Subgroups — C₆₇×#34977 #30714228

Order 17152.

2962 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2610	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2563	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2548	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2541	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2500	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2431	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2414	C₂	1	4	yes	maximal, complemented
11	C₆₇×#2388	C₂	1	4	yes	maximal, complemented
12	C₄×#127×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₄×#124×C₆₇	C₂	1	2	yes	maximal, complemented
14	C₄×#121×C₆₇	C₂	1	4	yes	maximal, complemented
15	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×#41×C₆₇	C₂²	1	2	yes	—
18	#532×C₆₇	—	2	8	no	—
19	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#127×C₆₇	C₂²	1	2	yes	—
22	C₂×#124×C₆₇	C₂²	1	2	yes	—
23	C₂×#124×C₆₇	C₂²	1	2	yes	—
24	C₂×#121×C₆₇	C₂²	1	2	yes	—
25	C₂×#121×C₆₇	C₂²	1	4	yes	—
26	C₂×#121×C₆₇	C₂²	1	2	yes	—
27	C₄×#41×C₆₇	—	2	8	no	—
28	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
36	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
37	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#117×C₆₇	C₂²	1	2	yes	—
39	C₂×#117×C₆₇	C₂²	1	2	yes	—
40	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
41	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
42	#395×C₆₇	C₂²	1	4	yes	—
43	#394×C₆₇	C₂²	1	8	yes	—
44	#394×C₆₇	C₂²	1	4	yes	—
45	#393×C₆₇	C₂²	1	4	yes	—
46	#392×C₆₇	C₂²	1	4	yes	—
47	#392×C₆₇	C₂²	1	8	yes	—
48	#391×C₆₇	C₂²	1	2	yes	—
49	#390×C₆₇	C₂²	1	2	yes	—
50	#390×C₆₇	C₂²	1	2	yes	—
51	#390×C₆₇	C₂²	1	4	yes	—
52	#390×C₆₇	C₂²	1	2	yes	—
53	#390×C₆₇	C₂²	1	8	yes	—
54	#389×C₆₇	C₂²	1	4	yes	—
55	#388×C₆₇	C₂²	1	8	yes	—
56	#388×C₆₇	C₂²	1	8	yes	—
57	#386×C₆₇	C₂²	1	8	yes	—
58	#384×C₆₇	C₂²	1	2	yes	—
59	#382×C₆₇	C₂²	1	2	yes	—
60	#380×C₆₇	C₂²	1	4	yes	—
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
62	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
63	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
66	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
67	#376×C₆₇	C₂²	1	2	yes	—
68	C₄³×C₆₇	C₂²	1	2	yes	—
69	C₂×#41×C₆₇	—	2	4	no	—
70	C₂×#41×C₆₇	—	2	4	no	—
71	C₂×#41×C₆₇	—	2	4	no	—
72	C₂×#41×C₆₇	—	2	8	no	—
73	C₂×#41×C₆₇	—	4	8	no	—
74	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
76	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
77	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
78	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
79	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
80	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
82	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
83	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
84	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
85	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
86	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
87	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
90	C₄⋊Q₈×C₆₇	—	2	4	no	—
91	#127×C₆₇	—	2	8	no	—
92	#127×C₆₇	—	2	4	no	—
93	#124×C₆₇	—	2	8	no	—
94	#124×C₆₇	—	2	8	no	—
95	#121×C₆₇	—	2	8	no	—
96	#121×C₆₇	—	2	8	no	—
97	#121×C₆₇	—	2	16	no	—
98	C₄×Q₈×C₆₇	—	2	4	no	—
99	C₄×Q₈×C₆₇	—	2	4	no	—
100	C₄×D₄×C₆₇	—	2	8	no	—
101	C₄×D₄×C₆₇	—	2	8	no	—
102	C₄×D₄×C₆₇	—	2	4	no	—
103	C₄×D₄×C₆₇	—	2	4	no	—
104	C₄×D₄×C₆₇	—	2	8	no	—
105	C₄×D₄×C₆₇	—	2	8	no	—
