Subgroups — C₂×D₄×D₈×C₆₇ #30740290

Order 17152.

3710 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	C₂³×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×D₄²×C₆₇	C₂	1	2	yes	maximal, complemented
3	D₄×D₈×C₆₇	C₂	1	16	yes	maximal, direct product factor
4	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#466×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#459×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#447×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₆₇	—	2	2	no	—
11	C₂³×D₄×C₆₇	C₂²	1	4	yes	complemented
12	C₂²×D₈×C₆₇	—	2	16	no	—
13	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₆₇	C₂²	1	4	yes	complemented
15	C₂²×D₈×C₆₇	C₂²	1	2	yes	complemented
16	C₂²×D₈×C₆₇	C₂²	1	2	yes	complemented
17	C₂²×D₈×C₆₇	C₂²	1	16	yes	complemented
18	C₂³×C₈×C₆₇	C₂²	1	2	yes	complemented
19	D₄²×C₆₇	—	2	8	no	—
20	D₄²×C₆₇	C₂²	1	16	yes	complemented
21	C₂×#127×C₆₇	C₂²	1	2	yes	complemented
22	C₂×#121×C₆₇	—	2	2	no	—
23	C₂×#121×C₆₇	C₂²	1	4	yes	complemented
24	C₂×#120×C₆₇	—	2	4	no	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
27	#493×C₆₇	C₂²	1	8	yes	complemented
28	#466×C₆₇	C₂²	1	16	yes	complemented
29	#459×C₆₇	C₂²	1	16	yes	complemented
30	#447×C₆₇	C₂²	1	32	yes	complemented
31	C₄×D₈×C₆₇	C₂²	1	8	yes	complemented
32	C₈×D₄×C₆₇	C₂²	1	8	yes	complemented
33	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
36	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
37	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	C₂⁵×C₆₇	—	2	4	no	—
40	C₂²×D₄×C₆₇	—	2	16	no	—
41	C₂²×D₄×C₆₇	—	2	4	no	—
42	C₂²×D₄×C₆₇	—	2	16	no	—
43	C₂²×D₄×C₆₇	—	2	4	no	—
44	C₂²×D₄×C₆₇	—	2	2	no	—
45	C₂²×D₄×C₆₇	—	2	8	no	—
46	C₂²×D₄×C₆₇	—	2	2	no	—
47	C₂²×D₄×C₆₇	—	4	8	no	—
48	C₂²×D₄×C₆₇	—	4	16	no	—
49	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
51	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
52	C₂²×D₄×C₆₇	C₂³	1	16	yes	complemented
53	C₂³×C₄×C₆₇	—	2	2	no	—
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂×D₈×C₆₇	—	2	16	no	—
56	C₂×D₈×C₆₇	—	2	64	no	—
57	C₂×D₈×C₆₇	—	2	8	no	—
58	C₂×D₈×C₆₇	—	4	4	no	—
59	C₂×D₈×C₆₇	D₄	1	8	yes	direct product factor
60	C₂×D₈×C₆₇	C₂³	1	4	yes	—
61	C₂×D₈×C₆₇	C₂³	1	16	yes	complemented
62	C₂×D₈×C₆₇	C₂³	1	8	yes	complemented
63	C₂×D₈×C₆₇	C₂³	1	8	yes	complemented
64	C₂²×C₈×C₆₇	—	2	8	no	—
65	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₆₇	C₂³	1	8	yes	complemented
69	#127×C₆₇	—	2	4	no	—
70	#127×C₆₇	C₂³	1	8	yes	complemented
71	#121×C₆₇	—	2	8	no	—
72	#121×C₆₇	—	2	16	no	—
73	#121×C₆₇	C₂³	1	16	yes	complemented
74	#120×C₆₇	—	2	32	no	—
75	C₄×D₄×C₆₇	—	2	4	no	—
76	C₄×D₄×C₆₇	D₄	1	4	yes	—
77	C₄×D₄×C₆₇	C₂³	1	4	yes	—
78	C₄×D₄×C₆₇	C₂³	1	8	yes	complemented
79	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
