Subgroups — C₆₇×#25887 #30705138

Order 17152.

1386 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#550×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#486×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#475×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2547	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2034	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1982	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×#103×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	C₂²	1	2	yes	—
11	#550×C₆₇	—	2	4	no	—
12	#550×C₆₇	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₆₇	—	2	1	no	—
15	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#124×C₆₇	—	2	1	no	—
17	C₂×#121×C₆₇	—	2	2	no	—
18	C₂×#121×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₆₇	—	2	1	no	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	#486×C₆₇	—	2	4	no	—
23	#475×C₆₇	—	2	4	no	—
24	C₂×#106×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#103×C₆₇	C₂²	1	2	yes	—
27	C₂×#103×C₆₇	C₂²	1	2	yes	—
28	C₂×#102×C₆₇	C₂²	1	4	yes	—
29	C₂×#102×C₆₇	C₂²	1	4	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	1	yes	characteristic
43	C₂²×D₄×C₆₇	—	2	2	no	—
44	C₂²×D₄×C₆₇	—	2	2	no	—
45	C₂²×D₄×C₆₇	—	2	1	no	—
46	C₂²×D₄×C₆₇	—	2	1	no	—
47	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₆₇	—	2	2	no	—
50	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
53	C₄⋊Q₈×C₆₇	—	2	2	no	—
54	C₄⋊Q₈×C₆₇	—	2	2	no	—
55	#127×C₆₇	—	2	8	no	—
56	#127×C₆₇	—	2	2	no	—
57	#127×C₆₇	D₄	1	4	yes	—
58	#124×C₆₇	—	4	4	no	—
59	#121×C₆₇	—	2	16	no	—
60	#121×C₆₇	—	2	4	no	—
61	#121×C₆₇	—	2	4	no	—
62	C₄×Q₈×C₆₇	—	2	2	no	—
63	C₄×Q₈×C₆₇	D₄	1	4	yes	—
64	C₄×D₄×C₆₇	—	2	8	no	—
65	C₄×D₄×C₆₇	—	2	2	no	—
66	C₄×D₄×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	4	no	—
75	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
76	C₂×C₄²×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	#106×C₆₇	—	2	4	no	—
80	C₄⋊C₈×C₆₇	—	2	2	no	—
81	#103×C₆₇	—	2	4	no	—
82	#103×C₆₇	—	2	4	no	—
83	#102×C₆₇	—	2	8	no	—
84	#102×C₆₇	—	4	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	2	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	1	no	—
95	C₂×Q₈×C₆₇	—	2	2	no	—
96	C₂×Q₈×C₆₇	—	4	2	no	—
97	C₂×Q₈×C₆₇	—	4	2	no	—
98	C₂×Q₈×C₆₇	—	4	2	no	—
99	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
100	C₂×Q₈×C₆₇	#41	1	4	yes	—
101	C₂×D₄×C₆₇	—	2	8	no	—
102	C₂×D₄×C₆₇	—	2	8	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	—	2	4	no	—
105	C₂×D₄×C₆₇	—	2	4	no	—
106	C₂×D₄×C₆₇	—	2	2	no	—
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	4	no	—
110	C₂×D₄×C₆₇	—	2	4	no	—
111	C₂×D₄×C₆₇	—	4	8	no	—
112	C₂×D₄×C₆₇	—	4	2	no	—
113	C₂×D₄×C₆₇	—	4	2	no	—
114	C₂×D₄×C₆₇	—	8	2	no	—
115	C₂×D₄×C₆₇	—	8	4	no	—
116	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
117	C₂²×C₄×C₆₇	—	2	1	no	—
118	C₂²×C₄×C₆₇	—	2	8	no	—
119	C₂²×C₄×C₆₇	—	2	1	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	4	no	—
123	C₂²×C₄×C₆₇	—	2	4	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	4	8	no	—
127	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	—	2	2	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	4	no	—
