Subgroups — C₆₇×#25882 #30705133

Order 17152.

1334 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#547×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#487×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#474×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2547	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2036	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1981	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1954	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1788	C₂	1	4	yes	maximal, complemented
9	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	C₂²	1	2	yes	—
11	#547×C₆₇	—	2	4	no	—
12	#547×C₆₇	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₆₇	—	2	1	no	—
16	C₂×#123×C₆₇	—	2	2	no	—
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂×#121×C₆₇	—	2	1	no	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
22	#487×C₆₇	—	2	4	no	—
23	#474×C₆₇	—	2	4	no	—
24	C₂×#107×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#103×C₆₇	C₂²	1	4	yes	—
27	C₂×#103×C₆₇	C₂²	1	4	yes	—
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#397×C₆₇	—	2	2	no	—
33	#391×C₆₇	—	2	1	no	—
34	#380×C₆₇	C₂²	1	4	yes	—
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	#340×C₆₇	C₂²	1	2	yes	—
37	C₂×#41×C₆₇	—	2	4	no	—
38	C₂×#41×C₆₇	—	2	4	no	—
39	C₂×#41×C₆₇	—	2	8	no	—
40	C₂×#41×C₆₇	—	4	4	no	—
41	C₂²×Q₈×C₆₇	—	2	1	no	—
42	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
43	C₂²×D₄×C₆₇	—	2	2	no	—
44	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
45	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	—	2	1	no	—
47	C₂³×C₄×C₆₇	—	2	2	no	—
48	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₄⋊Q₈×C₆₇	—	2	2	no	—
52	C₄⋊Q₈×C₆₇	—	2	2	no	—
53	C₄⋊Q₈×C₆₇	—	2	2	no	—
54	C₄⋊Q₈×C₆₇	D₄	1	4	yes	—
55	#124×C₆₇	—	4	4	no	—
56	#123×C₆₇	—	2	16	no	—
57	C₂²⋊Q₈×C₆₇	—	2	4	no	—
58	C₂²⋊Q₈×C₆₇	—	2	4	no	—
59	#121×C₆₇	—	2	8	no	—
60	C₄×D₄×C₆₇	—	2	2	no	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	C₄×D₄×C₆₇	—	2	2	no	—
63	C₄×D₄×C₆₇	D₄	1	4	yes	—
64	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
76	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
77	C₂×C₄²×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	#107×C₆₇	—	2	4	no	—
80	C₄⋊C₈×C₆₇	—	2	2	no	—
81	#103×C₆₇	—	2	8	no	—
82	#103×C₆₇	—	4	4	no	—
83	#102×C₆₇	—	2	4	no	—
84	#102×C₆₇	—	2	4	no	—
85	C₂²⋊C₈×C₆₇	—	4	4	no	—
86	C₄×C₈×C₆₇	—	2	2	no	—
87	#95×C₆₇	—	2	1	no	—
88	#95×C₆₇	—	2	4	no	—
89	C₂⁴×C₆₇	—	2	1	no	—
90	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
91	#41×C₆₇	—	4	16	no	—
92	#41×C₆₇	—	4	8	no	—
93	#41×C₆₇	—	4	8	no	—
94	C₂×Q₈×C₆₇	—	2	2	no	—
95	C₂×Q₈×C₆₇	—	2	4	no	—
96	C₂×Q₈×C₆₇	—	2	4	no	—
97	C₂×Q₈×C₆₇	—	4	2	no	—
98	C₂×Q₈×C₆₇	—	4	2	no	—
99	C₂×Q₈×C₆₇	—	4	2	no	—
100	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
101	C₂×D₄×C₆₇	—	2	8	no	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	1	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	4	8	no	—
108	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
109	C₂×D₄×C₆₇	#41	1	4	yes	—
110	C₂²×C₄×C₆₇	—	2	8	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	4	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	4	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	4	8	no	—
121	C₂²×C₄×C₆₇	—	4	2	no	—
