Subgroups — C₆₇×#26040 #30705291

Order 17152.

914 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#474×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2551	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2086	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2057	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1862	C₂	1	2	yes	maximal
9	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
17	#474×C₆₇	—	2	8	no	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×#107×C₆₇	C₂²	1	2	yes	—
20	C₂×#107×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#102×C₆₇	C₂²	1	2	yes	—
24	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#395×C₆₇	—	2	2	no	—
28	#389×C₆₇	—	2	1	no	—
29	#389×C₆₇	C₂²	1	2	yes	—
30	#389×C₆₇	C₂²	1	2	yes	—
31	#385×C₆₇	—	2	2	no	—
32	#384×C₆₇	C₂²	1	1	yes	characteristic
33	#384×C₆₇	C₂²	1	1	yes	characteristic
34	#380×C₆₇	—	2	1	no	—
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	#340×C₆₇	C₂²	1	1	yes	characteristic
38	#340×C₆₇	C₂²	1	1	yes	characteristic
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	C₂²×Q₈×C₆₇	—	2	2	no	—
43	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	—	2	1	no	—
45	C₂³×C₄×C₆₇	—	2	2	no	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₇	—	2	4	no	—
50	C₄⋊Q₈×C₆₇	—	4	2	no	—
51	C₄⋊Q₈×C₆₇	—	4	4	no	—
52	C₄×D₄×C₆₇	—	2	4	no	—
53	C₄×D₄×C₆₇	—	2	2	no	—
54	C₄×D₄×C₆₇	—	4	2	no	—
55	C₄×D₄×C₆₇	Q₈	1	4	yes	complemented
56	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
64	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	—	2	1	no	—
74	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#107×C₆₇	—	2	4	no	—
77	#107×C₆₇	—	2	4	no	—
78	#107×C₆₇	—	4	2	no	—
79	C₄⋊C₈×C₆₇	—	2	4	no	—
80	C₄⋊C₈×C₆₇	—	2	2	no	—
81	#102×C₆₇	—	2	4	no	—
82	#102×C₆₇	D₄	1	4	yes	—
83	#102×C₆₇	Q₈	1	4	yes	complemented
84	C₄×C₈×C₆₇	—	2	2	no	—
85	#95×C₆₇	—	2	1	no	—
86	#95×C₆₇	—	2	2	no	—
87	#95×C₆₇	—	2	2	no	—
88	#95×C₆₇	D₄	1	2	yes	—
89	C₂⁴×C₆₇	—	2	1	no	—
90	C₂×Q₈×C₆₇	—	4	4	no	—
91	C₂×Q₈×C₆₇	—	4	4	no	—
92	C₂×Q₈×C₆₇	—	8	2	no	—
93	C₂×D₄×C₆₇	—	2	1	no	—
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	—	2	1	no	—
96	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
97	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
98	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
99	C₂²×C₄×C₆₇	—	2	1	no	—
100	C₂²×C₄×C₆₇	—	2	1	no	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	4	4	no	—
106	C₂²×C₄×C₆₇	—	4	2	no	—
107	C₂²×C₄×C₆₇	—	4	2	no	—
108	C₂²×C₄×C₆₇	—	4	4	no	—
109	C₂²×C₄×C₆₇	—	8	2	no	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
115	C₂×C₈×C₆₇	—	2	1	no	—
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	1	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	2	2	no	—
120	C₂×C₈×C₆₇	—	2	2	no	—
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	1	no	—
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	—	2	1	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	4	4	no	—
133	C₄⋊C₄×C₆₇	—	4	4	no	—
134	C₄⋊C₄×C₆₇	—	4	2	no	—
135	C₄⋊C₄×C₆₇	—	4	4	no	—
136	C₄⋊C₄×C₆₇	—	4	2	no	—
137	C₄⋊C₄×C₆₇	—	8	1	no	—
138	C₄⋊C₄×C₆₇	—	8	1	no	—
139	C₄⋊C₄×C₆₇	D₈	1	4	yes	—
140	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
141	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
142	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
143	C₂²⋊C₄×C₆₇	—	2	4	no	—
144	C₂²⋊C₄×C₆₇	—	4	4	no	—
