Subgroups — C₆₇×#25886 #30705137

Order 17152.

1314 conjugacy classes of subgroups.

The subgroup lattice has 458 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×#476×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2563	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2035	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×#123×C₆₇	—	2	2	no	—
16	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
17	C₂×#121×C₆₇	—	2	1	no	—
18	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
21	#487×C₆₇	—	2	4	no	—
22	#476×C₆₇	—	2	4	no	—
23	C₂×#106×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#103×C₆₇	C₂²	1	4	yes	—
26	C₂×#103×C₆₇	C₂²	1	4	yes	—
27	C₂×#102×C₆₇	C₂²	1	2	yes	—
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#393×C₆₇	—	2	2	no	—
32	#390×C₆₇	—	2	1	no	—
33	#382×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	#123×C₆₇	—	2	16	no	—
56	C₂²⋊Q₈×C₆₇	—	2	4	no	—
57	C₂²⋊Q₈×C₆₇	—	2	4	no	—
58	#121×C₆₇	—	2	8	no	—
59	C₄×D₄×C₆₇	—	2	2	no	—
60	C₄×D₄×C₆₇	—	2	2	no	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	C₄×D₄×C₆₇	D₄	1	4	yes	—
63	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
76	C₂×C₄²×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	#106×C₆₇	—	2	4	no	—
79	C₄⋊C₈×C₆₇	—	2	2	no	—
80	#103×C₆₇	—	2	8	no	—
81	#103×C₆₇	—	4	4	no	—
82	#102×C₆₇	—	2	4	no	—
83	#102×C₆₇	—	2	4	no	—
84	C₂²⋊C₈×C₆₇	—	4	4	no	—
85	C₄×C₈×C₆₇	—	2	2	no	—
86	#95×C₆₇	—	2	1	no	—
87	#95×C₆₇	—	2	4	no	—
88	C₂⁴×C₆₇	—	2	1	no	—
89	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
90	#41×C₆₇	—	4	16	no	—
91	#41×C₆₇	—	4	8	no	—
92	#41×C₆₇	—	4	8	no	—
93	C₂×Q₈×C₆₇	—	2	2	no	—
94	C₂×Q₈×C₆₇	—	2	4	no	—
95	C₂×Q₈×C₆₇	—	2	4	no	—
96	C₂×Q₈×C₆₇	—	4	2	no	—
97	C₂×Q₈×C₆₇	—	8	2	no	—
98	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
99	C₂×D₄×C₆₇	—	2	8	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	2	1	no	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	4	8	no	—
106	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
107	C₂×D₄×C₆₇	#41	1	4	yes	—
108	C₂²×C₄×C₆₇	—	2	8	no	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	4	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	4	8	no	—
119	C₂²×C₄×C₆₇	—	4	2	no	—
120	C₂²×C₄×C₆₇	—	4	2	no	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	4	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₄⋊C₄×C₆₇	—	2	8	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	4	no	—
134	C₄⋊C₄×C₆₇	—	2	1	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	4	2	no	—
140	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
141	C₂²⋊C₄×C₆₇	—	2	8	no	—
142	C₂²⋊C₄×C₆₇	—	2	4	no	—
143	C₂²⋊C₄×C₆₇	—	2	4	no	—
144	C₂²⋊C₄×C₆₇	—	8	4	no	—
145	C₄²×C₆₇	—	4	2	no	—
146	C₄²×C₆₇	C₂×D₄	1	2	yes	—
147	C₄²×C₆₇	C₂×D₄	1	2	yes	—
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	—	2	4	no	—
150	C₂³×C₆₇	—	2	2	no	—
151	C₂³×C₆₇	—	2	4	no	—
152	C₂³×C₆₇	—	2	2	no	—
153	C₂³×C₆₇	—	2	2	no	—
154	C₂³×C₆₇	—	2	4	no	—
155	C₂³×C₆₇	—	4	1	no	—