122	C₂²×C₄×C₆₇	—	4	2	no	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	4	no	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	8	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	2	4	no	—
135	C₄⋊C₄×C₆₇	—	2	4	no	—
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	4	2	no	—
142	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
143	C₂²⋊C₄×C₆₇	—	2	8	no	—
144	C₂²⋊C₄×C₆₇	—	2	4	no	—
145	C₂²⋊C₄×C₆₇	—	2	4	no	—
146	C₂²⋊C₄×C₆₇	—	4	8	no	—
147	C₄²×C₆₇	—	4	2	no	—
148	C₄²×C₆₇	C₂×D₄	1	2	yes	—
149	C₄²×C₆₇	C₂×D₄	1	2	yes	—
150	C₂³×C₆₇	—	2	2	no	—
151	C₂³×C₆₇	—	2	4	no	—
152	C₂³×C₆₇	—	2	2	no	—
153	C₂³×C₆₇	—	2	4	no	—
154	C₂³×C₆₇	—	2	2	no	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	2	4	no	—
157	C₂³×C₆₇	—	4	1	no	—
158	C₂³×C₆₇	—	4	1	no	—
159	C₂³×C₆₇	—	4	2	no	—
160	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₆₇	—	2	8	no	—
162	Q₈×C₆₇	—	4	2	no	—
163	Q₈×C₆₇	—	4	2	no	—
164	Q₈×C₆₇	—	8	4	no	—
165	D₄×C₆₇	—	2	2	no	—
166	D₄×C₆₇	—	2	4	no	—
167	D₄×C₆₇	—	4	16	no	—
168	D₄×C₆₇	—	4	4	no	—
169	D₄×C₆₇	#124	1	4	yes	complemented
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	—	2	4	no	—
174	C₂×C₄×C₆₇	—	2	4	no	—
175	C₂×C₄×C₆₇	—	4	16	no	—
176	C₂×C₄×C₆₇	—	4	4	no	—
177	C₂×C₄×C₆₇	—	4	1	no	—
178	C₂×C₄×C₆₇	—	4	1	no	—
179	C₂×C₄×C₆₇	—	4	2	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	4	no	—
182	C₂×C₄×C₆₇	—	4	8	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	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	#120	1	2	yes	—
190	C₂×C₄×C₆₇	#120	1	2	yes	—
191	C₂×C₄×C₆₇	#135	1	2	yes	—
192	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
193	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
196	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
197	C₈×C₆₇	—	4	2	no	—
198	C₈×C₆₇	—	4	2	no	—
199	C₂²×C₆₇	—	2	4	no	—
200	C₂²×C₆₇	—	4	4	no	—
201	C₂²×C₆₇	—	4	2	no	—
202	C₂²×C₆₇	—	4	2	no	—
203	C₂²×C₆₇	—	4	4	no	—
204	C₂²×C₆₇	—	4	4	no	—
205	C₂²×C₆₇	—	4	2	no	—
206	C₂²×C₆₇	—	4	2	no	—
207	C₂²×C₆₇	—	4	2	no	—
208	C₂²×C₆₇	—	4	4	no	—
209	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
210	C₂²×C₆₇	#546	1	2	yes	—
211	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
212	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₂²×C₆₇	#578	1	2	yes	—
214	C₄×C₆₇	—	2	2	no	—
215	C₄×C₆₇	—	4	2	no	—
216	C₄×C₆₇	—	4	4	no	—
217	C₄×C₆₇	—	4	2	no	—
218	C₄×C₆₇	—	8	2	no	—
219	C₄×C₆₇	—	8	2	no	—
220	C₄×C₆₇	#449	1	4	yes	—
221	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
222	C₄×C₆₇	#535	1	1	yes	characteristic
223	C₄×C₆₇	#546	1	2	yes	—
224	#25882	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂×C₆₇	—	4	4	no	—
226	C₂×C₆₇	—	8	2	no	—
227	C₂×C₆₇	#2435	1	1	yes	characteristic
228	C₂×C₆₇	C₂×#449	1	1	yes	characteristic
229	C₂×C₆₇	#3011	1	1	yes	characteristic
230	C₂×C₆₇	#3302	1	2	yes	—
231	C₂×C₆₇	#3330	1	2	yes	—
232	C₂×#547	C₂×C₆₇	1	1	yes	characteristic
233	C₂×#487	C₂×C₆₇	1	1	yes	characteristic, complemented
234	C₂×#474	C₂×C₆₇	1	1	yes	characteristic, complemented
235	#2547	C₂×C₆₇	1	1	yes	characteristic, complemented
236	#2036	C₂×C₆₇	1	4	yes	complemented
237	#1981	C₂×C₆₇	1	1	yes	characteristic, complemented
238	#1954	C₂×C₆₇	1	1	yes	characteristic, complemented
239	#1788	C₂×C₆₇	1	4	yes	complemented
240	C₄×#102	C₂×C₆₇	1	1	yes	characteristic, complemented