83	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	#107×C₆₇	C₂³	1	4	yes	complemented
86	C₄⋊C₈×C₆₇	C₂³	1	4	yes	complemented
87	#102×C₆₇	C₂³	1	16	yes	complemented
88	#102×C₆₇	C₂³	1	8	yes	complemented
89	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	complemented
90	C₄×C₈×C₆₇	C₂³	1	4	yes	complemented
91	C₂⁴×C₆₇	—	2	2	no	—
92	C₂⁴×C₆₇	—	2	4	no	—
93	C₂⁴×C₆₇	—	2	16	no	—
94	C₂⁴×C₆₇	—	4	16	no	—
95	C₂⁴×C₆₇	—	4	16	no	—
96	C₂⁴×C₆₇	—	8	8	no	—
97	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
98	C₂×D₄×C₆₇	—	2	32	no	—
99	C₂×D₄×C₆₇	—	2	16	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	8	no	—
102	C₂×D₄×C₆₇	—	2	16	no	—
103	C₂×D₄×C₆₇	—	2	4	no	—
104	C₂×D₄×C₆₇	—	2	8	no	—
105	C₂×D₄×C₆₇	—	2	8	no	—
106	C₂×D₄×C₆₇	—	2	4	no	—
107	C₂×D₄×C₆₇	—	2	8	no	—
108	C₂×D₄×C₆₇	—	4	16	no	—
109	C₂×D₄×C₆₇	—	4	32	no	—
110	C₂×D₄×C₆₇	—	4	16	no	—
111	C₂×D₄×C₆₇	—	4	16	no	—
112	C₂×D₄×C₆₇	—	4	64	no	—
113	C₂×D₄×C₆₇	—	4	8	no	—
114	C₂×D₄×C₆₇	—	8	4	no	—
115	C₂×D₄×C₆₇	D₈	1	8	yes	direct product factor
116	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
117	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	complemented
118	C₂×D₄×C₆₇	C₂⁴	1	8	yes	—
119	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	2	4	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	8	no	—
124	C₂²×C₄×C₆₇	—	4	8	no	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
131	D₈×C₆₇	—	2	32	no	—
132	D₈×C₆₇	—	2	16	no	—
133	D₈×C₆₇	—	4	8	no	—
134	D₈×C₆₇	C₂×D₄	1	16	yes	direct product factor
135	C₂×C₈×C₆₇	—	2	4	no	—
136	C₂×C₈×C₆₇	—	2	2	no	—
137	C₂×C₈×C₆₇	—	2	16	no	—
138	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	complemented
139	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
140	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
141	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
142	C₄⋊C₄×C₆₇	—	2	2	no	—
143	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
144	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
145	C₂²⋊C₄×C₆₇	—	2	8	no	—
146	C₂²⋊C₄×C₆₇	—	2	16	no	—
147	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
148	C₄²×C₆₇	C₂×D₄	1	2	yes	—
149	C₄²×C₆₇	C₂⁴	1	2	yes	—
150	C₂³×C₆₇	—	2	4	no	—
151	C₂³×C₆₇	—	2	4	no	—
152	C₂³×C₆₇	—	2	8	no	—
153	C₂³×C₆₇	—	2	4	no	—
154	C₂³×C₆₇	—	2	8	no	—
155	C₂³×C₆₇	—	4	32	no	—
156	C₂³×C₆₇	—	4	4	no	—
157	C₂³×C₆₇	—	4	4	no	—
158	C₂³×C₆₇	—	4	8	no	—
159	C₂³×C₆₇	—	4	8	no	—
160	C₂³×C₆₇	—	4	32	no	—
161	C₂³×C₆₇	—	4	8	no	—
162	C₂³×C₆₇	—	8	16	no	—
163	C₂³×C₆₇	—	8	64	no	—
164	C₂³×C₆₇	C₂×D₈	1	8	yes	complemented
165	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
166	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
167	D₄×C₆₇	—	2	8	no	—
168	D₄×C₆₇	—	2	8	no	—
169	D₄×C₆₇	—	2	4	no	—
170	D₄×C₆₇	—	2	8	no	—
171	D₄×C₆₇	—	4	32	no	—
172	D₄×C₆₇	—	4	32	no	—