156	C₂³×C₆₇	—	4	1	no	—
157	C₂³×C₆₇	—	4	2	no	—
158	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₆₇	—	2	8	no	—
160	Q₈×C₆₇	—	4	2	no	—
161	Q₈×C₆₇	—	4	2	no	—
162	Q₈×C₆₇	—	8	4	no	—
163	D₄×C₆₇	—	2	2	no	—
164	D₄×C₆₇	—	2	4	no	—
165	D₄×C₆₇	—	4	16	no	—
166	D₄×C₆₇	—	4	4	no	—
167	D₄×C₆₇	#124	1	4	yes	—
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	4	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	4	16	no	—
174	C₂×C₄×C₆₇	—	4	4	no	—
175	C₂×C₄×C₆₇	—	4	1	no	—
176	C₂×C₄×C₆₇	—	4	1	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	4	no	—
180	C₂×C₄×C₆₇	—	4	8	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	4	no	—
186	C₂×C₄×C₆₇	#120	1	2	yes	—
187	C₂×C₄×C₆₇	#120	1	2	yes	—
188	C₂×C₄×C₆₇	#135	1	2	yes	—
189	C₂×C₄×C₆₇	#137	1	2	yes	—
190	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
193	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
194	C₈×C₆₇	—	4	2	no	—
195	C₈×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	2	4	no	—
197	C₂²×C₆₇	—	4	4	no	—
198	C₂²×C₆₇	—	4	2	no	—
199	C₂²×C₆₇	—	4	2	no	—
200	C₂²×C₆₇	—	4	4	no	—
201	C₂²×C₆₇	—	4	4	no	—
202	C₂²×C₆₇	—	4	2	no	—
203	C₂²×C₆₇	—	4	2	no	—
204	C₂²×C₆₇	—	4	2	no	—
205	C₂²×C₆₇	—	4	4	no	—
206	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
207	C₂²×C₆₇	#546	1	2	yes	—
208	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
209	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
210	C₂²×C₆₇	#577	1	2	yes	—
211	C₄×C₆₇	—	2	2	no	—
212	C₄×C₆₇	—	4	2	no	—
213	C₄×C₆₇	—	4	4	no	—
214	C₄×C₆₇	—	4	2	no	—
215	C₄×C₆₇	—	8	2	no	—
216	C₄×C₆₇	—	8	2	no	—
217	C₄×C₆₇	#452	1	4	yes	—
218	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
219	C₄×C₆₇	#535	1	1	yes	characteristic
220	C₄×C₆₇	#546	1	2	yes	—
221	#25886	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂×C₆₇	—	4	4	no	—
223	C₂×C₆₇	—	8	2	no	—
224	C₂×C₆₇	#2435	1	1	yes	characteristic
225	C₂×C₆₇	C₂×#452	1	1	yes	characteristic
226	C₂×C₆₇	#3010	1	1	yes	characteristic
227	C₂×C₆₇	#3303	1	2	yes	—
228	C₂×C₆₇	#3331	1	2	yes	—
229	C₂×#547	C₂×C₆₇	1	1	yes	characteristic
230	C₂×#487	C₂×C₆₇	1	1	yes	characteristic, complemented
231	C₂×#476	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#2563	C₂×C₆₇	1	1	yes	characteristic, complemented
233	#2035	C₂×C₆₇	1	4	yes	complemented
234	#1981	C₂×C₆₇	1	1	yes	characteristic, complemented
235	#1954	C₂×C₆₇	1	1	yes	characteristic, complemented
236	#1788	C₂×C₆₇	1	4	yes	complemented
237	C₄×#102	C₂×C₆₇	1	1	yes	characteristic, complemented
238	C₆₇	#25886	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²×#41	C₂²×C₆₇	1	2	yes	—
240	#547	—	2	4	no	—
241	#547	—	2	4	no	—
242	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
243	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
244	C₂×#123	—	2	2	no	—
245	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
246	C₂×#121	—	2	1	no	—
247	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
248	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