135	C₂×C₈×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	4	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	—	2	2	no	—
143	C₄⋊C₄×C₆₇	—	4	2	no	—
144	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
145	C₂²⋊C₄×C₆₇	—	2	8	no	—
146	C₂²⋊C₄×C₆₇	—	2	4	no	—
147	C₂²⋊C₄×C₆₇	—	4	8	no	—
148	C₄²×C₆₇	—	2	4	no	—
149	C₄²×C₆₇	—	4	2	no	—
150	C₄²×C₆₇	C₂×D₄	1	2	yes	—
151	C₄²×C₆₇	C₂×D₄	1	2	yes	—
152	C₂³×C₆₇	—	2	2	no	—
153	C₂³×C₆₇	—	2	4	no	—
154	C₂³×C₆₇	—	2	4	no	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	2	2	no	—
157	C₂³×C₆₇	—	2	4	no	—
158	C₂³×C₆₇	—	4	4	no	—
159	C₂³×C₆₇	—	4	2	no	—
160	C₂³×C₆₇	—	4	2	no	—
161	C₂³×C₆₇	—	4	4	no	—
162	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
163	Q₈×C₆₇	—	2	2	no	—
164	Q₈×C₆₇	—	2	4	no	—
165	Q₈×C₆₇	—	8	4	no	—
166	Q₈×C₆₇	#124	1	4	yes	complemented
167	D₄×C₆₇	—	2	8	no	—
168	D₄×C₆₇	—	4	16	no	—
169	D₄×C₆₇	—	4	2	no	—
170	D₄×C₆₇	—	4	2	no	—
171	D₄×C₆₇	—	4	4	no	—
172	D₄×C₆₇	—	4	8	no	—
173	D₄×C₆₇	—	8	4	no	—
174	D₄×C₆₇	—	8	8	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	4	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	4	16	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	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	#120	1	2	yes	—
193	C₂×C₄×C₆₇	#120	1	2	yes	—
194	C₂×C₄×C₆₇	#135	1	2	yes	—
195	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
196	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
199	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
200	C₈×C₆₇	—	4	2	no	—
201	C₈×C₆₇	—	4	2	no	—
202	C₂²×C₆₇	—	4	4	no	—
203	C₂²×C₆₇	—	4	4	no	—
204	C₂²×C₆₇	—	4	2	no	—
205	C₂²×C₆₇	—	4	2	no	—
206	C₂²×C₆₇	—	4	4	no	—
207	C₂²×C₆₇	—	4	2	no	—
208	C₂²×C₆₇	—	4	4	no	—
209	C₂²×C₆₇	—	4	4	no	—
210	C₂²×C₆₇	—	8	4	no	—
211	C₂²×C₆₇	—	8	4	no	—
212	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
213	C₂²×C₆₇	#546	1	2	yes	—
214	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
215	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₂²×C₆₇	#577	1	2	yes	—
217	C₄×C₆₇	—	2	4	no	—
218	C₄×C₆₇	—	2	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	#449	1	4	yes	—
224	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
225	C₄×C₆₇	#535	1	1	yes	characteristic
226	C₄×C₆₇	#546	1	2	yes	—
227	#25887	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂×C₆₇	—	8	2	no	—
229	C₂×C₆₇	—	8	4	no	—
230	C₂×C₆₇	#2435	1	1	yes	characteristic
231	C₂×C₆₇	C₂×#449	1	1	yes	characteristic
232	C₂×C₆₇	#3009	1	1	yes	characteristic
233	C₂×C₆₇	#3304	1	2	yes	—
234	C₂×C₆₇	#3332	1	2	yes	—
235	C₂×#550	C₂×C₆₇	1	1	yes	characteristic
236	C₂×#486	C₂×C₆₇	1	1	yes	characteristic, complemented
237	C₂×#475	C₂×C₆₇	1	1	yes	characteristic, complemented
238	#2547	C₂×C₆₇	1	1	yes	characteristic, complemented
239	#2034	C₂×C₆₇	1	4	yes	complemented
240	#1982	C₂×C₆₇	1	1	yes	characteristic, complemented
241	#1954	C₂×C₆₇	1	1	yes	characteristic, complemented
242	#1788	C₂×C₆₇	1	4	yes	complemented
243	C₄×#103	C₂×C₆₇	1	1	yes	characteristic, complemented
244	C₆₇	#25887	1	1	yes	characteristic, direct product factor, Sylow
245	C₂²×#41	C₂²×C₆₇	1	2	yes	—
246	#550	—	2	4	no	—
247	#550	—	2	4	no	—
248	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
249	C₂×#127	—	2	1	no	—
250	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