173	D₄×C₆₇	—	4	8	no	—
174	D₄×C₆₇	—	4	16	no	—
175	D₄×C₆₇	—	8	8	no	—
176	D₄×C₆₇	C₂×D₈	1	16	yes	direct product factor
177	D₄×C₆₇	C₂²×D₄	1	8	yes	complemented
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	8	no	—
181	C₂×C₄×C₆₇	—	2	4	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	4	4	no	—
185	C₂×C₄×C₆₇	—	4	16	no	—
186	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	complemented
187	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
188	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
189	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
191	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
192	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
193	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
194	C₈×C₆₇	—	2	4	no	—
195	C₈×C₆₇	—	2	2	no	—
196	C₈×C₆₇	C₂²×D₄	1	4	yes	complemented
197	C₂²×C₆₇	—	2	4	no	—
198	C₂²×C₆₇	—	2	16	no	—
199	C₂²×C₆₇	—	2	4	no	—
200	C₂²×C₆₇	—	2	8	no	—
201	C₂²×C₆₇	—	4	16	no	—
202	C₂²×C₆₇	—	4	4	no	—
203	C₂²×C₆₇	—	4	4	no	—
204	C₂²×C₆₇	—	4	8	no	—
205	C₂²×C₆₇	—	4	8	no	—
206	C₂²×C₆₇	—	4	8	no	—
207	C₂²×C₆₇	—	8	64	no	—
208	C₂²×C₆₇	—	8	16	no	—
209	C₂²×C₆₇	D₄²	1	2	yes	—
210	C₂²×C₆₇	C₂²×D₈	1	2	yes	—
211	C₂²×C₆₇	C₂²×D₈	1	8	yes	complemented
212	C₂²×C₆₇	#576	1	2	yes	—
213	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
214	C₄×C₆₇	—	2	4	no	—
215	C₄×C₆₇	—	2	2	no	—
216	C₄×C₆₇	—	4	4	no	—
217	C₄×C₆₇	D₄²	1	2	yes	—
218	C₄×C₆₇	C₂²×D₈	1	4	yes	complemented
219	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
220	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
221	C₂×D₄×D₈	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂×C₆₇	—	2	8	no	—
223	C₂×C₆₇	—	4	8	no	—
224	C₂×C₆₇	—	8	8	no	—
225	C₂×C₆₇	D₄×D₈	1	4	yes	direct product factor
226	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
227	C₂×C₆₇	C₂³×D₈	1	1	yes	characteristic
228	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
229	C₂³×D₈	C₂×C₆₇	1	2	yes	complemented
230	C₂×D₄²	C₂×C₆₇	1	2	yes	complemented
231	D₄×D₈	C₂×C₆₇	1	16	yes	direct product factor
232	C₂×#493	C₂×C₆₇	1	1	yes	characteristic, complemented
233	C₂×#466	C₂×C₆₇	1	2	yes	complemented
234	C₂×#459	C₂×C₆₇	1	2	yes	complemented
235	C₂×#447	C₂×C₆₇	1	4	yes	complemented
236	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
237	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
238	C₆₇	C₂×D₄×D₈	1	1	yes	characteristic, direct product factor, Sylow
239	C₂³×D₄	—	2	2	no	—
240	C₂³×D₄	C₂²×C₆₇	1	4	yes	complemented
241	C₂²×D₈	—	2	16	no	—
242	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
243	C₂²×D₈	C₂²×C₆₇	1	2	yes	complemented
244	C₂²×D₈	C₂²×C₆₇	1	4	yes	complemented
245	C₂²×D₈	C₂²×C₆₇	1	2	yes	complemented
246	C₂²×D₈	C₂²×C₆₇	1	16	yes	complemented
247	C₂³×C₈	C₂²×C₆₇	1	2	yes	complemented
