Subgroups — #31795916

Order 17536.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578312	C₂	1	1	yes	maximal, characteristic, complemented
2	#22578310	C₂	1	1	yes	maximal, characteristic, complemented
3	#22577790	C₂	1	1	yes	maximal, characteristic
4	#17984149	—	2	1	no	—
5	#17984147	—	2	1	no	—
6	C₂×#14555498	—	2	1	no	—
7	C₂×#14555495	—	2	1	no	—
8	#17984076	C₂²	1	1	yes	characteristic
9	#17983999	C₄	1	1	yes	characteristic, complemented
10	#127×C₁₃₇	C₄	1	1	yes	characteristic, complemented
11	#14555519	—	4	1	no	—
12	C₂²⋊#10587739	—	4	1	no	—
13	C₂×Dic₂₇₄	D₄	1	1	yes	characteristic
14	C₂×D₅₄₈	D₄	1	1	yes	characteristic
15	#14555498	—	4	2	no	—
16	#14555495	—	4	2	no	—
17	C₂×#11764040	—	2	1	no	—
18	#14555487	—	4	1	no	—
19	C₂×D₄×C₁₃₇	—	2	1	no	—
20	C₂×D₄×C₁₃₇	—	4	1	no	—
21	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×#10587739	—	8	1	no	—
23	#11764047	—	8	1	no	—
24	C₂²×D₁₃₇	—	2	1	no	—
25	Dic₂₇₄	—	4	1	no	—
26	D₅₄₈	—	4	1	no	—
27	C₂×Dic₁₃₇	—	2	1	no	—
28	#11764040	—	2	2	no	—
29	C₂³×C₁₃₇	—	4	1	no	—
30	D₄×C₁₃₇	—	4	1	no	—
31	D₄×C₁₃₇	—	4	1	no	—
32	D₄×C₁₃₇	—	8	2	no	—
33	C₂×C₄×C₁₃₇	—	2	1	no	—
34	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₁₃₇	—	8	2	no	—
36	C₂²×C₁₃₇	—	8	1	no	—
37	C₂²×C₁₃₇	#99	1	1	yes	characteristic
38	C₂×D₁₃₇	—	4	1	no	—
39	C₂×D₁₃₇	—	8	1	no	—
40	#10587739	—	16	1	no	—
41	C₄×C₁₃₇	—	2	1	no	—
42	C₄×C₁₃₇	—	4	1	no	—
43	Dic₁₃₇	—	4	1	no	—
44	C₂×C₁₃₇	—	2	1	no	—
45	C₂×C₁₃₇	—	16	1	no	—
46	C₂×C₁₃₇	#353	1	1	yes	characteristic
47	D₁₃₇	—	8	1	no	—
48	C₁₃₇	#1406	1	1	yes	characteristic, complemented, Sylow
49	#1406	—	137	1	no	maximal, Sylow
50	#496	—	137	1	no	—
51	#355	—	137	1	no	—
52	#354	—	137	1	no	—
53	C₂×SD₁₆	—	274	1	no	—
54	C₂×D₈	—	274	1	no	—
55	#127	#10587739	1	1	yes	characteristic, complemented
56	#124	—	137	1	no	—
57	#101	—	274	1	no	—
58	#99	—	274	1	no	—
59	#97	—	137	1	no	—
60	C₂×Q₈	—	137	1	no	—
61	C₂×D₄	—	2	1	no	—
62	C₂×D₄	—	4	1	no	—
63	C₂×D₄	—	137	1	no	—
64	SD₁₆	—	548	2	no	—
65	D₈	—	548	2	no	—
66	M₁₆	—	548	1	no	—
67	C₂×C₈	—	274	1	no	—
68	C₂²⋊C₄	—	548	1	no	—
69	C₂²⋊C₄	—	548	1	no	—
70	C₄²	C₂×#10587739	1	1	yes	characteristic
71	C₂³	—	4	1	no	—
72	C₂³	—	274	1	no	—
73	Q₈	—	548	1	no	—
74	D₄	—	4	1	no	—
75	D₄	—	4	1	no	—
76	D₄	—	8	2	no	—
77	D₄	—	548	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	274	1	no	—
80	C₂×C₄	—	1096	1	no	—
81	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
82	C₈	—	274	2	no	—
83	C₈	—	1096	1	no	—
84	C₂²	—	8	2	no	—
85	C₂²	—	8	1	no	—
86	C₂²	—	548	1	no	—
87	C₂²	—	1096	1	no	—
88	C₂²	#17984146	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	—	548	1	no	—
92	C₄	—	2192	1	no	—
93	C₂	—	2	1	no	—
94	C₂	—	16	1	no	—
95	C₂	—	1096	1	no	—
96	C₂	#22578307	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578307
commutator	C₄²×C₁₃₇	C₂×C₄
Frattini	C₄²	C₂×#10587739
Fitting	#127×C₁₃₇	C₄
socle	C₂×C₁₃₇	#353
prime radical	#22577790	C₂
cosocle	#17984076	C₂²
hypercenter	#127	#10587739
hypocenter	C₁₃₇	#1406
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #22578312	C₂	semidirect
2. #22578310	C₂	semidirect
9. #17983999	C₄	semidirect
10. #127×C₁₃₇	C₄	semidirect
48. C₁₃₇	#1406	semidirect
55. #127	#10587739	semidirect

Maximal subgroups

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

subgroup	index
#22578312	2
#22578310	2
#22577790	2
#1406	137