Subgroups — C₂²⋊C₄×D₈×C₆₇ #30701697

Order 17152.

1830 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
3	D₄×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1970	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1879	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1808	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1786	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
10	C₂²×D₈×C₆₇	—	2	8	no	—
11	C₂²×D₈×C₆₇	C₄	1	8	yes	complemented
12	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
13	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
17	C₂²×C₂²⋊C₄×C₆₇	—	2	2	no	—
18	C₄×D₈×C₆₇	—	2	8	no	—
19	C₂×#107×C₆₇	C₂²	1	2	yes	complemented
20	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
21	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
22	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
24	#390×C₆₇	C₂²	1	4	yes	complemented
25	#386×C₆₇	—	2	4	no	—
26	#380×C₆₇	C₂²	1	2	yes	—
27	#379×C₆₇	—	2	2	no	—
28	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	#340×C₆₇	C₂²	1	2	yes	complemented
30	#336×C₆₇	C₂²	1	2	yes	complemented
31	C₂⁵×C₆₇	—	2	2	no	—
32	C₂²×D₄×C₆₇	—	2	8	no	—
33	C₂²×D₄×C₆₇	—	2	2	no	—
34	C₂²×D₄×C₆₇	—	2	4	no	—
35	C₂²×D₄×C₆₇	—	4	4	no	—
36	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
37	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
38	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
39	C₂³×C₄×C₆₇	—	2	4	no	—
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×D₈×C₆₇	—	2	8	no	—
42	C₂×D₈×C₆₇	—	2	32	no	—
43	C₂×D₈×C₆₇	—	2	4	no	—
44	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×D₈×C₆₇	C₂×C₄	1	8	yes	complemented
46	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	complemented
47	C₂×D₈×C₆₇	D₄	1	8	yes	—
48	C₂²×C₈×C₆₇	—	2	4	no	—
49	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₄×D₄×C₆₇	—	2	8	no	—
54	C₄×D₄×C₆₇	—	4	4	no	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
56	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
58	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
60	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
61	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
64	#107×C₆₇	—	2	4	no	—
65	#102×C₆₇	—	2	8	no	—
66	#102×C₆₇	—	4	4	no	—
67	C₂²⋊C₈×C₆₇	—	2	2	no	—
68	C₄×C₈×C₆₇	—	2	4	no	—
69	#95×C₆₇	—	2	2	no	—
70	#95×C₆₇	D₄	1	2	yes	—
71	C₂⁴×C₆₇	—	2	2	no	—
72	C₂⁴×C₆₇	—	2	2	no	—
73	C₂⁴×C₆₇	—	2	8	no	—
74	C₂⁴×C₆₇	—	4	8	no	—
75	C₂⁴×C₆₇	—	4	4	no	—
76	C₂⁴×C₆₇	—	4	4	no	—
77	C₂⁴×C₆₇	—	8	4	no	—
78	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×D₄×C₆₇	—	2	16	no	—
80	C₂×D₄×C₆₇	—	2	8	no	—
81	C₂×D₄×C₆₇	—	2	4	no	—
82	C₂×D₄×C₆₇	—	2	2	no	—
83	C₂×D₄×C₆₇	—	2	2	no	—
84	C₂×D₄×C₆₇	—	2	4	no	—
85	C₂×D₄×C₆₇	—	2	4	no	—
86	C₂×D₄×C₆₇	—	4	8	no	—
87	C₂×D₄×C₆₇	—	4	16	no	—
88	C₂×D₄×C₆₇	—	4	8	no	—
89	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
90	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