248	D₄²	—	2	8	no	—
249	D₄²	C₂²×C₆₇	1	16	yes	complemented
250	C₂×#127	C₂²×C₆₇	1	2	yes	complemented
251	C₂×#121	—	2	2	no	—
252	C₂×#121	C₂²×C₆₇	1	4	yes	complemented
253	C₂×#120	—	2	4	no	—
254	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
255	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
256	#493	C₂²×C₆₇	1	8	yes	complemented
257	#466	C₂²×C₆₇	1	16	yes	complemented
258	#459	C₂²×C₆₇	1	16	yes	complemented
259	#447	C₂²×C₆₇	1	32	yes	complemented
260	C₄×D₈	C₂²×C₆₇	1	8	yes	complemented
261	C₈×D₄	C₂²×C₆₇	1	8	yes	complemented
262	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic, complemented
263	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
264	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
265	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
266	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
267	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
268	C₂⁵	—	2	4	no	—
269	C₂²×D₄	—	2	16	no	—
270	C₂²×D₄	—	2	4	no	—
271	C₂²×D₄	—	2	16	no	—
272	C₂²×D₄	—	2	4	no	—
273	C₂²×D₄	—	2	2	no	—
274	C₂²×D₄	—	2	8	no	—
275	C₂²×D₄	—	2	2	no	—
276	C₂²×D₄	—	4	8	no	—
277	C₂²×D₄	—	4	16	no	—
278	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
279	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
280	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
281	C₂²×D₄	C₂³×C₆₇	1	16	yes	complemented
282	C₂³×C₄	—	2	2	no	—
283	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
284	C₂×D₈	—	2	16	no	—
285	C₂×D₈	—	2	64	no	—
286	C₂×D₈	—	2	8	no	—
287	C₂×D₈	—	4	4	no	—
288	C₂×D₈	D₄×C₆₇	1	8	yes	direct product factor
289	C₂×D₈	C₂³×C₆₇	1	4	yes	—
290	C₂×D₈	C₂³×C₆₇	1	16	yes	complemented
291	C₂×D₈	C₂³×C₆₇	1	8	yes	complemented
292	C₂×D₈	C₂³×C₆₇	1	8	yes	complemented
293	C₂²×C₈	—	2	8	no	—
294	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
295	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₆₇	1	8	yes	complemented
298	#127	—	2	4	no	—
299	#127	C₂³×C₆₇	1	8	yes	complemented
300	#121	—	2	8	no	—
301	#121	—	2	16	no	—
302	#121	C₂³×C₆₇	1	16	yes	complemented
303	#120	—	2	32	no	—
304	C₄×D₄	—	2	4	no	—
305	C₄×D₄	D₄×C₆₇	1	4	yes	—
306	C₄×D₄	C₂³×C₆₇	1	4	yes	—
307	C₄×D₄	C₂³×C₆₇	1	8	yes	complemented
308	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
309	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₂²⋊C₄	—	2	4	no	—
312	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
313	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
314	#107	C₂³×C₆₇	1	4	yes	complemented
315	C₄⋊C₈	C₂³×C₆₇	1	4	yes	complemented
316	#102	C₂³×C₆₇	1	8	yes	complemented
317	#102	C₂³×C₆₇	1	16	yes	complemented
318	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	complemented
319	C₄×C₈	C₂³×C₆₇	1	4	yes	complemented
320	C₂⁴	—	2	2	no	—
