Subgroups — #22569217

Order 8576.

406 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11761627	C₂	1	1	yes	maximal, characteristic
2	#16863526	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863523	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863467	C₂	1	2	yes	maximal, complemented
5	#16863464	C₂	1	2	yes	maximal, complemented
6	D₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863452	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863450	C₂	1	2	yes	maximal, complemented
9	C₄×#11761611	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14523953	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11761627	C₄	1	1	yes	characteristic
13	C₂×#11761611	C₂²	1	1	yes	characteristic
14	C₂×#11761611	C₂²	1	1	yes	characteristic
15	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
16	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
17	#14524008	C₄	1	1	yes	characteristic
18	D₄×Dic₆₇	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
20	(C₂²⋊D₆₇)⋊C₄	C₄	1	2	yes	—
21	#14524001	C₄	1	1	yes	characteristic
22	#14523999	C₄	1	2	yes	—
23	C₂×C₄×Dic₆₇	C₂²	1	2	yes	—
24	C₄²×D₆₇	C₂²	1	1	yes	characteristic
25	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
26	Dic₆₇⋊C₈	C₂²	1	2	yes	—
27	Dic₆₇⋊C₈	C₂²	1	2	yes	—
28	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
29	(C₂×D₆₇)⋊C₈	C₂²	1	2	yes	—
30	#14523953	C₂²	1	2	yes	—
31	#14523953	C₂²	1	2	yes	—
32	#14523953	C₂²	1	1	yes	characteristic
33	#14523953	C₂²	1	2	yes	—
34	#14523953	C₂²	1	1	yes	characteristic
35	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
37	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
40	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	#11761627	—	2	4	no	—
42	C₂×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₆₇	C₂×C₄	1	2	yes	—
44	C₂²×Dic₆₇	C₂×C₄	1	2	yes	—
45	C₂²×Dic₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
49	#11761611	—	2	2	no	—
50	#11761611	—	2	2	no	—
51	C₂×C₆₇⋊C₈	—	2	4	no	—
52	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
53	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
54	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
56	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
57	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
58	C₂²⋊Dic₆₇	C₂×C₄	1	2	yes	—
59	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
63	C₄×Dic₆₇	C₂×C₄	1	2	yes	—
64	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
65	C₄×Dic₆₇	C₂³	1	2	yes	—
66	C₂×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₈×C₆₇	—	2	4	no	—
69	C₂×C₈×C₆₇	C₂³	1	2	yes	—
70	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
74	C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₂³×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂²×D₆₇	C₂²×C₄	1	1	yes	characteristic
77	D₄×C₆₇	M₁₆	1	4	yes	complemented
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₂²⋊D₆₇	—	2	4	no	—
85	C₂×Dic₆₇	—	2	4	no	—
86	C₂×Dic₆₇	—	4	2	no	—
87	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
91	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
92	C₄×D₆₇	—	2	2	no	—
93	C₄×D₆₇	—	2	1	no	—
94	C₄×D₆₇	—	2	1	no	—
95	C₄×D₆₇	—	4	1	no	—
96	Dic₁₃₄	—	2	2	no	—
97	C₈×C₆₇	—	2	2	no	—
98	C₈×C₆₇	—	2	1	no	—
99	C₈×C₆₇	#41	1	2	yes	complemented
100	C₆₇⋊C₈	—	2	2	no	—
101	C₆₇⋊C₈	—	2	1	no	—
102	C₆₇⋊C₈	#41	1	2	yes	complemented
103	C₂²×C₆₇	—	2	2	no	—
104	C₂²×C₆₇	—	2	2	no	—
105	C₂²×C₆₇	C₂×M₁₆	1	4	yes	—
106	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×D₆₇	—	2	1	no	—
108	C₂×D₆₇	—	2	1	no	—
109	C₂×D₆₇	—	2	1	no	—
110	C₄×C₆₇	—	2	2	no	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	C₂×M₁₆	1	2	yes	—
113	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
114	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
115	Dic₆₇	—	2	1	no	—
116	Dic₆₇	—	2	2	no	—
117	Dic₆₇	—	2	1	no	—
118	Dic₆₇	—	2	1	no	—
119	Dic₆₇	—	2	2	no	—
120	C₂×C₆₇	—	2	4	no	—
121	C₂×C₆₇	C₄×#41	1	1	yes	characteristic
122	C₂×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
