Subgroups — C₆₇×#26111 #30705362

Order 17152.

770 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2571	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2101	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2087	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1947	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1862	C₂	1	2	yes	maximal
8	C₆₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1804	C₂	1	2	yes	maximal
10	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#102×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#400×C₆₇	—	2	2	no	—
23	#389×C₆₇	—	2	1	no	—
24	#389×C₆₇	C₂²	1	2	yes	—
25	#389×C₆₇	C₂²	1	2	yes	—
26	#385×C₆₇	—	2	2	no	—
27	#384×C₆₇	C₂²	1	1	yes	characteristic
28	#384×C₆₇	C₂²	1	1	yes	characteristic
29	#382×C₆₇	—	2	1	no	—
30	#382×C₆₇	C₂²	1	2	yes	—
31	#380×C₆₇	—	2	1	no	—
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	#340×C₆₇	C₂²	1	1	yes	characteristic
35	#340×C₆₇	C₂²	1	1	yes	characteristic
36	#340×C₆₇	C₂²	1	2	yes	—
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	—	2	1	no	—
43	C₂³×C₄×C₆₇	—	2	2	no	—
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₄×D₄×C₆₇	—	2	4	no	—
48	C₄×D₄×C₆₇	—	2	2	no	—
49	C₄×D₄×C₆₇	—	4	2	no	—
50	C₄×D₄×C₆₇	Q₈	1	4	yes	—
51	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
58	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₄²×C₆₇	—	2	1	no	—
68	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	#107×C₆₇	—	4	2	no	—
71	C₄⋊C₈×C₆₇	—	2	2	no	—
72	C₄⋊C₈×C₆₇	—	4	2	no	—
73	#102×C₆₇	—	2	2	no	—
74	#102×C₆₇	—	2	4	no	—
75	#102×C₆₇	—	2	2	no	—
76	C₄×C₈×C₆₇	—	2	2	no	—
77	#95×C₆₇	—	2	1	no	—
78	#95×C₆₇	—	2	2	no	—
79	#95×C₆₇	—	2	2	no	—
80	#95×C₆₇	—	2	2	no	—
81	#95×C₆₇	—	2	2	no	—
82	#95×C₆₇	D₄	1	2	yes	—
83	C₂⁴×C₆₇	—	2	1	no	—
84	C₂×D₄×C₆₇	—	2	1	no	—
85	C₂×D₄×C₆₇	—	2	1	no	—
86	C₂×D₄×C₆₇	—	2	1	no	—
87	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
88	C₂×D₄×C₆₇	#41	1	2	yes	—
89	C₂×D₄×C₆₇	#41	1	2	yes	—
90	C₂²×C₄×C₆₇	—	2	1	no	—
91	C₂²×C₄×C₆₇	—	2	4	no	—
92	C₂²×C₄×C₆₇	—	2	1	no	—
93	C₂²×C₄×C₆₇	—	2	1	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	4	4	no	—
97	C₂²×C₄×C₆₇	—	4	4	no	—
98	C₂²×C₄×C₆₇	—	4	2	no	—
99	C₂²×C₄×C₆₇	—	4	2	no	—
100	C₂²×C₄×C₆₇	—	8	2	no	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	1	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	1	no	—
111	C₂×C₈×C₆₇	—	2	1	no	—
112	C₂×C₈×C₆₇	—	2	1	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	1	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	1	no	—
118	C₄⋊C₄×C₆₇	—	2	1	no	—
119	C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	1	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	4	4	no	—
127	C₄⋊C₄×C₆₇	—	4	2	no	—
128	C₄⋊C₄×C₆₇	—	4	2	no	—
129	C₄⋊C₄×C₆₇	—	8	1	no	—
130	C₄⋊C₄×C₆₇	—	8	1	no	—
131	C₄⋊C₄×C₆₇	—	8	2	no	—
132	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
133	C₂²⋊C₄×C₆₇	—	2	4	no	—
134	C₂²⋊C₄×C₆₇	—	4	4	no	—
135	C₄²×C₆₇	—	2	2	no	—
136	C₄²×C₆₇	—	2	2	no	—
137	C₄²×C₆₇	—	4	2	no	—
138	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
139	C₄²×C₆₇	#41	1	2	yes	—
140	C₂³×C₆₇	—	2	2	no	—
141	C₂³×C₆₇	—	2	2	no	—
142	C₂³×C₆₇	—	2	2	no	—
143	C₂³×C₆₇	—	4	1	no	—
144	C₂³×C₆₇	—	4	1	no	—
145	C₂³×C₆₇	—	4	1	no	—
146	C₂³×C₆₇	—	4	1	no	—
147	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
148	D₄×C₆₇	—	2	2	no	—
149	D₄×C₆₇	—	2	2	no	—
150	D₄×C₆₇	—	2	2	no	—
151	D₄×C₆₇	#125	1	4	yes	—
152	C₂×C₄×C₆₇	—	2	1	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	4	1	no	—
162	C₂×C₄×C₆₇	—	4	1	no	—
163	C₂×C₄×C₆₇	—	4	8	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	4	1	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	4	1	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	—	8	8	no	—
177	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
178	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
179	C₂×C₄×C₆₇	#135	1	2	yes	—
180	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
181	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
186	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