321	C₂⁴	—	2	4	no	—
322	C₂⁴	—	2	16	no	—
323	C₂⁴	—	4	16	no	—
324	C₂⁴	—	4	16	no	—
325	C₂⁴	—	8	8	no	—
326	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
327	C₂×D₄	—	2	32	no	—
328	C₂×D₄	—	2	16	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	8	no	—
331	C₂×D₄	—	2	16	no	—
332	C₂×D₄	—	2	4	no	—
333	C₂×D₄	—	2	8	no	—
334	C₂×D₄	—	2	8	no	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	8	no	—
337	C₂×D₄	—	4	16	no	—
338	C₂×D₄	—	4	32	no	—
339	C₂×D₄	—	4	16	no	—
340	C₂×D₄	—	4	16	no	—
341	C₂×D₄	—	4	64	no	—
342	C₂×D₄	—	4	8	no	—
343	C₂×D₄	—	8	4	no	—
344	C₂×D₄	D₈×C₆₇	1	8	yes	direct product factor
345	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
346	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	complemented
347	C₂×D₄	C₂⁴×C₆₇	1	8	yes	—
348	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	8	no	—
353	C₂²×C₄	—	4	8	no	—
354	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
356	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
357	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
359	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
360	D₈	—	2	32	no	—
361	D₈	—	2	16	no	—
362	D₈	—	4	8	no	—
363	D₈	C₂×D₄×C₆₇	1	16	yes	direct product factor
364	C₂×C₈	—	2	4	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	16	no	—
367	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	complemented
368	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
369	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
370	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
373	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
374	C₂²⋊C₄	—	2	8	no	—
375	C₂²⋊C₄	—	2	16	no	—
376	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
377	C₄²	C₂×D₄×C₆₇	1	2	yes	—
378	C₄²	C₂⁴×C₆₇	1	2	yes	—
379	C₂³	—	2	4	no	—
380	C₂³	—	2	4	no	—
381	C₂³	—	2	8	no	—
382	C₂³	—	2	4	no	—
383	C₂³	—	2	8	no	—
384	C₂³	—	4	32	no	—
385	C₂³	—	4	4	no	—
386	C₂³	—	4	4	no	—
387	C₂³	—	4	8	no	—
388	C₂³	—	4	8	no	—
389	C₂³	—	4	32	no	—
390	C₂³	—	4	8	no	—
391	C₂³	—	8	16	no	—
392	C₂³	—	8	64	no	—
393	C₂³	C₂×D₈×C₆₇	1	8	yes	complemented
394	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
395	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
396	D₄	—	2	8	no	—
397	D₄	—	2	8	no	—
398	D₄	—	2	4	no	—
399	D₄	—	2	8	no	—
400	D₄	—	4	32	no	—
401	D₄	—	4	32	no	—
402	D₄	—	4	8	no	—
403	D₄	—	4	16	no	—
404	D₄	—	8	8	no	—
405	D₄	C₂×D₈×C₆₇	1	16	yes	direct product factor
406	D₄	C₂²×D₄×C₆₇	1	8	yes	complemented
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	8	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	4	4	no	—
414	C₂×C₄	—	4	16	no	—
415	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	complemented
416	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