91	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	4	no	—
95	C₂²×C₄×C₆₇	—	4	8	no	—
96	C₂²×C₄×C₆₇	—	4	8	no	—
97	C₂²×C₄×C₆₇	—	8	4	no	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₆₇	#41	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
104	D₈×C₆₇	—	2	16	no	—
105	D₈×C₆₇	—	2	8	no	—
106	D₈×C₆₇	C₂²⋊C₄	1	16	yes	direct product factor
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	8	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	4	no	—
117	C₂²⋊C₄×C₆₇	—	2	2	no	—
118	C₂²⋊C₄×C₆₇	—	4	8	no	—
119	C₂²⋊C₄×C₆₇	—	4	4	no	—
120	C₂²⋊C₄×C₆₇	—	4	8	no	—
121	C₂²⋊C₄×C₆₇	—	8	4	no	—
122	C₂²⋊C₄×C₆₇	D₈	1	4	yes	direct product factor
123	C₄²×C₆₇	—	2	2	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₂³×C₆₇	—	2	2	no	—
126	C₂³×C₆₇	—	2	2	no	—
127	C₂³×C₆₇	—	2	4	no	—
128	C₂³×C₆₇	—	2	8	no	—
129	C₂³×C₆₇	—	2	4	no	—
130	C₂³×C₆₇	—	4	16	no	—
131	C₂³×C₆₇	—	4	2	no	—
132	C₂³×C₆₇	—	4	2	no	—
133	C₂³×C₆₇	—	4	4	no	—
134	C₂³×C₆₇	—	4	4	no	—
135	C₂³×C₆₇	—	4	2	no	—
136	C₂³×C₆₇	—	4	2	no	—
137	C₂³×C₆₇	—	4	4	no	—
138	C₂³×C₆₇	—	4	16	no	—
139	C₂³×C₆₇	—	8	8	no	—
140	C₂³×C₆₇	—	8	16	no	—
141	C₂³×C₆₇	—	8	16	no	—
142	C₂³×C₆₇	C₄×D₄	1	2	yes	—
143	C₂³×C₆₇	C₂×D₈	1	2	yes	—
144	C₂³×C₆₇	#135	1	2	yes	—
145	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
146	D₄×C₆₇	—	2	4	no	—
147	D₄×C₆₇	—	2	4	no	—
148	D₄×C₆₇	—	2	4	no	—
149	D₄×C₆₇	—	2	8	no	—
150	D₄×C₆₇	—	4	16	no	—
151	D₄×C₆₇	—	4	4	no	—
152	D₄×C₆₇	C₂×C₂²⋊C₄	1	8	yes	complemented
153	C₂×C₄×C₆₇	—	2	2	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	4	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	4	4	no	—
163	C₂×C₄×C₆₇	—	4	4	no	—
164	C₂×C₄×C₆₇	—	4	4	no	—
165	C₂×C₄×C₆₇	—	8	16	no	—
166	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	complemented
168	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
170	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
172	C₈×C₆₇	—	2	2	no	—
173	C₈×C₆₇	—	4	2	no	—
174	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	complemented
175	C₂²×C₆₇	—	2	2	no	—
176	C₂²×C₆₇	—	2	2	no	—
177	C₂²×C₆₇	—	2	4	no	—
178	C₂²×C₆₇	—	2	4	no	—
179	C₂²×C₆₇	—	2	8	no	—
180	C₂²×C₆₇	—	4	4	no	—
181	C₂²×C₆₇	—	4	4	no	—
182	C₂²×C₆₇	—	4	8	no	—
183	C₂²×C₆₇	—	4	8	no	—
184	C₂²×C₆₇	—	4	4	no	—
185	C₂²×C₆₇	—	4	4	no	—
186	C₂²×C₆₇	—	4	4	no	—
187	C₂²×C₆₇	—	8	32	no	—
188	C₂²×C₆₇	—	8	4	no	—
189	C₂²×C₆₇	—	8	4	no	—
190	C₂²×C₆₇	C₄×D₈	1	4	yes	complemented
191	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂²×C₆₇	D₄²	1	2	yes	—
193	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
194	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
195	C₂²×C₆₇	#576	1	2	yes	—
196	C₄×C₆₇	—	2	2	no	—
197	C₄×C₆₇	—	2	4	no	—
198	C₄×C₆₇	—	4	2	no	—
199	C₄×C₆₇	—	8	4	no	—
200	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
201	C₄×C₆₇	#518	1	1	yes	characteristic
202	C₄×C₆₇	#546	1	2	yes	—
203	C₂²⋊C₄×D₈	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂×C₆₇	—	2	4	no	—