145	C₄²×C₆₇	—	2	2	no	—
146	C₄²×C₆₇	—	2	2	no	—
147	C₄²×C₆₇	—	4	2	no	—
148	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
149	C₄²×C₆₇	#41	1	2	yes	—
150	C₂³×C₆₇	—	2	2	no	—
151	C₂³×C₆₇	—	2	2	no	—
152	C₂³×C₆₇	—	2	2	no	—
153	C₂³×C₆₇	—	4	1	no	—
154	C₂³×C₆₇	—	4	1	no	—
155	C₂³×C₆₇	—	4	1	no	—
156	C₂³×C₆₇	—	4	1	no	—
157	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
158	Q₈×C₆₇	—	8	8	no	—
159	D₄×C₆₇	—	2	2	no	—
160	D₄×C₆₇	—	2	2	no	—
161	D₄×C₆₇	—	2	2	no	—
162	D₄×C₆₇	C₄⋊Q₈	1	4	yes	complemented
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	2	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	1	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	—	4	1	no	—
177	C₂×C₄×C₆₇	—	4	1	no	—
178	C₂×C₄×C₆₇	—	4	1	no	—
179	C₂×C₄×C₆₇	—	4	8	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	8	8	no	—
187	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
188	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
189	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
190	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
191	C₂×C₄×C₆₇	#137	1	2	yes	—
192	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
198	C₈×C₆₇	—	2	2	no	—
199	C₈×C₆₇	—	2	2	no	—
200	C₈×C₆₇	—	4	2	no	—
201	C₂²×C₆₇	—	2	4	no	—
202	C₂²×C₆₇	—	4	2	no	—
203	C₂²×C₆₇	—	4	2	no	—
204	C₂²×C₆₇	—	4	2	no	—
205	C₂²×C₆₇	—	4	2	no	—
206	C₂²×C₆₇	—	4	2	no	—
207	C₂²×C₆₇	—	4	2	no	—
208	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
209	C₂²×C₆₇	D₄²	1	1	yes	characteristic
210	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
211	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
212	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
213	C₂²×C₆₇	#576	1	1	yes	characteristic
214	C₂²×C₆₇	#578	1	1	yes	characteristic
215	C₄×C₆₇	—	2	2	no	—
216	C₄×C₆₇	—	2	4	no	—
217	C₄×C₆₇	—	4	2	no	—
218	C₄×C₆₇	—	4	2	no	—
219	C₄×C₆₇	—	4	2	no	—
220	C₄×C₆₇	—	4	2	no	—
221	C₄×C₆₇	—	8	2	no	—
222	C₄×C₆₇	—	8	2	no	—
223	C₄×C₆₇	#474	1	4	yes	complemented
224	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
225	C₄×C₆₇	#544	1	1	yes	characteristic
226	C₄×C₆₇	#547	1	1	yes	characteristic
227	C₄×C₆₇	#554	1	1	yes	characteristic
228	#26040	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂×C₆₇	—	4	4	no	—
230	C₂×C₆₇	#2440	1	1	yes	characteristic
231	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
232	C₂×C₆₇	#3083	1	1	yes	characteristic
233	C₂×C₆₇	#3276	1	1	yes	characteristic
234	C₂×C₆₇	D₄×D₈	1	1	yes	characteristic
235	C₂×C₆₇	Q₈×D₈	1	1	yes	characteristic
236	C₂×C₆₇	#3385	1	1	yes	characteristic
237	C₂×#474	C₂×C₆₇	1	2	yes	—
238	#2551	C₂×C₆₇	1	1	yes	characteristic, complemented
239	D₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
240	#2086	C₂×C₆₇	1	2	yes	complemented
241	#2057	C₂×C₆₇	1	2	yes	complemented
242	#1947	C₂×C₆₇	1	1	yes	characteristic, complemented
243	#1939	C₂×C₆₇	1	1	yes	characteristic, complemented
244	#1862	C₂×C₆₇	1	2	yes	—
245	#1845	C₂×C₆₇	1	1	yes	characteristic, complemented
246	#1802	C₂×C₆₇	1	1	yes	characteristic
247	C₄×#102	C₂×C₆₇	1	1	yes	characteristic
248	C₆₇	#26040	1	1	yes	characteristic, direct product factor, Sylow
249	C₂×C₄⋊Q₈	—	2	1	no	—
250	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
251	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
252	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
253	C₂²×C₄⋊C₄	—	2	1	no	—