417	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
418	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
419	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
420	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
421	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
422	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
423	C₈	—	2	4	no	—
424	C₈	—	2	2	no	—
425	C₈	C₂²×D₄×C₆₇	1	4	yes	complemented
426	C₂²	—	2	16	no	—
427	C₂²	—	2	4	no	—
428	C₂²	—	2	4	no	—
429	C₂²	—	2	8	no	—
430	C₂²	—	4	16	no	—
431	C₂²	—	4	4	no	—
432	C₂²	—	4	4	no	—
433	C₂²	—	4	8	no	—
434	C₂²	—	4	8	no	—
435	C₂²	—	4	8	no	—
436	C₂²	—	8	64	no	—
437	C₂²	—	8	16	no	—
438	C₂²	D₄²×C₆₇	1	2	yes	—
439	C₂²	C₂²×D₈×C₆₇	1	2	yes	—
440	C₂²	C₂²×D₈×C₆₇	1	8	yes	complemented
441	C₂²	#576×C₆₇	1	2	yes	—
442	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
443	C₄	—	2	2	no	—
444	C₄	—	2	4	no	—
445	C₄	—	4	4	no	—
446	C₄	D₄²×C₆₇	1	2	yes	—
447	C₄	C₂²×D₈×C₆₇	1	4	yes	complemented
448	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
449	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
450	C₂	—	2	8	no	—
451	C₂	—	4	8	no	—
452	C₂	—	8	8	no	—
453	C₂	D₄×D₈×C₆₇	1	4	yes	direct product factor
454	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
455	C₂	C₂³×D₈×C₆₇	1	1	yes	characteristic
456	C₂	C₂×#576×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×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
1. C₂³×D₈×C₆₇	C₂	semidirect
2. C₂×D₄²×C₆₇	C₂	semidirect
3. D₄×D₈×C₆₇	C₂	direct (N × H)
4. C₂×#493×C₆₇	C₂	semidirect
5. C₂×#466×C₆₇	C₂	semidirect
6. C₂×#459×C₆₇	C₂	semidirect
7. C₂×#447×C₆₇	C₂	semidirect
8. C₂×C₄×D₈×C₆₇	C₂	semidirect
9. C₂×C₈×D₄×C₆₇	C₂	semidirect
11. C₂³×D₄×C₆₇	C₂²	semidirect
14. C₂²×D₈×C₆₇	C₂²	semidirect
15. C₂²×D₈×C₆₇	C₂²	semidirect
16. C₂²×D₈×C₆₇	C₂²	semidirect
17. C₂²×D₈×C₆₇	C₂²	semidirect
18. C₂³×C₈×C₆₇	C₂²	semidirect
20. D₄²×C₆₇	C₂²	semidirect
21. C₂×#127×C₆₇	C₂²	semidirect
23. C₂×#121×C₆₇	C₂²	semidirect
26. C₂×C₄×D₄×C₆₇	C₂²	semidirect
27. #493×C₆₇	C₂²	semidirect
28. #466×C₆₇	C₂²	semidirect
29. #459×C₆₇	C₂²	semidirect
30. #447×C₆₇	C₂²	semidirect
31. C₄×D₈×C₆₇	C₂²	semidirect
32. C₈×D₄×C₆₇	C₂²	semidirect
33. C₂×#107×C₆₇	C₂²	semidirect
34. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
35. C₂×#102×C₆₇	C₂²	semidirect
36. C₂×#102×C₆₇	C₂²	semidirect
37. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
38. C₂×C₄×C₈×C₆₇	C₂²	semidirect
52. C₂²×D₄×C₆₇	C₂³	semidirect
59. C₂×D₈×C₆₇	D₄	direct (N × H)
61. C₂×D₈×C₆₇	C₂³	semidirect
62. C₂×D₈×C₆₇	C₂³	semidirect
63. C₂×D₈×C₆₇	C₂³	semidirect
68. C₂²×C₈×C₆₇	C₂³	semidirect
70. #127×C₆₇	C₂³	semidirect
73. #121×C₆₇	C₂³	semidirect
78. C₄×D₄×C₆₇	C₂³	semidirect
85. #107×C₆₇	C₂³	semidirect
86. C₄⋊C₈×C₆₇	C₂³	semidirect
87. #102×C₆₇	C₂³	semidirect
