Subgroups — #31720224

Order 17152.

1810 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569150	C₂	1	12	yes	maximal, complemented
2	#22569144	C₂	1	12	yes	maximal, complemented
3	#22568773	C₂	1	3	yes	maximal
4	C₂×#16863352	C₂	1	3	yes	maximal
5	C₆₇×#3458	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	3	yes	—
7	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	#16863527	C₂²	1	48	yes	complemented
9	Q₈×C₆₇⋊C₈	C₂²	1	16	yes	complemented
10	#16863423	C₂²	1	24	yes	—
11	#16863420	C₂²	1	12	yes	—
12	#16863352	C₂²	1	12	yes	—
13	#16863352	C₂²	1	12	yes	—
14	C₂×#14523954	C₂²	1	6	yes	—
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	6	yes	—
16	#544×C₆₇	C₄	1	12	yes	—
17	#543×C₆₇	C₄	1	4	yes	—
18	#533×C₆₇	C₂²	1	12	yes	—
19	C₂×#117×C₆₇	C₂²	1	3	yes	—
20	C₂×#11761612	—	2	12	no	—
21	C₂×#11761612	C₂³	1	12	yes	—
22	C₂²×C₆₇⋊C₈	Q₈	1	8	yes	complemented
23	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
24	C₂²×C₆₇⋊C₈	C₂³	1	3	yes	—
25	C₂²×C₆₇⋊C₈	C₂³	1	3	yes	—
26	C₄⋊(C₆₇⋊C₈)	C₂³	1	48	yes	—
27	#14523954	C₂³	1	24	yes	—
28	C₄×C₆₇⋊C₈	C₂³	1	12	yes	—
29	C₄×C₆₇⋊C₈	C₂³	1	12	yes	—
30	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₆₇	C₂×C₄	1	24	yes	—
32	#125×C₆₇	C₂×C₄	1	24	yes	—
33	C₂²⋊Q₈×C₆₇	C₂×C₄	1	48	yes	—
34	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
35	#117×C₆₇	C₂×C₄	1	12	yes	—
36	#117×C₆₇	C₂³	1	12	yes	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
38	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
40	#11761612	—	2	24	no	—
41	C₂×C₆₇⋊C₈	—	2	12	no	—
42	C₂×C₆₇⋊C₈	C₂×Q₈	1	12	yes	—
43	C₂×C₆₇⋊C₈	C₂×Q₈	1	12	yes	—
44	C₂×C₆₇⋊C₈	C₂⁴	1	12	yes	—
45	C₂×C₆₇⋊C₈	C₂⁴	1	1	yes	characteristic
46	C₂×C₆₇⋊C₈	C₂⁴	1	3	yes	—
47	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	—
49	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
51	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
52	C₂²×C₄×C₆₇	#41	1	4	yes	—
53	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
54	C₄⋊C₄×C₆₇	C₂²×C₄	1	24	yes	—
55	C₄⋊C₄×C₆₇	C₂²×C₄	1	48	yes	—
56	C₂²⋊C₄×C₆₇	C₂²×C₄	1	24	yes	—
57	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
58	C₄²×C₆₇	C₂⁴	1	12	yes	—
59	C₂³×C₆₇	—	4	2	no	—
60	C₂³×C₆₇	C₄×Q₈	1	4	yes	—
61	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
62	Q₈×C₆₇	—	4	16	no	—
63	C₂×C₄×C₆₇	—	4	12	no	—
64	C₂×C₄×C₆₇	C₄×Q₈	1	12	yes	—
65	C₂×C₄×C₆₇	C₂³×C₄	1	24	yes	—
66	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
67	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
68	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
69	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
70	C₆₇⋊C₈	—	2	12	no	—
71	C₆₇⋊C₈	C₂²×Q₈	1	2	yes	—
72	C₆₇⋊C₈	C₂²×Q₈	1	6	yes	—
73	C₂²×C₆₇	—	2	6	no	—
74	C₂²×C₆₇	—	4	4	no	—
75	C₂²×C₆₇	C₂×C₄×Q₈	1	6	yes	—
76	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
77	C₄×C₆₇	—	2	24	no	—
78	C₄×C₆₇	C₂×C₄×Q₈	1	6	yes	—
79	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
80	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
81	#60191	—	67	1	no	maximal, Sylow
82	C₂×C₆₇	—	2	6	no	—
83	C₂×C₆₇	C₂²×C₄×Q₈	1	1	yes	characteristic
84	C₂×C₆₇	#3573	1	2	yes	—
85	#3458	D₆₇	1	1	yes	characteristic
86	#2961	—	67	12	no	—