241	C₆₇	#25882	1	1	yes	characteristic, direct product factor, Sylow
242	C₂²×#41	C₂²×C₆₇	1	2	yes	—
243	#547	—	2	4	no	—
244	#547	—	2	4	no	—
245	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
247	C₂×#124	—	2	1	no	—
248	C₂×#123	—	2	2	no	—
249	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
250	C₂×#121	—	2	1	no	—
251	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
252	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
253	C₂²×C₄⋊C₄	—	2	1	no	—
254	#487	—	2	4	no	—
255	#474	—	2	4	no	—
256	C₂×#107	C₂²×C₆₇	1	2	yes	—
257	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×#103	C₂²×C₆₇	1	4	yes	—
259	C₂×#103	C₂²×C₆₇	1	4	yes	—
260	C₂×#102	C₂²×C₆₇	1	2	yes	—
261	C₂×#102	C₂²×C₆₇	1	2	yes	—
262	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
263	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
264	#397	—	2	2	no	—
265	#391	—	2	1	no	—
266	#380	C₂²×C₆₇	1	4	yes	—
267	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
268	#340	C₂²×C₆₇	1	2	yes	—
269	C₂×#41	—	2	4	no	—
270	C₂×#41	—	2	4	no	—
271	C₂×#41	—	2	8	no	—
272	C₂×#41	—	4	4	no	—
273	C₂²×Q₈	—	2	1	no	—
274	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
275	C₂²×D₄	—	2	2	no	—
276	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
277	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
278	C₂³×C₄	—	2	1	no	—
279	C₂³×C₄	—	2	2	no	—
280	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
281	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
282	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
283	C₄⋊Q₈	—	2	2	no	—
284	C₄⋊Q₈	—	2	2	no	—
285	C₄⋊Q₈	—	2	2	no	—
286	C₄⋊Q₈	D₄×C₆₇	1	4	yes	—
287	#124	—	4	4	no	—
288	#123	—	2	16	no	—
289	C₂²⋊Q₈	—	2	4	no	—
290	C₂²⋊Q₈	—	2	4	no	—
291	#121	—	2	8	no	—
292	C₄×D₄	—	2	2	no	—
293	C₄×D₄	—	2	2	no	—
294	C₄×D₄	—	2	2	no	—
295	C₄×D₄	D₄×C₆₇	1	4	yes	—
296	C₂×C₄⋊C₄	—	2	8	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
303	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
304	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
305	C₂×C₂²⋊C₄	—	2	2	no	—
306	C₂×C₂²⋊C₄	—	2	1	no	—
307	C₂×C₂²⋊C₄	—	2	4	no	—
308	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
309	C₂×C₄²	—	2	1	no	—
310	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
311	#107	—	2	4	no	—
312	C₄⋊C₈	—	2	2	no	—
313	#103	—	2	8	no	—
314	#103	—	4	4	no	—
315	#102	—	2	4	no	—
316	#102	—	2	4	no	—
317	C₂²⋊C₈	—	4	4	no	—
318	C₄×C₈	—	2	2	no	—
319	#95	—	2	1	no	—
320	#95	—	2	4	no	—
321	C₂⁴	—	2	1	no	—
322	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
323	#41	—	4	16	no	—
324	#41	—	4	8	no	—
325	#41	—	4	8	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	4	no	—
328	C₂×Q₈	—	2	4	no	—
329	C₂×Q₈	—	4	2	no	—
330	C₂×Q₈	—	4	2	no	—
331	C₂×Q₈	—	4	2	no	—
332	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
333	C₂×D₄	—	2	8	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	4	8	no	—
340	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
341	C₂×D₄	#41×C₆₇	1	4	yes	—
342	C₂²×C₄	—	2	8	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	4	8	no	—
353	C₂²×C₄	—	4	2	no	—
354	C₂²×C₄	—	4	2	no	—
355	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
356	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	4	no	—
364	C₄⋊C₄	—	2	8	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	4	no	—
