Subgroups — #31795918

Order 17536.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578312	C₂	1	1	yes	maximal, characteristic
2	#22578310	C₂	1	1	yes	maximal, characteristic
3	#22577795	C₂	1	1	yes	maximal, characteristic
4	#17984149	—	2	1	no	—
5	#17984147	—	2	1	no	—
6	C₂×#14555500	—	2	1	no	—
7	C₂×#14555497	—	2	1	no	—
8	#17984076	C₂²	1	1	yes	characteristic
9	#17983999	C₄	1	1	yes	characteristic, complemented
10	C₄⋊Q₈×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	#14555500	—	4	2	no	—
16	#14555497	—	4	2	no	—
17	C₂×#11764040	—	2	1	no	—
18	#14555487	—	4	1	no	—
19	C₂×Q₈×C₁₃₇	—	2	1	no	—
20	C₄⋊C₄×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	Q₈×C₁₃₇	—	4	1	no	—
30	Q₈×C₁₃₇	—	4	1	no	—
31	C₂×C₄×C₁₃₇	—	2	1	no	—
32	C₂×C₄×C₁₃₇	—	4	1	no	—
33	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₁₃₇	#99	1	1	yes	characteristic
35	C₂×D₁₃₇	—	4	1	no	—
36	C₂×D₁₃₇	—	8	1	no	—
37	#10587739	—	16	1	no	—
38	C₄×C₁₃₇	—	2	1	no	—
39	C₄×C₁₃₇	—	4	1	no	—
40	C₄×C₁₃₇	—	8	1	no	—
41	Dic₁₃₇	—	4	1	no	—
42	C₂×C₁₃₇	—	2	1	no	—
43	C₂×C₁₃₇	#353	1	1	yes	characteristic
44	D₁₃₇	—	8	1	no	—
45	C₁₃₇	#1407	1	1	yes	characteristic, complemented, Sylow
46	#1407	—	137	1	no	maximal, Sylow
47	#497	—	137	1	no	—
48	#355	—	137	1	no	—
49	#354	—	137	1	no	—
50	C₂×Q₁₆	—	274	1	no	—
51	C₂×SD₁₆	—	274	1	no	—
52	C₄⋊Q₈	#10587739	1	1	yes	characteristic, complemented
53	#124	—	137	1	no	—
54	#101	—	274	1	no	—
55	#99	—	274	1	no	—
56	#97	—	137	1	no	—
57	C₂×Q₈	—	2	1	no	—
58	C₂×Q₈	—	137	1	no	—
59	C₂×D₄	—	137	1	no	—
60	Q₁₆	—	548	2	no	—
61	SD₁₆	—	548	2	no	—
62	M₁₆	—	548	1	no	—
63	C₂×C₈	—	274	1	no	—
64	C₄⋊C₄	—	4	1	no	—
65	C₂²⋊C₄	—	548	1	no	—
66	C₂²⋊C₄	—	548	1	no	—
67	C₄²	C₂×#10587739	1	1	yes	characteristic
68	C₂³	—	274	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	1	no	—
71	Q₈	—	548	1	no	—
72	D₄	—	548	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	274	1	no	—
76	C₂×C₄	—	1096	1	no	—
77	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
78	C₈	—	274	2	no	—
79	C₈	—	1096	1	no	—
80	C₂²	—	548	1	no	—
81	C₂²	—	1096	1	no	—
82	C₂²	#17984146	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	8	1	no	—
86	C₄	—	548	1	no	—
87	C₄	—	2192	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	1096	1	no	—
90	C₂	#22578307	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578307
commutator	C₄²×C₁₃₇	C₂×C₄
Frattini	C₄²	C₂×#10587739
Fitting	C₄⋊Q₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	#353
prime radical	C₂×D₅₄₈	D₄
cosocle	#17984076	C₂²
hypercenter	C₄⋊Q₈	#10587739
hypocenter	C₁₃₇	#1407
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
9. #17983999	C₄	semidirect
10. C₄⋊Q₈×C₁₃₇	C₄	semidirect
45. C₁₃₇	#1407	semidirect
52. C₄⋊Q₈	#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
#22577795	2
#1407	137