187	C₈×C₆₇	—	4	1	no	—
188	C₈×C₆₇	—	4	2	no	—
189	C₈×C₆₇	—	4	1	no	—
190	C₂²×C₆₇	—	2	4	no	—
191	C₂²×C₆₇	—	4	2	no	—
192	C₂²×C₆₇	—	4	2	no	—
193	C₂²×C₆₇	—	4	2	no	—
194	C₂²×C₆₇	—	4	2	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	4	2	no	—
197	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
198	C₂²×C₆₇	#546	1	1	yes	characteristic
199	C₂²×C₆₇	#548	1	1	yes	characteristic
200	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
201	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
202	C₂²×C₆₇	#576	1	1	yes	characteristic
203	C₂²×C₆₇	#578	1	1	yes	characteristic
204	C₄×C₆₇	—	2	2	no	—
205	C₄×C₆₇	—	4	2	no	—
206	C₄×C₆₇	—	4	2	no	—
207	C₄×C₆₇	—	4	2	no	—
208	C₄×C₆₇	—	8	2	no	—
209	C₄×C₆₇	—	8	1	no	—
210	C₄×C₆₇	—	8	1	no	—
211	C₄×C₆₇	—	8	2	no	—
212	C₄×C₆₇	#478	1	4	yes	—
213	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
214	C₄×C₆₇	#544	1	1	yes	characteristic
215	C₄×C₆₇	#552	1	1	yes	characteristic
216	C₄×C₆₇	#553	1	1	yes	characteristic
217	#26111	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂×C₆₇	—	4	4	no	—
219	C₂×C₆₇	#2442	1	1	yes	characteristic
220	C₂×C₆₇	C₂×#478	1	1	yes	characteristic
221	C₂×C₆₇	#3083	1	1	yes	characteristic
222	C₂×C₆₇	#3302	1	1	yes	characteristic
223	C₂×C₆₇	#3325	1	1	yes	characteristic
224	C₂×C₆₇	#3361	1	1	yes	characteristic
225	C₂×C₆₇	#3367	1	1	yes	characteristic
226	#2571	C₂×C₆₇	1	1	yes	characteristic, complemented
227	D₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
228	#2101	C₂×C₆₇	1	2	yes	complemented
229	#2087	C₂×C₆₇	1	2	yes	complemented
230	#1947	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#1939	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#1862	C₂×C₆₇	1	2	yes	—
233	#1834	C₂×C₆₇	1	1	yes	characteristic, complemented
234	#1804	C₂×C₆₇	1	2	yes	—
235	#1802	C₂×C₆₇	1	1	yes	characteristic
236	C₄×#102	C₂×C₆₇	1	1	yes	characteristic
237	C₆₇	#26111	1	1	yes	characteristic, direct product factor, Sylow
238	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
239	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
240	C₂²×C₄⋊C₄	—	2	1	no	—
241	C₂×#107	C₂²×C₆₇	1	2	yes	—
242	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
243	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
244	C₂×#102	C₂²×C₆₇	1	2	yes	—
245	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
247	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
248	#400	—	2	2	no	—
249	#389	—	2	1	no	—
250	#389	C₂²×C₆₇	1	2	yes	—
251	#389	C₂²×C₆₇	1	2	yes	—
252	#385	—	2	2	no	—
253	#384	C₂²×C₆₇	1	1	yes	characteristic
254	#384	C₂²×C₆₇	1	1	yes	characteristic
255	#382	—	2	1	no	—
256	#382	C₂²×C₆₇	1	2	yes	—
257	#380	—	2	1	no	—
258	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
259	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
260	#340	C₂²×C₆₇	1	2	yes	—
261	#340	C₂²×C₆₇	1	2	yes	—
262	#340	C₂²×C₆₇	1	2	yes	—
263	#340	C₂²×C₆₇	1	1	yes	characteristic
264	#340	C₂²×C₆₇	1	2	yes	—
265	#340	C₂²×C₆₇	1	1	yes	characteristic
266	#336	C₂²×C₆₇	1	2	yes	—
267	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
268	C₂³×C₄	—	2	1	no	—
269	C₂³×C₄	—	2	2	no	—
270	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
271	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
272	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
273	C₄×D₄	—	2	4	no	—
274	C₄×D₄	—	2	2	no	—
275	C₄×D₄	—	4	2	no	—
276	C₄×D₄	Q₈×C₆₇	1	4	yes	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	—	4	4	no	—
284	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
285	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
287	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
291	C₂×C₂²⋊C₄	—	2	1	no	—
292	C₂×C₂²⋊C₄	—	2	2	no	—
293	C₂×C₄²	—	2	1	no	—
294	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
295	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
296	#107	—	4	2	no	—
297	C₄⋊C₈	—	2	2	no	—
298	C₄⋊C₈	—	4	2	no	—
299	#102	—	2	4	no	—
300	#102	—	2	2	no	—
301	#102	—	2	2	no	—
302	C₄×C₈	—	2	2	no	—
303	#95	—	2	1	no	—
304	#95	—	2	2	no	—
305	#95	—	2	2	no	—
306	#95	—	2	2	no	—
307	#95	—	2	2	no	—
