Subgroups — C₆₇×#25888 #30705139

Order 17152.

1282 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#487×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#477×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2563	C₂	1	1	yes	maximal, characteristic
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
8	C₆₇×#1787	C₂	1	4	yes	maximal, complemented
9	C₄×#103×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
11	D₄×Q₈×C₆₇	—	2	4	no	—
12	D₄×Q₈×C₆₇	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₆₇	—	2	1	no	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	#487×C₆₇	—	2	4	no	—
22	#477×C₆₇	—	2	4	no	—
23	C₂×#107×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#103×C₆₇	C₂²	1	2	yes	—
26	C₂×#103×C₆₇	C₂²	1	4	yes	—
27	C₂×#103×C₆₇	C₂²	1	2	yes	—
28	C₂×#103×C₆₇	C₂²	1	4	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	C₂²×Q₈×C₆₇	—	2	4	no	—
38	C₂²×Q₈×C₆₇	—	2	4	no	—
39	C₂²×Q₈×C₆₇	—	2	2	no	—
40	C₂²×Q₈×C₆₇	—	2	1	no	—
41	C₂²×Q₈×C₆₇	—	2	8	no	—
42	C₂²×Q₈×C₆₇	—	4	4	no	—
43	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
44	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	—	2	2	no	—
47	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₆₇	—	2	8	no	—
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	C₂²⋊Q₈×C₆₇	—	2	16	no	—
56	C₂²⋊Q₈×C₆₇	—	2	4	no	—
57	C₂²⋊Q₈×C₆₇	—	2	4	no	—
58	C₄×Q₈×C₆₇	—	2	2	no	—
59	C₄×Q₈×C₆₇	D₄	1	4	yes	—
60	C₄×D₄×C₆₇	—	2	8	no	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	C₄×D₄×C₆₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	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	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	4	no	—
74	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
75	C₂×C₄²×C₆₇	—	2	1	no	—
76	C₂×C₄²×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	#107×C₆₇	—	2	4	no	—
79	C₄⋊C₈×C₆₇	—	2	2	no	—
80	#103×C₆₇	—	2	4	no	—
81	#103×C₆₇	—	2	8	no	—
82	#103×C₆₇	—	2	4	no	—
83	#103×C₆₇	—	4	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	C₂×D₄	1	2	yes	—
89	C₂×Q₈×C₆₇	—	2	8	no	—
90	C₂×Q₈×C₆₇	—	2	1	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	—	2	4	no	—
94	C₂×Q₈×C₆₇	—	2	4	no	—
95	C₂×Q₈×C₆₇	—	4	8	no	—
96	C₂×Q₈×C₆₇	—	4	16	no	—
97	C₂×Q₈×C₆₇	—	4	8	no	—
98	C₂×Q₈×C₆₇	—	4	8	no	—
99	C₂×Q₈×C₆₇	—	4	2	no	—
100	C₂×Q₈×C₆₇	—	8	2	no	—
101	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
102	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
103	C₂×Q₈×C₆₇	#41	1	4	yes	—
104	C₂×D₄×C₆₇	—	2	2	no	—
105	C₂×D₄×C₆₇	—	2	2	no	—
106	C₂×D₄×C₆₇	Q₁₆	1	4	yes	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
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	2	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	4	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	4	8	no	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	4	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	2	8	no	—
128	C₄⋊C₄×C₆₇	—	2	4	no	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	—	2	4	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	1	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	4	2	no	—
138	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
139	C₂²⋊C₄×C₆₇	—	2	8	no	—
140	C₂²⋊C₄×C₆₇	—	2	4	no	—
141	C₂²⋊C₄×C₆₇	—	8	4	no	—
142	C₄²×C₆₇	—	2	4	no	—
143	C₄²×C₆₇	—	4	2	no	—
144	C₄²×C₆₇	C₂×D₄	1	2	yes	—
145	C₄²×C₆₇	C₂×D₄	1	2	yes	—
146	C₂³×C₆₇	—	2	2	no	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	—	2	4	no	—
150	C₂³×C₆₇	C₂×Q₁₆	1	4	yes	—
151	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
152	Q₈×C₆₇	—	2	2	no	—
153	Q₈×C₆₇	—	2	4	no	—
154	Q₈×C₆₇	—	2	8	no	—
155	Q₈×C₆₇	—	4	16	no	—
156	Q₈×C₆₇	—	4	2	no	—
157	Q₈×C₆₇	—	4	2	no	—
158	Q₈×C₆₇	—	8	4	no	—
159	Q₈×C₆₇	#124	1	4	yes	—
160	D₄×C₆₇	—	2	8	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	4	16	no	—
169	C₂×C₄×C₆₇	—	4	4	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	—	4	4	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	4	no	—
179	C₂×C₄×C₆₇	#120	1	2	yes	—
