Finite Group Databasebeta

Subgroups — #31798003

Order 17536.

162 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 59 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×#17984185	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17984184	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578506	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578500	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊#14555522	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14555522	C₂	1	1	yes	maximal, characteristic, complemented
7	#22578430	C₂	1	1	yes	maximal, characteristic
8	#17984185	—	2	2	no	—
9	#17984184	—	2	2	no	—
10	C₂×#14555522	C₂²	1	1	yes	characteristic
11	C₂×#14555522	C₂²	1	1	yes	characteristic
12	C₂×#14555522	C₂²	1	1	yes	characteristic
13	C₂×#14555522	C₂²	1	1	yes	characteristic
14	C₂×#14555518	C₂²	1	1	yes	characteristic
15	C₂²×#11764047	C₂²	1	1	yes	characteristic
16	#17984155	C₄	1	1	yes	characteristic
17	#17984145	C₄	1	1	yes	characteristic
18	C₂²⋊#11764047	C₄	1	1	yes	characteristic
19	C₄×#11764047	C₂²	1	1	yes	characteristic
20	#17984048	C₄	1	1	yes	characteristic
21	#14555522	—	2	1	no	—
22	#14555522	—	2	1	no	—
23	#14555522	—	2	1	no	—
24	#14555522	D₄	1	2	yes	complemented
25	#14555518	—	2	1	no	—
26	#14555518	—	2	1	no	—
27	C₂×#11764047	—	2	1	no	—
28	C₂×#11764047	—	2	1	no	—
29	C₂×#11764047	C₂×C₄	1	1	yes	characteristic
30	C₂×#11764047	C₂×C₄	1	1	yes	characteristic
31	C₂×#11764047	C₂×C₄	1	1	yes	characteristic
32	C₂×#11764047	C₂³	1	1	yes	characteristic
33	C₂×C₂²⋊D₁₃₇	C₈	1	1	yes	characteristic
34	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
36	#14555489	C₈	1	1	yes	characteristic
37	#14555487	C₈	1	1	yes	characteristic
38	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₁₃₇	C₈	1	1	yes	characteristic
40	#11764047	—	2	1	no	—
41	#11764047	—	2	1	no	—
42	#11764047	—	2	1	no	—
43	#11764047	C₂×D₄	1	1	yes	characteristic
44	#11764047	#41	1	1	yes	characteristic
45	C₂²×D₁₃₇	C₂×C₈	1	1	yes	characteristic
46	C₂²⋊D₁₃₇	—	2	2	no	—
47	C₂×Dic₁₃₇	—	2	1	no	—
48	C₂×Dic₁₃₇	—	2	1	no	—
49	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
50	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
51	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
52	C₄×D₁₃₇	—	2	1	no	—
53	C₄×D₁₃₇	—	2	1	no	—
54	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₁₃₇	—	2	1	no	—
58	C₂²×C₁₃₇	—	2	1	no	—
59	C₂²×C₁₃₇	—	2	1	no	—
60	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
61	C₂×D₁₃₇	—	2	1	no	—
62	C₂×D₁₃₇	—	2	1	no	—
63	C₂×D₁₃₇	—	2	1	no	—
64	C₄×C₁₃₇	—	2	1	no	—
65	C₄×C₁₃₇	—	2	1	no	—
66	Dic₁₃₇	—	2	1	no	—
67	Dic₁₃₇	M₃₂	1	2	yes	complemented
68	Dic₁₃₇	C₄×D₄	1	1	yes	characteristic
69	Dic₁₃₇	#131	1	1	yes	characteristic
