Finite Group Databasebeta

Subgroups — #31780327

Order 17536.

156 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577749	C₂	1	1	yes	maximal, characteristic
2	#22577698	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577534	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577155	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577151	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577147	C₂	1	1	yes	maximal, characteristic
7	#460×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#17984088	C₂²	1	1	yes	characteristic
9	#17984021	—	2	1	no	—
10	C₄⋊#11764040	—	2	1	no	—
11	C₄⋊#11764040	C₂²	1	1	yes	characteristic
12	C₄⋊#11764040	C₂²	1	1	yes	characteristic
13	#17984012	—	2	1	no	—
14	#17984009	—	2	1	no	—
15	#17984009	—	2	1	no	—
16	#17983999	—	2	1	no	—
17	C₄×#11764040	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₁₃₇	—	2	1	no	—
19	#127×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	#102×C₁₃₇	—	2	1	no	—
23	C₂×Dic₂₇₄	—	2	1	no	—
24	C₂×#11764040	—	2	1	no	—
25	C₂×#11764040	—	2	1	no	—
26	C₂×#11764040	D₄	1	1	yes	characteristic
27	C₂×#11764040	D₄	1	1	yes	characteristic
28	C₄⋊Dic₁₃₇	—	2	1	no	—
29	C₄⋊Dic₁₃₇	—	2	1	no	—
30	C₂×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂×D₄×C₁₃₇	—	2	1	no	—
32	C₂×D₄×C₁₃₇	—	2	1	no	—
33	C₂×D₄×C₁₃₇	—	2	1	no	—
34	SD₁₆×C₁₃₇	—	4	2	no	—
35	C₂×C₈×C₁₃₇	—	2	1	no	—
36	C₄⋊C₄×C₁₃₇	—	2	1	no	—
37	C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
38	C₄²×C₁₃₇	—	2	1	no	—
39	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
40	Dic₂₇₄	—	8	1	no	—
41	C₂×Dic₁₃₇	—	4	1	no	—
42	#11764040	—	2	1	no	—
43	#11764040	—	2	1	no	—
44	#11764040	—	4	1	no	—
45	#11764040	—	4	1	no	—
46	C₂³×C₁₃₇	—	4	1	no	—
47	Q₈×C₁₃₇	—	2	1	no	—
48	Q₈×C₁₃₇	—	2	1	no	—
49	D₄×C₁₃₇	—	4	1	no	—
50	D₄×C₁₃₇	—	4	1	no	—
51	D₄×C₁₃₇	—	4	1	no	—
52	D₄×C₁₃₇	—	4	1	no	—
53	D₄×C₁₃₇	—	8	1	no	—
54	C₂×C₄×C₁₃₇	—	2	1	no	—
55	C₂×C₄×C₁₃₇	—	2	1	no	—
56	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₇	#41	1	1	yes	characteristic
59	C₈×C₁₃₇	—	4	1	no	—
60	C₂²×C₁₃₇	—	8	1	no	—
61	C₂²×C₁₃₇	—	8	1	no	—
62	C₂²×C₁₃₇	—	8	1	no	—
63	C₂²×C₁₃₇	#121	1	1	yes	characteristic
64	C₄×C₁₃₇	—	2	1	no	—
65	C₄×C₁₃₇	—	4	1	no	—
66	C₄×C₁₃₇	—	4	1	no	—
67	C₄×C₁₃₇	#135	1	1	yes	characteristic
68	C₄×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₁₃₇	#137	1	1	yes	characteristic
70	C₄×C₁₃₇	#137	1	1	yes	characteristic
71	Dic₁₃₇	—	8	1	no	—
72	C₂×C₁₃₇	—	16	1	no	—
73	C₂×C₁₃₇	#464	1	1	yes	characteristic
74	C₂×C₁₃₇	#468	1	1	yes	characteristic
75	C₂×C₁₃₇	#472	1	1	yes	characteristic
76	C₁₃₇	#1698	1	1	yes	characteristic, complemented, Sylow
77	#1698	—	137	1	no	maximal, Sylow
78	#486	—	137	1	no	—
79	#477	—	137	1	no	—
80	#460	D₁₃₇	1	1	yes	characteristic
81	#446	—	137	1	no	—
82	#335	—	137	1	no	—
83	#333	—	137	1	no	—
84	#331	—	137	1	no	—
85	C₂×SD₁₆	—	2	1	no	—
86	C₄⋊Q₈	—	137	1	no	—
87	#127	C₂×D₁₃₇	1	1	yes	characteristic
88	C₄×Q₈	C₂×D₁₃₇	1	1	yes	characteristic
89	#107	—	274	1	no	—
90	C₄⋊C₈	—	274	1	no	—
91	C₄⋊C₈	—	137	1	no	—
92	C₄⋊C₈	—	137	1	no	—
93	C₄⋊C₈	C₂×D₁₃₇	1	1	yes	characteristic
94	#103	—	274	1	no	—
95	#102	—	2	1	no	—
96	#102	—	274	1	no	—
97	#102	—	274	1	no	—
98	#97	—	274	1	no	—
99	C₄×C₈	—	137	1	no	—
100	C₂×Q₈	—	274	1	no	—
101	C₂×Q₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	—	2	1	no	—
105	SD₁₆	—	4	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	274	1	no	—
108	C₂×C₈	—	274	1	no	—
109	C₂×C₈	—	137	1	no	—
110	C₂×C₈	—	137	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	274	1	no	—
113	C₄⋊C₄	—	274	1	no	—
114	C₄⋊C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
115	C₄²	—	2	1	no	—
116	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
117	C₂³	—	4	1	no	—
118	Q₈	—	2	1	no	—
119	Q₈	—	2	1	no	—
120	Q₈	—	1096	1	no	—
121	D₄	—	4	1	no	—
122	D₄	—	4	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	8	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	548	1	no	—
129	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
131	C₂×C₄	#14555509	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	274	1	no	—
134	C₈	—	274	1	no	—
135	C₈	—	548	1	no	—
136	C₈	—	548	1	no	—
137	C₂²	—	8	1	no	—
138	C₂²	—	8	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	#17984060	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	1	no	—
143	C₄	—	4	1	no	—
144	C₄	—	1096	1	no	—
145	C₄	#17984109	1	1	yes	characteristic
146	C₄	#17984114	1	1	yes	characteristic
147	C₄	#17984117	1	1	yes	characteristic
148	C₄	#17984118	1	1	yes	characteristic
149	C₂	—	16	1	no	—
150	C₂	#22577623	1	1	yes	characteristic
151	C₂	#22577647	1	1	yes	characteristic
152	C₂	#22577671	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17984060
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#460×C₁₃₇	C₂
socle	C₂²×C₁₃₇	#121
prime radical	#127×C₁₃₇	C₂²
hypercenter	#460	D₁₃₇
hypocenter	C₁₃₇	#1698
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #460

Complements

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

N	H ≅ G/N	type
2. #22577698	C₂	semidirect
3. #22577534	C₂	semidirect
4. #22577155	C₂	semidirect
5. #22577151	C₂	semidirect
76. C₁₃₇	#1698	semidirect

Maximal subgroups

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

subgroup	index
#22577749	2
#22577698	2
#22577534	2
#22577155	2
#22577151	2
#22577147	2
#460×C₁₃₇	2
#1698	137