Finite Group Databasebeta

Subgroups — #31782668

Order 17536.

146 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22577833	C₂	1	1	yes	maximal, characteristic
2	#22577521	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577232	C₂	1	1	yes	maximal, characteristic, complemented
4	#22577210	C₂	1	1	yes	maximal, characteristic, complemented
5	#22577209	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14555485	C₂	1	1	yes	maximal, characteristic, complemented
7	#443×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#17984110	C₂²	1	1	yes	characteristic
9	#17984094	—	2	1	no	—
10	#17984034	C₄	1	1	yes	characteristic, complemented
11	#17984032	C₄	1	1	yes	characteristic, complemented
12	#17984031	C₄	1	1	yes	characteristic, complemented
13	#17984029	C₄	1	1	yes	characteristic, complemented
14	C₂×#14555485	C₂²	1	1	yes	characteristic
15	C₂×#14555485	C₂²	1	1	yes	characteristic
16	#17984024	C₂²	1	1	yes	characteristic
17	#17984016	—	2	1	no	—
18	#135×C₁₃₇	C₂²	1	1	yes	characteristic
19	#131×C₁₃₇	—	2	1	no	—
20	#108×C₁₃₇	C₂²	1	1	yes	characteristic
21	#104×C₁₃₇	—	2	1	no	—
22	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
23	#14555510	—	2	1	no	—
24	Dic₅₄₈	C₂×C₄	1	1	yes	characteristic
25	#14555499	—	2	1	no	—
26	D₁₀₉₆	C₂×C₄	1	1	yes	characteristic
27	#14555494	—	2	1	no	—
28	#14555493	—	4	1	no	—
29	C₈×D₁₃₇	—	4	1	no	—
30	#14555485	—	2	1	no	—
31	#14555485	C₂×C₄	1	1	yes	characteristic
32	#14555485	C₂×C₄	1	1	yes	characteristic
33	#41×C₁₃₇	—	2	1	no	—
34	Q₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	SD₁₆×C₁₃₇	—	2	1	no	—
36	D₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	M₁₆×C₁₃₇	—	2	1	no	—
38	M₁₆×C₁₃₇	—	4	1	no	—
39	C₂×C₈×C₁₃₇	—	4	1	no	—
40	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
42	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
43	Dic₂₇₄	—	2	1	no	—
44	D₅₄₈	—	2	1	no	—
45	C₂²⋊D₁₃₇	—	4	1	no	—
46	C₄×D₁₃₇	—	4	1	no	—
47	#11764040	—	4	1	no	—
48	Q₈×C₁₃₇	—	2	1	no	—
49	D₄×C₁₃₇	—	2	1	no	—
50	D₄×C₁₃₇	—	4	1	no	—
51	C₂×C₄×C₁₃₇	—	2	1	no	—
52	C₂×C₄×C₁₃₇	—	4	1	no	—
53	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
54	C₈×C₁₃₇	—	4	1	no	—
55	C₈×C₁₃₇	D₈	1	1	yes	characteristic
56	C₈×C₁₃₇	D₈	1	1	yes	characteristic
57	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₈×C₁₃₇	#41	1	1	yes	characteristic
59	C₂²×C₁₃₇	—	4	1	no	—
60	C₂²×C₁₃₇	#135	1	1	yes	characteristic
61	C₂×D₁₃₇	—	4	1	no	—
62	C₄×C₁₃₇	—	2	2	no	—
63	C₄×C₁₃₇	—	4	1	no	—
64	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
65	C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
66	Dic₁₃₇	—	4	1	no	—
67	C₂×C₁₃₇	—	2	1	no	—
68	C₂×C₁₃₇	—	8	1	no	—
69	C₂×C₁₃₇	C₄×D₈	1	1	yes	characteristic
70	D₁₃₇	—	8	1	no	—
71	C₁₃₇	#2178	1	1	yes	characteristic, complemented, Sylow
72	#2178	—	137	1	no	maximal, Sylow
73	#508	—	137	1	no	—
74	#443	—	137	1	no	—
75	#443	D₁₃₇	1	1	yes	characteristic, complemented
76	#368	—	137	1	no	—
77	#359	—	137	1	no	—
78	#359	—	137	1	no	—
79	C₄×C₁₆	—	137	1	no	—
80	#135	—	137	1	no	—
81	#135	C₂×D₁₃₇	1	1	yes	characteristic
82	#131	—	2	1	no	—
83	#131	—	274	1	no	—
84	Q₃₂	—	137	1	no	—
85	SD₃₂	—	137	1	no	—
86	SD₃₂	—	137	1	no	—
87	D₁₆	—	137	1	no	—
88	C₂×C₁₆	—	137	1	no	—
89	C₂×C₁₆	—	137	1	no	—
90	#108	—	137	1	no	—
91	#108	C₂×D₁₃₇	1	1	yes	characteristic
92	#104	—	2	1	no	—
93	#104	—	274	1	no	—
94	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
95	#41	—	2	1	no	—
96	#41	—	274	1	no	—
97	Q₁₆	—	137	1	no	—
98	Q₁₆	C₄×D₁₃₇	1	1	yes	characteristic
99	SD₁₆	—	2	1	no	—
100	SD₁₆	—	274	1	no	—
101	D₈	—	137	1	no	—
102	D₈	C₄×D₁₃₇	1	1	yes	characteristic
103	M₁₆	—	2	1	no	—
104	M₁₆	—	4	1	no	—
105	M₁₆	—	274	1	no	—
106	M₁₆	—	548	1	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	—	548	1	no	—
109	C₂×C₈	C₂²⋊D₁₃₇	1	1	yes	characteristic
110	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
111	C₄²	C₂²⋊D₁₃₇	1	1	yes	characteristic
112	C₁₆	—	274	1	no	—
113	C₁₆	—	137	1	no	—
114	C₁₆	—	137	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	274	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	4	1	no	—
119	D₄	—	274	1	no	—
120	D₄	—	548	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	548	1	no	—
124	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	548	1	no	—
127	C₈	#14555495	1	1	yes	characteristic
128	C₈	#14555495	1	1	yes	characteristic
129	C₈	C₂×C₄×D₁₃₇	1	1	yes	characteristic
130	C₈	#14555510	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	548	1	no	—
133	C₂²	#17984108	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	4	1	no	—
136	C₄	—	548	1	no	—
137	C₄	C₄×C₂²⋊D₁₃₇	1	1	yes	characteristic
138	C₄	C₂×#14555495	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	8	1	no	—
141	C₂	—	1096	1	no	—
142	C₂	C₄×#14555495	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14555495
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	#443×C₁₃₇	C₂
socle	C₂×C₁₃₇	C₄×D₈
prime radical	#22577833	C₂
cosocle	C₂×C₈×C₁₃₇	C₂³
hypercenter	#443	D₁₃₇
hypocenter	C₁₃₇	#2178
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #22577521	C₂	semidirect
3. #22577232	C₂	semidirect
4. #22577210	C₂	semidirect
5. #22577209	C₂	semidirect
6. C₄×#14555485	C₂	semidirect
7. #443×C₁₃₇	C₂	semidirect
10. #17984034	C₄	semidirect
11. #17984032	C₄	semidirect
12. #17984031	C₄	semidirect
13. #17984029	C₄	semidirect
71. C₁₃₇	#2178	semidirect
75. #443	D₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#22577833	2
#22577521	2
#22577232	2
#22577210	2
#22577209	2
C₄×#14555485	2
#443×C₁₃₇	2
#2178	137