Finite Group Databasebeta

Subgroups — #31797973

Order 17536.

220 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22578522	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578522	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578504	C₂	1	4	yes	maximal, complemented
4	C₂×#17984176	C₂	1	1	yes	maximal, characteristic
5	#17984185	—	2	1	no	—
6	#17984185	C₂²	1	2	yes	complemented
7	#17984184	—	2	1	no	—
8	#17984184	—	2	1	no	—
9	#17984183	—	2	1	no	—
10	#17984183	C₂²	1	2	yes	complemented
11	#17984176	—	2	1	no	—
12	#17984176	—	2	1	no	—
13	#17984176	C₄	1	2	yes	—
14	#17984176	C₂²	1	2	yes	—
15	C₂×#14555520	—	2	1	no	—
16	C₂×#14555520	C₄	1	1	yes	characteristic
17	C₂×#14555518	—	2	1	no	—
18	C₂×#14555518	C₂²	1	1	yes	characteristic
19	C₂×#14555511	C₄	1	1	yes	characteristic
20	#14555522	—	2	2	no	—
21	#14555522	—	2	2	no	—
22	#14555520	—	2	2	no	—
23	#14555520	—	2	2	no	—
24	#14555520	—	2	1	no	—
25	#14555520	C₂×C₄	1	2	yes	—
26	#14555518	—	2	2	no	—
27	#14555518	—	2	2	no	—
28	#14555518	C₂×C₄	1	1	yes	characteristic
29	#14555518	D₄	1	1	yes	characteristic
30	#14555518	D₄	1	1	yes	characteristic
31	#14555518	C₂³	1	1	yes	characteristic
32	C₂×#11764047	—	2	1	no	—
33	C₂×#11764047	D₄	1	1	yes	characteristic
34	C₂×#11764047	D₄	1	1	yes	characteristic
35	#14555511	—	2	1	no	—
36	#14555511	—	2	1	no	—
37	#14555511	C₈	1	2	yes	—
38	#14555511	C₂×C₄	1	2	yes	—
39	C₂×D₅₄₈	—	2	1	no	—
40	C₂×D₅₄₈	C₈	1	1	yes	characteristic
41	C₂×C₄×D₁₃₇	—	2	1	no	—
42	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₃₇	C₈	1	1	yes	characteristic
44	#11764047	—	2	2	no	—
45	#11764047	C₂²⋊C₄	1	1	yes	characteristic
46	#11764047	C₂²⋊C₄	1	1	yes	characteristic
47	#11764047	C₂×D₄	1	1	yes	characteristic
48	#11764047	C₂×D₄	1	1	yes	characteristic
49	C₂²×D₁₃₇	—	2	1	no	—
50	C₂²×D₁₃₇	M₁₆	1	1	yes	characteristic
51	D₅₄₈	—	2	2	no	—
52	D₅₄₈	—	2	2	no	—
53	D₅₄₈	—	2	1	no	—
54	D₅₄₈	C₂×C₈	1	2	yes	—
55	C₂×Dic₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₄×D₁₃₇	—	2	2	no	—
57	C₄×D₁₃₇	—	2	2	no	—
58	C₄×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₄×D₁₃₇	C₂×C₈	1	1	yes	characteristic
60	C₄×D₁₃₇	M₁₆	1	1	yes	characteristic
61	C₄×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
62	Q₈×C₁₃₇	—	2	1	no	—
63	Q₈×C₁₃₇	C₂×C₈	1	2	yes	—
64	C₂×C₄×C₁₃₇	—	2	1	no	—
65	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
67	C₂×D₁₃₇	—	2	2	no	—
68	C₂×D₁₃₇	—	4	1	no	—
69	C₂×D₁₃₇	—	4	2	no	—
70	C₂×D₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
71	C₂×D₁₃₇	C₂×M₁₆	1	1	yes	characteristic
72	C₄×C₁₃₇	—	2	2	no	—
73	C₄×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
74	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
75	Dic₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
76	Dic₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₁₃₇	—	2	1	no	—