251	C₂×#124	—	2	1	no	—
252	C₂×#121	—	2	2	no	—
253	C₂×#121	C₂²×C₆₇	1	2	yes	—
254	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
255	C₂×C₄×D₄	—	2	1	no	—
256	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
257	#486	—	2	4	no	—
258	#475	—	2	4	no	—
259	C₂×#106	C₂²×C₆₇	1	2	yes	—
260	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×#103	C₂²×C₆₇	1	2	yes	—
262	C₂×#103	C₂²×C₆₇	1	2	yes	—
263	C₂×#102	C₂²×C₆₇	1	4	yes	—
264	C₂×#102	C₂²×C₆₇	1	4	yes	—
265	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
266	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
267	#397	—	2	2	no	—
268	#391	—	2	1	no	—
269	#380	C₂²×C₆₇	1	4	yes	—
270	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
271	#340	C₂²×C₆₇	1	2	yes	—
272	C₂×#41	—	2	4	no	—
273	C₂×#41	—	2	4	no	—
274	C₂×#41	—	2	8	no	—
275	C₂×#41	—	4	4	no	—
276	C₂²×Q₈	—	2	1	no	—
277	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
278	C₂²×D₄	—	2	2	no	—
279	C₂²×D₄	—	2	2	no	—
280	C₂²×D₄	—	2	1	no	—
281	C₂²×D₄	—	2	1	no	—
282	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
283	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
284	C₂³×C₄	—	2	2	no	—
285	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
286	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
287	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
288	C₄⋊Q₈	—	2	2	no	—
289	C₄⋊Q₈	—	2	2	no	—
290	#127	—	2	8	no	—
291	#127	—	2	2	no	—
292	#127	D₄×C₆₇	1	4	yes	—
293	#124	—	4	4	no	—
294	#121	—	2	16	no	—
295	#121	—	2	4	no	—
296	#121	—	2	4	no	—
297	C₄×Q₈	—	2	2	no	—
298	C₄×Q₈	D₄×C₆₇	1	4	yes	—
299	C₄×D₄	—	2	8	no	—
300	C₄×D₄	—	2	2	no	—
301	C₄×D₄	—	2	2	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
306	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
307	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
308	C₂×C₂²⋊C₄	—	2	2	no	—
309	C₂×C₂²⋊C₄	—	2	4	no	—
310	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
311	C₂×C₄²	—	2	1	no	—
312	C₂×C₄²	—	2	1	no	—
313	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
314	#106	—	2	4	no	—
315	C₄⋊C₈	—	2	2	no	—
316	#103	—	2	4	no	—
317	#103	—	2	4	no	—
318	#102	—	2	8	no	—
319	#102	—	4	4	no	—
320	C₂²⋊C₈	—	4	4	no	—
321	C₄×C₈	—	2	2	no	—
322	#95	—	2	1	no	—
323	#95	—	2	4	no	—
324	C₂⁴	—	2	2	no	—
325	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
326	#41	—	4	16	no	—
327	#41	—	4	8	no	—
328	#41	—	4	8	no	—
329	C₂×Q₈	—	2	1	no	—
330	C₂×Q₈	—	2	2	no	—
331	C₂×Q₈	—	4	2	no	—
332	C₂×Q₈	—	4	2	no	—
333	C₂×Q₈	—	4	2	no	—
334	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
335	C₂×Q₈	#41×C₆₇	1	4	yes	—
336	C₂×D₄	—	2	8	no	—
337	C₂×D₄	—	2	8	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	4	no	—
340	C₂×D₄	—	2	4	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	4	no	—
345	C₂×D₄	—	2	4	no	—
346	C₂×D₄	—	4	8	no	—
347	C₂×D₄	—	4	2	no	—
348	C₂×D₄	—	4	2	no	—
349	C₂×D₄	—	8	2	no	—
350	C₂×D₄	—	8	4	no	—
351	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	8	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	4	8	no	—
362	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
363	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
365	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	4	no	—