205	C₂×C₆₇	—	4	8	no	—
206	C₂×C₆₇	—	8	4	no	—
207	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
209	C₂×C₆₇	#2948	1	1	yes	characteristic
210	C₂×C₆₇	D₄×D₈	1	2	yes	—
211	C₂×C₆₇	#3280	1	2	yes	—
212	C₂³×D₈	C₂×C₆₇	1	1	yes	characteristic
213	C₂×C₄×D₈	C₂×C₆₇	1	2	yes	complemented
214	D₄×C₂²⋊C₄	C₂×C₆₇	1	2	yes	complemented
215	#1970	C₂×C₆₇	1	2	yes	complemented
216	#1879	C₂×C₆₇	1	4	yes	complemented
217	#1808	C₂×C₆₇	1	1	yes	characteristic, complemented
218	#1786	C₂×C₆₇	1	2	yes	complemented
219	C₈×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
220	C₆₇	C₂²⋊C₄×D₈	1	1	yes	characteristic, direct product factor, Sylow
221	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
222	C₂²×D₈	—	2	8	no	—
223	C₂²×D₈	C₄×C₆₇	1	8	yes	complemented
224	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
225	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
226	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
227	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
229	C₂²×C₂²⋊C₄	—	2	2	no	—
230	C₄×D₈	—	2	8	no	—
231	C₂×#107	C₂²×C₆₇	1	2	yes	complemented
232	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
233	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
234	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
236	#390	C₂²×C₆₇	1	4	yes	complemented
237	#386	—	2	4	no	—
238	#380	C₂²×C₆₇	1	2	yes	—
239	#379	—	2	2	no	—
240	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
241	#340	C₂²×C₆₇	1	2	yes	complemented
242	#336	C₂²×C₆₇	1	2	yes	complemented
243	C₂⁵	—	2	2	no	—
244	C₂²×D₄	—	2	8	no	—
245	C₂²×D₄	—	2	2	no	—
246	C₂²×D₄	—	2	4	no	—
247	C₂²×D₄	—	4	4	no	—
248	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
249	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
250	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
251	C₂³×C₄	—	2	4	no	—
252	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
253	C₂×D₈	—	2	8	no	—
254	C₂×D₈	—	2	32	no	—
255	C₂×D₈	—	2	4	no	—
256	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
257	C₂×D₈	C₂×C₄×C₆₇	1	8	yes	complemented
258	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	complemented
259	C₂×D₈	D₄×C₆₇	1	8	yes	—
260	C₂²×C₈	—	2	4	no	—
261	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
262	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
263	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
264	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
265	C₄×D₄	—	2	8	no	—
266	C₄×D₄	—	4	4	no	—
267	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
268	C₂×C₂²⋊C₄	—	2	2	no	—
269	C₂×C₂²⋊C₄	—	2	4	no	—
270	C₂×C₂²⋊C₄	—	2	4	no	—
271	C₂×C₂²⋊C₄	—	2	4	no	—
272	C₂×C₂²⋊C₄	—	4	8	no	—
273	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
274	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
276	#107	—	2	4	no	—
277	#102	—	2	8	no	—
278	#102	—	4	4	no	—
279	C₂²⋊C₈	—	2	2	no	—
280	C₄×C₈	—	2	4	no	—
281	#95	—	2	2	no	—
282	#95	D₄×C₆₇	1	2	yes	—
