Subgroups — #31784070

Order 17536.

840 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₄₈	C₂	1	6	yes	maximal, complemented
2	#22577340	C₂	1	4	yes	maximal, complemented
3	#22577313	C₂	1	4	yes	maximal, complemented
4	#376×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₄₈	C₂²	1	6	yes	—
6	C₂²×C₄×D₁₃₇	C₂²	1	4	yes	—
7	C₄×D₅₄₈	—	2	24	no	—
8	C₂×C₄⋊Dic₁₃₇	C₂²	1	6	yes	—
9	C₂×#14555487	C₂²	1	12	yes	—
10	C₂×C₄²×C₁₃₇	C₂²	1	3	yes	—
11	#95×C₁₃₇	C₂²	1	4	yes	—
12	C₂³×D₁₃₇	C₂³	1	4	yes	—
13	C₂×D₅₄₈	—	2	24	no	—
14	C₂×D₅₄₈	—	2	12	no	—
15	C₂²×Dic₁₃₇	C₂³	1	4	yes	—
16	C₂×C₄×D₁₃₇	—	2	24	no	—
17	C₄⋊Dic₁₃₇	—	2	12	no	—
18	#14555487	—	2	24	no	—
19	C₂²×C₄×C₁₃₇	C₂³	1	6	yes	—
20	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
21	C₄²×C₁₃₇	D₄	1	12	yes	complemented
22	C₂²×D₁₃₇	—	2	12	no	—
23	C₂²×D₁₃₇	—	2	12	no	—
24	C₂²×D₁₃₇	#41	1	8	yes	—
25	D₅₄₈	—	2	48	no	—
26	C₂×Dic₁₃₇	—	2	12	no	—
27	C₄×D₁₃₇	—	4	16	no	—
28	C₂³×C₁₃₇	C₂⁴	1	1	yes	characteristic
29	C₂×C₄×C₁₃₇	—	2	12	no	—
30	C₂×C₄×C₁₃₇	C₂×D₄	1	12	yes	—
31	C₂×C₄×C₁₃₇	C₂×D₄	1	6	yes	—
32	C₂²×C₁₃₇	C₂²×D₄	1	3	yes	—
33	C₂²×C₁₃₇	C₂×#41	1	4	yes	—
34	C₂×D₁₃₇	—	2	24	no	—
35	C₂×D₁₃₇	—	4	12	no	—
36	C₂×D₁₃₇	—	4	4	no	—
37	C₄×C₁₃₇	—	2	12	no	—
38	C₄×C₁₃₇	#127	1	4	yes	—
39	Dic₁₃₇	—	4	4	no	—
40	C₂×C₁₃₇	#525	1	6	yes	—
41	C₂×C₁₃₇	C₂×#127	1	1	yes	characteristic
42	D₁₃₇	—	4	8	no	—
43	C₁₃₇	#2397	1	1	yes	characteristic, complemented, Sylow
44	#2397	—	137	1	no	maximal, Sylow
45	C₂×C₄×D₄	—	137	6	no	—
46	#399	—	137	4	no	—
47	#392	—	137	4	no	—
48	#376	D₁₃₇	1	1	yes	characteristic, complemented
49	C₂²×D₄	—	137	6	no	—
50	C₂³×C₄	—	137	4	no	—
51	C₄×D₄	—	274	24	no	—
52	C₂×C₄⋊C₄	—	137	6	no	—
53	C₂×C₂²⋊C₄	—	137	12	no	—
54	C₂×C₄²	C₂×D₁₃₇	1	3	yes	—
55	#95	C₂×D₁₃₇	1	4	yes	—
56	C₂⁴	—	137	4	no	—
57	C₂×D₄	—	274	24	no	—
58	C₂×D₄	—	274	12	no	—
59	C₂²×C₄	—	274	24	no	—
60	C₂²×C₄	—	137	4	no	—
61	C₂²×C₄	C₂²×D₁₃₇	1	6	yes	—
62	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
63	C₄⋊C₄	—	274	12	no	—
64	C₂²⋊C₄	—	274	24	no	—
65	C₄²	D₅₄₈	1	12	yes	complemented
66	C₂³	—	274	12	no	—
67	C₂³	—	274	12	no	—
68	C₂³	—	137	8	no	—
69	C₂³	C₂³×D₁₃₇	1	1	yes	characteristic
70	D₄	—	274	48	no	—
71	C₂×C₄	—	2	12	no	—
72	C₂×C₄	—	274	12	no	—
73	C₂×C₄	—	548	16	no	—
74	C₂×C₄	C₂×D₅₄₈	1	12	yes	—
75	C₂×C₄	C₂×D₅₄₈	1	6	yes	—
76	C₂²	—	274	24	no	—
77	C₂²	—	548	4	no	—
78	C₂²	—	548	12	no	—
79	C₂²	C₂²×D₅₄₈	1	3	yes	—
80	C₂²	#41×D₁₃₇	1	4	yes	—
81	C₄	—	2	12	no	—
82	C₄	—	548	4	no	—
83	C₄	#17984085	1	4	yes	—
84	C₂	—	548	8	no	—
85	C₂	#22577915	1	6	yes	—
86	C₂	C₂×#17984085	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₇
commutator, socle, cosocle	C₂³×C₁₃₇	C₂⁴
Frattini	C₂³	C₂³×D₁₃₇
Fitting	#376×C₁₃₇	C₂
hypercenter	#376	D₁₃₇
hypocenter	C₁₃₇	#2397
LCS	G ▷ C₂³×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×D₅₄₈	C₂	semidirect
2. #22577340	C₂	semidirect
3. #22577313	C₂	semidirect
4. #376×C₁₃₇	C₂	semidirect
21. C₄²×C₁₃₇	D₄	semidirect
43. C₁₃₇	#2397	semidirect
48. #376	D₁₃₇	semidirect
65. C₄²	D₅₄₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅₄₈	2
#22577340	2
#22577313	2
#376×C₁₃₇	2
#2397	137