371	C₄⋊C₄	—	2	4	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	4	2	no	—
379	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
380	C₂²⋊C₄	—	2	8	no	—
381	C₂²⋊C₄	—	2	4	no	—
382	C₂²⋊C₄	—	4	8	no	—
383	C₄²	—	2	4	no	—
384	C₄²	—	4	2	no	—
385	C₄²	C₂×D₄×C₆₇	1	2	yes	—
386	C₄²	C₂×D₄×C₆₇	1	2	yes	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	2	4	no	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	2	4	no	—
393	C₂³	—	4	4	no	—
394	C₂³	—	4	2	no	—
395	C₂³	—	4	2	no	—
396	C₂³	—	4	4	no	—
397	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
398	Q₈	—	2	2	no	—
399	Q₈	—	2	4	no	—
400	Q₈	—	8	4	no	—
401	Q₈	#124×C₆₇	1	4	yes	complemented
402	D₄	—	2	8	no	—
403	D₄	—	4	16	no	—
404	D₄	—	4	4	no	—
405	D₄	—	4	8	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	2	no	—
408	D₄	—	8	4	no	—
409	D₄	—	8	8	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	4	16	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	4	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	#120×C₆₇	1	2	yes	—
428	C₂×C₄	#120×C₆₇	1	2	yes	—
429	C₂×C₄	#135×C₆₇	1	2	yes	—
430	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
431	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
432	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
434	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
435	C₈	—	4	2	no	—
436	C₈	—	4	2	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	4	4	no	—
440	C₂²	—	4	4	no	—
441	C₂²	—	4	4	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	4	no	—
445	C₂²	—	8	4	no	—
446	C₂²	—	8	4	no	—
447	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
448	C₂²	#546×C₆₇	1	2	yes	—
449	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
450	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
451	C₂²	#577×C₆₇	1	2	yes	—
452	C₄	—	2	2	no	—
453	C₄	—	2	4	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	2	no	—
456	C₄	—	8	2	no	—
457	C₄	—	8	2	no	—
458	C₄	#449×C₆₇	1	4	yes	—
459	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
460	C₄	#535×C₆₇	1	1	yes	characteristic
461	C₄	#546×C₆₇	1	2	yes	—
462	C₂	—	8	2	no	—
463	C₂	—	8	4	no	—
464	C₂	C₆₇×#2435	1	1	yes	characteristic
465	C₂	C₂×#449×C₆₇	1	1	yes	characteristic
466	C₂	C₆₇×#3009	1	1	yes	characteristic
467	C₂	C₆₇×#3304	1	2	yes	—
468	C₂	C₆₇×#3332	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#550×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×#486×C₆₇	C₂	semidirect
3. C₂×#475×C₆₇	C₂	semidirect
4. C₆₇×#2547	C₂	semidirect
5. C₆₇×#2034	C₂	semidirect
6. C₆₇×#1982	C₂	semidirect
7. C₆₇×#1954	C₂	semidirect
8. C₆₇×#1788	C₂	semidirect
9. C₄×#103×C₆₇	C₂	semidirect
166. Q₈×C₆₇	#124	semidirect
227. #25887	C₆₇	direct (N × H)
236. C₂×#486	C₂×C₆₇	semidirect
237. C₂×#475	C₂×C₆₇	semidirect
238. #2547	C₂×C₆₇	semidirect
239. #2034	C₂×C₆₇	semidirect
240. #1982	C₂×C₆₇	semidirect
241. #1954	C₂×C₆₇	semidirect
242. #1788	C₂×C₆₇	semidirect
243. C₄×#103	C₂×C₆₇	semidirect
244. C₆₇	#25887	direct (N × H)
401. Q₈	#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×#550×C₆₇	2
C₂×#486×C₆₇	2
C₂×#475×C₆₇	2
C₆₇×#2547	2
C₆₇×#2034	2
C₆₇×#1982	2
C₆₇×#1954	2
C₆₇×#1788	2
C₄×#103×C₆₇	2
#25887	67

Subgroups — C67×#25887 #30705138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25887 #30705138