106	C₄×D₄×C₆₇	—	4	16	no	—
107	#117×C₆₇	—	2	4	no	—
108	#117×C₆₇	—	2	8	no	—
109	#117×C₆₇	—	2	8	no	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
118	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
123	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
124	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
125	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
126	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
127	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
128	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
129	C₂×C₄²×C₆₇	—	2	8	no	—
130	C₂×C₄²×C₆₇	D₄	1	8	yes	complemented
131	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
132	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
133	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
134	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
135	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
136	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
137	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
138	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
140	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
141	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
142	#95×C₆₇	C₂³	1	4	yes	—
143	#95×C₆₇	C₂³	1	2	yes	—
144	#95×C₆₇	C₂³	1	2	yes	—
145	#95×C₆₇	C₂³	1	8	yes	—
146	#95×C₆₇	C₂³	1	8	yes	—
147	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
148	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
150	#41×C₆₇	—	2	32	no	—
151	#41×C₆₇	—	4	32	no	—
152	C₂×Q₈×C₆₇	—	2	2	no	—
153	C₂×Q₈×C₆₇	—	2	4	no	—
154	C₂×Q₈×C₆₇	—	2	2	no	—
155	C₂×Q₈×C₆₇	—	4	4	no	—
156	C₂×Q₈×C₆₇	#41	1	4	yes	complemented
157	C₂×D₄×C₆₇	—	2	4	no	—
158	C₂×D₄×C₆₇	—	2	4	no	—
159	C₂×D₄×C₆₇	—	2	8	no	—
160	C₂×D₄×C₆₇	—	2	16	no	—
161	C₂×D₄×C₆₇	—	2	2	no	—
162	C₂×D₄×C₆₇	—	2	4	no	—
163	C₂×D₄×C₆₇	—	2	4	no	—
164	C₂×D₄×C₆₇	—	2	4	no	—
165	C₂×D₄×C₆₇	—	2	4	no	—
166	C₂×D₄×C₆₇	—	2	2	no	—
167	C₂×D₄×C₆₇	—	2	16	no	—
168	C₂×D₄×C₆₇	—	2	8	no	—
169	C₂×D₄×C₆₇	—	2	4	no	—
170	C₂×D₄×C₆₇	—	2	4	no	—
171	C₂×D₄×C₆₇	—	4	8	no	—
172	C₂×D₄×C₆₇	—	4	4	no	—
173	C₂×D₄×C₆₇	—	4	8	no	—
174	C₂×D₄×C₆₇	#41	1	4	yes	complemented
175	C₂²×C₄×C₆₇	—	2	4	no	—
176	C₂²×C₄×C₆₇	—	2	4	no	—
177	C₂²×C₄×C₆₇	—	2	4	no	—
178	C₂²×C₄×C₆₇	—	2	4	no	—
179	C₂²×C₄×C₆₇	—	2	2	no	—
180	C₂²×C₄×C₆₇	—	2	2	no	—
181	C₂²×C₄×C₆₇	—	2	4	no	—
182	C₂²×C₄×C₆₇	—	2	4	no	—
183	C₂²×C₄×C₆₇	—	2	4	no	—
184	C₂²×C₄×C₆₇	—	2	8	no	—
185	C₂²×C₄×C₆₇	—	2	8	no	—
186	C₂²×C₄×C₆₇	—	2	8	no	—
187	C₂²×C₄×C₆₇	—	2	16	no	—
188	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
189	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
190	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
191	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
192	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
194	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
195	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
196	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
197	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
198	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
199	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
201	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
202	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
203	C₄⋊C₄×C₆₇	—	2	4	no	—
204	C₄⋊C₄×C₆₇	—	2	4	no	—
205	C₄⋊C₄×C₆₇	—	2	2	no	—
206	C₄⋊C₄×C₆₇	—	2	2	no	—
207	C₄⋊C₄×C₆₇	—	2	4	no	—
