Finite Group Databasebeta

Subgroups — #31795996

Order 17536.

208 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 98 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	#22578419	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11764047	C₂	1	1	yes	maximal, characteristic
3	#22578245	C₂	1	2	yes	maximal, complemented
4	#22578245	C₂	1	2	yes	maximal, complemented
5	M₁₆×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11764047	C₄	1	2	yes	—
7	C₄⋊#11764047	C₄	1	2	yes	—
8	C₂²⋊#11764047	C₄	1	2	yes	—
9	C₄×#11764047	C₄	1	2	yes	—
10	C₄×#11764047	C₂²	1	1	yes	characteristic
11	C₄×#11764047	C₂²	1	1	yes	characteristic
12	C₄×#11764047	C₂²	1	2	yes	—
13	C₂×#14555494	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₁₃₇	C₂²	1	1	yes	characteristic
15	#17983998	C₄	1	2	yes	—
16	C₈×Dic₁₃₇	C₂²	1	2	yes	—
17	C₂×M₁₆×C₁₃₇	C₄	1	1	yes	characteristic
18	C₂×#11764047	—	2	4	no	—
19	C₂×#11764047	C₂×C₄	1	2	yes	—
20	C₂×#11764047	C₂×C₄	1	4	yes	—
21	C₂×#11764047	C₂×C₄	1	2	yes	—
22	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
24	#14555494	—	2	2	no	—
25	C₂×#11764040	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₁₃₇	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
31	M₁₆×C₁₃₇	—	2	2	no	—
32	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
33	#11764047	—	2	2	no	—
34	#11764047	—	2	4	no	—
35	#11764047	M₁₆	1	4	yes	complemented
36	C₂×Dic₁₃₇	—	2	2	no	—
37	C₂×Dic₁₃₇	—	2	2	no	—
38	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
39	C₂×Dic₁₃₇	C₄²	1	1	yes	characteristic
40	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
42	#11764040	—	2	2	no	—
43	C₂³×C₁₃₇	C₄²	1	1	yes	characteristic
44	C₂×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
45	C₂×C₄×C₁₃₇	M₁₆	1	2	yes	—
46	C₂×C₄×C₁₃₇	M₁₆	1	2	yes	—
47	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
48	C₈×C₁₃₇	—	2	2	no	—
49	C₂²×C₁₃₇	—	2	1	no	—
50	C₂²×C₁₃₇	—	2	1	no	—
51	C₂²×C₁₃₇	#97	1	2	yes	—
52	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
53	C₄×C₁₃₇	#97	1	2	yes	—
54	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
55	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
56	Dic₁₃₇	—	2	1	no	—
57	Dic₁₃₇	—	2	1	no	—
58	Dic₁₃₇	C₂×M₁₆	1	2	yes	—
59	Dic₁₃₇	#131	1	2	yes	—
60	C₂×C₁₃₇	—	2	2	no	—
61	C₂×C₁₃₇	C₂×#97	1	1	yes	characteristic
62	C₂×C₁₃₇	C₄×M₁₆	1	1	yes	characteristic
63	C₂×C₁₃₇	#405	1	1	yes	characteristic
64	C₁₃₇	#1451	1	1	yes	characteristic, complemented, Sylow
65	#1451	—	137	1	no	maximal, Sylow
66	#431	—	137	1	no	—
67	C₄×M₁₆	—	137	1	no	—
68	C₂×C₄×C₈	—	137	1	no	—
69	#322	—	137	2	no	—
70	#322	—	137	2	no	—
71	C₂×M₁₆	—	137	1	no	—
72	C₂×M₁₆	#10587739	1	1	yes	characteristic
73	C₂²×C₈	—	137	2	no	—
74	C₂×C₄²	—	137	1	no	—
75	C₄⋊C₈	—	137	2	no	—
76	C₂²⋊C₈	—	137	2	no	—
77	#97	—	137	2	no	—
78	C₄×C₈	—	137	1	no	—
79	C₄×C₈	—	137	1	no	—
80	C₄×C₈	—	137	2	no	—
81	C₄×C₈	—	137	2	no	—
82	C₄×C₈	—	137	2	no	—
83	C₂²×C₄	—	137	1	no	—
84	C₂²×C₄	—	137	1	no	—
85	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	274	2	no	—
88	C₂×C₈	—	274	4	no	—
89	C₂×C₈	—	137	2	no	—
90	C₂×C₈	—	137	4	no	—
91	C₂×C₈	—	137	2	no	—
92	C₂×C₈	—	137	2	no	—
93	C₂×C₈	C₂×#10587739	1	2	yes	—
94	C₄²	—	137	1	no	—
95	C₄²	—	137	1	no	—
96	C₄²	—	137	1	no	—
97	C₄²	—	137	1	no	—
98	C₂³	C₄×#10587739	1	1	yes	characteristic
99	C₂×C₄	—	274	2	no	—
100	C₂×C₄	—	274	2	no	—
101	C₂×C₄	—	137	1	no	—
102	C₂×C₄	—	137	1	no	—
103	C₂×C₄	—	137	1	no	—
104	C₂×C₄	—	137	1	no	—
105	C₂×C₄	C₄×#10587739	1	1	yes	characteristic
106	C₂×C₄	#14555519	1	2	yes	—
107	C₂×C₄	#14555520	1	2	yes	—
108	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	274	2	no	—
111	C₈	—	274	4	no	—
112	C₈	—	274	2	no	—
113	C₈	—	137	4	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#17984143	1	2	yes	—
117	C₂²	C₂×C₄×#10587739	1	1	yes	characteristic
118	C₄	—	274	1	no	—
119	C₄	—	274	1	no	—
120	C₄	—	137	2	no	—
121	C₄	—	137	2	no	—
122	C₄	#17984143	1	2	yes	—
123	C₄	C₂×#14555519	1	1	yes	characteristic
124	C₄	C₂×#14555520	1	1	yes	characteristic
125	C₂	—	2	2	no	—
126	C₂	C₂×#17984143	1	1	yes	characteristic
127	C₂	M₁₆×#10587739	1	1	yes	characteristic
128	C₂	#22578369	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂×C₄²
Fitting	C₂×M₁₆×C₁₃₇	C₄
prime radical	C₂³×C₁₃₇	C₄²
cosocle	C₄×Dic₁₃₇	C₂³
hypercenter	C₂×M₁₆	#10587739
hypocenter	C₁₃₇	#1451
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22578245	C₂	semidirect
4. #22578245	C₂	semidirect
35. #11764047	M₁₆	semidirect
64. C₁₃₇	#1451	semidirect

Maximal subgroups

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

subgroup	index
#22578419	2
C₂×C₄×#11764047	2
#22578245	2
M₁₆×Dic₁₃₇	2
#1451	137