Finite Group Databasebeta

Subgroups — #31796010

Order 17536.

178 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 78 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	#22578420	C₂	1	1	yes	maximal, characteristic
2	#22578419	C₂	1	1	yes	maximal, characteristic
3	#22578245	C₂	1	1	yes	maximal, characteristic, complemented
4	#22578245	C₂	1	1	yes	maximal, characteristic, complemented
5	#22578244	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#11764047	C₂	1	1	yes	maximal, characteristic, complemented
7	#22577453	C₂	1	1	yes	maximal, characteristic
8	#17984155	C₄	1	2	yes	—
9	C₄⋊#11764047	C₄	1	2	yes	—
10	C₂²⋊#11764047	C₄	1	2	yes	—
11	C₂²⋊#11764047	C₄	1	2	yes	—
12	C₄×#11764047	C₂²	1	1	yes	characteristic
13	C₄×#11764047	C₂²	1	1	yes	characteristic
14	C₄×#11764047	C₂²	1	1	yes	characteristic
15	C₄×#11764047	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
17	#17984017	C₄	1	1	yes	characteristic
18	#17984017	C₄	1	1	yes	characteristic
19	C₂²⋊#11764040	C₄	1	1	yes	characteristic
20	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
21	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
22	C₂²⋊C₈×C₁₃₇	C₄	1	1	yes	characteristic
23	C₂×#11764047	C₂×C₄	1	2	yes	—
24	C₂×#11764047	C₂×C₄	1	2	yes	—
25	C₂×#11764047	C₂×C₄	1	2	yes	—
26	C₂×#11764047	C₂×C₄	1	2	yes	—
27	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
30	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
38	#11764047	—	2	2	no	—
39	#11764047	—	2	2	no	—
40	#11764047	—	2	2	no	—
41	#11764047	—	2	2	no	—
42	C₂×Dic₁₃₇	—	2	2	no	—
43	C₂×Dic₁₃₇	—	2	2	no	—
44	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
45	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
46	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
48	#11764040	—	2	1	no	—
49	#11764040	—	2	1	no	—
50	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
51	C₂×C₄×C₁₃₇	—	2	1	no	—
52	C₂×C₄×C₁₃₇	—	2	1	no	—
53	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
54	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₈×C₁₃₇	—	2	1	no	—
56	C₈×C₁₃₇	—	2	1	no	—
57	C₂²×C₁₃₇	—	2	1	no	—
58	C₂²×C₁₃₇	—	2	1	no	—
59	C₂²×C₁₃₇	—	2	1	no	—
60	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
61	C₄×C₁₃₇	—	2	1	no	—
62	C₄×C₁₃₇	#131	1	1	yes	characteristic
63	C₄×C₁₃₇	#131	1	1	yes	characteristic
64	Dic₁₃₇	—	2	1	no	—
65	Dic₁₃₇	—	2	1	no	—
66	Dic₁₃₇	#131	1	2	yes	—
67	Dic₁₃₇	#131	1	2	yes	—
68	C₂×C₁₃₇	—	4	1	no	—
69	C₂×C₁₃₇	#405	1	1	yes	characteristic
70	C₂×C₁₃₇	#405	1	1	yes	characteristic
71	C₂×C₁₃₇	#405	1	1	yes	characteristic
72	C₁₃₇	#1454	1	1	yes	characteristic, complemented, Sylow
73	#1454	—	137	1	no	maximal, Sylow
74	#431	—	137	1	no	—
75	#431	—	137	1	no	—
76	#431	—	137	1	no	—
77	#322	—	137	1	no	—
78	#322	—	137	1	no	—
79	#322	—	137	1	no	—
80	C₈²	—	137	1	no	—
81	C₂×C₄²	—	137	1	no	—
82	C₄⋊C₈	—	137	2	no	—
83	C₄⋊C₈	—	137	2	no	—
84	C₄⋊C₈	—	137	1	no	—
85	C₄⋊C₈	—	137	1	no	—
86	C₂²⋊C₈	—	137	2	no	—
87	C₂²⋊C₈	—	137	2	no	—
88	C₂²⋊C₈	—	137	1	no	—
89	C₂²⋊C₈	#10587739	1	1	yes	characteristic
90	C₄×C₈	—	137	1	no	—
91	C₄×C₈	—	137	1	no	—
92	C₄×C₈	—	137	1	no	—
93	C₄×C₈	—	137	1	no	—
94	C₄×C₈	—	137	1	no	—
95	C₄×C₈	—	137	1	no	—
96	C₂²×C₄	—	137	1	no	—
97	C₂²×C₄	—	137	1	no	—
98	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
99	C₂×C₈	—	137	2	no	—
100	C₂×C₈	—	137	2	no	—
101	C₂×C₈	—	137	2	no	—
102	C₂×C₈	—	137	2	no	—
103	C₂×C₈	—	137	1	no	—
104	C₂×C₈	—	137	1	no	—
105	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
106	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
107	C₄²	—	137	1	no	—
108	C₄²	—	137	1	no	—
109	C₄²	—	137	1	no	—
110	C₄²	—	137	1	no	—
111	C₂³	C₄×#10587739	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	274	2	no	—
115	C₂×C₄	—	274	2	no	—
116	C₂×C₄	—	137	1	no	—
117	C₂×C₄	—	137	1	no	—
118	C₂×C₄	—	137	1	no	—
119	C₂×C₄	—	137	1	no	—
120	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
121	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
122	C₈	—	2	1	no	—
123	C₈	—	2	1	no	—
124	C₈	—	274	2	no	—
125	C₈	—	274	2	no	—
126	C₈	—	274	2	no	—
127	C₈	—	274	2	no	—
128	C₈	—	274	1	no	—
129	C₈	—	274	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	274	1	no	—
136	C₄	—	274	1	no	—
137	C₄	—	137	2	no	—
138	C₄	—	137	2	no	—
139	C₄	#17984175	1	1	yes	characteristic
140	C₄	#17984175	1	1	yes	characteristic
141	C₂	—	4	1	no	—
142	C₂	#22578366	1	1	yes	characteristic
143	C₂	#22578367	1	1	yes	characteristic
144	C₂	#22578369	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂×C₄²
Fitting	C₂²⋊C₈×C₁₃₇	C₄
prime radical	C₂³×C₁₃₇	C₄²
cosocle	C₄×Dic₁₃₇	C₂³
hypercenter	C₂²⋊C₈	#10587739
hypocenter	C₁₃₇	#1454
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. #22578245	C₂	semidirect
4. #22578245	C₂	semidirect
5. #22578244	C₂	semidirect
6. C₈×#11764047	C₂	semidirect
72. C₁₃₇	#1454	semidirect

Maximal subgroups

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

subgroup	index
#22578420	2
#22578419	2
#22578245	2
#22578244	2
C₈×#11764047	2
#22577453	2
#1454	137