254	#474	—	2	8	no	—
255	C₂×#107	C₂²×C₆₇	1	2	yes	—
256	C₂×#107	C₂²×C₆₇	1	2	yes	—
257	C₂×#107	C₂²×C₆₇	1	2	yes	—
258	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
259	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
260	C₂×#102	C₂²×C₆₇	1	2	yes	—
261	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
262	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
263	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
264	#395	—	2	2	no	—
265	#389	—	2	1	no	—
266	#389	C₂²×C₆₇	1	2	yes	—
267	#389	C₂²×C₆₇	1	2	yes	—
268	#385	—	2	2	no	—
269	#384	C₂²×C₆₇	1	1	yes	characteristic
270	#384	C₂²×C₆₇	1	1	yes	characteristic
271	#380	—	2	1	no	—
272	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
273	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
274	#340	C₂²×C₆₇	1	1	yes	characteristic
275	#340	C₂²×C₆₇	1	2	yes	—
276	#340	C₂²×C₆₇	1	1	yes	characteristic
277	#340	C₂²×C₆₇	1	2	yes	—
278	#336	C₂²×C₆₇	1	2	yes	—
279	C₂²×Q₈	—	2	2	no	—
280	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
281	C₂³×C₄	—	2	1	no	—
282	C₂³×C₄	—	2	2	no	—
283	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
284	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
285	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
286	C₄⋊Q₈	—	2	4	no	—
287	C₄⋊Q₈	—	4	2	no	—
288	C₄⋊Q₈	—	4	4	no	—
289	C₄×D₄	—	2	4	no	—
290	C₄×D₄	—	2	2	no	—
291	C₄×D₄	—	4	2	no	—
292	C₄×D₄	Q₈×C₆₇	1	4	yes	complemented
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	2	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	4	4	no	—
301	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
302	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
304	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
305	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
306	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₂²⋊C₄	—	2	2	no	—
310	C₂×C₄²	—	2	1	no	—
311	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
312	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
313	#107	—	2	4	no	—
314	#107	—	2	4	no	—
315	#107	—	4	2	no	—
316	C₄⋊C₈	—	2	4	no	—
317	C₄⋊C₈	—	2	2	no	—
318	#102	—	2	4	no	—
319	#102	D₄×C₆₇	1	4	yes	—
320	#102	Q₈×C₆₇	1	4	yes	complemented
321	C₄×C₈	—	2	2	no	—
322	#95	—	2	1	no	—
323	#95	—	2	2	no	—
324	#95	—	2	2	no	—
325	#95	D₄×C₆₇	1	2	yes	—
326	C₂⁴	—	2	1	no	—
327	C₂×Q₈	—	4	4	no	—
328	C₂×Q₈	—	4	4	no	—
329	C₂×Q₈	—	8	2	no	—
330	C₂×D₄	—	2	1	no	—
331	C₂×D₄	—	2	1	no	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
334	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
335	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	4	4	no	—
343	C₂²×C₄	—	4	2	no	—
344	C₂²×C₄	—	4	2	no	—
345	C₂²×C₄	—	4	4	no	—
346	C₂²×C₄	—	8	2	no	—
347	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
348	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
350	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
351	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
360	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	4	4	no	—
370	C₄⋊C₄	—	4	4	no	—
371	C₄⋊C₄	—	4	2	no	—
372	C₄⋊C₄	—	4	4	no	—
373	C₄⋊C₄	—	4	2	no	—
374	C₄⋊C₄	—	8	1	no	—
375	C₄⋊C₄	—	8	1	no	—
376	C₄⋊C₄	D₈×C₆₇	1	4	yes	—
377	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
378	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
379	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
380	C₂²⋊C₄	—	2	4	no	—