78	C₂×C₁₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
79	D₁₃₇	—	2	1	no	—
80	D₁₃₇	—	2	1	no	—
81	D₁₃₇	—	4	2	no	—
82	C₁₃₇	#2117	1	1	yes	characteristic, complemented, Sylow
83	#2117	—	137	1	no	maximal, Sylow
84	C₂×#131	—	137	1	no	—
85	C₂×M₃₂	—	137	1	no	—
86	C₂×M₃₂	—	137	1	no	—
87	#350	—	137	4	no	—
88	C₂×#41	—	137	1	no	—
89	#131	—	274	1	no	—
90	#131	—	274	1	no	—
91	#131	—	137	2	no	—
92	#131	—	137	2	no	—
93	C₂×M₁₆	—	274	1	no	—
94	C₂×M₁₆	—	137	1	no	—
95	C₂²×C₈	—	274	1	no	—
96	C₂²×C₈	—	137	1	no	—
97	M₃₂	—	274	1	no	—
98	M₃₂	—	274	1	no	—
99	M₃₂	—	274	1	no	—
100	M₃₂	—	137	2	no	—
101	C₂×C₁₆	—	274	1	no	—
102	C₂×C₁₆	—	137	2	no	—
103	#41	—	274	1	no	—
104	#41	—	274	1	no	—
105	#41	—	137	2	no	—
106	#41	—	137	2	no	—
107	C₂×Q₈	#11764049	1	1	yes	characteristic
108	C₂×D₄	—	274	1	no	—
109	C₂×D₄	—	137	1	no	—
110	C₂²×C₄	—	274	1	no	—
111	C₂²×C₄	—	137	1	no	—
112	M₁₆	—	274	1	no	—
113	M₁₆	—	274	2	no	—
114	M₁₆	—	274	2	no	—
115	M₁₆	—	137	2	no	—
116	C₂×C₈	—	274	1	no	—
117	C₂×C₈	—	274	2	no	—
118	C₂×C₈	—	274	2	no	—
119	C₂×C₈	—	137	1	no	—
120	C₂×C₈	—	137	1	no	—
121	C₂×C₈	—	137	1	no	—
122	C₂×C₈	—	137	1	no	—
123	C₂×C₈	—	137	1	no	—
124	C₂×C₈	—	137	1	no	—
125	C₁₆	—	274	2	no	—
126	C₁₆	—	274	2	no	—
127	C₂³	—	274	1	no	—
128	C₂³	—	137	1	no	—
129	Q₈	—	2	1	no	—
130	Q₈	C₂×#11764049	1	2	yes	—
131	D₄	—	274	2	no	—
132	D₄	—	274	2	no	—
133	D₄	—	274	1	no	—
134	D₄	—	137	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	274	2	no	—
137	C₂×C₄	—	274	2	no	—
138	C₂×C₄	—	137	1	no	—
139	C₂×C₄	—	137	1	no	—
140	C₂×C₄	—	137	1	no	—
141	C₂×C₄	—	137	1	no	—
142	C₂×C₄	—	137	1	no	—
143	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
144	C₈	—	274	2	no	—
145	C₈	—	137	1	no	—
146	C₈	—	137	1	no	—
147	C₈	—	137	1	no	—
148	C₈	—	137	1	no	—
149	C₂²	—	274	2	no	—
150	C₂²	—	548	1	no	—
151	C₂²	—	548	2	no	—
152	C₂²	—	137	1	no	—
153	C₂²	—	137	1	no	—
154	C₂²	C₂²⋊#11764049	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	—	137	1	no	—
157	C₄	—	137	1	no	—
158	C₄	C₂²⋊#11764049	1	1	yes	characteristic
159	C₄	C₂²×#11764049	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	274	1	no	—
162	C₂	—	274	1	no	—
163	C₂	—	548	2	no	—
164	C₂	C₂×C₂²⋊#11764049	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#11764049
commutator	C₄×C₁₃₇	C₂²×C₈
Frattini	C₄	C₂²×#11764049
Fitting	C₂×Q₈×C₁₃₇	C₈
socle	C₂×C₁₃₇	C₂×C₂²⋊C₈
prime radical	C₂×#14555511	C₄
cosocle	#14555518	C₂³
hypercenter	C₂×Q₈	#11764049
hypocenter	C₁₃₇	#2117
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #22578522	C₂	semidirect
2. #22578522	C₂	semidirect
3. #22578504	C₂	semidirect
6. #17984185	C₂²	semidirect
10. #17984183	C₂²	semidirect
82. C₁₃₇	#2117	semidirect

Maximal subgroups

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

subgroup	index
#22578522	2
#22578504	2
C₂×#17984176	2
#2117	137