Subgroups — #22569198

Order 8576.

372 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863930	C₂	1	1	yes	maximal, characteristic
2	#16863525	C₂	1	2	yes	maximal, complemented
3	#16863466	C₂	1	2	yes	maximal, complemented
4	#16863464	C₂	1	2	yes	maximal, complemented
5	(C₂²⋊D₆₇)⋊C₈	C₂	1	2	yes	maximal, complemented
6	#16863455	C₂	1	2	yes	maximal, complemented
7	#16863350	C₂	1	2	yes	maximal, complemented
8	C₂×#14523954	C₂	1	1	yes	maximal, characteristic, complemented
9	#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11761611	C₂²	1	2	yes	—
11	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
12	C₂×C₈×D₆₇	C₂²	1	2	yes	—
13	#14524037	C₄	1	2	yes	—
14	#14524035	C₄	1	1	yes	characteristic
15	#14524031	C₄	1	1	yes	characteristic
16	#14524026	C₄	1	2	yes	—
17	#14524022	C₄	1	1	yes	characteristic
18	#14524017	C₄	1	1	yes	characteristic
19	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
20	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
22	#14524000	C₂²	1	2	yes	—
23	Dic₆₇⋊C₈	C₂²	1	2	yes	—
24	Dic₆₇⋊C₈	C₂²	1	2	yes	—
25	(C₂×D₆₇)⋊C₈	C₂²	1	2	yes	—
26	(C₂×D₆₇)⋊C₈	C₂²	1	2	yes	—
27	#14523954	C₂²	1	2	yes	—
28	#14523954	C₂²	1	1	yes	characteristic
29	#14523954	C₂²	1	1	yes	characteristic
30	#14523953	C₂²	1	2	yes	—
31	C₈×Dic₆₇	C₂²	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
34	C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×C₈×C₆₇	C₂²	1	2	yes	—
37	C₂×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
38	C₂×D₂₆₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₆₇	C₂³	1	2	yes	—
41	#11761611	—	2	4	no	—
42	C₂×C₆₇⋊C₈	—	2	4	no	—
43	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
44	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
45	C₈×D₆₇	—	2	4	no	—
46	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
47	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
48	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
49	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
50	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
51	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
52	C₂²⋊Dic₆₇	C₂×C₄	1	2	yes	—
53	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
57	C₄×Dic₆₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂³	1	2	yes	—
62	C₂×C₈×C₆₇	C₂³	1	2	yes	—
63	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₆₇	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	4	1	no	—
72	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
78	C₂²⋊D₆₇	—	2	4	no	—
79	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
80	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
81	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
82	D₂₆₈	—	4	1	no	—
83	C₄×D₆₇	—	2	2	no	—
84	C₄×D₆₇	—	2	2	no	—
85	Dic₁₃₄	—	4	1	no	—
86	C₈×C₆₇	—	2	2	no	—
87	C₈×C₆₇	—	2	2	no	—
88	C₆₇⋊C₈	—	2	2	no	—
89	C₆₇⋊C₈	#41	1	4	yes	—
90	C₂²×C₆₇	—	2	1	no	—
91	C₂²×C₆₇	—	2	1	no	—
92	C₂²×C₆₇	#131	1	2	yes	—
93	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×D₆₇	—	2	2	no	—
95	C₂×D₆₇	—	2	2	no	—
96	C₂×D₆₇	—	2	2	no	—
97	C₄×C₆₇	—	2	1	no	—
98	C₄×C₆₇	—	2	1	no	—
99	C₄×C₆₇	—	2	1	no	—
100	C₄×C₆₇	—	2	1	no	—
101	C₄×C₆₇	—	2	1	no	—
102	C₄×C₆₇	C₂×#41	1	2	yes	—
103	Dic₆₇	—	2	2	no	—
104	Dic₆₇	—	2	2	no	—
105	Dic₆₇	—	2	2	no	—
106	C₂×C₆₇	—	2	2	no	—