249	C₂²×C₄⋊C₄	—	2	1	no	—
250	#487	—	2	4	no	—
251	#476	—	2	4	no	—
252	C₂×#106	C₂²×C₆₇	1	2	yes	—
253	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
254	C₂×#103	C₂²×C₆₇	1	4	yes	—
255	C₂×#103	C₂²×C₆₇	1	4	yes	—
256	C₂×#102	C₂²×C₆₇	1	2	yes	—
257	C₂×#102	C₂²×C₆₇	1	2	yes	—
258	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
259	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
260	#393	—	2	2	no	—
261	#390	—	2	1	no	—
262	#382	—	2	1	no	—
263	#380	C₂²×C₆₇	1	4	yes	—
264	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
265	#340	C₂²×C₆₇	1	2	yes	—
266	C₂×#41	—	2	4	no	—
267	C₂×#41	—	2	4	no	—
268	C₂×#41	—	2	8	no	—
269	C₂×#41	—	4	4	no	—
270	C₂²×Q₈	—	2	1	no	—
271	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
272	C₂²×D₄	—	2	2	no	—
273	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
274	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
275	C₂³×C₄	—	2	1	no	—
276	C₂³×C₄	—	2	2	no	—
277	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
278	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
279	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
280	C₄⋊Q₈	—	2	2	no	—
281	C₄⋊Q₈	—	2	2	no	—
282	C₄⋊Q₈	—	2	2	no	—
283	C₄⋊Q₈	D₄×C₆₇	1	4	yes	—
284	#123	—	2	16	no	—
285	C₂²⋊Q₈	—	2	4	no	—
286	C₂²⋊Q₈	—	2	4	no	—
287	#121	—	2	8	no	—
288	C₄×D₄	—	2	2	no	—
289	C₄×D₄	—	2	2	no	—
290	C₄×D₄	—	2	2	no	—
291	C₄×D₄	D₄×C₆₇	1	4	yes	—
292	C₂×C₄⋊C₄	—	2	8	no	—
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	1	no	—
297	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
299	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	—	2	1	no	—
303	C₂×C₂²⋊C₄	—	2	4	no	—
304	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
305	C₂×C₄²	—	2	1	no	—
306	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
307	#106	—	2	4	no	—
308	C₄⋊C₈	—	2	2	no	—
309	#103	—	2	8	no	—
310	#103	—	4	4	no	—
311	#102	—	2	4	no	—
312	#102	—	2	4	no	—
313	C₂²⋊C₈	—	4	4	no	—
314	C₄×C₈	—	2	2	no	—
315	#95	—	2	1	no	—
316	#95	—	2	4	no	—
317	C₂⁴	—	2	1	no	—
318	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
319	#41	—	4	16	no	—
320	#41	—	4	8	no	—
321	#41	—	4	8	no	—
322	C₂×Q₈	—	2	2	no	—
323	C₂×Q₈	—	2	4	no	—
324	C₂×Q₈	—	2	4	no	—
325	C₂×Q₈	—	4	2	no	—
326	C₂×Q₈	—	8	2	no	—
327	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
328	C₂×D₄	—	2	8	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	4	8	no	—
335	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
336	C₂×D₄	#41×C₆₇	1	4	yes	—
337	C₂²×C₄	—	2	8	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	4	no	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	4	8	no	—
348	C₂²×C₄	—	4	2	no	—
349	C₂²×C₄	—	4	2	no	—
350	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
351	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
353	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	4	no	—
359	C₄⋊C₄	—	2	8	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	4	no	—
362	C₄⋊C₄	—	2	4	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