367	C₄⋊C₄	—	2	4	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	4	2	no	—
374	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
375	C₂²⋊C₄	—	2	8	no	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	2	4	no	—
378	C₂²⋊C₄	—	4	8	no	—
379	C₄²	—	4	2	no	—
380	C₄²	C₂×D₄×C₆₇	1	2	yes	—
381	C₄²	C₂×D₄×C₆₇	1	2	yes	—
382	C₂³	—	2	2	no	—
383	C₂³	—	2	4	no	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	4	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	4	1	no	—
390	C₂³	—	4	1	no	—
391	C₂³	—	4	2	no	—
392	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
393	Q₈	—	2	8	no	—
394	Q₈	—	4	2	no	—
395	Q₈	—	4	2	no	—
396	Q₈	—	8	4	no	—
397	D₄	—	2	2	no	—
398	D₄	—	2	4	no	—
399	D₄	—	4	16	no	—
400	D₄	—	4	4	no	—
401	D₄	#124×C₆₇	1	4	yes	complemented
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	4	16	no	—
408	C₂×C₄	—	4	4	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	8	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	4	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	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	#120×C₆₇	1	2	yes	—
422	C₂×C₄	#120×C₆₇	1	2	yes	—
423	C₂×C₄	#135×C₆₇	1	2	yes	—
424	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
425	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
426	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
428	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
429	C₈	—	4	2	no	—
430	C₈	—	4	2	no	—
431	C₂²	—	2	4	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	4	4	no	—
434	C₂²	—	4	4	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	2	no	—
437	C₂²	—	4	4	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
442	C₂²	#546×C₆₇	1	2	yes	—
443	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
444	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
445	C₂²	#578×C₆₇	1	2	yes	—
446	C₄	—	2	2	no	—
447	C₄	—	4	2	no	—
448	C₄	—	4	4	no	—
449	C₄	—	4	2	no	—
450	C₄	—	8	2	no	—
451	C₄	—	8	2	no	—
452	C₄	#449×C₆₇	1	4	yes	—
453	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
454	C₄	#535×C₆₇	1	1	yes	characteristic
455	C₄	#546×C₆₇	1	2	yes	—
456	C₂	—	4	4	no	—
457	C₂	—	8	2	no	—
458	C₂	C₆₇×#2435	1	1	yes	characteristic
459	C₂	C₂×#449×C₆₇	1	1	yes	characteristic
460	C₂	C₆₇×#3011	1	1	yes	characteristic
461	C₂	C₆₇×#3302	1	2	yes	—
462	C₂	C₆₇×#3330	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#547×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#487×C₆₇	C₂	semidirect
3. C₂×#474×C₆₇	C₂	semidirect
4. C₆₇×#2547	C₂	semidirect
5. C₆₇×#2036	C₂	semidirect
6. C₆₇×#1981	C₂	semidirect
7. C₆₇×#1954	C₂	semidirect
8. C₆₇×#1788	C₂	semidirect
9. C₄×#102×C₆₇	C₂	semidirect
169. D₄×C₆₇	#124	semidirect
224. #25882	C₆₇	direct (N × H)
233. C₂×#487	C₂×C₆₇	semidirect
234. C₂×#474	C₂×C₆₇	semidirect
235. #2547	C₂×C₆₇	semidirect
236. #2036	C₂×C₆₇	semidirect
237. #1981	C₂×C₆₇	semidirect
238. #1954	C₂×C₆₇	semidirect
239. #1788	C₂×C₆₇	semidirect
240. C₄×#102	C₂×C₆₇	semidirect
241. C₆₇	#25882	direct (N × H)
401. D₄	#124×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#547×C₆₇	2
C₂×#487×C₆₇	2
C₂×#474×C₆₇	2
C₆₇×#2547	2
C₆₇×#2036	2
C₆₇×#1981	2
C₆₇×#1954	2
C₆₇×#1788	2
C₄×#102×C₆₇	2
#25882	67

Subgroups — C67×#25882 #30705133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25882 #30705133