180	C₂×C₄×C₆₇	#120	1	2	yes	—
181	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
182	C₂×C₄×C₆₇	#137	1	2	yes	—
183	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
186	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
187	C₈×C₆₇	—	4	2	no	—
188	C₈×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	2	4	no	—
190	C₂²×C₆₇	—	2	8	no	—
191	C₂²×C₆₇	—	4	4	no	—
192	C₂²×C₆₇	—	4	2	no	—
193	C₂²×C₆₇	—	4	2	no	—
194	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
195	C₂²×C₆₇	#546	1	2	yes	—
196	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
197	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
198	C₂²×C₆₇	#577	1	2	yes	—
199	C₄×C₆₇	—	2	4	no	—
200	C₄×C₆₇	—	2	2	no	—
201	C₄×C₆₇	—	4	2	no	—
202	C₄×C₆₇	—	4	4	no	—
203	C₄×C₆₇	—	4	2	no	—
204	C₄×C₆₇	—	8	2	no	—
205	C₄×C₆₇	—	8	2	no	—
206	C₄×C₆₇	#451	1	4	yes	—
207	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
208	C₄×C₆₇	#535	1	1	yes	characteristic
209	C₄×C₆₇	#546	1	2	yes	—
210	#25888	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂×C₆₇	—	4	4	no	—
212	C₂×C₆₇	#2435	1	1	yes	characteristic
213	C₂×C₆₇	C₂×#451	1	1	yes	characteristic
214	C₂×C₆₇	#3010	1	1	yes	characteristic
215	C₂×C₆₇	#3299	1	2	yes	—
216	C₂×C₆₇	#3329	1	2	yes	—
217	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
218	C₂×#487	C₂×C₆₇	1	1	yes	characteristic, complemented
219	C₂×#477	C₂×C₆₇	1	1	yes	characteristic, complemented
220	#2563	C₂×C₆₇	1	1	yes	characteristic
221	#2036	C₂×C₆₇	1	4	yes	complemented
222	#1981	C₂×C₆₇	1	1	yes	characteristic, complemented
223	#1954	C₂×C₆₇	1	1	yes	characteristic
224	#1787	C₂×C₆₇	1	4	yes	complemented
225	C₄×#103	C₂×C₆₇	1	1	yes	characteristic, complemented
226	C₆₇	#25888	1	1	yes	characteristic, direct product factor, Sylow
227	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
228	D₄×Q₈	—	2	4	no	—
229	D₄×Q₈	—	2	4	no	—
230	C₂×C₄⋊Q₈	—	2	1	no	—
231	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
232	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
233	C₂×C₂²⋊Q₈	—	2	2	no	—
234	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
235	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×C₄×D₄	—	2	1	no	—
237	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
238	#487	—	2	4	no	—
239	#477	—	2	4	no	—
240	C₂×#107	C₂²×C₆₇	1	2	yes	—
241	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
242	C₂×#103	C₂²×C₆₇	1	2	yes	—
243	C₂×#103	C₂²×C₆₇	1	4	yes	—
244	C₂×#103	C₂²×C₆₇	1	2	yes	—
245	C₂×#103	C₂²×C₆₇	1	4	yes	—
246	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
247	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
248	#393	—	2	2	no	—
249	#390	—	2	1	no	—
250	#382	—	2	1	no	—
251	#380	C₂²×C₆₇	1	4	yes	—
252	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
253	#340	C₂²×C₆₇	1	2	yes	—
254	C₂²×Q₈	—	2	4	no	—
255	C₂²×Q₈	—	2	4	no	—
256	C₂²×Q₈	—	2	2	no	—
257	C₂²×Q₈	—	2	1	no	—
258	C₂²×Q₈	—	2	8	no	—
259	C₂²×Q₈	—	4	4	no	—
260	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
261	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
262	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
263	C₂³×C₄	—	2	2	no	—
264	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
265	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
266	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
267	C₄⋊Q₈	—	2	8	no	—
268	C₄⋊Q₈	—	2	2	no	—
269	C₄⋊Q₈	—	2	2	no	—
270	C₄⋊Q₈	—	2	2	no	—
271	C₄⋊Q₈	D₄×C₆₇	1	4	yes	—
272	C₂²⋊Q₈	—	2	16	no	—
273	C₂²⋊Q₈	—	2	4	no	—
274	C₂²⋊Q₈	—	2	4	no	—
275	C₄×Q₈	—	2	2	no	—
276	C₄×Q₈	D₄×C₆₇	1	4	yes	—
277	C₄×D₄	—	2	8	no	—
278	C₄×D₄	—	2	2	no	—
279	C₄×D₄	—	2	2	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	2	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
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	2	yes	—
288	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
289	C₂×C₂²⋊C₄	—	2	2	no	—
290	C₂×C₂²⋊C₄	—	2	4	no	—
291	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
292	C₂×C₄²	—	2	1	no	—