381	C₂²⋊C₄	—	4	4	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	4	2	no	—
385	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
386	C₄²	#41×C₆₇	1	2	yes	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	4	1	no	—
391	C₂³	—	4	1	no	—
392	C₂³	—	4	1	no	—
393	C₂³	—	4	1	no	—
394	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
395	Q₈	—	8	8	no	—
396	D₄	—	2	2	no	—
397	D₄	—	2	2	no	—
398	D₄	—	2	2	no	—
399	D₄	C₄⋊Q₈×C₆₇	1	4	yes	complemented
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	8	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	8	8	no	—
424	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
425	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
426	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
427	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
428	C₂×C₄	#137×C₆₇	1	2	yes	—
429	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
430	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
431	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
433	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
434	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
435	C₈	—	2	2	no	—
436	C₈	—	2	2	no	—
437	C₈	—	4	2	no	—
438	C₂²	—	2	4	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	2	no	—
445	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
446	C₂²	D₄²×C₆₇	1	1	yes	characteristic
447	C₂²	D₄×Q₈×C₆₇	1	1	yes	characteristic
448	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
449	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
450	C₂²	#576×C₆₇	1	1	yes	characteristic
451	C₂²	#578×C₆₇	1	1	yes	characteristic
452	C₄	—	2	4	no	—
453	C₄	—	2	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	2	no	—
457	C₄	—	4	2	no	—
458	C₄	—	8	2	no	—
459	C₄	—	8	2	no	—
460	C₄	#474×C₆₇	1	4	yes	complemented
461	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
462	C₄	#544×C₆₇	1	1	yes	characteristic
463	C₄	#547×C₆₇	1	1	yes	characteristic
464	C₄	#554×C₆₇	1	1	yes	characteristic
465	C₂	—	4	4	no	—
466	C₂	C₆₇×#2440	1	1	yes	characteristic
467	C₂	C₂×#474×C₆₇	1	1	yes	characteristic
468	C₂	C₆₇×#3083	1	1	yes	characteristic
469	C₂	C₆₇×#3276	1	1	yes	characteristic
470	C₂	D₄×D₈×C₆₇	1	1	yes	characteristic
471	C₂	Q₈×D₈×C₆₇	1	1	yes	characteristic
472	C₂	C₆₇×#3385	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄×C₄⋊C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2551	C₂	semidirect
4. C₆₇×#2086	C₂	semidirect
5. C₆₇×#2057	C₂	semidirect
6. C₆₇×#1947	C₂	semidirect
7. C₆₇×#1939	C₂	semidirect
9. C₆₇×#1845	C₂	semidirect
55. C₄×D₄×C₆₇	Q₈	semidirect
83. #102×C₆₇	Q₈	semidirect
162. D₄×C₆₇	C₄⋊Q₈	semidirect
223. C₄×C₆₇	#474	semidirect
228. #26040	C₆₇	direct (N × H)
238. #2551	C₂×C₆₇	semidirect
240. #2086	C₂×C₆₇	semidirect
241. #2057	C₂×C₆₇	semidirect
242. #1947	C₂×C₆₇	semidirect
243. #1939	C₂×C₆₇	semidirect
245. #1845	C₂×C₆₇	semidirect
248. C₆₇	#26040	direct (N × H)
292. C₄×D₄	Q₈×C₆₇	semidirect
320. #102	Q₈×C₆₇	semidirect
399. D₄	C₄⋊Q₈×C₆₇	semidirect
460. C₄	#474×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#474×C₆₇	2
C₆₇×#2551	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#2086	2
C₆₇×#2057	2
C₆₇×#1947	2
C₆₇×#1939	2
C₆₇×#1862	2
C₆₇×#1845	2
C₆₇×#1802	2
C₄×#102×C₆₇	2
#26040	67

Subgroups — C67×#26040 #30705291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#26040 #30705291