	no	—
291	C₄×C₆₇	—	4	2	no	—
292	C₄×C₆₇	—	4	4	no	—
293	C₄×C₆₇	—	4	4	no	—
294	C₄×C₆₇	—	4	2	no	—
295	C₄×C₆₇	—	4	2	no	—
296	C₄×C₆₇	—	4	2	no	—
297	C₄×C₆₇	—	4	4	no	—
298	C₄×C₆₇	C₂×#123	1	4	yes	—
299	C₄×C₆₇	C₂×#123	1	4	yes	—
300	#35152	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
301	C₂×C₆₇	—	2	4	no	—
302	C₂×C₆₇	—	4	8	no	—
303	C₂×C₆₇	—	8	2	no	—
304	C₂×C₆₇	C₂²×#123	1	1	yes	characteristic
305	C₂×C₆₇	#3460	1	2	yes	—
306	C₂×C₆₇	#3484	1	1	yes	characteristic
307	C₂×C₆₇	#3485	1	1	yes	characteristic
308	C₂×C₆₇	#3506	1	2	yes	—
309	C₂×#535	C₂×C₆₇	1	1	yes	characteristic
310	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
311	#2849	C₂×C₆₇	1	4	yes	complemented
312	#2622	C₂×C₆₇	1	4	yes	complemented
313	#2447	C₂×C₆₇	1	4	yes	complemented
314	#2445	C₂×C₆₇	1	4	yes	complemented
315	#2422	C₂×C₆₇	1	4	yes	complemented
316	#2420	C₂×C₆₇	1	4	yes	complemented
317	#2317	C₂×C₆₇	1	1	yes	characteristic, complemented
318	#2311	C₂×C₆₇	1	1	yes	characteristic, complemented
319	#2309	C₂×C₆₇	1	1	yes	characteristic, complemented
320	#2297	C₂×C₆₇	1	1	yes	characteristic, complemented
321	#2296	C₂×C₆₇	1	1	yes	characteristic, complemented
322	C₆₇	#35152	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
324	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
325	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
326	#535	—	2	8	no	—
327	C₂×#127	C₂²×C₆₇	1	2	yes	—
328	C₂×#124	C₂²×C₆₇	1	2	yes	—
329	C₂×#123	C₂²×C₆₇	1	8	yes	complemented
330	C₂×#121	C₂²×C₆₇	1	2	yes	—
331	C₂×#121	C₂²×C₆₇	1	2	yes	—
332	C₂×#120	C₂²×C₆₇	1	4	yes	—
333	C₂×C₄×D₄	—	2	8	no	—
334	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
335	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
336	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
337	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
338	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
339	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
340	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
341	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
342	#397	C₂²×C₆₇	1	8	yes	complemented
343	#397	C₂²×C₆₇	1	8	yes	complemented
344	#394	C₂²×C₆₇	1	8	yes	complemented
345	#394	C₂²×C₆₇	1	8	yes	complemented
346	#391	C₂²×C₆₇	1	2	yes	complemented
347	#391	C₂²×C₆₇	1	2	yes	complemented
348	#390	C₂²×C₆₇	1	4	yes	—
349	#390	C₂²×C₆₇	1	4	yes	—
350	#390	C₂²×C₆₇	1	2	yes	complemented
351	#390	C₂²×C₆₇	1	2	yes	complemented
352	#389	C₂²×C₆₇	1	4	yes	complemented
353	#389	C₂²×C₆₇	1	2	yes	complemented
354	#389	C₂²×C₆₇	1	2	yes	complemented
355	#386	C₂²×C₆₇	1	4	yes	—
356	#386	C₂²×C₆₇	1	2	yes	—
357	#386	C₂²×C₆₇	1	2	yes	—
358	#386	C₂²×C₆₇	1	8	yes	complemented
359	#385	C₂²×C₆₇	1	4	yes	—
360	#385	C₂²×C₆₇	1	2	yes	—
361	#385	C₂²×C₆₇	1	2	yes	—
362	#385	C₂²×C₆₇	1	8	yes	complemented
363	#384	C₂²×C₆₇	1	2	yes	complemented
364	#382	C₂²×C₆₇	1	2	yes	complemented
365	#381	C₂²×C₆₇	1	8	yes	complemented
366	#380	C₂²×C₆₇	1	4	yes	complemented
367	#379	C₂²×C₆₇	1	4	yes	—
368	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
369	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
370	C₂⁵	C₂³×C₆₇	1	2	yes	—
371	C₂×#41	—	2	4	no	—
372	C₂×#41	—	2	4	no	—
373	C₂×#41	—	2	4	no	—
374	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
375	C₂²×D₄	—	2	4	no	—
376	C₂²×D₄	—	2	4	no	—
377	C₂²×D₄	D₄×C₆₇	1	8	yes	—
378	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
379	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
380	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
381	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
382	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
383	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
384	C₂³×C₄	—	2	8	no	—
385	C₂³×C₄	—	4	8	no	—
386	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
387	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
388	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
389	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
