Finite Group Databasebeta

Subgroups — #31795876

Order 17536.

166 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 38 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	#22578383	C₂	1	1	yes	maximal, characteristic
2	#22578376	C₂	1	1	yes	maximal, characteristic
3	#22577375	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555519	C₄	1	1	yes	characteristic, complemented
5	C₂²×#11764047	C₄	1	1	yes	characteristic, complemented
6	#17984167	—	2	1	no	—
7	#17984146	—	2	1	no	—
8	#17984146	C₄	1	2	yes	complemented
9	#17984145	—	2	1	no	—
10	C₂²⋊#11764047	—	2	1	no	—
11	C₂×#14555509	C₂²	1	1	yes	characteristic
12	#17984044	—	2	1	no	—
13	#17984003	C₄	1	1	yes	characteristic, complemented
14	#17984002	—	2	1	no	—
15	#99×C₁₃₇	C₄	1	1	yes	characteristic, complemented
16	#14555519	—	4	1	no	—
17	C₂×#11764047	—	2	1	no	—
18	C₂×#11764047	—	2	1	no	—
19	C₂×#11764047	—	4	1	no	—
20	C₄⋊#10587739	—	4	1	no	—
21	C₂²⋊#10587739	—	2	2	no	—
22	C₄×#10587739	—	4	1	no	—
23	#14555509	—	4	1	no	—
24	#14555509	—	4	1	no	—
25	C₂×Dic₂₇₄	Q₈	1	1	yes	characteristic
26	C₂×C₂²⋊D₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₁₃₇	D₄	1	1	yes	characteristic
28	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₁₃₇	D₄	1	1	yes	characteristic
30	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
31	#14555489	—	4	1	no	—
32	C₂²⋊Dic₁₃₇	—	2	1	no	—
33	C₄×Dic₁₃₇	—	4	1	no	—
34	C₂×D₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
36	C₂×#10587739	—	4	2	no	—
37	#11764047	—	4	1	no	—
38	#11764047	—	4	1	no	—
39	C₂²×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
40	Dic₂₇₄	—	4	1	no	—
41	C₂²⋊D₁₃₇	—	4	1	no	—
42	C₂²⋊D₁₃₇	—	4	1	no	—
43	C₂×Dic₁₃₇	—	2	1	no	—
44	C₂×Dic₁₃₇	—	2	1	no	—
45	C₂×Dic₁₃₇	—	2	1	no	—
46	C₂×Dic₁₃₇	—	2	1	no	—
47	C₂×Dic₁₃₇	—	4	1	no	—
48	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×Dic₁₃₇	C₄⋊C₄	1	1	yes	characteristic
51	C₄×D₁₃₇	—	2	1	no	—
52	C₄×D₁₃₇	—	2	1	no	—
53	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
54	C₂³×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
55	D₄×C₁₃₇	—	4	1	no	—
56	C₂×C₄×C₁₃₇	—	4	1	no	—
57	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₃₇	—	2	1	no	—
59	C₂²×C₁₃₇	—	2	1	no	—
60	C₂²×C₁₃₇	—	4	1	no	—
61	C₂²×C₁₃₇	—	4	1	no	—
62	C₂²×C₁₃₇	#95	1	1	yes	characteristic
63	C₂×D₁₃₇	—	2	1	no	—
64	C₂×D₁₃₇	—	4	1	no	—
65	#10587739	—	8	2	no	—
66	C₄×C₁₃₇	—	2	1	no	—
67	C₄×C₁₃₇	—	8	1	no	—
68	Dic₁₃₇	—	2	1	no	—
69	Dic₁₃₇	—	2	1	no	—
70	Dic₁₃₇	—	8	1	no	—
71	Dic₁₃₇	#99	1	1	yes	characteristic
72	Dic₁₃₇	#99	1	1	yes	characteristic
73	C₂×C₁₃₇	—	2	1	no	—
74	C₂×C₁₃₇	—	4	1	no	—
75	C₂×C₁₃₇	—	4	1	no	—
76	C₂×C₁₃₇	#342	1	1	yes	characteristic