208	C₄⋊C₄×C₆₇	—	2	4	no	—
209	C₄⋊C₄×C₆₇	—	2	4	no	—
210	C₄⋊C₄×C₆₇	—	4	8	no	—
211	C₄⋊C₄×C₆₇	#41	1	8	yes	complemented
212	C₂²⋊C₄×C₆₇	—	2	4	no	—
213	C₂²⋊C₄×C₆₇	—	2	4	no	—
214	C₂²⋊C₄×C₆₇	—	2	4	no	—
215	C₂²⋊C₄×C₆₇	—	2	8	no	—
216	C₂²⋊C₄×C₆₇	—	2	8	no	—
217	C₂²⋊C₄×C₆₇	—	2	8	no	—
218	C₂²⋊C₄×C₆₇	—	2	16	no	—
219	C₂²⋊C₄×C₆₇	—	2	16	no	—
220	C₂²⋊C₄×C₆₇	—	4	8	no	—
221	C₂²⋊C₄×C₆₇	—	4	8	no	—
222	C₄²×C₆₇	—	2	4	no	—
223	C₄²×C₆₇	—	2	4	no	—
224	C₄²×C₆₇	—	2	4	no	—
225	C₄²×C₆₇	—	2	8	no	—
226	C₄²×C₆₇	—	2	4	no	—
227	C₄²×C₆₇	—	2	4	no	—
228	C₄²×C₆₇	—	2	8	no	—
229	C₄²×C₆₇	C₂×D₄	1	4	yes	—
230	C₄²×C₆₇	C₂×D₄	1	4	yes	—
231	C₄²×C₆₇	C₂×D₄	1	4	yes	—
232	C₄²×C₆₇	C₂×D₄	1	4	yes	—
233	C₂³×C₆₇	—	2	2	no	—
234	C₂³×C₆₇	—	2	2	no	—
235	C₂³×C₆₇	—	2	1	no	—
236	C₂³×C₆₇	—	2	1	no	—
237	C₂³×C₆₇	—	2	2	no	—
238	C₂³×C₆₇	—	2	4	no	—
239	C₂³×C₆₇	—	2	4	no	—
240	C₂³×C₆₇	—	2	2	no	—
241	C₂³×C₆₇	—	2	4	no	—
242	C₂³×C₆₇	—	2	4	no	—
243	C₂³×C₆₇	—	2	8	no	—
244	C₂³×C₆₇	—	2	4	no	—
245	C₂³×C₆₇	—	2	8	no	—
246	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
247	C₂³×C₆₇	C₂×#41	1	4	yes	—
248	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
249	Q₈×C₆₇	—	2	8	no	—
250	Q₈×C₆₇	—	4	8	no	—
251	D₄×C₆₇	—	2	16	no	—
252	D₄×C₆₇	—	2	8	no	—
253	D₄×C₆₇	—	4	32	no	—
254	D₄×C₆₇	—	4	8	no	—
255	D₄×C₆₇	—	4	16	no	—
256	D₄×C₆₇	—	4	8	no	—
257	D₄×C₆₇	—	4	8	no	—
258	C₂×C₄×C₆₇	—	2	16	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	2	no	—
261	C₂×C₄×C₆₇	—	2	4	no	—
262	C₂×C₄×C₆₇	—	2	1	no	—
263	C₂×C₄×C₆₇	—	2	1	no	—
264	C₂×C₄×C₆₇	—	2	2	no	—
265	C₂×C₄×C₆₇	—	2	4	no	—
266	C₂×C₄×C₆₇	—	2	2	no	—
267	C₂×C₄×C₆₇	—	2	8	no	—
268	C₂×C₄×C₆₇	—	2	8	no	—
269	C₂×C₄×C₆₇	—	2	8	no	—
270	C₂×C₄×C₆₇	—	2	2	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	2	4	no	—
273	C₂×C₄×C₆₇	—	2	4	no	—
274	C₂×C₄×C₆₇	—	2	4	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	8	no	—
279	C₂×C₄×C₆₇	—	4	8	no	—
280	C₂×C₄×C₆₇	—	4	8	no	—
281	C₂×C₄×C₆₇	—	4	16	no	—
282	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
283	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
284	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
285	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
286	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
287	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
288	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
289	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
290	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
291	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
292	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
293	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
294	C₂²×C₆₇	—	2	2	no	—
295	C₂²×C₆₇	—	2	4	no	—
296	C₂²×C₆₇	—	2	2	no	—
297	C₂²×C₆₇	—	2	2	no	—
298	C₂²×C₆₇	—	2	8	no	—
299	C₂²×C₆₇	—	4	2	no	—
300	C₂²×C₆₇	—	4	2	no	—
301	C₂²×C₆₇	—	4	2	no	—
302	C₂²×C₆₇	—	4	2	no	—
303	C₂²×C₆₇	—	4	4	no	—
304	C₂²×C₆₇	—	4	4	no	—
305	C₂²×C₆₇	—	4	4	no	—
306	C₂²×C₆₇	—	4	4	no	—
307	C₂²×C₆₇	—	4	8	no	—
308	C₂²×C₆₇	—	4	8	no	—
309	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
310	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
311	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
312	C₂²×C₆₇	C₂²×#41	1	2	yes	—
313	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
314	C₄×C₆₇	—	2	2	no	—
315	C₄×C₆₇	—	2	4	no	—
316	C₄×C₆₇	—	2	2	no	—
317	C₄×C₆₇	—	2	2	no	—
318	C₄×C₆₇	—	2	8	no	—
319	C₄×C₆₇	—	4	2	no	—
320	C₄×C₆₇	—	4	2	no	—
321	C₄×C₆₇	—	4	4	no	—
