Finite Group Databasebeta

Subgroups — #31782971

Order 17536.

150 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 47 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	#22577640	C₂	1	1	yes	maximal, characteristic
2	#107×D₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577228	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577205	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577203	C₂	1	1	yes	maximal, characteristic, complemented
6	#22577182	C₂	1	1	yes	maximal, characteristic, complemented
7	#366×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₀₉₆	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
10	#17984063	—	2	1	no	—
11	C₄⋊C₄×D₁₃₇	—	2	1	no	—
12	C₂×#14555485	C₂²	1	1	yes	characteristic
13	#17984022	C₂²	1	1	yes	characteristic
14	#17984021	—	2	1	no	—
15	#17984008	—	2	1	no	—
16	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
17	#107×C₁₃₇	C₂²	1	1	yes	characteristic
18	#107×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×D₅₄₈	—	2	1	no	—
20	C₂×D₅₄₈	—	4	1	no	—
21	C₂×C₄×D₁₃₇	—	4	1	no	—
22	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
23	D₁₀₉₆	—	2	1	no	—
24	D₁₀₉₆	—	2	1	no	—
25	C₂×#11764040	D₄	1	1	yes	characteristic
26	C₈×D₁₃₇	—	2	1	no	—
27	C₈×D₁₃₇	—	2	1	no	—
28	C₄⋊Dic₁₃₇	—	2	1	no	—
29	#14555489	—	4	1	no	—
30	#14555487	—	4	1	no	—
31	#14555485	—	2	1	no	—
32	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₃₇	—	2	1	no	—
34	C₄⋊C₄×C₁₃₇	—	2	1	no	—
35	C₁₆×C₁₃₇	—	2	1	no	—
36	C₂²×D₁₃₇	—	4	1	no	—
37	C₂²×D₁₃₇	D₈	1	1	yes	characteristic
38	D₅₄₈	—	4	1	no	—
39	D₅₄₈	—	4	1	no	—
40	D₅₄₈	—	8	2	no	—
41	C₂×Dic₁₃₇	—	4	1	no	—
42	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
43	C₄×D₁₃₇	—	2	1	no	—
44	C₄×D₁₃₇	—	2	1	no	—
45	C₄×D₁₃₇	—	8	2	no	—
46	#11764040	—	2	1	no	—
47	C₂×C₄×C₁₃₇	—	4	1	no	—
48	C₂×C₄×C₁₃₇	—	4	1	no	—
49	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
50	C₈×C₁₃₇	#41	1	1	yes	characteristic
51	C₈×C₁₃₇	#41	1	1	yes	characteristic
52	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
53	C₂×D₁₃₇	—	2	1	no	—
54	C₂×D₁₃₇	—	2	1	no	—
55	C₂×D₁₃₇	—	8	1	no	—
56	C₂×D₁₃₇	—	8	1	no	—
57	C₂×D₁₃₇	—	8	1	no	—
58	C₂×D₁₃₇	D₁₆	1	2	yes	—
59	C₄×C₁₃₇	—	8	1	no	—
60	C₄×C₁₃₇	—	8	1	no	—
61	C₄×C₁₃₇	#123	1	1	yes	characteristic
62	C₄×C₁₃₇	#137	1	1	yes	characteristic
63	Dic₁₃₇	—	2	1	no	—
64	Dic₁₃₇	—	8	1	no	—
65	C₂×C₁₃₇	#480	1	1	yes	characteristic
66	C₂×C₁₃₇	C₂×D₁₆	1	1	yes	characteristic
67	C₂×C₁₃₇	#510	1	1	yes	characteristic
68	D₁₃₇	—	2	2	no	—
69	D₁₃₇	—	16	1	no	—
70	C₁₃₇	#2232	1	1	yes	characteristic, complemented, Sylow
71	#2232	—	137	1	no	maximal, Sylow
72	#466	—	137	1	no	—
73	C₂×#107	—	137	1	no	—
74	#366	—	137	1	no	—
75	#366	D₁₃₇	1	1	yes	characteristic, complemented
76	#357	—	137	1	no	—
77	#357	—	137	1	no	—
78	C₂²⋊C₁₆	—	137	1	no	—
79	C₂×D₈	—	137	1	no	—
80	C₂²×C₈	—	137	1	no	—
81	#121	—	274	1	no	—
82	C₂×C₄⋊C₄	—	274	1	no	—
83	C₂×C₁₆	—	137	1	no	—
84	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
85	#107	—	274	1	no	—
86	#107	—	137	1	no	—
87	#107	C₂×D₁₃₇	1	1	yes	characteristic
88	#107	C₂×D₁₃₇	1	1	yes	characteristic
89	#102	—	274	1	no	—
90	C₂×D₄	—	274	1	no	—
91	C₂×D₄	—	548	1	no	—
92	C₂²×C₄	—	548	1	no	—
93	C₂²×C₄	—	137	1	no	—
94	D₈	—	274	1	no	—
95	D₈	—	274	1	no	—
96	C₂×C₈	—	274	1	no	—
97	C₂×C₈	—	274	1	no	—
98	C₂×C₈	—	137	1	no	—
99	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	274	1	no	—
103	C₄⋊C₄	—	548	1	no	—
104	C₂²⋊C₄	—	548	1	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	—	274	1	no	—
107	C₂³	—	548	1	no	—
108	C₂³	—	137	1	no	—
109	D₄	—	548	1	no	—
110	D₄	—	548	1	no	—
111	D₄	—	1096	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	274	1	no	—
115	C₂×C₄	—	274	1	no	—
116	C₂×C₄	—	548	1	no	—
117	C₂×C₄	—	1096	2	no	—
118	C₂×C₄	—	137	1	no	—
119	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
120	C₈	—	274	1	no	—
121	C₈	#14555510	1	1	yes	characteristic
122	C₈	#14555511	1	1	yes	characteristic
123	C₂²	—	274	1	no	—
124	C₂²	—	274	1	no	—
125	C₂²	—	1096	1	no	—
126	C₂²	—	1096	1	no	—
127	C₂²	—	1096	1	no	—
128	C₂²	—	137	2	no	—
129	C₂²	D₈×D₁₃₇	1	1	yes	characteristic
130	C₄	—	8	1	no	—
131	C₄	—	8	1	no	—
132	C₄	—	274	1	no	—
133	C₄	—	1096	1	no	—
134	C₄	#17984073	1	1	yes	characteristic
135	C₄	#17984120	1	1	yes	characteristic
136	C₂	—	274	2	no	—
137	C₂	—	2192	1	no	—
138	C₂	#22577718	1	1	yes	characteristic
139	C₂	D₁₆×D₁₃₇	1	1	yes	characteristic
140	C₂	#22577845	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₃₇
commutator, cosocle	C₂×C₈×C₁₃₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	#366×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×D₈
prime radical	#22577640	C₂
hypercenter	#366	D₁₃₇
hypocenter	C₁₃₇	#2232
LCS	G ▷ C₂×C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
2. #107×D₁₃₇	C₂	semidirect
3. #22577228	C₂	semidirect
4. #22577205	C₂	semidirect
5. #22577203	C₂	semidirect
6. #22577182	C₂	semidirect
7. #366×C₁₃₇	C₂	semidirect
70. C₁₃₇	#2232	semidirect
75. #366	D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22577640	2
#107×D₁₃₇	2
#22577228	2
#22577205	2
#22577203	2
#22577182	2
#366×C₁₃₇	2
#2232	137