Finite Group Databasebeta

Subgroups — #31797104

Order 17536.

220 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 62 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	#22578459	C₂	1	1	yes	maximal, characteristic, complemented
2	D₁₃₇⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
3	#22578317	C₂	1	4	yes	maximal, complemented
4	C₂×#17984107	C₂	1	1	yes	maximal, characteristic
5	D₁₃₇⋊M₁₆	—	2	1	no	—
6	#17984163	—	2	1	no	—
7	#17984162	—	2	1	no	—
8	#17984161	C₂²	1	2	yes	complemented
9	#17984159	C₂²	1	2	yes	complemented
10	#17984158	—	2	1	no	—
11	C₂×#14555509	—	2	1	no	—
12	#17984107	—	2	1	no	—
13	#17984107	—	2	1	no	—
14	#17984107	C₄	1	2	yes	—
15	#17984107	C₂²	1	2	yes	—
16	C₂×Dic₅₄₈	C₄	1	1	yes	characteristic
17	C₂×#14555498	—	2	1	no	—
18	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₁₃₇	C₄	1	1	yes	characteristic
20	#14555520	—	2	2	no	—
21	#14555519	—	4	1	no	—
22	#14555518	—	4	1	no	—
23	#14555513	—	2	2	no	—
24	#14555509	—	2	2	no	—
25	#14555509	—	2	2	no	—
26	#14555509	—	4	1	no	—
27	#14555509	—	4	1	no	—
28	C₂×Dic₂₇₄	—	2	1	no	—
29	C₂×C₂²⋊D₁₃₇	—	4	1	no	—
30	C₂²×Dic₁₃₇	—	4	1	no	—
31	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
32	Dic₅₄₈	—	2	1	no	—
33	Dic₅₄₈	C₂×C₄	1	2	yes	—
34	#14555498	—	2	2	no	—
35	#14555498	—	2	2	no	—
36	C₂×#11764040	D₄	1	1	yes	characteristic
37	C₈×D₁₃₇	C₂×C₄	1	1	yes	characteristic
38	C₈×D₁₃₇	D₄	1	1	yes	characteristic
39	C₈×D₁₃₇	D₄	1	1	yes	characteristic
40	C₈×D₁₃₇	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₁₃₇	—	2	1	no	—
42	D₈×C₁₃₇	—	2	1	no	—
43	D₈×C₁₃₇	C₂×C₄	1	2	yes	—
44	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
45	#11764047	—	4	2	no	—
46	C₂²×D₁₃₇	SD₁₆	1	1	yes	characteristic
47	Dic₂₇₄	—	2	2	no	—
48	Dic₂₇₄	—	4	1	no	—
49	C₂²⋊D₁₃₇	—	4	2	no	—
50	C₂²⋊D₁₃₇	—	4	2	no	—
51	C₂×Dic₁₃₇	—	4	1	no	—
52	C₂×Dic₁₃₇	—	4	2	no	—
53	C₂×Dic₁₃₇	—	4	2	no	—
54	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
55	C₄×D₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₄×D₁₃₇	D₈	1	1	yes	characteristic
57	C₄×D₁₃₇	SD₁₆	1	1	yes	characteristic
58	C₄×D₁₃₇	C₂×D₄	1	1	yes	characteristic
59	#11764040	C₂²⋊C₄	1	1	yes	characteristic
60	#11764040	C₂×D₄	1	1	yes	characteristic
61	C₂³×C₁₃₇	—	4	1	no	—
62	D₄×C₁₃₇	—	2	2	no	—
63	D₄×C₁₃₇	—	4	1	no	—
64	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₁₃₇	—	4	2	no	—
68	C₂²×C₁₃₇	—	8	2	no	—
69	C₂²×C₁₃₇	#102	1	1	yes	characteristic
70	C₂×D₁₃₇	—	4	1	no	—
71	C₂×D₁₃₇	#102	1	1	yes	characteristic
72	C₂×D₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₁₃₇	#102	1	1	yes	characteristic
74	C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	Dic₁₃₇	—	4	2	no	—
76	Dic₁₃₇	#102	1	1	yes	characteristic
77	Dic₁₃₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₁₃₇	—	2	1	no	—
79	C₂×C₁₃₇	—	8	2	no	—
80	C₂×C₁₃₇	C₂×#102	1	1	yes	characteristic
81	D₁₃₇	—	2	1	no	—