322	C₄×C₆₇	—	4	4	no	—
323	C₄×C₆₇	#525	1	4	yes	—
324	C₄×C₆₇	#532	1	4	yes	—
325	C₄×C₆₇	#535	1	4	yes	—
326	#34977	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
327	C₂×C₆₇	—	4	2	no	—
328	C₂×C₆₇	—	4	4	no	—
329	C₂×C₆₇	—	8	4	no	—
330	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
331	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
332	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
333	C₂×C₆₇	#3481	1	2	yes	—
334	C₂×C₆₇	#3483	1	2	yes	—
335	C₂×#532	C₂×C₆₇	1	1	yes	characteristic, complemented
336	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
337	#2610	C₂×C₆₇	1	4	yes	complemented
338	#2567	C₂×C₆₇	1	1	yes	characteristic, complemented
339	#2563	C₂×C₆₇	1	1	yes	characteristic, complemented
340	#2548	C₂×C₆₇	1	1	yes	characteristic, complemented
341	#2541	C₂×C₆₇	1	1	yes	characteristic, complemented
342	#2500	C₂×C₆₇	1	2	yes	complemented
343	#2431	C₂×C₆₇	1	2	yes	complemented
344	#2414	C₂×C₆₇	1	4	yes	complemented
345	#2388	C₂×C₆₇	1	4	yes	complemented
346	C₄×#127	C₂×C₆₇	1	2	yes	complemented
347	C₄×#124	C₂×C₆₇	1	2	yes	complemented
348	C₄×#121	C₂×C₆₇	1	4	yes	complemented
349	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
350	C₆₇	#34977	1	1	yes	characteristic, direct product factor, Sylow
351	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
352	C₂²×#41	C₂²×C₆₇	1	2	yes	—
353	#532	—	2	8	no	—
354	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
355	C₂×#127	C₂²×C₆₇	1	2	yes	—
356	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
357	C₂×#124	C₂²×C₆₇	1	2	yes	—
358	C₂×#124	C₂²×C₆₇	1	2	yes	—
359	C₂×#121	C₂²×C₆₇	1	2	yes	—
360	C₂×#121	C₂²×C₆₇	1	4	yes	—
361	C₂×#121	C₂²×C₆₇	1	2	yes	—
362	C₄×#41	—	2	8	no	—
363	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
364	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
365	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
366	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
367	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
368	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
369	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
370	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
371	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
372	C₂×#117	C₂²×C₆₇	1	2	yes	—
373	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
374	C₂×#117	C₂²×C₆₇	1	2	yes	—
375	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
376	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
377	#395	C₂²×C₆₇	1	4	yes	—
378	#394	C₂²×C₆₇	1	4	yes	—
379	#394	C₂²×C₆₇	1	8	yes	—
380	#393	C₂²×C₆₇	1	4	yes	—
381	#392	C₂²×C₆₇	1	4	yes	—
382	#392	C₂²×C₆₇	1	8	yes	—
383	#391	C₂²×C₆₇	1	2	yes	—
384	#390	C₂²×C₆₇	1	2	yes	—
385	#390	C₂²×C₆₇	1	8	yes	—
386	#390	C₂²×C₆₇	1	2	yes	—
387	#390	C₂²×C₆₇	1	4	yes	—
388	#390	C₂²×C₆₇	1	2	yes	—
389	#389	C₂²×C₆₇	1	4	yes	—
390	#388	C₂²×C₆₇	1	8	yes	—
391	#388	C₂²×C₆₇	1	8	yes	—
392	#386	C₂²×C₆₇	1	8	yes	—
393	#384	C₂²×C₆₇	1	2	yes	—
394	#382	C₂²×C₆₇	1	2	yes	—
395	#380	C₂²×C₆₇	1	4	yes	—
396	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
397	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
398	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
399	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
400	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
401	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
402	#376	C₂²×C₆₇	1	2	yes	—
403	C₄³	C₂²×C₆₇	1	2	yes	—