293	C₂×C₄²	—	2	1	no	—
294	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
295	#107	—	2	4	no	—
296	C₄⋊C₈	—	2	2	no	—
297	#103	—	2	4	no	—
298	#103	—	2	8	no	—
299	#103	—	2	4	no	—
300	#103	—	4	4	no	—
301	C₂²⋊C₈	—	4	4	no	—
302	C₄×C₈	—	2	2	no	—
303	#95	—	2	1	no	—
304	#95	—	2	4	no	—
305	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
306	C₂×Q₈	—	2	8	no	—
307	C₂×Q₈	—	2	1	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	4	no	—
311	C₂×Q₈	—	2	4	no	—
312	C₂×Q₈	—	4	8	no	—
313	C₂×Q₈	—	4	16	no	—
314	C₂×Q₈	—	4	8	no	—
315	C₂×Q₈	—	4	8	no	—
316	C₂×Q₈	—	4	2	no	—
317	C₂×Q₈	—	8	2	no	—
318	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
319	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
320	C₂×Q₈	#41×C₆₇	1	4	yes	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	Q₁₆×C₆₇	1	4	yes	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	8	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	4	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	4	8	no	—
335	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
338	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	4	no	—
344	C₄⋊C₄	—	2	8	no	—
345	C₄⋊C₄	—	2	4	no	—
346	C₄⋊C₄	—	2	4	no	—
347	C₄⋊C₄	—	2	4	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
356	C₂²⋊C₄	—	2	8	no	—
357	C₂²⋊C₄	—	2	4	no	—
358	C₂²⋊C₄	—	8	4	no	—
359	C₄²	—	2	4	no	—
360	C₄²	—	4	2	no	—
361	C₄²	C₂×D₄×C₆₇	1	2	yes	—
362	C₄²	C₂×D₄×C₆₇	1	2	yes	—
363	C₂³	—	2	2	no	—
364	C₂³	—	2	4	no	—
365	C₂³	—	2	2	no	—
366	C₂³	—	2	4	no	—
367	C₂³	C₂×Q₁₆×C₆₇	1	4	yes	—
368	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
369	Q₈	—	2	2	no	—
370	Q₈	—	2	4	no	—
371	Q₈	—	2	8	no	—
372	Q₈	—	4	16	no	—
373	Q₈	—	4	2	no	—
374	Q₈	—	4	2	no	—
375	Q₈	—	8	4	no	—
376	Q₈	#124×C₆₇	1	4	yes	—
377	D₄	—	2	8	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	4	16	no	—
386	C₂×C₄	—	4	4	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	4	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	4	no	—
396	C₂×C₄	#120×C₆₇	1	2	yes	—
397	C₂×C₄	#120×C₆₇	1	2	yes	—
398	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
399	C₂×C₄	#137×C₆₇	1	2	yes	—
400	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
401	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
402	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
403	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
404	C₈	—	4	2	no	—
405	C₈	—	4	2	no	—
406	C₂²	—	2	8	no	—
407	C₂²	—	2	4	no	—
408	C₂²	—	4	2	no	—
409	C₂²	—	4	4	no	—
410	C₂²	—	4	2	no	—
411	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
412	C₂²	#546×C₆₇	1	2	yes	—
413	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
414	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
415	C₂²	#577×C₆₇	1	2	yes	—
416	C₄	—	2	2	no	—
417	C₄	—	2	4	no	—
418	C₄	—	4	2	no	—
419	C₄	—	4	2	no	—
420	C₄	—	4	4	no	—
421	C₄	—	8	2	no	—
422	C₄	—	8	2	no	—
423	C₄	#451×C₆₇	1	4	yes	—
424	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
425	C₄	#535×C₆₇	1	1	yes	characteristic
426	C₄	#546×C₆₇	1	2	yes	—
427	C₂	—	4	4	no	—
428	C₂	C₆₇×#2435	1	1	yes	characteristic
429	C₂	C₂×#451×C₆₇	1	1	yes	characteristic
430	C₂	C₆₇×#3010	1	1	yes	characteristic
431	C₂	C₆₇×#3299	1	2	yes	—
432	C₂	C₆₇×#3329	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	D₄
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×#477×C₆₇	C₂	semidirect
5. C₆₇×#2036	C₂	semidirect
6. C₆₇×#1981	C₂	semidirect
8. C₆₇×#1787	C₂	semidirect
9. C₄×#103×C₆₇	C₂	semidirect
210. #25888	C₆₇	direct (N × H)
218. C₂×#487	C₂×C₆₇	semidirect
219. C₂×#477	C₂×C₆₇	semidirect
221. #2036	C₂×C₆₇	semidirect
222. #1981	C₂×C₆₇	semidirect
224. #1787	C₂×C₆₇	semidirect
225. C₄×#103	C₂×C₆₇	semidirect
226. C₆₇	#25888	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×D₄×Q₈×C₆₇	2
C₂×#487×C₆₇	2
C₂×#477×C₆₇	2
C₆₇×#2563	2
C₆₇×#2036	2
C₆₇×#1981	2
C₆₇×#1954	2
C₆₇×#1787	2
C₄×#103×C₆₇	2
#25888	67

Subgroups — C67×#25888 #30705139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25888 #30705139