Subgroups — #31795794

Order 17536.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22578295	C₂	1	2	yes	maximal
2	#22577771	C₂	1	1	yes	maximal, characteristic
3	#17984162	—	2	2	no	—
4	#17984080	C₄	1	1	yes	characteristic
5	#17984012	—	2	2	no	—
6	#17983999	C₂²	1	1	yes	characteristic
7	C₄⋊Q₈×C₁₃₇	C₄	1	1	yes	characteristic
8	#14555513	—	2	4	no	—
9	C₂×#11764040	D₄	1	2	yes	—
10	C₄⋊Dic₁₃₇	—	2	1	no	—
11	#14555489	—	4	1	no	—
12	C₂×Q₈×C₁₃₇	—	2	1	no	—
13	C₄⋊C₄×C₁₃₇	—	4	1	no	—
14	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₁₃₇	—	4	1	no	—
16	#11764040	—	2	2	no	—
17	Q₈×C₁₃₇	—	4	2	no	—
18	C₂×C₄×C₁₃₇	—	2	1	no	—
19	C₂×C₄×C₁₃₇	—	4	1	no	—
20	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₂²×C₁₃₇	#100	1	1	yes	characteristic
22	C₄×C₁₃₇	—	4	1	no	—
23	C₄×C₁₃₇	—	8	1	no	—
24	C₄×C₁₃₇	#104	1	2	yes	—
25	Dic₁₃₇	—	8	1	no	—
26	C₂×C₁₃₇	—	2	1	no	—
27	C₂×C₁₃₇	#329	1	1	yes	characteristic
28	C₁₃₇	#1356	1	1	yes	characteristic, complemented, Sylow
29	#1356	—	137	1	no	maximal, Sylow
30	#491	—	137	1	no	—
31	#347	—	137	2	no	—
32	C₄⋊Q₈	#10587739	1	1	yes	characteristic
33	#125	—	137	1	no	—
34	M₃₂	—	274	2	no	—
35	#103	—	274	2	no	—
36	#97	—	137	1	no	—
37	C₂×Q₈	—	2	1	no	—
38	C₂×C₈	—	137	2	no	—
39	C₄⋊C₄	—	4	1	no	—
40	C₄⋊C₄	—	274	1	no	—
41	C₄⋊C₄	—	548	1	no	—
42	C₄²	C₂×#10587739	1	1	yes	characteristic
43	C₁₆	—	274	4	no	—
44	Q₈	—	4	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	548	1	no	—
48	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
49	C₈	—	274	2	no	—
50	C₂²	#17984148	1	1	yes	characteristic
51	C₄	—	4	1	no	—
52	C₄	—	8	1	no	—
53	C₄	—	1096	1	no	—
54	C₄	#17984152	1	2	yes	—
55	C₂	—	2	1	no	—
56	C₂	#22578258	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22578258
commutator	C₄²×C₁₃₇	C₂×C₄
Frattini	C₄²	C₂×#10587739
Fitting	C₄⋊Q₈×C₁₃₇	C₄
socle	C₂×C₁₃₇	#329
prime radical	C₂²×C₁₃₇	#100
cosocle	#17983999	C₂²
hypercenter	C₄⋊Q₈	#10587739
hypocenter	C₁₃₇	#1356
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂×C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
28. C₁₃₇	#1356	semidirect

Maximal subgroups

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

subgroup	index
#22578295	2
#22577771	2
#1356	137