404	C₂×#41	—	2	4	no	—
405	C₂×#41	—	2	4	no	—
406	C₂×#41	—	2	4	no	—
407	C₂×#41	—	2	8	no	—
408	C₂×#41	—	4	8	no	—
409	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
410	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
411	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
412	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
413	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
414	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
415	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
416	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
417	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
418	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
419	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
420	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
421	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
422	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
423	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
424	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
425	C₄⋊Q₈	—	2	4	no	—
426	#127	—	2	8	no	—
427	#127	—	2	4	no	—
428	#124	—	2	8	no	—
429	#124	—	2	8	no	—
430	#121	—	2	8	no	—
431	#121	—	2	8	no	—
432	#121	—	2	16	no	—
433	C₄×Q₈	—	2	4	no	—
434	C₄×Q₈	—	2	4	no	—
435	C₄×D₄	—	2	8	no	—
436	C₄×D₄	—	2	8	no	—
437	C₄×D₄	—	2	4	no	—
438	C₄×D₄	—	2	4	no	—
439	C₄×D₄	—	2	8	no	—
440	C₄×D₄	—	2	8	no	—
441	C₄×D₄	—	4	16	no	—
442	#117	—	2	4	no	—
443	#117	—	2	8	no	—
444	#117	—	2	8	no	—
445	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
447	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
448	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
449	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
450	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
451	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
452	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
453	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
454	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
455	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
456	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
457	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
458	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
459	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
460	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	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×C₄²	—	2	8	no	—
465	C₂×C₄²	D₄×C₆₇	1	8	yes	complemented
466	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
467	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
468	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
469	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
470	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
471	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
472	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
473	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
474	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
475	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
476	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
477	#95	C₂³×C₆₇	1	4	yes	—
478	#95	C₂³×C₆₇	1	2	yes	—
479	#95	C₂³×C₆₇	1	2	yes	—
480	#95	C₂³×C₆₇	1	8	yes	—
481	#95	C₂³×C₆₇	1	8	yes	—
482	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
483	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