123	C₂×C₆₇	#568	1	1	yes	characteristic
124	D₆₇	—	4	1	no	—
125	#2970	—	67	1	no	maximal, Sylow
126	C₆₇	#2970	1	1	yes	characteristic, complemented, Sylow
127	C₄×#41	—	67	1	no	—
128	#446	—	67	1	no	—
129	#446	—	67	1	no	—
130	#435	—	67	2	no	—
131	#435	—	67	2	no	—
132	C₈×D₄	—	67	1	no	—
133	C₈×D₄	D₆₇	1	1	yes	characteristic, complemented
134	#432	—	67	1	no	—
135	#432	—	67	2	no	—
136	C₄×M₁₆	—	67	1	no	—
137	C₂×#97	—	67	2	no	—
138	C₂×#41	—	67	1	no	—
139	C₂×M₁₆	—	67	1	no	—
140	C₂×M₁₆	—	67	1	no	—
141	C₂²×C₈	—	67	2	no	—
142	C₂²×C₈	C₂×D₆₇	1	2	yes	—
143	C₄×Q₈	—	67	1	no	—
144	C₄×Q₈	—	67	1	no	—
145	C₄×D₄	—	67	1	no	—
146	C₄×D₄	—	67	2	no	—
147	C₄×D₄	—	67	2	no	—
148	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
149	#117	—	67	1	no	—
150	#117	—	67	2	no	—
151	C₂×C₄²	—	67	2	no	—
152	C₂×C₄²	—	67	1	no	—
153	C₄⋊C₈	—	67	2	no	—
154	C₄⋊C₈	—	67	2	no	—
155	C₄⋊C₈	—	67	1	no	—
156	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
157	C₂²⋊C₈	—	67	2	no	—
158	C₂²⋊C₈	—	67	2	no	—
159	C₂²⋊C₈	C₂×D₆₇	1	2	yes	—
160	#97	—	67	1	no	—
161	#97	—	67	2	no	—
162	#97	—	67	2	no	—
163	#97	—	67	2	no	—
164	#97	—	67	1	no	—
165	C₄×C₈	—	67	1	no	—
166	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
167	#41	—	134	4	no	—
168	C₂×Q₈	—	67	1	no	—
169	C₂×D₄	—	67	2	no	—
170	C₂×D₄	C₄×D₆₇	1	1	yes	characteristic
171	C₂²×C₄	—	67	2	no	—
172	C₂²×C₄	—	67	1	no	—
173	C₂²×C₄	—	67	2	no	—
174	C₂²×C₄	—	67	1	no	—
175	C₂²×C₄	—	67	1	no	—
176	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
177	M₁₆	—	134	2	no	—
178	M₁₆	—	134	2	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	67	1	no	—
181	C₂×C₈	—	67	1	no	—
182	C₂×C₈	—	67	2	no	—
183	C₂×C₈	—	134	4	no	—
184	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
185	C₂×C₈	C₂²×D₆₇	1	2	yes	—
186	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
187	C₄⋊C₄	—	67	2	no	—
188	C₄⋊C₄	—	67	2	no	—
189	C₄⋊C₄	—	67	1	no	—
190	C₄⋊C₄	C₄×D₆₇	1	1	yes	characteristic
191	C₂²⋊C₄	—	67	2	no	—
192	C₂²⋊C₄	—	67	2	no	—
193	C₂²⋊C₄	C₄×D₆₇	1	2	yes	—
194	C₄²	—	67	2	no	—
195	C₄²	—	67	1	no	—
196	C₄²	—	67	1	no	—
197	C₄²	—	67	2	no	—
198	C₄²	—	67	1	no	—
199	C₄²	—	67	2	no	—
200	C₄²	C₂²×D₆₇	1	1	yes	characteristic
201	C₂³	—	67	1	no	—
202	C₂³	C₂×C₄×D₆₇	1	2	yes	—
203	Q₈	—	134	2	no	—
204	D₄	—	134	4	no	—
205	D₄	#11761611	1	4	yes	complemented
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	268	1	no	—
209	C₂×C₄	—	268	2	no	—
210	C₂×C₄	—	67	1	no	—
211	C₂×C₄	—	67	1	no	—
212	C₂×C₄	—	67	1	no	—
213	C₂×C₄	—	67	2	no	—
214	C₂×C₄	—	67	2	no	—
215	C₂×C₄	—	134	4	no	—
216	C₂×C₄	—	134	2	no	—
217	C₂×C₄	—	134	1	no	—
218	C₂×C₄	—	134	1	no	—
219	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₈	—	67	2	no	—
226	C₈	—	134	2	no	—
227	C₈	—	134	1	no	—
228	C₈	#11761627	1	2	yes	complemented
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	134	1	no	—
232	C₂²	—	134	1	no	—
233	C₂²	—	134	1	no	—
234	C₂²	C₂×#11761611	1	4	yes	—
235	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	134	1	no	—
239	C₄	—	134	1	no	—
240	C₄	—	134	2	no	—
241	C₄	—	134	1	no	—
242	C₄	—	134	2	no	—
243	C₄	C₂×#11761611	1	2	yes	—
244	C₄	#41×D₆₇	1	1	yes	characteristic
245	C₄	C₂×#11761627	1	1	yes	characteristic
246	C₂	—	2	4	no	—
247	C₂	—	268	1	no	—
248	C₂	C₄×#11761627	1	1	yes	characteristic
249	C₂	C₂²×#11761611	1	1	yes	characteristic
250	C₂	#16864161	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	C₈×D₄×C₆₇	C₂
hypercenter	C₈×D₄	D₆₇
hypocenter	C₆₇	#2970
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #16863526	C₂	semidirect
3. #16863523	C₂	semidirect
4. #16863467	C₂	semidirect
5. #16863464	C₂	semidirect
6. D₄×C₆₇⋊C₈	C₂	semidirect
7. #16863452	C₂	semidirect
8. #16863450	C₂	semidirect
9. C₄×#11761611	C₂	semidirect
10. C₂×#14523953	C₂	semidirect
11. C₈×D₄×C₆₇	C₂	semidirect
77. D₄×C₆₇	M₁₆	semidirect
99. C₈×C₆₇	#41	semidirect
102. C₆₇⋊C₈	#41	semidirect
126. C₆₇	#2970	semidirect
133. C₈×D₄	D₆₇	semidirect
205. D₄	#11761611	semidirect
228. C₈	#11761627	semidirect