107	C₂×C₆₇	C₄×#41	1	1	yes	characteristic
108	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
109	C₂×C₆₇	#568	1	1	yes	characteristic
110	D₆₇	—	4	2	no	—
111	#2968	—	67	1	no	maximal, Sylow
112	C₆₇	#2968	1	1	yes	characteristic, complemented, Sylow
113	#529	—	67	1	no	—
114	#446	—	67	2	no	—
115	#435	—	67	2	no	—
116	#435	—	67	2	no	—
117	C₈×D₄	—	67	2	no	—
118	#433	—	67	2	no	—
119	#431	D₆₇	1	1	yes	characteristic, complemented
120	#405	—	67	2	no	—
121	C₂×#97	—	67	1	no	—
122	C₂×M₁₆	—	67	2	no	—
123	C₂²×C₈	—	67	2	no	—
124	C₂²×C₈	—	67	2	no	—
125	#126	—	67	2	no	—
126	#125	—	67	1	no	—
127	#124	—	67	1	no	—
128	#123	—	67	2	no	—
129	C₂²⋊Q₈	—	67	1	no	—
130	#121	—	67	1	no	—
131	C₄×Q₈	—	67	1	no	—
132	C₄×D₄	—	67	2	no	—
133	C₄×D₄	—	67	1	no	—
134	#117	—	67	2	no	—
135	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
136	C₄⋊C₈	—	67	2	no	—
137	C₄⋊C₈	—	67	2	no	—
138	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
139	C₂²⋊C₈	—	67	2	no	—
140	C₂²⋊C₈	—	67	2	no	—
141	C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
142	C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
143	#97	—	67	2	no	—
144	#97	—	67	1	no	—
145	#97	—	67	1	no	—
146	#97	—	67	2	no	—
147	C₄×C₈	—	67	2	no	—
148	C₄×C₈	C₂×D₆₇	1	2	yes	—
149	C₂×Q₈	—	67	1	no	—
150	C₂×D₄	—	67	2	no	—
151	C₂×D₄	—	67	1	no	—
152	C₂²×C₄	—	67	2	no	—
153	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
156	M₁₆	—	134	4	no	—
157	C₂×C₈	—	67	2	no	—
158	C₂×C₈	—	67	2	no	—
159	C₂×C₈	—	134	4	no	—
160	C₂×C₈	—	134	4	no	—
161	C₂×C₈	C₂²×D₆₇	1	2	yes	—
162	C₂×C₈	C₂²×D₆₇	1	2	yes	—
163	C₄⋊C₄	—	67	2	no	—
164	C₄⋊C₄	—	67	2	no	—
165	C₄⋊C₄	—	67	1	no	—
166	C₄⋊C₄	—	67	2	no	—
167	C₄⋊C₄	—	67	1	no	—
168	C₄⋊C₄	—	67	2	no	—
169	C₂²⋊C₄	—	67	2	no	—
170	C₂²⋊C₄	—	67	2	no	—
171	C₂²⋊C₄	—	67	2	no	—
172	C₂²⋊C₄	—	67	2	no	—
173	C₂²⋊C₄	—	67	2	no	—
174	C₄²	—	67	2	no	—
175	C₄²	C₄×D₆₇	1	1	yes	characteristic
176	C₄²	C₄×D₆₇	1	1	yes	characteristic
177	C₄²	C₂²×D₆₇	1	1	yes	characteristic
178	C₄²	C₂²×D₆₇	1	1	yes	characteristic
179	C₂³	—	67	2	no	—
180	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
181	Q₈	—	268	1	no	—
182	D₄	—	268	1	no	—
183	D₄	—	134	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	67	2	no	—
188	C₂×C₄	—	67	2	no	—
189	C₂×C₄	—	67	2	no	—
190	C₂×C₄	—	134	2	no	—
191	C₂×C₄	—	134	2	no	—
192	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	2	2	no	—
200	C₈	—	67	4	no	—
201	C₈	—	134	2	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	134	2	no	—
205	C₂²	—	134	2	no	—
206	C₂²	—	134	2	no	—
207	C₂²	#14524049	1	2	yes	—
208	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	134	2	no	—
215	C₄	—	134	2	no	—
216	C₄	—	134	2	no	—
217	C₄	#41×D₆₇	1	2	yes	—
218	C₂	—	2	2	no	—
219	C₂	—	268	2	no	—
220	C₂	#16863864	1	1	yes	characteristic
221	C₂	C₂×#14524049	1	1	yes	characteristic
222	C₂	#16864162	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#431×C₆₇	C₂
hypercenter	#431	D₆₇
hypocenter	C₆₇	#2968
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #16863525	C₂	semidirect
3. #16863466	C₂	semidirect
4. #16863464	C₂	semidirect
5. (C₂²⋊D₆₇)⋊C₈	C₂	semidirect
6. #16863455	C₂	semidirect
7. #16863350	C₂	semidirect
8. C₂×#14523954	C₂	semidirect
9. #431×C₆₇	C₂	semidirect
112. C₆₇	#2968	semidirect
119. #431	D₆₇	semidirect