484	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
485	#41	—	2	32	no	—
486	#41	—	4	32	no	—
487	C₂×Q₈	—	2	2	no	—
488	C₂×Q₈	—	2	4	no	—
489	C₂×Q₈	—	2	2	no	—
490	C₂×Q₈	—	4	4	no	—
491	C₂×Q₈	#41×C₆₇	1	4	yes	complemented
492	C₂×D₄	—	2	4	no	—
493	C₂×D₄	—	2	4	no	—
494	C₂×D₄	—	2	8	no	—
495	C₂×D₄	—	2	16	no	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	2	4	no	—
498	C₂×D₄	—	2	4	no	—
499	C₂×D₄	—	2	4	no	—
500	C₂×D₄	—	2	4	no	—
501	C₂×D₄	—	2	2	no	—
502	C₂×D₄	—	2	16	no	—
503	C₂×D₄	—	2	8	no	—
504	C₂×D₄	—	2	4	no	—
505	C₂×D₄	—	2	4	no	—
506	C₂×D₄	—	4	8	no	—
507	C₂×D₄	—	4	4	no	—
508	C₂×D₄	—	4	8	no	—
509	C₂×D₄	#41×C₆₇	1	4	yes	complemented
510	C₂²×C₄	—	2	4	no	—
511	C₂²×C₄	—	2	4	no	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	2	4	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	2	2	no	—
516	C₂²×C₄	—	2	4	no	—
517	C₂²×C₄	—	2	4	no	—
518	C₂²×C₄	—	2	4	no	—
519	C₂²×C₄	—	2	8	no	—
520	C₂²×C₄	—	2	8	no	—
521	C₂²×C₄	—	2	8	no	—
522	C₂²×C₄	—	2	16	no	—
523	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
524	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
525	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
526	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
527	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
528	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
529	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
530	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
531	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
532	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
533	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
534	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
535	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
536	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
537	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
538	C₄⋊C₄	—	2	4	no	—
539	C₄⋊C₄	—	2	4	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	—	2	4	no	—
543	C₄⋊C₄	—	2	4	no	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	4	8	no	—
546	C₄⋊C₄	#41×C₆₇	1	8	yes	complemented
547	C₂²⋊C₄	—	2	4	no	—
548	C₂²⋊C₄	—	2	4	no	—
549	C₂²⋊C₄	—	2	4	no	—
550	C₂²⋊C₄	—	2	8	no	—
551	C₂²⋊C₄	—	2	8	no	—
552	C₂²⋊C₄	—	2	8	no	—
553	C₂²⋊C₄	—	2	16	no	—
554	C₂²⋊C₄	—	2	16	no	—
555	C₂²⋊C₄	—	4	8	no	—
556	C₂²⋊C₄	—	4	8	no	—
557	C₄²	—	2	4	no	—
558	C₄²	—	2	4	no	—
559	C₄²	—	2	4	no	—
560	C₄²	—	2	8	no	—
561	C₄²	—	2	4	no	—
562	C₄²	—	2	4	no	—
563	C₄²	—	2	8	no	—
564	C₄²	C₂×D₄×C₆₇	1	4	yes	—
565	C₄²	C₂×D₄×C₆₇	1	4	yes	—
566	C₄²	C₂×D₄×C₆₇	1	4	yes	—
567	C₄²	C₂×D₄×C₆₇	1	4	yes	—
568	C₂³	—	2	2	no	—
569	C₂³	—	2	2	no	—
570	C₂³	—	2	1	no	—
571	C₂³	—	2	1	no	—
572	C₂³	—	2	2	no	—
573	C₂³	—	2	4	no	—
574	C₂³	—	2	4	no	—
575	C₂³	—	2	2	no	—
576	C₂³	—	2	4	no	—
577	C₂³	—	2	4	no	—
578	C₂³	—	2	8	no	—
579	C₂³	—	2	4	no	—
580	C₂³	—	2	8	no	—
581	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
582	C₂³	C₂×#41×C₆₇	1	4	yes	—
583	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
584	Q₈	—	2	8	no	—
585	Q₈	—	4	8	no	—
586	D₄	—	2	16	no	—
587	D₄	—	2	8	no	—
588	D₄	—	4	32	no	—
589	D₄	—	4	8	no	—
590	D₄	—	4	16	no	—
591	D₄	—	4	8	no	—
592	D₄	—	4	8	no	—
593	C₂×C₄	—	2	16	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	8	no	—
603	C₂×C₄	—	2	8	no	—
604	C₂×C₄	—	2	8	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	4	no	—
609	C₂×C₄	—	2	4	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	4	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	8	no	—
