Subgroups — #31797124

Order 17536.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984162	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14555513	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#14555513	C₂	1	4	yes	maximal, complemented
4	#22577359	C₂	1	1	yes	maximal, characteristic
5	#17984162	—	2	2	no	—
6	C₂×#14555513	—	2	2	no	—
7	C₂×#14555513	C₂²	1	2	yes	—
8	C₂×#14555513	C₂²	1	2	yes	—
9	C₂×#14555494	C₄	1	1	yes	characteristic
10	C₂²×#11764040	C₂²	1	1	yes	characteristic
11	#17983999	C₄	1	2	yes	—
12	C₄×#11764040	C₂²	1	2	yes	—
13	#117×C₁₃₇	C₄	1	1	yes	characteristic
14	#14555513	—	2	2	no	—
15	#14555513	—	2	2	no	—
16	#14555494	—	2	2	no	—
17	C₂×#11764040	C₂×C₄	1	2	yes	—
18	C₂×#11764040	C₂×C₄	1	1	yes	characteristic
19	C₂×#11764040	D₄	1	2	yes	—
20	C₂×#11764040	Q₈	1	2	yes	—
21	C₂×#11764040	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₁₃₇	C₈	1	2	yes	—
24	C₂²⋊C₄×C₁₃₇	C₈	1	2	yes	—
25	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
26	#11764040	—	2	2	no	—
27	#11764040	C₄⋊C₄	1	2	yes	—
28	#11764040	C₂×D₄	1	1	yes	characteristic
29	#11764040	C₂×Q₈	1	1	yes	characteristic
30	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₃₇	C₂×C₈	1	4	yes	—
33	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	M₁₆	1	2	yes	—
35	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₁₃₇	—	2	2	no	—
37	C₂²×C₁₃₇	C₄⋊C₈	1	2	yes	—
38	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₁₃₇	—	2	4	no	—
40	C₄×C₁₃₇	C₄⋊C₈	1	2	yes	—
41	C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₁₃₇	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₁₃₇	—	2	2	no	—
44	C₂×C₁₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₃₇	#504	1	2	yes	—
46	C₁₃₇	#2151	1	1	yes	characteristic, complemented, Sylow
47	#2151	—	137	1	no	maximal, Sylow
48	C₂×M₃₂	—	137	1	no	—
49	C₂²×C₁₆	—	137	1	no	—
50	#405	—	137	1	no	—
51	C₄⋊C₁₆	—	137	4	no	—
52	C₂×M₁₆	—	137	1	no	—
53	C₂²×C₈	—	137	1	no	—
54	#117	#10587739	1	1	yes	characteristic
55	M₃₂	—	274	2	no	—
56	C₂×C₁₆	—	274	2	no	—
57	C₂×C₁₆	—	137	2	no	—
58	C₂×C₁₆	—	137	2	no	—
59	#97	—	137	2	no	—
60	C₄×C₈	—	137	2	no	—
61	C₂²×C₄	C₂×#10587739	1	1	yes	characteristic
62	M₁₆	—	274	2	no	—
63	C₂×C₈	—	137	2	no	—
64	C₂×C₈	—	137	2	no	—
65	C₂×C₈	—	137	2	no	—
66	C₂×C₈	—	137	1	no	—
67	C₂×C₈	—	137	1	no	—
68	C₄⋊C₄	#11764047	1	2	yes	—
69	C₂²⋊C₄	#11764047	1	2	yes	—
70	C₄²	C₂×#10587739	1	2	yes	—
71	C₁₆	—	274	2	no	—
72	C₁₆	—	274	2	no	—
73	C₂³	C₂×#11764047	1	1	yes	characteristic
74	C₂×C₄	C₄⋊#10587739	1	2	yes	—
75	C₂×C₄	C₂×#11764047	1	4	yes	—
76	C₂×C₄	C₂×#11764047	1	1	yes	characteristic
77	C₂×C₄	#14555520	1	2	yes	—
78	C₂×C₄	C₂²×#10587739	1	1	yes	characteristic
79	C₈	—	274	2	no	—
80	C₈	—	137	1	no	—
81	C₈	—	137	2	no	—
82	C₈	—	137	1	no	—
83	C₂²	—	2	2	no	—
84	C₂²	C₄⋊#11764047	1	2	yes	—
85	C₂²	C₂²×#11764047	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	C₄⋊#11764047	1	2	yes	—
88	C₄	C₂×C₄⋊#10587739	1	1	yes	characteristic
89	C₄	C₂×#14555520	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊#11764047	1	1	yes	characteristic
92	C₂	#22578461	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10587739
commutator, socle	C₂²×C₁₃₇	C₂²×C₈
Fitting	#117×C₁₃₇	C₄
prime radical	C₂³×C₁₃₇	C₂×C₈
cosocle	C₂×#11764040	C₂³
hypercenter	#117	#10587739
hypocenter	C₁₃₇	#2151
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. C₄⋊#14555513	C₂	semidirect
46. C₁₃₇	#2151	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
C₂²×#14555513	2
C₄⋊#14555513	2
#22577359	2
#2151	137