Subgroups — #30764813

Order 17152.

330 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	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#22558126	C₂	1	2	yes	maximal, complemented
4	C₈×C₆₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₄	1	2	yes	—
7	C₂²×C₆₇⋊C₁₆	C₄	1	2	yes	—
8	#16863172	C₄	1	2	yes	—
9	C₄×C₆₇⋊C₁₆	C₄	1	2	yes	—
10	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
13	#431×C₆₇	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#322×C₆₇	C₄	1	2	yes	—
16	C₈²×C₆₇	C₂²	1	2	yes	—
17	#14523983	C₈	1	8	yes	complemented
18	C₂×C₆₇⋊C₁₆	C₈	1	8	yes	complemented
19	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₄⋊C₈×C₆₇	C₈	1	4	yes	—
26	C₂²⋊C₈×C₆₇	C₈	1	4	yes	—
27	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₆₇⋊C₁₆	C₂×C₈	1	4	yes	—
33	C₆₇⋊C₁₆	C₂×C₈	1	4	yes	—
34	C₆₇⋊C₁₆	C₂×C₈	1	8	yes	—
35	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
40	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₆₇	C₂×C₈	1	8	yes	—
42	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₆₇	C₄²	1	1	yes	characteristic
45	C₄²×C₆₇	C₂×C₈	1	2	yes	—
46	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂³×C₆₇	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₇	—	2	4	no	—
49	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
50	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
51	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₆₇	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
55	C₈×C₆₇	—	2	8	no	—
56	C₈×C₆₇	C₄×C₈	1	2	yes	—
57	C₈×C₆₇	C₄×C₈	1	2	yes	—
58	C₈×C₆₇	C₂²×C₈	1	2	yes	—
59	C₈×C₆₇	C₂²×C₈	1	2	yes	—
60	C₂²×C₆₇	—	2	2	no	—
61	C₂²×C₆₇	C₈²	1	2	yes	—
62	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄×C₆₇	—	2	2	no	—
64	C₄×C₆₇	C₈²	1	2	yes	—
65	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
66	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
67	C₄×C₆₇	#504	1	4	yes	—
68	#11399	—	67	1	no	maximal, Sylow
69	C₂×C₆₇	—	2	2	no	—
70	C₂×C₆₇	C₂×C₈²	1	1	yes	characteristic
71	C₂×C₆₇	#2108	1	2	yes	—
72	C₄×M₃₂	—	67	1	no	—
73	C₂×C₄×C₁₆	—	67	1	no	—
74	#1450	D₆₇	1	1	yes	characteristic
75	#1311	—	67	2	no	—
76	C₈×C₁₆	—	67	2	no	—
77	C₆₇	#11399	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₃₂	—	67	2	no	—
79	C₂²×C₁₆	—	67	2	no	—
80	#431	Dic₆₇	1	2	yes	—
81	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
82	#346	—	67	2	no	—
83	C₄×C₁₆	—	67	2	no	—
84	C₄×C₁₆	—	67	1	no	—
85	C₄×C₁₆	—	67	2	no	—
86	C₄×C₁₆	—	67	1	no	—
87	#322	Dic₆₇	1	2	yes	—
88	C₈²	C₂×D₆₇	1	2	yes	—
89	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
90	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
91	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
92	M₃₂	—	67	8	no	—
93	C₂×C₁₆	—	67	8	no	—
94	C₂×C₁₆	—	67	2	no	—
95	C₂×C₁₆	—	67	2	no	—
96	C₂×C₁₆	—	67	4	no	—
97	C₄⋊C₈	C₆₇⋊C₈	1	4	yes	—
98	C₂²⋊C₈	C₆₇⋊C₈	1	4	yes	—
99	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
100	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
101	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
102	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
103	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
104	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
106	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
107	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
108	C₂×C₈	C₈×D₆₇	1	4	yes	—
109	C₂×C₈	C₈×D₆₇	1	4	yes	—
110	C₂×C₈	C₂×C₆₇⋊C₈	1	8	yes	—
111	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
112	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
113	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
114	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
115	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
116	C₁₆	—	67	4	no	—
117	C₁₆	—	67	4	no	—
118	C₁₆	—	67	8	no	—
119	C₂³	C₄×C₆₇⋊C₈	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	C₈×Dic₆₇	1	2	yes	—
122	C₂×C₄	C₈×Dic₆₇	1	2	yes	—
123	C₂×C₄	C₄×C₆₇⋊C₈	1	2	yes	—
124	C₂×C₄	C₄×C₆₇⋊C₈	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
126	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
127	C₈	—	2	8	no	—
128	C₈	C₈×Dic₆₇	1	2	yes	—
129	C₈	C₈×Dic₆₇	1	2	yes	—
130	C₈	C₂×C₈×D₆₇	1	2	yes	—
131	C₈	C₂×C₈×D₆₇	1	2	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	C₈×C₆₇⋊C₈	1	2	yes	—
134	C₂²	C₂×C₄×C₆₇⋊C₈	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	C₈×C₆₇⋊C₈	1	2	yes	—
137	C₄	C₂×C₈×Dic₆₇	1	1	yes	characteristic
138	C₄	C₂×C₈×Dic₆₇	1	1	yes	characteristic
139	C₄	#16863813	1	4	yes	—
140	C₂	—	2	2	no	—
141	C₂	C₂×C₈×C₆₇⋊C₈	1	1	yes	characteristic
142	C₂	#22561806	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₆₇
commutator	C₂×C₆₇	C₂×C₈²
Fitting	C₆₇×#1450	C₂
socle	C₂²×C₆₇	C₂×C₄×C₈
prime radical	C₂³×C₆₇	C₄×C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1450	D₆₇
hypocenter	C₆₇	#11399
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #22558126	C₂	semidirect
4. C₈×C₆₇⋊C₁₆	C₂	semidirect
17. #14523983	C₈	semidirect
18. C₂×C₆₇⋊C₁₆	C₈	semidirect
77. C₆₇	#11399	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523983	2
C₂×C₄×C₆₇⋊C₁₆	2
#22558126	2
C₈×C₆₇⋊C₁₆	2
C₆₇×#1450	2
#11399	67