283	C₂⁴	—	2	2	no	—
284	C₂⁴	—	2	2	no	—
285	C₂⁴	—	2	8	no	—
286	C₂⁴	—	4	8	no	—
287	C₂⁴	—	4	4	no	—
288	C₂⁴	—	4	4	no	—
289	C₂⁴	—	8	4	no	—
290	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
291	C₂×D₄	—	2	16	no	—
292	C₂×D₄	—	2	8	no	—
293	C₂×D₄	—	2	4	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	2	4	no	—
297	C₂×D₄	—	2	4	no	—
298	C₂×D₄	—	4	8	no	—
299	C₂×D₄	—	4	16	no	—
300	C₂×D₄	—	4	8	no	—
301	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
302	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
303	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	2	no	—
306	C₂²×C₄	—	2	4	no	—
307	C₂²×C₄	—	4	8	no	—
308	C₂²×C₄	—	4	8	no	—
309	C₂²×C₄	—	8	4	no	—
310	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
311	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
314	C₂²×C₄	#41×C₆₇	1	2	yes	—
315	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
316	D₈	—	2	16	no	—
317	D₈	—	2	8	no	—
318	D₈	C₂²⋊C₄×C₆₇	1	16	yes	direct product factor
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	8	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
325	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
326	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	2	4	no	—
329	C₂²⋊C₄	—	2	2	no	—
330	C₂²⋊C₄	—	4	8	no	—
331	C₂²⋊C₄	—	4	4	no	—
332	C₂²⋊C₄	—	4	8	no	—
333	C₂²⋊C₄	—	8	4	no	—
334	C₂²⋊C₄	D₈×C₆₇	1	4	yes	direct product factor
335	C₄²	—	2	2	no	—
336	C₄²	—	2	2	no	—
337	C₂³	—	2	2	no	—
338	C₂³	—	2	2	no	—
339	C₂³	—	2	4	no	—
340	C₂³	—	2	8	no	—
341	C₂³	—	2	4	no	—
342	C₂³	—	4	16	no	—
343	C₂³	—	4	2	no	—
344	C₂³	—	4	2	no	—
345	C₂³	—	4	4	no	—
346	C₂³	—	4	4	no	—
347	C₂³	—	4	2	no	—
348	C₂³	—	4	2	no	—
349	C₂³	—	4	4	no	—
350	C₂³	—	4	16	no	—
351	C₂³	—	8	8	no	—
352	C₂³	—	8	16	no	—
353	C₂³	—	8	16	no	—
354	C₂³	C₄×D₄×C₆₇	1	2	yes	—
355	C₂³	C₂×D₈×C₆₇	1	2	yes	—
356	C₂³	#135×C₆₇	1	2	yes	—
357	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
358	D₄	—	2	4	no	—
359	D₄	—	2	4	no	—
360	D₄	—	2	4	no	—
361	D₄	—	2	8	no	—
362	D₄	—	4	16	no	—
363	D₄	—	4	4	no	—
364	D₄	C₂×C₂²⋊C₄×C₆₇	1	8	yes	complemented
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	4	4	no	—
375	C₂×C₄	—	4	4	no	—
376	C₂×C₄	—	4	4	no	—
377	C₂×C₄	—	8	16	no	—
378	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
379	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	complemented
380	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
382	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
383	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
384	C₈	—	2	2	no	—
385	C₈	—	4	2	no	—
386	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	complemented
387	C₂²	—	2	4	no	—
388	C₂²	—	2	4	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	8	no	—
392	C₂²	—	4	8	no	—
393	C₂²	—	4	8	no	—
394	C₂²	—	4	4	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	4	4	no	—
397	C₂²	—	4	4	no	—
398	C₂²	—	4	4	no	—
399	C₂²	—	8	32	no	—
400	C₂²	—	8	4	no	—
