Subgroups — #30746254

Order 17152.

574 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₆₇⋊C₈	C₂	1	2	yes	maximal
2	C₆₇⋊#1273	C₂	1	4	yes	maximal
3	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₆₇⋊C₈	C₄	1	4	yes	—
5	C₂×C₄×C₆₇⋊C₈	C₄	1	4	yes	—
6	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
10	C₄³×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
12	C₄×C₆₇⋊C₈	—	2	8	no	—
13	C₄×C₆₇⋊C₈	—	2	8	no	—
14	C₄×C₆₇⋊C₈	C₈	1	16	yes	complemented
15	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
16	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₆₇	—	2	4	no	—
22	C₄×C₈×C₆₇	—	2	4	no	—
23	C₄×C₈×C₆₇	C₈	1	8	yes	complemented
24	C₂×C₆₇⋊C₈	—	2	8	no	—
25	C₂×C₆₇⋊C₈	—	2	8	no	—
26	C₂×C₆₇⋊C₈	C₂×C₈	1	16	yes	—
27	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
29	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
31	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₆₇	—	2	4	no	—
33	C₂×C₈×C₆₇	—	2	4	no	—
34	C₂×C₈×C₆₇	C₂×C₈	1	8	yes	—
35	C₄²×C₆₇	C₂×C₈	1	8	yes	—
36	C₄²×C₆₇	C₂×C₈	1	4	yes	—
37	C₄²×C₆₇	M₁₆	1	8	yes	—
38	C₄²×C₆₇	M₁₆	1	4	yes	—
39	C₄²×C₆₇	#41	1	4	yes	—
40	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
41	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
42	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
43	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
44	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
45	C₂×C₄×C₆₇	#97	1	4	yes	—
46	C₂×C₄×C₆₇	#97	1	4	yes	—
47	C₂×C₄×C₆₇	#97	1	4	yes	—
48	C₂×C₄×C₆₇	#117	1	4	yes	—
49	C₂×C₄×C₆₇	#117	1	2	yes	—
50	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
51	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
52	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
53	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
54	C₈×C₆₇	—	2	8	no	—
55	C₆₇⋊C₈	—	2	16	no	—
56	C₂²×C₆₇	#376	1	1	yes	characteristic
57	C₂²×C₆₇	C₂×C₄×C₈	1	2	yes	—
58	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
59	C₂²×C₆₇	C₂×#97	1	2	yes	—
60	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
61	C₄×C₆₇	C₈²	1	4	yes	—
62	C₄×C₆₇	#322	1	4	yes	—
63	C₄×C₆₇	#322	1	8	yes	—
64	C₄×C₆₇	#431	1	8	yes	—
65	C₄×C₆₇	#431	1	4	yes	—
66	#8131	—	67	1	no	maximal, Sylow
67	C₂×C₆₇	C₂×C₈²	1	1	yes	characteristic
68	C₂×C₆₇	C₂×#322	1	1	yes	characteristic
69	C₂×C₆₇	C₂×#322	1	2	yes	—
70	C₂×C₆₇	#1744	1	2	yes	—
71	C₂×C₆₇	#1744	1	1	yes	characteristic
72	C₄²×C₈	—	67	2	no	—
73	C₄²×C₈	D₆₇	1	1	yes	characteristic
74	#1273	—	67	4	no	—
75	C₆₇	#8131	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄×C₈	—	67	4	no	—
77	C₂×C₄×C₈	—	67	4	no	—
78	C₂×C₄×C₈	—	67	4	no	—
79	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
80	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
81	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
82	C₄³	C₂×D₆₇	1	1	yes	characteristic
83	C₂²×C₈	—	67	8	no	—
84	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
85	C₂×C₄²	C₄×D₆₇	1	2	yes	—
86	C₂×C₄²	C₄×D₆₇	1	2	yes	—
87	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
88	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
89	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
90	C₄×C₈	—	2	4	no	—
91	C₄×C₈	—	2	4	no	—
92	C₄×C₈	—	67	16	no	—
93	C₄×C₈	—	134	8	no	—
94	C₄×C₈	—	134	8	no	—
95	C₄×C₈	C₆₇⋊C₈	1	8	yes	complemented
96	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
97	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
98	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
99	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
100	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	67	16	no	—
104	C₂×C₈	—	134	8	no	—
105	C₂×C₈	—	134	8	no	—
106	C₂×C₈	C₂×C₆₇⋊C₈	1	8	yes	—
107	C₄²	C₈×D₆₇	1	8	yes	—
108	C₄²	C₂×C₆₇⋊C₈	1	4	yes	—
109	C₄²	#11761611	1	8	yes	—
110	C₄²	#11761612	1	4	yes	—
111	C₄²	#11761628	1	4	yes	—
112	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
113	C₂×C₄	C₈×Dic₆₇	1	4	yes	—
114	C₂×C₄	C₈×Dic₆₇	1	4	yes	—
115	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
116	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
117	C₂×C₄	#14523953	1	4	yes	—
118	C₂×C₄	#14523953	1	4	yes	—
119	C₂×C₄	#14523954	1	4	yes	—
120	C₂×C₄	#14524000	1	4	yes	—
121	C₂×C₄	#14524002	1	2	yes	—
122	C₂×C₄	C₂×C₈×D₆₇	1	4	yes	—
123	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
124	C₂×C₄	C₂×#11761611	1	4	yes	—
125	C₂×C₄	C₂×#11761612	1	2	yes	—
126	C₈	—	2	8	no	—
127	C₈	—	134	16	no	—
128	C₂²	#16863242	1	1	yes	characteristic
129	C₂²	C₂×C₈×Dic₆₇	1	2	yes	—
130	C₂²	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
131	C₂²	C₂×#14523953	1	2	yes	—
132	C₂²	C₂×#14523954	1	1	yes	characteristic
133	C₄	C₈×C₆₇⋊C₈	1	4	yes	—
134	C₄	#16863117	1	4	yes	—
135	C₄	#16863118	1	8	yes	—
136	C₄	#16863447	1	8	yes	—
137	C₄	#16863449	1	4	yes	—
138	C₂	C₂×C₈×C₆₇⋊C₈	1	1	yes	characteristic
139	C₂	C₂×#16863117	1	1	yes	characteristic
140	C₂	C₂×#16863118	1	2	yes	—
141	C₂	#22559954	1	2	yes	—
142	C₂	#22559955	1	1	yes	characteristic

	subgroup	G/H
center	C₄³	C₂×D₆₇
commutator	C₂×C₆₇	C₂×C₈²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄²×C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	#8131
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
14. C₄×C₆₇⋊C₈	C₈	semidirect
23. C₄×C₈×C₆₇	C₈	semidirect
75. C₆₇	#8131	semidirect
95. C₄×C₈	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₆₇⋊C₈	2
C₆₇⋊#1273	2
C₄²×C₈×C₆₇	2
#8131	67