Finite Group Databasebeta

Subgroups — #31797197

Order 17536.

142 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 44 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	#22578449	C₂	1	1	yes	maximal, characteristic
2	#22578443	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578328	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578322	C₂	1	1	yes	maximal, characteristic, complemented
5	#22578320	C₂	1	1	yes	maximal, characteristic
6	#22578294	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577845	C₂	1	1	yes	maximal, characteristic
8	#17984176	—	2	1	no	—
9	Q₈×#10587739	—	2	1	no	—
10	#17984163	C₂²	1	1	yes	characteristic
11	#17984158	C₂²	1	1	yes	characteristic
12	#17984156	C₂²	1	1	yes	characteristic
13	#17984153	—	2	1	no	—
14	#17984151	—	2	1	no	—
15	#17984142	C₂²	1	1	yes	characteristic
16	#17984111	C₂²	1	1	yes	characteristic
17	Q₁₆×D₁₃₇	C₂²	1	1	yes	characteristic
18	#17984034	C₄	1	1	yes	characteristic, complemented
19	#17984033	C₄	1	1	yes	characteristic, complemented
20	#17984031	C₄	1	1	yes	characteristic, complemented
21	#17984027	C₂²	1	1	yes	characteristic
22	Q₃₂×C₁₃₇	C₄	1	1	yes	characteristic, complemented
23	#14555520	—	2	1	no	—
24	#14555520	—	4	1	no	—
25	#14555518	—	4	1	no	—
26	#14555518	D₄	1	1	yes	characteristic
27	C₄⋊#10587739	—	2	1	no	—
28	C₄⋊#10587739	—	4	1	no	—
29	C₄×#10587739	—	4	1	no	—
30	C₄×#10587739	D₄	1	1	yes	characteristic
31	#14555513	—	2	1	no	—
32	#14555511	—	2	1	no	—
33	Q₈×D₁₃₇	—	2	1	no	—
34	Dic₅₄₈	C₂×C₄	1	1	yes	characteristic
35	#14555500	—	2	1	no	—
36	#14555497	—	2	1	no	—
37	D₁₀₉₆	C₂×C₄	1	1	yes	characteristic
38	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
39	#14555485	C₂×C₄	1	1	yes	characteristic
40	Q₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
41	Q₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×#10587739	—	2	1	no	—
44	C₂×#10587739	—	4	1	no	—
45	#11764047	—	2	1	no	—
46	#11764047	—	4	1	no	—
47	Dic₂₇₄	—	2	1	no	—
48	Dic₂₇₄	—	4	1	no	—
49	D₅₄₈	—	2	1	no	—
50	D₅₄₈	—	4	1	no	—
51	C₄×D₁₃₇	—	4	1	no	—
52	C₄×D₁₃₇	—	4	1	no	—
53	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
54	#11764040	#41	1	1	yes	characteristic
55	Q₈×C₁₃₇	—	2	1	no	—
56	Q₈×C₁₃₇	—	2	1	no	—
57	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×D₁₃₇	—	4	1	no	—
59	C₂×D₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	#10587739	—	4	1	no	—
61	#10587739	—	8	1	no	—
62	C₄×C₁₃₇	—	4	1	no	—
63	C₄×C₁₃₇	—	4	1	no	—
64	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
65	Dic₁₃₇	—	4	1	no	—
66	Dic₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₁₃₇	#442	1	1	yes	characteristic
68	D₁₃₇	—	2	1	no	—
69	D₁₃₇	—	8	1	no	—
70	C₁₃₇	#2180	1	1	yes	characteristic, complemented, Sylow
71	#2180	—	137	1	no	maximal, Sylow
72	#510	—	137	1	no	—
73	#444	—	137	1	no	—
74	#441	—	137	1	no	—
75	#365	—	137	1	no	—
76	#362	—	137	1	no	—
77	#360	—	137	1	no	—
78	#347	—	137	1	no	—
79	#135	—	137	1	no	—
80	C₂×Q₁₆	—	137	1	no	—
81	#131	—	274	1	no	—
82	C₄×Q₈	—	274	1	no	—
83	Q₃₂	—	137	1	no	—
84	Q₃₂	#10587739	1	1	yes	characteristic, complemented
85	SD₃₂	—	137	1	no	—
86	SD₃₂	—	137	1	no	—
87	M₃₂	—	137	1	no	—
88	M₃₂	—	137	1	no	—
89	#108	—	137	1	no	—
90	#106	—	137	1	no	—
91	#104	—	274	1	no	—
92	#103	—	274	1	no	—
93	#97	—	137	1	no	—
94	#41	—	274	1	no	—
95	C₂×Q₈	—	274	1	no	—
96	Q₁₆	—	274	1	no	—
97	Q₁₆	—	137	1	no	—
98	Q₁₆	C₂×#10587739	1	1	yes	characteristic
99	Q₁₆	C₂×#10587739	1	1	yes	characteristic
100	SD₁₆	—	274	1	no	—
101	D₈	—	137	1	no	—
102	M₁₆	—	274	1	no	—
103	M₁₆	—	548	1	no	—
104	C₂×C₈	—	548	1	no	—
105	C₂×C₈	—	137	1	no	—
106	C₂×C₈	—	137	1	no	—
107	C₄⋊C₄	—	274	1	no	—
108	C₄⋊C₄	—	548	1	no	—
109	C₄²	—	548	1	no	—
110	C₄²	—	137	1	no	—
111	C₁₆	—	274	1	no	—
112	C₁₆	—	137	1	no	—
113	C₁₆	C₂×#10587739	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	274	1	no	—
117	Q₈	—	548	1	no	—
118	D₄	—	274	1	no	—
119	D₄	—	548	1	no	—
120	C₂×C₄	—	274	1	no	—
121	C₂×C₄	—	548	1	no	—
122	C₂×C₄	—	548	1	no	—
123	C₂×C₄	—	548	1	no	—
124	C₂×C₄	—	137	1	no	—
125	C₈	—	274	1	no	—
126	C₈	—	548	1	no	—
127	C₈	—	137	1	no	—
128	C₈	C₂²×#10587739	1	1	yes	characteristic
129	C₂²	—	548	1	no	—
130	C₂²	—	137	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	548	1	no	—
134	C₄	—	548	1	no	—
135	C₄	—	1096	1	no	—
136	C₄	—	137	1	no	—
137	C₄	D₄×#10587739	1	1	yes	characteristic
138	C₂	—	274	1	no	—
139	C₂	—	1096	1	no	—
140	C₂	#22578444	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578444
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₈	C₂²×#10587739
Fitting	Q₃₂×C₁₃₇	C₄
socle	C₂×C₁₃₇	#442
prime radical	#17984111	C₂²
cosocle	C₈×D₁₃₇	C₂³
hypercenter	Q₃₂	#10587739
hypocenter	C₁₃₇	#2180
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
2. #22578443	C₂	semidirect
3. #22578328	C₂	semidirect
4. #22578322	C₂	semidirect
6. #22578294	C₂	semidirect
18. #17984034	C₄	semidirect
19. #17984033	C₄	semidirect
20. #17984031	C₄	semidirect
22. Q₃₂×C₁₃₇	C₄	semidirect
70. C₁₃₇	#2180	semidirect
84. Q₃₂	#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578449	2
#22578443	2
#22578328	2
#22578322	2
#22578320	2
#22578294	2
#22577845	2
#2180	137