308	#95	D₄×C₆₇	1	2	yes	—
309	C₂⁴	—	2	1	no	—
310	C₂×D₄	—	2	1	no	—
311	C₂×D₄	—	2	1	no	—
312	C₂×D₄	—	2	1	no	—
313	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
314	C₂×D₄	#41×C₆₇	1	2	yes	—
315	C₂×D₄	#41×C₆₇	1	2	yes	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	4	no	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	4	4	no	—
323	C₂²×C₄	—	4	4	no	—
324	C₂²×C₄	—	4	2	no	—
325	C₂²×C₄	—	4	2	no	—
326	C₂²×C₄	—	8	2	no	—
327	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
328	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
329	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
330	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
331	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	1	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	4	4	no	—
353	C₄⋊C₄	—	4	2	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	—	8	1	no	—
356	C₄⋊C₄	—	8	1	no	—
357	C₄⋊C₄	—	8	2	no	—
358	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
359	C₂²⋊C₄	—	2	4	no	—
360	C₂²⋊C₄	—	4	4	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	4	2	no	—
364	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
365	C₄²	#41×C₆₇	1	2	yes	—
366	C₂³	—	2	2	no	—
367	C₂³	—	2	2	no	—
368	C₂³	—	2	2	no	—
369	C₂³	—	4	1	no	—
370	C₂³	—	4	1	no	—
371	C₂³	—	4	1	no	—
372	C₂³	—	4	1	no	—
373	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
374	D₄	—	2	2	no	—
375	D₄	—	2	2	no	—
376	D₄	—	2	2	no	—
377	D₄	#125×C₆₇	1	4	yes	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	8	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	8	8	no	—
403	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
404	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
405	C₂×C₄	#135×C₆₇	1	2	yes	—
406	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
407	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
409	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
410	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
411	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
412	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
413	C₈	—	4	1	no	—
414	C₈	—	4	1	no	—
415	C₈	—	4	2	no	—
416	C₂²	—	2	4	no	—
417	C₂²	—	4	2	no	—
418	C₂²	—	4	2	no	—
419	C₂²	—	4	2	no	—
420	C₂²	—	4	2	no	—
421	C₂²	—	4	2	no	—
422	C₂²	—	4	2	no	—
423	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
424	C₂²	#546×C₆₇	1	1	yes	characteristic
425	C₂²	#548×C₆₇	1	1	yes	characteristic
426	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
427	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
428	C₂²	#576×C₆₇	1	1	yes	characteristic
429	C₂²	#578×C₆₇	1	1	yes	characteristic
430	C₄	—	2	2	no	—
431	C₄	—	4	2	no	—
432	C₄	—	4	2	no	—
433	C₄	—	4	2	no	—
434	C₄	—	8	2	no	—
435	C₄	—	8	1	no	—
436	C₄	—	8	2	no	—
437	C₄	—	8	1	no	—
438	C₄	#478×C₆₇	1	4	yes	—
439	C₄	C₂×#125×C₆₇	1	1	yes	characteristic
440	C₄	#544×C₆₇	1	1	yes	characteristic
441	C₄	#552×C₆₇	1	1	yes	characteristic
442	C₄	#553×C₆₇	1	1	yes	characteristic
443	C₂	—	4	4	no	—
444	C₂	C₆₇×#2442	1	1	yes	characteristic
445	C₂	C₂×#478×C₆₇	1	1	yes	characteristic
446	C₂	C₆₇×#3083	1	1	yes	characteristic
447	C₂	C₆₇×#3302	1	1	yes	characteristic
448	C₂	C₆₇×#3325	1	1	yes	characteristic
449	C₂	C₆₇×#3361	1	1	yes	characteristic
450	C₂	C₆₇×#3367	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
1. C₆₇×#2571	C₂	semidirect
3. C₆₇×#2101	C₂	semidirect
4. C₆₇×#2087	C₂	semidirect
5. C₆₇×#1947	C₂	semidirect
6. C₆₇×#1939	C₂	semidirect
8. C₆₇×#1834	C₂	semidirect
217. #26111	C₆₇	direct (N × H)
226. #2571	C₂×C₆₇	semidirect
228. #2101	C₂×C₆₇	semidirect
229. #2087	C₂×C₆₇	semidirect
230. #1947	C₂×C₆₇	semidirect
231. #1939	C₂×C₆₇	semidirect
233. #1834	C₂×C₆₇	semidirect
237. C₆₇	#26111	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2571	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#2101	2
C₆₇×#2087	2
C₆₇×#1947	2
C₆₇×#1939	2
C₆₇×#1862	2
C₆₇×#1834	2
C₆₇×#1804	2
C₆₇×#1802	2
C₄×#102×C₆₇	2
#26111	67

Subgroups — C67×#26111 #30705362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#26111 #30705362