87	#2960	—	67	12	no	—
88	#2905	—	67	3	no	—
89	C₂×#405	—	67	3	no	—
90	C₆₇	#60191	1	1	yes	characteristic, complemented, Sylow
91	C₂²×M₁₆	—	67	3	no	—
92	C₂³×C₈	—	67	1	no	—
93	#544	Dic₆₇	1	12	yes	—
94	#543	Dic₆₇	1	4	yes	—
95	#533	C₂×D₆₇	1	12	yes	—
96	C₂×#117	C₂×D₆₇	1	3	yes	—
97	#446	—	67	48	no	—
98	C₈×Q₈	—	67	16	no	—
99	#424	—	67	24	no	—
100	#423	—	67	12	no	—
101	#405	—	67	12	no	—
102	#405	—	67	12	no	—
103	C₂×#97	—	67	6	no	—
104	C₂×C₄×C₈	—	67	6	no	—
105	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
106	C₂×M₁₆	—	67	12	no	—
107	C₂×M₁₆	—	134	12	no	—
108	C₂²×C₈	—	67	8	no	—
109	C₂²×C₈	—	67	3	no	—
110	C₂²×C₈	—	67	3	no	—
111	C₂²×C₈	—	67	6	no	—
112	C₄⋊Q₈	C₂×Dic₆₇	1	24	yes	—
113	#125	C₂×Dic₆₇	1	24	yes	—
114	C₂²⋊Q₈	C₂×Dic₆₇	1	48	yes	—
115	C₄×Q₈	C₂²×D₆₇	1	16	yes	—
116	#117	C₂×Dic₆₇	1	12	yes	—
117	#117	C₂²×D₆₇	1	12	yes	—
118	C₂×C₄⋊C₄	C₂×Dic₆₇	1	6	yes	—
119	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	6	yes	—
120	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
121	C₄⋊C₈	—	67	48	no	—
122	#97	—	67	24	no	—
123	C₄×C₈	—	67	12	no	—
124	C₄×C₈	—	67	12	no	—
125	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
126	C₂×Q₈	C₂²×Dic₆₇	1	16	yes	—
127	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
128	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
129	C₂²×C₄	Q₈×D₆₇	1	4	yes	—
130	C₂²×C₄	#11761629	1	4	yes	—
131	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
132	M₁₆	—	134	24	no	—
133	C₂×C₈	—	67	12	no	—
134	C₂×C₈	—	67	12	no	—
135	C₂×C₈	—	67	1	no	—
136	C₂×C₈	—	67	3	no	—
137	C₂×C₈	—	67	12	no	—
138	C₂×C₈	—	134	12	no	—
139	C₄⋊C₄	C₂²×Dic₆₇	1	24	yes	—
140	C₄⋊C₄	C₂²×Dic₆₇	1	48	yes	—
141	C₂²⋊C₄	C₂²×Dic₆₇	1	24	yes	—
142	C₄²	C₂²×Dic₆₇	1	12	yes	—
143	C₄²	C₂³×D₆₇	1	12	yes	—
144	C₂³	—	4	2	no	—
145	C₂³	Q₈×Dic₆₇	1	4	yes	—
146	C₂³	C₂³×Dic₆₇	1	3	yes	—
147	Q₈	—	4	16	no	—
148	C₂×C₄	—	4	12	no	—
149	C₂×C₄	Q₈×Dic₆₇	1	12	yes	—
150	C₂×C₄	C₂³×Dic₆₇	1	24	yes	—
151	C₂×C₄	C₂³×Dic₆₇	1	3	yes	—
152	C₂×C₄	C₂×Q₈×D₆₇	1	6	yes	—
153	C₂×C₄	C₂×#11761629	1	6	yes	—
154	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
155	C₈	—	67	2	no	—
156	C₈	—	67	6	no	—
157	C₈	—	134	12	no	—
158	C₂²	—	2	6	no	—
159	C₂²	—	4	4	no	—
160	C₂²	C₂×Q₈×Dic₆₇	1	6	yes	—
161	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
162	C₄	—	2	24	no	—
163	C₄	C₂×Q₈×Dic₆₇	1	6	yes	—
164	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
165	C₄	C₂²×#11761629	1	1	yes	characteristic
166	C₂	—	2	6	no	—
167	C₂	C₂²×Q₈×Dic₆₇	1	1	yes	characteristic
168	C₂	#22574921	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁴×C₄
Fitting	C₆₇×#3458	C₂
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	#3458	D₆₇
hypocenter	C₆₇	#60191
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #3458

Complements

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

N	H ≅ G/N	type
1. #22569150	C₂	semidirect
2. #22569144	C₂	semidirect
8. #16863527	C₂²	semidirect
9. Q₈×C₆₇⋊C₈	C₂²	semidirect
22. C₂²×C₆₇⋊C₈	Q₈	semidirect
90. C₆₇	#60191	semidirect

Maximal subgroups

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

subgroup	index
#22569150	2
#22569144	2
#22568773	2
C₂×#16863352	2
C₆₇×#3458	2
#60191	67