614	C₂×C₄	—	4	8	no	—
615	C₂×C₄	—	4	16	no	—
616	C₂×C₄	—	4	8	no	—
617	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
618	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
619	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
620	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
621	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
622	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
623	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
624	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
625	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
626	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
627	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
628	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	4	no	—
631	C₂²	—	2	2	no	—
632	C₂²	—	2	2	no	—
633	C₂²	—	2	8	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	4	4	no	—
639	C₂²	—	4	4	no	—
640	C₂²	—	4	4	no	—
641	C₂²	—	4	4	no	—
642	C₂²	—	4	8	no	—
643	C₂²	—	4	8	no	—
644	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
645	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
646	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
647	C₂²	C₂²×#41×C₆₇	1	2	yes	—
648	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
649	C₄	—	2	2	no	—
650	C₄	—	2	4	no	—
651	C₄	—	2	2	no	—
652	C₄	—	2	2	no	—
653	C₄	—	2	8	no	—
654	C₄	—	4	2	no	—
655	C₄	—	4	4	no	—
656	C₄	—	4	2	no	—
657	C₄	—	4	4	no	—
658	C₄	#525×C₆₇	1	4	yes	—
659	C₄	#532×C₆₇	1	4	yes	—
660	C₄	#535×C₆₇	1	4	yes	—
661	C₂	—	4	2	no	—
662	C₂	—	4	4	no	—
663	C₂	—	8	4	no	—
664	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
665	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
666	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
667	C₂	C₆₇×#3481	1	2	yes	—
668	C₂	C₆₇×#3483	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#532×C₆₇	C₂	semidirect
2. C₂×C₄×#41×C₆₇	C₂	semidirect
3. C₆₇×#2610	C₂	semidirect
4. C₆₇×#2567	C₂	semidirect
5. C₆₇×#2563	C₂	semidirect
6. C₆₇×#2548	C₂	semidirect
7. C₆₇×#2541	C₂	semidirect
8. C₆₇×#2500	C₂	semidirect
9. C₆₇×#2431	C₂	semidirect
10. C₆₇×#2414	C₂	semidirect
11. C₆₇×#2388	C₂	semidirect
12. C₄×#127×C₆₇	C₂	semidirect
13. C₄×#124×C₆₇	C₂	semidirect
14. C₄×#121×C₆₇	C₂	semidirect
15. C₄×#117×C₆₇	C₂	semidirect
130. C₂×C₄²×C₆₇	D₄	semidirect
156. C₂×Q₈×C₆₇	#41	semidirect
174. C₂×D₄×C₆₇	#41	semidirect
211. C₄⋊C₄×C₆₇	#41	semidirect
326. #34977	C₆₇	direct (N × H)
335. C₂×#532	C₂×C₆₇	semidirect
336. C₂×C₄×#41	C₂×C₆₇	semidirect
337. #2610	C₂×C₆₇	semidirect
338. #2567	C₂×C₆₇	semidirect
339. #2563	C₂×C₆₇	semidirect
340. #2548	C₂×C₆₇	semidirect
341. #2541	C₂×C₆₇	semidirect
342. #2500	C₂×C₆₇	semidirect
343. #2431	C₂×C₆₇	semidirect
344. #2414	C₂×C₆₇	semidirect
345. #2388	C₂×C₆₇	semidirect
346. C₄×#127	C₂×C₆₇	semidirect
347. C₄×#124	C₂×C₆₇	semidirect
348. C₄×#121	C₂×C₆₇	semidirect
349. C₄×#117	C₂×C₆₇	semidirect
350. C₆₇	#34977	direct (N × H)
465. C₂×C₄²	D₄×C₆₇	semidirect
491. C₂×Q₈	#41×C₆₇	semidirect
509. C₂×D₄	#41×C₆₇	semidirect
546. C₄⋊C₄	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#532×C₆₇	2
C₂×C₄×#41×C₆₇	2
C₆₇×#2610	2
C₆₇×#2567	2
C₆₇×#2563	2
C₆₇×#2548	2
C₆₇×#2541	2
C₆₇×#2500	2
C₆₇×#2431	2
C₆₇×#2414	2
C₆₇×#2388	2
C₄×#127×C₆₇	2
C₄×#124×C₆₇	2
C₄×#121×C₆₇	2
C₄×#117×C₆₇	2
#34977	67

Subgroups — C67×#34977 #30714228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34977 #30714228