Subgroups — #31797010

Order 17536.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984162	C₂	1	2	yes	maximal
2	#22578295	C₂	1	4	yes	maximal, complemented
3	#22577359	C₂	1	1	yes	maximal, characteristic
4	#17984162	C₄	1	4	yes	complemented
5	#17984162	C₂²	1	4	yes	—
6	C₂×#14555513	C₄	1	4	yes	complemented
7	C₂×#14555494	C₄	1	1	yes	characteristic
8	C₂²×#11764040	C₂²	1	1	yes	characteristic
9	#17983999	C₂²	1	2	yes	—
10	C₄×#11764040	C₄	1	2	yes	—
11	#117×C₁₃₇	C₄	1	1	yes	characteristic
12	#14555513	C₂×C₄	1	8	yes	—
13	#14555494	—	2	2	no	—
14	C₂×#11764040	C₂×C₄	1	2	yes	—
15	C₂×#11764040	C₂×C₄	1	2	yes	—
16	C₂×#11764040	C₂×C₄	1	2	yes	—
17	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
18	C₂×#11764040	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₄×C₁₃₇	—	2	1	no	—
21	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
22	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
23	#11764040	—	2	2	no	—
24	#11764040	C₄²	1	1	yes	characteristic
25	#11764040	C₄²	1	1	yes	characteristic
26	#11764040	C₂²×C₄	1	2	yes	—
27	C₂³×C₁₃₇	M₁₆	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	—	2	2	no	—
29	C₂×C₄×C₁₃₇	C₄²	1	2	yes	—
30	C₂×C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	M₁₆	1	2	yes	—
32	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₁₃₇	—	4	1	no	—
34	C₂²×C₁₃₇	#97	1	2	yes	—
35	C₂²×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
36	C₄×C₁₃₇	—	4	2	no	—
37	C₄×C₁₃₇	#97	1	2	yes	—
38	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
39	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
40	C₂×C₁₃₇	—	2	1	no	—
41	C₂×C₁₃₇	—	2	2	no	—
42	C₂×C₁₃₇	C₂×#97	1	1	yes	characteristic
43	C₁₃₇	#2110	1	1	yes	characteristic, complemented, Sylow
44	#2110	—	137	1	no	maximal, Sylow
45	C₂×M₃₂	—	137	2	no	—
46	#405	—	137	1	no	—
47	#347	—	137	4	no	—
48	C₂×M₁₆	—	137	1	no	—
49	C₂²×C₈	—	137	1	no	—
50	#117	#10587739	1	1	yes	characteristic
51	M₃₂	—	137	4	no	—
52	M₃₂	—	137	4	no	—
53	C₂×C₁₆	—	137	4	no	—
54	#97	—	137	2	no	—
55	C₄×C₈	—	137	2	no	—
56	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
57	M₁₆	—	274	2	no	—
58	C₂×C₈	—	137	2	no	—
59	C₂×C₈	—	137	2	no	—
60	C₂×C₈	—	137	2	no	—
61	C₂×C₈	—	137	1	no	—
62	C₂×C₈	—	137	1	no	—
63	C₄⋊C₄	—	2	1	no	—
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	C₂×#10587739	1	2	yes	—
66	C₁₆	—	137	8	no	—
67	C₂³	#14555519	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	C₄×#10587739	1	2	yes	—
70	C₂×C₄	#14555519	1	1	yes	characteristic
71	C₂×C₄	#14555520	1	2	yes	—
72	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
73	C₈	—	274	2	no	—
74	C₈	—	137	2	no	—
75	C₈	—	137	1	no	—
76	C₈	—	137	1	no	—
77	C₂²	—	4	1	no	—
78	C₂²	#17984143	1	2	yes	—
79	C₂²	C₂×#14555519	1	1	yes	characteristic
80	C₄	—	4	2	no	—
81	C₄	#17984143	1	2	yes	—
82	C₄	C₂×C₄×#10587739	1	1	yes	characteristic
83	C₄	C₂×#14555520	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×#17984143	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₄×#10587739
commutator	C₄×C₁₃₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#10587739
Fitting	#117×C₁₃₇	C₄
socle	C₂×C₁₃₇	C₂×#97
prime radical	C₂³×C₁₃₇	M₁₆
cosocle	C₂×#11764040	C₂³
hypercenter	#117	#10587739
hypocenter	C₁₃₇	#2110
LCS	G ▷ C₄×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. #22578295	C₂	semidirect
4. #17984162	C₄	semidirect
6. C₂×#14555513	C₄	semidirect
43. C₁₃₇	#2110	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984162	2
#22578295	2
#22577359	2
#2110	137