88. #102×C₆₇	C₂³	semidirect
89. C₂²⋊C₈×C₆₇	C₂³	semidirect
90. C₄×C₈×C₆₇	C₂³	semidirect
115. C₂×D₄×C₆₇	D₈	direct (N × H)
117. C₂×D₄×C₆₇	C₂×D₄	semidirect
134. D₈×C₆₇	C₂×D₄	direct (N × H)
138. C₂×C₈×C₆₇	C₂×D₄	semidirect
164. C₂³×C₆₇	C₂×D₈	semidirect
176. D₄×C₆₇	C₂×D₈	direct (N × H)
177. D₄×C₆₇	C₂²×D₄	semidirect
186. C₂×C₄×C₆₇	C₂×D₈	semidirect
196. C₈×C₆₇	C₂²×D₄	semidirect
211. C₂²×C₆₇	C₂²×D₈	semidirect
218. C₄×C₆₇	C₂²×D₈	semidirect
221. C₂×D₄×D₈	C₆₇	direct (N × H)
225. C₂×C₆₇	D₄×D₈	direct (N × H)
229. C₂³×D₈	C₂×C₆₇	semidirect
230. C₂×D₄²	C₂×C₆₇	semidirect
231. D₄×D₈	C₂×C₆₇	direct (N × H)
232. C₂×#493	C₂×C₆₇	semidirect
233. C₂×#466	C₂×C₆₇	semidirect
234. C₂×#459	C₂×C₆₇	semidirect
235. C₂×#447	C₂×C₆₇	semidirect
236. C₂×C₄×D₈	C₂×C₆₇	semidirect
237. C₂×C₈×D₄	C₂×C₆₇	semidirect
238. C₆₇	C₂×D₄×D₈	direct (N × H)
240. C₂³×D₄	C₂²×C₆₇	semidirect
243. C₂²×D₈	C₂²×C₆₇	semidirect
244. C₂²×D₈	C₂²×C₆₇	semidirect
245. C₂²×D₈	C₂²×C₆₇	semidirect
246. C₂²×D₈	C₂²×C₆₇	semidirect
247. C₂³×C₈	C₂²×C₆₇	semidirect
249. D₄²	C₂²×C₆₇	semidirect
250. C₂×#127	C₂²×C₆₇	semidirect
252. C₂×#121	C₂²×C₆₇	semidirect
255. C₂×C₄×D₄	C₂²×C₆₇	semidirect
256. #493	C₂²×C₆₇	semidirect
257. #466	C₂²×C₆₇	semidirect
258. #459	C₂²×C₆₇	semidirect
259. #447	C₂²×C₆₇	semidirect
260. C₄×D₈	C₂²×C₆₇	semidirect
261. C₈×D₄	C₂²×C₆₇	semidirect
262. C₂×#107	C₂²×C₆₇	semidirect
263. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
264. C₂×#102	C₂²×C₆₇	semidirect
265. C₂×#102	C₂²×C₆₇	semidirect
266. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
267. C₂×C₄×C₈	C₂²×C₆₇	semidirect
281. C₂²×D₄	C₂³×C₆₇	semidirect
288. C₂×D₈	D₄×C₆₇	direct (N × H)
290. C₂×D₈	C₂³×C₆₇	semidirect
291. C₂×D₈	C₂³×C₆₇	semidirect
292. C₂×D₈	C₂³×C₆₇	semidirect
297. C₂²×C₈	C₂³×C₆₇	semidirect
299. #127	C₂³×C₆₇	semidirect
302. #121	C₂³×C₆₇	semidirect
307. C₄×D₄	C₂³×C₆₇	semidirect
314. #107	C₂³×C₆₇	semidirect
315. C₄⋊C₈	C₂³×C₆₇	semidirect
316. #102	C₂³×C₆₇	semidirect
317. #102	C₂³×C₆₇	semidirect
318. C₂²⋊C₈	C₂³×C₆₇	semidirect
319. C₄×C₈	C₂³×C₆₇	semidirect
344. C₂×D₄	D₈×C₆₇	direct (N × H)
346. C₂×D₄	C₂×D₄×C₆₇	semidirect
363. D₈	C₂×D₄×C₆₇	direct (N × H)
367. C₂×C₈	C₂×D₄×C₆₇	semidirect
393. C₂³	C₂×D₈×C₆₇	semidirect
405. D₄	C₂×D₈×C₆₇	direct (N × H)
406. D₄	C₂²×D₄×C₆₇	semidirect
415. C₂×C₄	C₂×D₈×C₆₇	semidirect
425. C₈	C₂²×D₄×C₆₇	semidirect
440. C₂²	C₂²×D₈×C₆₇	semidirect
447. C₄	C₂²×D₈×C₆₇	semidirect
453. C₂	D₄×D₈×C₆₇	direct (N × H)

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×D₄²×C₆₇	2
D₄×D₈×C₆₇	2
C₂×#493×C₆₇	2
C₂×#466×C₆₇	2
C₂×#459×C₆₇	2
C₂×#447×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×D₄×D₈	67

Subgroups — C2×D4×D8×C67 #30740290

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×D₈×C₆₇ #30740290