Subgroups — #31795757

Order 17536.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14555513	C₂	1	1	yes	maximal, characteristic
2	#22578291	C₂	1	1	yes	maximal, characteristic
3	#22577534	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555513	—	2	1	no	—
5	C₂×#14555513	—	2	1	no	—
6	C₄⋊#11764040	—	2	1	no	—
7	C₄⋊#11764040	C₄	1	1	yes	characteristic
8	#17983999	—	2	1	no	—
9	C₄×#11764040	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₁₃₇	C₄	1	1	yes	characteristic
11	#14555513	—	2	2	no	—
12	#14555513	—	4	1	no	—
13	C₂×#11764040	—	2	1	no	—
14	C₂×#11764040	D₄	1	1	yes	characteristic
15	C₂×#11764040	D₄	1	1	yes	characteristic
16	C₂×Q₈×C₁₃₇	C₈	1	1	yes	characteristic
17	C₄⋊C₄×C₁₃₇	—	2	1	no	—
18	C₄⋊C₄×C₁₃₇	C₈	1	1	yes	characteristic
19	C₄²×C₁₃₇	—	2	1	no	—
20	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	#11764040	—	2	1	no	—
22	#11764040	—	4	1	no	—
23	#11764040	SD₁₆	1	1	yes	characteristic
24	#11764040	Q₁₆	1	1	yes	characteristic
25	Q₈×C₁₃₇	—	4	1	no	—
26	C₂×C₄×C₁₃₇	—	2	1	no	—
27	C₂×C₄×C₁₃₇	—	2	1	no	—
28	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₃₇	M₁₆	1	1	yes	characteristic
31	C₂²×C₁₃₇	C₂²⋊C₈	1	1	yes	characteristic
32	C₄×C₁₃₇	—	2	1	no	—
33	C₄×C₁₃₇	—	2	1	no	—
34	C₄×C₁₃₇	—	4	1	no	—
35	C₄×C₁₃₇	—	4	1	no	—
36	C₄×C₁₃₇	#103	1	1	yes	characteristic
37	C₄×C₁₃₇	#104	1	1	yes	characteristic
38	C₂×C₁₃₇	Q₈⋊C₈	1	1	yes	characteristic
39	C₂×C₁₃₇	#349	1	1	yes	characteristic
40	C₂×C₁₃₇	#350	1	1	yes	characteristic
41	C₁₃₇	#1338	1	1	yes	characteristic, complemented, Sylow
42	#1338	—	137	1	no	maximal, Sylow
43	#446	—	137	1	no	—
44	C₄⋊C₁₆	—	137	1	no	—
45	#346	—	137	1	no	—
46	C₄×Q₈	#10587739	1	1	yes	characteristic
47	C₂×C₁₆	—	274	1	no	—
48	C₂×C₁₆	—	274	1	no	—
49	C₄⋊C₈	—	274	1	no	—
50	C₄⋊C₈	—	137	1	no	—
51	#97	—	274	1	no	—
52	C₄×C₈	—	137	1	no	—
53	C₂×Q₈	#11764047	1	1	yes	characteristic
54	C₂×C₈	—	274	1	no	—
55	C₂×C₈	—	137	1	no	—
56	C₂×C₈	—	137	1	no	—
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	#11764047	1	1	yes	characteristic
59	C₄²	—	2	1	no	—
60	C₄²	C₂×#10587739	1	1	yes	characteristic
61	C₁₆	—	274	2	no	—
62	C₁₆	—	548	1	no	—
63	Q₈	—	4	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	C₂²⋊#10587739	1	1	yes	characteristic
67	C₂×C₄	C₂×#11764047	1	1	yes	characteristic
68	C₂×C₄	#14555519	1	1	yes	characteristic
69	C₈	—	274	1	no	—
70	C₈	—	548	1	no	—
71	C₈	—	137	1	no	—
72	C₈	—	137	1	no	—
73	C₂²	C₂²⋊#11764047	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	#17984151	1	1	yes	characteristic
79	C₄	#17984153	1	1	yes	characteristic
80	C₂	Q₈⋊#11764047	1	1	yes	characteristic
81	C₂	#22578299	1	1	yes	characteristic
82	C₂	#22578301	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#10587739
commutator	C₂×C₄×C₁₃₇	C₂×C₈
Frattini	C₄²	C₂×#10587739
Fitting	C₄×Q₈×C₁₃₇	C₄
socle, prime radical	C₂²×C₁₃₇	C₂²⋊C₈
cosocle	C₄×#11764040	C₂²
hypercenter	C₄×Q₈	#10587739
hypocenter	C₁₃₇	#1338
LCS	G ▷ C₂×C₄×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
41. C₁₃₇	#1338	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14555513	2
#22578291	2
#22577534	2
#1338	137