77	D₁₃₇	—	4	1	no	—
78	C₁₃₇	#1391	1	1	yes	characteristic, complemented, Sylow
79	#1391	—	137	1	no	maximal, Sylow
80	#410	—	137	1	no	—
81	#410	—	137	1	no	—
82	#408	—	137	1	no	—
83	C₂×#41	—	137	1	no	—
84	C₂×M₁₆	—	137	1	no	—
85	C₂²×C₈	—	137	1	no	—
86	#117	—	274	1	no	—
87	#117	—	274	1	no	—
88	#99	—	274	1	no	—
89	#99	—	274	1	no	—
90	#99	—	137	2	no	—
91	#99	—	137	1	no	—
92	#99	#10587739	1	1	yes	characteristic, complemented
93	C₂²⋊C₈	—	274	1	no	—
94	C₂²⋊C₈	—	274	1	no	—
95	#41	—	548	1	no	—
96	#41	—	548	1	no	—
97	C₂×Q₈	—	137	1	no	—
98	C₂×D₄	—	137	1	no	—
99	C₂×D₄	—	137	1	no	—
100	C₂×D₄	C₂×#10587739	1	1	yes	characteristic
101	C₂²×C₄	—	137	1	no	—
102	C₂²×C₄	—	137	1	no	—
103	C₂²×C₄	—	137	1	no	—
104	M₁₆	—	548	1	no	—
105	C₂×C₈	—	274	1	no	—
106	C₂×C₈	—	274	1	no	—
107	C₂×C₈	—	548	1	no	—
108	C₄⋊C₄	—	548	1	no	—
109	C₄⋊C₄	—	548	1	no	—
110	C₂²⋊C₄	—	2	1	no	—
111	C₂²⋊C₄	—	274	2	no	—
112	C₂²⋊C₄	—	274	1	no	—
113	C₄²	—	548	1	no	—
114	C₄²	—	548	1	no	—
115	C₂³	—	137	1	no	—
116	C₂³	C₄×#10587739	1	1	yes	characteristic
117	C₂³	C₂²⋊#10587739	1	1	yes	characteristic
118	Q₈	—	548	1	no	—
119	D₄	—	4	1	no	—
120	D₄	—	548	1	no	—
121	D₄	—	548	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	274	1	no	—
124	C₂×C₄	—	274	1	no	—
125	C₂×C₄	—	274	1	no	—
126	C₂×C₄	—	274	1	no	—
127	C₂×C₄	—	274	1	no	—
128	C₂×C₄	—	274	1	no	—
129	C₂×C₄	—	548	1	no	—
130	C₂×C₄	—	548	2	no	—
131	C₂×C₄	—	137	1	no	—
132	C₂×C₄	—	137	1	no	—
133	C₂×C₄	—	137	1	no	—
134	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
135	C₈	—	548	1	no	—
136	C₈	—	548	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	274	1	no	—
142	C₂²	—	548	1	no	—
143	C₂²	(C₂×#10587739)⋊C₄	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	8	1	no	—
146	C₄	—	274	1	no	—
147	C₄	—	274	1	no	—
148	C₄	—	1096	1	no	—
149	C₄	—	1096	2	no	—
150	C₄	—	137	1	no	—
151	C₄	—	137	1	no	—
152	C₂	—	2	1	no	—
153	C₂	—	4	1	no	—
154	C₂	—	4	1	no	—
155	C₂	—	548	1	no	—
156	C₂	#22578280	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578280
commutator	C₂³×C₁₃₇	C₄²
Frattini	C₂×D₄	C₂×#10587739
Fitting	#99×C₁₃₇	C₄
socle	C₂×C₁₃₇	#342
prime radical, cosocle	C₂×#14555509	C₂²
hypercenter	#99	#10587739
hypocenter	C₁₃₇	#1391
LCS	G ▷ C₂³×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #99

Complements

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

N	H ≅ G/N	type
4. C₂×#14555519	C₄	semidirect
5. C₂²×#11764047	C₄	semidirect
8. #17984146	C₄	semidirect
13. #17984003	C₄	semidirect
15. #99×C₁₃₇	C₄	semidirect
78. C₁₃₇	#1391	semidirect
92. #99	#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578383	2
#22578376	2
#22577375	2
#1391	137