390	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
391	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
392	#127	—	2	8	no	—
393	#124	—	2	8	no	—
394	#123	—	4	16	no	—
395	#121	—	2	8	no	—
396	#121	—	2	8	no	—
397	#120	—	2	16	no	—
398	C₄×D₄	—	2	16	no	—
399	C₄×D₄	—	2	16	no	—
400	#117	—	2	4	no	—
401	#117	—	2	4	no	—
402	#117	—	2	4	no	—
403	C₂×C₄⋊C₄	—	2	4	no	—
404	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
413	C₂×C₂²⋊C₄	—	2	8	no	—
414	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
425	C₂×C₄²	—	2	4	no	—
426	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
427	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
428	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
429	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
430	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
431	#95	C₂³×C₆₇	1	4	yes	—
432	#95	C₂³×C₆₇	1	4	yes	—
433	#95	C₂³×C₆₇	1	4	yes	—
434	#95	C₂³×C₆₇	1	4	yes	—
435	#95	C₂³×C₆₇	1	8	yes	—
436	#95	C₂³×C₆₇	1	8	yes	—
437	C₂⁴	—	2	4	no	—
438	C₂⁴	—	2	4	no	—
439	C₂⁴	—	4	4	no	—
440	C₂⁴	—	4	4	no	—
441	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
442	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
443	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
444	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
445	#41	—	4	16	no	—
446	C₂×Q₈	—	2	2	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	—	2	2	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	4	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	2	8	no	—
454	C₂×D₄	—	2	32	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	—	4	8	no	—
463	C₂×D₄	—	4	8	no	—
464	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
465	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	4	no	—
470	C₂²×C₄	—	2	16	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	—	2	16	no	—
475	C₂²×C₄	—	4	8	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	—	4	2	no	—
478	C₂²×C₄	—	4	32	no	—
479	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
480	C₂²×C₄	#41×C₆₇	1	4	yes	—
481	C₂²×C₄	#41×C₆₇	1	4	yes	—
482	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
484	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
485	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
486	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
487	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
488	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
489	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
490	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	8	no	—
499	C₄⋊C₄	—	4	8	no	—
500	C₄⋊C₄	—	4	4	no	—
501	C₂²⋊C₄	—	2	16	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	8	no	—
504	C₂²⋊C₄	—	2	8	no	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	8	no	—
508	C₂²⋊C₄	—	4	8	no	—
509	C₂²⋊C₄	—	4	8	no	—
510	C₂²⋊C₄	—	4	8	no	—
511	C₂²⋊C₄	—	4	4	no	—
512	C₄²	—	2	4	no	—
513	C₄²	—	2	4	no	—
514	C₄²	—	2	4	no	—
515	C₄²	—	2	4	no	—
516	C₄²	—	2	4	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	2	no	—
519	C₂³	—	2	16	no	—
520	C₂³	—	2	4	no	—
521	C₂³	—	2	4	no	—
522	C₂³	—	2	8	no	—
523	C₂³	—	2	8	no	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	4	no	—
528	C₂³	—	2	4	no	—
529	C₂³	—	4	8	no	—
530	C₂³	—	4	8	no	—
531	C₂³	—	4	16	no	—
532	C₂³	—	4	16	no	—
533	C₂³	C₂²×D₄×C₆₇	1	8	yes	—
534	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
535	C₂³	C₂×#41×C₆₇	1	2	yes	—
536	C₂³	C₂×#41×C₆₇	1	2	yes	—
537	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
538	Q₈	—	4	4	no	—
539	D₄	—	2	16	no	—
540	D₄	—	2	16	no	—
541	D₄	—	4	8	no	—
542	D₄	—	4	4	no	—
543	D₄	—	8	8	no	—
544	D₄	—	8	8	no	—
545	C₂×C₄	—	2	8	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	8	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	4	32	no	—
567	C₂×C₄	—	4	8	no	—
568	C₂×C₄	—	4	4	no	—
569	C₂×C₄	—	4	8	no	—
570	C₂×C₄	#123×C₆₇	1	16	yes	complemented
571	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
572	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