370	C₂²⋊C₄	—	2	8	no	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₂²⋊C₄	—	8	4	no	—
374	C₄²	—	4	2	no	—
375	C₄²	C₂×D₄×C₆₇	1	2	yes	—
376	C₄²	C₂×D₄×C₆₇	1	2	yes	—
377	C₂³	—	2	2	no	—
378	C₂³	—	2	4	no	—
379	C₂³	—	2	2	no	—
380	C₂³	—	2	4	no	—
381	C₂³	—	2	2	no	—
382	C₂³	—	2	2	no	—
383	C₂³	—	2	4	no	—
384	C₂³	—	4	1	no	—
385	C₂³	—	4	1	no	—
386	C₂³	—	4	2	no	—
387	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
388	Q₈	—	2	8	no	—
389	Q₈	—	4	2	no	—
390	Q₈	—	4	2	no	—
391	Q₈	—	8	4	no	—
392	D₄	—	2	2	no	—
393	D₄	—	2	4	no	—
394	D₄	—	4	16	no	—
395	D₄	—	4	4	no	—
396	D₄	#124×C₆₇	1	4	yes	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	4	16	no	—
403	C₂×C₄	—	4	4	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	8	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	4	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	4	no	—
415	C₂×C₄	#120×C₆₇	1	2	yes	—
416	C₂×C₄	#120×C₆₇	1	2	yes	—
417	C₂×C₄	#135×C₆₇	1	2	yes	—
418	C₂×C₄	#137×C₆₇	1	2	yes	—
419	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
420	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
421	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
422	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
423	C₈	—	4	2	no	—
424	C₈	—	4	2	no	—
425	C₂²	—	2	4	no	—
426	C₂²	—	4	2	no	—
427	C₂²	—	4	4	no	—
428	C₂²	—	4	4	no	—
429	C₂²	—	4	2	no	—
430	C₂²	—	4	2	no	—
431	C₂²	—	4	4	no	—
432	C₂²	—	4	4	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
436	C₂²	#546×C₆₇	1	2	yes	—
437	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
438	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
439	C₂²	#577×C₆₇	1	2	yes	—
440	C₄	—	2	2	no	—
441	C₄	—	4	2	no	—
442	C₄	—	4	4	no	—
443	C₄	—	4	2	no	—
444	C₄	—	8	2	no	—
445	C₄	—	8	2	no	—
446	C₄	#452×C₆₇	1	4	yes	—
447	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
448	C₄	#535×C₆₇	1	1	yes	characteristic
449	C₄	#546×C₆₇	1	2	yes	—
450	C₂	—	4	4	no	—
451	C₂	—	8	2	no	—
452	C₂	C₆₇×#2435	1	1	yes	characteristic
453	C₂	C₂×#452×C₆₇	1	1	yes	characteristic
454	C₂	C₆₇×#3010	1	1	yes	characteristic
455	C₂	C₆₇×#3303	1	2	yes	—
456	C₂	C₆₇×#3331	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×#476×C₆₇	C₂	semidirect
4. C₆₇×#2563	C₂	semidirect
5. C₆₇×#2035	C₂	semidirect
6. C₆₇×#1981	C₂	semidirect
7. C₆₇×#1954	C₂	semidirect
8. C₆₇×#1788	C₂	semidirect
9. C₄×#102×C₆₇	C₂	semidirect
221. #25886	C₆₇	direct (N × H)
230. C₂×#487	C₂×C₆₇	semidirect
231. C₂×#476	C₂×C₆₇	semidirect
232. #2563	C₂×C₆₇	semidirect
233. #2035	C₂×C₆₇	semidirect
234. #1981	C₂×C₆₇	semidirect
235. #1954	C₂×C₆₇	semidirect
236. #1788	C₂×C₆₇	semidirect
237. C₄×#102	C₂×C₆₇	semidirect
238. C₆₇	#25886	direct (N × H)

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×#476×C₆₇	2
C₆₇×#2563	2
C₆₇×#2035	2
C₆₇×#1981	2
C₆₇×#1954	2
C₆₇×#1788	2
C₄×#102×C₆₇	2
#25886	67

Subgroups — C67×#25886 #30705137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25886 #30705137