82	D₁₃₇	—	2	1	no	—
83	C₁₃₇	#2143	1	1	yes	characteristic, complemented, Sylow
84	#2143	—	137	1	no	maximal, Sylow
85	C₂×#135	—	137	1	no	—
86	C₂×M₃₂	—	137	1	no	—
87	C₂×#108	—	137	1	no	—
88	#359	—	137	4	no	—
89	C₂×#41	—	274	1	no	—
90	#135	—	274	1	no	—
91	#135	—	274	1	no	—
92	#135	—	137	2	no	—
93	#135	—	137	2	no	—
94	C₂×Q₁₆	—	137	1	no	—
95	C₂×SD₁₆	—	274	1	no	—
96	C₂×D₈	#10587739	1	1	yes	characteristic
97	C₂×M₁₆	—	274	1	no	—
98	C₂²×C₈	—	137	1	no	—
99	M₃₂	—	274	1	no	—
100	M₃₂	—	274	1	no	—
101	C₂×C₁₆	—	137	2	no	—
102	#108	—	274	1	no	—
103	#108	—	137	2	no	—
104	#41	—	274	2	no	—
105	#41	—	274	2	no	—
106	#41	—	548	1	no	—
107	#41	—	548	1	no	—
108	C₂×Q₈	—	274	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	548	1	no	—
111	C₂²×C₄	—	548	1	no	—
112	C₂²×C₄	—	137	1	no	—
113	Q₁₆	—	274	1	no	—
114	Q₁₆	—	137	2	no	—
115	SD₁₆	—	274	2	no	—
116	SD₁₆	—	274	2	no	—
117	D₈	—	2	1	no	—
118	D₈	C₂×#10587739	1	2	yes	—
119	M₁₆	—	274	2	no	—
120	M₁₆	—	548	1	no	—
121	C₂×C₈	—	548	1	no	—
122	C₂×C₈	—	137	1	no	—
123	C₂×C₈	—	137	1	no	—
124	C₂×C₈	—	137	1	no	—
125	C₂×C₈	—	137	1	no	—
126	C₂×C₈	—	137	1	no	—
127	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
128	C₁₆	—	274	2	no	—
129	C₂³	—	4	1	no	—
130	C₂³	—	137	1	no	—
131	Q₈	—	274	2	no	—
132	Q₈	—	548	1	no	—
133	D₄	—	2	2	no	—
134	D₄	—	4	1	no	—
135	D₄	—	548	2	no	—
136	D₄	—	548	2	no	—
137	C₂×C₄	—	548	1	no	—
138	C₂×C₄	—	548	2	no	—
139	C₂×C₄	—	548	2	no	—
140	C₂×C₄	—	137	1	no	—
141	C₂×C₄	—	137	1	no	—
142	C₂×C₄	—	137	1	no	—
143	C₂×C₄	—	137	1	no	—
144	C₂×C₄	—	137	1	no	—
145	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
146	C₈	—	548	2	no	—
147	C₈	—	137	1	no	—
148	C₈	—	137	1	no	—
149	C₈	C₂²⋊#10587739	1	1	yes	characteristic
150	C₈	C₂²×#10587739	1	1	yes	characteristic
151	C₂²	—	4	2	no	—
152	C₂²	—	8	2	no	—
153	C₂²	—	548	1	no	—
154	C₂²	—	137	1	no	—
155	C₂²	—	137	1	no	—
156	C₂²	#17984150	1	1	yes	characteristic
157	C₄	—	548	2	no	—
158	C₄	—	137	1	no	—
159	C₄	—	137	1	no	—
160	C₄	#17984150	1	1	yes	characteristic
161	C₄	C₂×C₂²⋊#10587739	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	8	2	no	—
164	C₂	—	274	1	no	—
165	C₂	—	274	1	no	—
166	C₂	C₂×#17984150	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#17984150
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₈	C₂²×#10587739
Fitting	C₂×D₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂×#102
prime radical	C₂×#17984107	C₂
cosocle	C₈×D₁₃₇	C₂³
hypercenter	C₂×D₈	#10587739
hypocenter	C₁₃₇	#2143
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #22578459	C₂	semidirect
2. D₁₃₇⋊#108	C₂	semidirect
3. #22578317	C₂	semidirect
8. #17984161	C₂²	semidirect
9. #17984159	C₂²	semidirect
83. C₁₃₇	#2143	semidirect

Maximal subgroups

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

subgroup	index
#22578459	2
D₁₃₇⋊#108	2
#22578317	2
C₂×#17984107	2
#2143	137