573	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
574	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
575	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
576	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
577	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
578	C₂²	—	2	4	no	—
579	C₂²	—	2	2	no	—
580	C₂²	—	2	2	no	—
581	C₂²	—	2	4	no	—
582	C₂²	—	2	8	no	—
583	C₂²	—	2	8	no	—
584	C₂²	—	2	8	no	—
585	C₂²	—	4	32	no	—
586	C₂²	—	4	8	no	—
587	C₂²	—	4	8	no	—
588	C₂²	—	4	2	no	—
589	C₂²	—	4	4	no	—
590	C₂²	—	4	2	no	—
591	C₂²	—	4	2	no	—
592	C₂²	—	4	4	no	—
593	C₂²	C₂×#123×C₆₇	1	4	yes	—
594	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
595	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
596	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
597	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
598	C₂²	#585×C₆₇	1	2	yes	—
599	C₄	—	2	4	no	—
600	C₄	—	4	2	no	—
601	C₄	—	4	4	no	—
602	C₄	—	4	4	no	—
603	C₄	—	4	2	no	—
604	C₄	—	4	2	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	4	no	—
607	C₄	C₂×#123×C₆₇	1	4	yes	—
608	C₄	C₂×#123×C₆₇	1	4	yes	—
609	C₂	—	2	4	no	—
610	C₂	—	4	8	no	—
611	C₂	—	8	2	no	—
612	C₂	C₂²×#123×C₆₇	1	1	yes	characteristic
613	C₂	C₆₇×#3460	1	2	yes	—
614	C₂	C₆₇×#3484	1	1	yes	characteristic
615	C₂	C₆₇×#3485	1	1	yes	characteristic
616	C₂	C₆₇×#3506	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#535×C₆₇	C₂
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
2. C₂²×C₄×D₄×C₆₇	C₂	semidirect
3. C₆₇×#2849	C₂	semidirect
4. C₆₇×#2622	C₂	semidirect
5. C₆₇×#2447	C₂	semidirect
6. C₆₇×#2445	C₂	semidirect
7. C₆₇×#2422	C₂	semidirect
8. C₆₇×#2420	C₂	semidirect
9. C₆₇×#2317	C₂	semidirect
10. C₆₇×#2311	C₂	semidirect
11. C₆₇×#2309	C₂	semidirect
12. C₆₇×#2297	C₂	semidirect
13. C₆₇×#2296	C₂	semidirect
20. C₂×#123×C₆₇	C₂²	semidirect
33. #397×C₆₇	C₂²	semidirect
34. #397×C₆₇	C₂²	semidirect
35. #394×C₆₇	C₂²	semidirect
36. #394×C₆₇	C₂²	semidirect
37. #391×C₆₇	C₂²	semidirect
38. #391×C₆₇	C₂²	semidirect
41. #390×C₆₇	C₂²	semidirect
42. #390×C₆₇	C₂²	semidirect
43. #389×C₆₇	C₂²	semidirect
44. #389×C₆₇	C₂²	semidirect
45. #389×C₆₇	C₂²	semidirect
49. #386×C₆₇	C₂²	semidirect
53. #385×C₆₇	C₂²	semidirect
54. #384×C₆₇	C₂²	semidirect
55. #382×C₆₇	C₂²	semidirect
56. #381×C₆₇	C₂²	semidirect
57. #380×C₆₇	C₂²	semidirect
59. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
60. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
261. C₂×C₄×C₆₇	#123	semidirect
300. #35152	C₆₇	direct (N × H)
310. C₂²×C₄×D₄	C₂×C₆₇	semidirect
311. #2849	C₂×C₆₇	semidirect
312. #2622	C₂×C₆₇	semidirect
313. #2447	C₂×C₆₇	semidirect
314. #2445	C₂×C₆₇	semidirect
315. #2422	C₂×C₆₇	semidirect
316. #2420	C₂×C₆₇	semidirect
317. #2317	C₂×C₆₇	semidirect
318. #2311	C₂×C₆₇	semidirect
319. #2309	C₂×C₆₇	semidirect
320. #2297	C₂×C₆₇	semidirect
321. #2296	C₂×C₆₇	semidirect
322. C₆₇	#35152	direct (N × H)
329. C₂×#123	C₂²×C₆₇	semidirect
342. #397	C₂²×C₆₇	semidirect
343. #397	C₂²×C₆₇	semidirect
344. #394	C₂²×C₆₇	semidirect
345. #394	C₂²×C₆₇	semidirect
346. #391	C₂²×C₆₇	semidirect
347. #391	C₂²×C₆₇	semidirect
350. #390	C₂²×C₆₇	semidirect
351. #390	C₂²×C₆₇	semidirect
352. #389	C₂²×C₆₇	semidirect
353. #389	C₂²×C₆₇	semidirect
354. #389	C₂²×C₆₇	semidirect
358. #386	C₂²×C₆₇	semidirect
362. #385	C₂²×C₆₇	semidirect
363. #384	C₂²×C₆₇	semidirect
364. #382	C₂²×C₆₇	semidirect
365. #381	C₂²×C₆₇	semidirect
366. #380	C₂²×C₆₇	semidirect
368. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
369. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
570. C₂×C₄	#123×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#535×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2849	2
C₆₇×#2622	2
C₆₇×#2447	2
C₆₇×#2445	2
C₆₇×#2422	2
C₆₇×#2420	2
C₆₇×#2317	2
C₆₇×#2311	2
C₆₇×#2309	2
C₆₇×#2297	2
C₆₇×#2296	2
#35152	67

Subgroups — C67×#35152 #30714403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35152 #30714403