70	C₂×C₁₃₇	—	4	1	no	—
71	C₂×C₁₃₇	C₈×D₄	1	1	yes	characteristic
72	C₂×C₁₃₇	C₂×M₃₂	1	1	yes	characteristic
73	C₂×C₁₃₇	#504	1	1	yes	characteristic
74	D₁₃₇	—	4	1	no	—
75	C₁₃₇	#2169	1	1	yes	characteristic, complemented, Sylow
76	#2169	—	137	1	no	maximal, Sylow
77	C₂×M₃₂	—	137	1	no	—
78	C₂×M₃₂	—	137	1	no	—
79	C₈×D₄	—	137	1	no	—
80	C₄⋊C₁₆	—	137	1	no	—
81	C₂²⋊C₁₆	—	137	1	no	—
82	C₂²⋊C₁₆	—	137	1	no	—
83	C₄×C₁₆	—	137	1	no	—
84	C₂²×C₈	—	137	1	no	—
85	C₂²×C₈	—	137	1	no	—
86	C₄×D₄	—	137	1	no	—
87	M₃₂	—	274	2	no	—
88	M₃₂	—	274	2	no	—
89	C₂×C₁₆	—	137	1	no	—
90	C₂×C₁₆	—	137	1	no	—
91	C₂×C₁₆	—	137	1	no	—
92	C₂×C₁₆	—	137	1	no	—
93	C₄⋊C₈	—	137	1	no	—
94	C₂²⋊C₈	—	137	1	no	—
95	C₂²⋊C₈	—	137	1	no	—
96	C₄×C₈	—	137	1	no	—
97	C₂×D₄	—	137	1	no	—
98	C₂²×C₄	—	137	1	no	—
99	C₂²×C₄	—	137	1	no	—
100	C₂×C₈	—	274	1	no	—
101	C₂×C₈	—	274	1	no	—
102	C₂×C₈	—	274	1	no	—
103	C₂×C₈	—	274	1	no	—
104	C₂×C₈	—	137	1	no	—
105	C₂×C₈	—	137	1	no	—
106	C₂×C₈	—	137	1	no	—
107	C₂×C₈	—	137	1	no	—
108	C₄⋊C₄	—	137	1	no	—
109	C₂²⋊C₄	—	137	1	no	—
110	C₂²⋊C₄	#11764049	1	1	yes	characteristic
111	C₄²	—	137	1	no	—
112	C₁₆	—	274	1	no	—
113	C₁₆	—	274	1	no	—
114	C₁₆	—	274	1	no	—
115	C₁₆	—	137	2	no	—
116	C₂³	—	137	1	no	—
117	C₂³	C₂×#11764049	1	1	yes	characteristic
118	D₄	—	274	2	no	—
119	C₂×C₄	—	274	1	no	—
120	C₂×C₄	—	274	1	no	—
121	C₂×C₄	—	274	1	no	—
122	C₂×C₄	—	274	1	no	—
123	C₂×C₄	—	137	1	no	—
124	C₂×C₄	—	137	1	no	—
125	C₂×C₄	—	137	1	no	—
126	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
127	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
128	C₈	—	274	1	no	—
129	C₈	—	274	1	no	—
130	C₈	—	274	1	no	—
131	C₈	—	137	1	no	—
132	C₈	—	137	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	274	1	no	—
137	C₂²	—	274	1	no	—
138	C₂²	—	274	1	no	—
139	C₂²	C₂²×#11764049	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	274	1	no	—
143	C₄	—	137	1	no	—
144	C₄	—	137	1	no	—
145	C₄	—	137	2	no	—
146	C₂	—	4	1	no	—
147	C₂	—	548	1	no	—
148	C₂	D₄×#11764049	1	1	yes	characteristic
149	C₂	#22578522	1	1	yes	characteristic
150	C₂	#22578524	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11764049
commutator, socle	C₂²×C₁₃₇	C₂²×C₈
Fitting	C₂²⋊C₄×C₁₃₇	C₈
prime radical	C₂×C₂²⋊D₁₃₇	C₈
cosocle	C₂×#11764047	C₂³
hypercenter	C₂²⋊C₄	#11764049
hypocenter	C₁₃₇	#2169
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#17984185	C₂	semidirect
2. C₂×#17984184	C₂	semidirect
3. #22578506	C₂	semidirect
4. #22578500	C₂	semidirect
5. C₂²⋊#14555522	C₂	semidirect
6. C₄×#14555522	C₂	semidirect
24. #14555522	D₄	semidirect
67. Dic₁₃₇	M₃₂	semidirect
75. C₁₃₇	#2169	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984185	2
C₂×#17984184	2
#22578506	2
#22578500	2
C₂²⋊#14555522	2
C₄×#14555522	2
#22578430	2
#2169	137