Finite Group Databasebeta

Subgroups — #31779390

Order 17536.

212 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₂×#17984080	C₂	1	1	yes	maximal, characteristic
2	#22577460	C₂	1	1	yes	maximal, characteristic
3	#22577144	C₂	1	4	yes	maximal, complemented
4	C₄×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#17984080	—	2	1	no	—
6	#17984080	—	2	1	no	—
7	#17984080	C₄	1	2	yes	—
8	#17984080	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₃₇	C₄	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃₇	C₄	1	1	yes	characteristic
11	C₂×#14555489	—	2	1	no	—
12	C₂×#14555489	—	2	1	no	—
13	C₄⋊#11764040	—	2	1	no	—
14	C₂²⋊#11764040	—	2	1	no	—
15	#17983999	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₃₇	—	2	1	no	—
17	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
18	#97×C₁₃₇	C₂²	1	2	yes	—
19	C₄×C₈×C₁₃₇	—	2	1	no	—
20	C₂²×Dic₁₃₇	—	2	1	no	—
21	C₂²×Dic₁₃₇	—	2	1	no	—
22	C₂×#11764040	—	2	2	no	—
23	C₄⋊Dic₁₃₇	—	2	1	no	—
24	C₄⋊Dic₁₃₇	—	2	1	no	—
25	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
26	C₄⋊Dic₁₃₇	C₂×C₄	1	2	yes	—
27	#14555489	—	2	2	no	—
28	#14555489	—	2	2	no	—
29	#14555489	—	2	2	no	—
30	#14555489	—	2	2	no	—
31	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
34	M₁₆×C₁₃₇	—	2	4	no	—
35	C₂×C₈×C₁₃₇	—	2	2	no	—
36	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
38	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
39	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₂×Dic₁₃₇	—	2	2	no	—
41	C₂×Dic₁₃₇	—	2	2	no	—
42	C₂×Dic₁₃₇	—	4	2	no	—
43	C₂×Dic₁₃₇	—	4	2	no	—
44	#11764040	—	4	2	no	—
45	C₂³×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₃₇	—	2	2	no	—
47	C₂×C₄×C₁₃₇	—	2	2	no	—
48	C₂×C₄×C₁₃₇	—	4	1	no	—
49	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
55	C₈×C₁₃₇	—	2	4	no	—
56	C₂²×C₁₃₇	—	2	1	no	—
57	C₂²×C₁₃₇	—	2	1	no	—
58	C₂²×C₁₃₇	#104	1	2	yes	—
59	C₂²×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₃₇	—	2	1	no	—
61	C₄×C₁₃₇	—	2	1	no	—
62	C₄×C₁₃₇	—	2	1	no	—
63	C₄×C₁₃₇	—	2	1	no	—
64	C₄×C₁₃₇	—	2	1	no	—
65	C₄×C₁₃₇	#101	1	2	yes	—
66	Dic₁₃₇	—	4	2	no	—
67	Dic₁₃₇	—	4	2	no	—
68	C₂×C₁₃₇	—	2	2	no	—
69	C₂×C₁₃₇	C₂×#101	1	1	yes	characteristic
70	C₂×C₁₃₇	C₂×#104	1	1	yes	characteristic
71	C₂×C₁₃₇	#421	1	1	yes	characteristic
72	C₁₃₇	#1528	1	1	yes	characteristic, complemented, Sylow
73	#1528	—	137	1	no	maximal, Sylow
74	C₂×#125	—	137	1	no	—
75	#432	—	137	1	no	—
76	C₄×M₁₆	D₁₃₇	1	1	yes	characteristic
77	#330	—	137	4	no	—
78	C₂×M₁₆	—	2	1	no	—
79	#125	—	274	1	no	—
80	#125	—	274	1	no	—
81	#125	—	137	2	no	—
82	#125	—	137	2	no	—
83	C₂×C₄⋊C₄	—	274	1	no	—
84	C₂×C₄⋊C₄	—	274	1	no	—
85	C₂×C₄⋊C₄	—	137	1	no	—
86	C₂×C₄⋊C₄	—	137	1	no	—
87	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
88	C₄⋊C₈	—	274	1	no	—
89	C₂²⋊C₈	—	274	1	no	—
90	#97	—	137	2	no	—
91	#97	C₂×D₁₃₇	1	2	yes	—
92	C₄×C₈	—	2	1	no	—
93	C₂²×C₄	—	274	1	no	—
94	C₂²×C₄	—	274	1	no	—
95	C₂²×C₄	C₄×D₁₃₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₃₇	1	1	yes	characteristic
97	C₂²×C₄	D₅₄₈	1	1	yes	characteristic
98	M₁₆	—	2	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	274	2	no	—
101	C₄⋊C₄	—	274	2	no	—
102	C₄⋊C₄	—	274	2	no	—
103	C₄⋊C₄	—	274	1	no	—
104	C₄⋊C₄	—	274	1	no	—
105	C₄⋊C₄	—	274	2	no	—
106	C₄⋊C₄	—	274	2	no	—
107	C₄⋊C₄	—	137	2	no	—
108	C₄⋊C₄	—	137	2	no	—
109	C₄²	C₄×D₁₃₇	1	1	yes	characteristic
110	C₄²	C₂²⋊D₁₃₇	1	1	yes	characteristic
111	C₄²	D₅₄₈	1	1	yes	characteristic
112	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
113	C₂³	#14555487	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	274	2	no	—
118	C₂×C₄	—	274	2	no	—
119	C₂×C₄	—	548	2	no	—
120	C₂×C₄	—	548	2	no	—
121	C₂×C₄	#14555487	1	1	yes	characteristic
122	C₂×C₄	#14555487	1	1	yes	characteristic
123	C₂×C₄	#14555487	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
126	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₈	—	548	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#17984016	1	2	yes	—
132	C₂²	C₂×#14555487	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	548	2	no	—
139	C₄	—	548	2	no	—
140	C₄	#17984006	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#17984006	1	1	yes	characteristic
143	C₂	C₂×#17984016	1	1	yes	characteristic
144	C₂	#22577419	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14555487
commutator	C₂×C₄×C₁₃₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃₇
Fitting	C₄×M₁₆×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₃₇	C₂²⋊C₄
cosocle	C₄²×C₁₃₇	C₂³
hypercenter	C₄×M₁₆	D₁₃₇
hypocenter	C₁₃₇	#1528
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22577144	C₂	semidirect
72. C₁₃₇	#1528	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984080	2
#22577460	2
#22577144	2
C₄×M₁₆×C₁₃₇	2
#1528	137