401	C₂²	—	8	4	no	—
402	C₂²	C₄×D₈×C₆₇	1	4	yes	complemented
403	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
404	C₂²	D₄²×C₆₇	1	2	yes	—
405	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
406	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
407	C₂²	#576×C₆₇	1	2	yes	—
408	C₄	—	2	4	no	—
409	C₄	—	2	2	no	—
410	C₄	—	4	2	no	—
411	C₄	—	8	4	no	—
412	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
413	C₄	#518×C₆₇	1	1	yes	characteristic
414	C₄	#546×C₆₇	1	2	yes	—
415	C₂	—	2	4	no	—
416	C₂	—	4	8	no	—
417	C₂	—	8	4	no	—
418	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
419	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
420	C₂	C₆₇×#2948	1	1	yes	characteristic
421	C₂	D₄×D₈×C₆₇	1	2	yes	—
422	C₂	C₆₇×#3280	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂³×D₈×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄×C₂²⋊C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₈×C₆₇	C₂	semidirect
3. D₄×C₂²⋊C₄×C₆₇	C₂	semidirect
4. C₆₇×#1970	C₂	semidirect
5. C₆₇×#1879	C₂	semidirect
6. C₆₇×#1808	C₂	semidirect
7. C₆₇×#1786	C₂	semidirect
8. C₈×C₂²⋊C₄×C₆₇	C₂	semidirect
11. C₂²×D₈×C₆₇	C₄	semidirect
16. C₂×C₄×D₄×C₆₇	C₂²	semidirect
19. C₂×#107×C₆₇	C₂²	semidirect
20. C₂×#102×C₆₇	C₂²	semidirect
21. C₂×#102×C₆₇	C₂²	semidirect
23. C₂×C₄×C₈×C₆₇	C₂²	semidirect
24. #390×C₆₇	C₂²	semidirect
29. #340×C₆₇	C₂²	semidirect
30. #336×C₆₇	C₂²	semidirect
36. C₂²×D₄×C₆₇	C₂×C₄	semidirect
45. C₂×D₈×C₆₇	C₂×C₄	semidirect
46. C₂×D₈×C₆₇	C₂×C₄	semidirect
49. C₂²×C₈×C₆₇	C₂×C₄	semidirect
106. D₈×C₆₇	C₂²⋊C₄	direct (N × H)
122. C₂²⋊C₄×C₆₇	D₈	direct (N × H)
152. D₄×C₆₇	C₂×C₂²⋊C₄	semidirect
167. C₂×C₄×C₆₇	C₂×D₈	semidirect
174. C₈×C₆₇	C₂×C₂²⋊C₄	semidirect
190. C₂²×C₆₇	C₄×D₈	semidirect
203. C₂²⋊C₄×D₈	C₆₇	direct (N × H)
213. C₂×C₄×D₈	C₂×C₆₇	semidirect
214. D₄×C₂²⋊C₄	C₂×C₆₇	semidirect
215. #1970	C₂×C₆₇	semidirect
216. #1879	C₂×C₆₇	semidirect
217. #1808	C₂×C₆₇	semidirect
218. #1786	C₂×C₆₇	semidirect
219. C₈×C₂²⋊C₄	C₂×C₆₇	semidirect
220. C₆₇	C₂²⋊C₄×D₈	direct (N × H)
223. C₂²×D₈	C₄×C₆₇	semidirect
228. C₂×C₄×D₄	C₂²×C₆₇	semidirect
231. C₂×#107	C₂²×C₆₇	semidirect
232. C₂×#102	C₂²×C₆₇	semidirect
233. C₂×#102	C₂²×C₆₇	semidirect
235. C₂×C₄×C₈	C₂²×C₆₇	semidirect
236. #390	C₂²×C₆₇	semidirect
241. #340	C₂²×C₆₇	semidirect
242. #336	C₂²×C₆₇	semidirect
248. C₂²×D₄	C₂×C₄×C₆₇	semidirect
257. C₂×D₈	C₂×C₄×C₆₇	semidirect
258. C₂×D₈	C₂×C₄×C₆₇	semidirect
261. C₂²×C₈	C₂×C₄×C₆₇	semidirect
318. D₈	C₂²⋊C₄×C₆₇	direct (N × H)
334. C₂²⋊C₄	D₈×C₆₇	direct (N × H)
364. D₄	C₂×C₂²⋊C₄×C₆₇	semidirect
379. C₂×C₄	C₂×D₈×C₆₇	semidirect
386. C₈	C₂×C₂²⋊C₄×C₆₇	semidirect
402. C₂²	C₄×D₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₈×C₆₇	2
C₂×C₄×D₈×C₆₇	2
D₄×C₂²⋊C₄×C₆₇	2
C₆₇×#1970	2
C₆₇×#1879	2
C₆₇×#1808	2
C₆₇×#1786	2
C₈×C₂²⋊C₄×C₆₇	2
C₂²⋊C₄×D₈	67

Subgroups — C22⋊C4×D8×C67 #30701697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊C₄×D₈×C₆₇ #30701697