Subgroups — C₄×#17984024 #31780691

Order 17536.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984024	C₂	1	2	yes	maximal
2	C₄×#14555494	C₂	1	2	yes	maximal
3	#22577168	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555494	C₂²	1	4	yes	—
6	#17984024	C₄	1	8	yes	direct product factor
7	#17983999	—	2	2	no	—
8	C₄×#11764040	—	2	2	no	—
9	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₃₇	C₄	1	4	yes	—
13	#14555494	—	2	4	no	—
14	#14555494	C₂×C₄	1	8	yes	—
15	C₂×#11764040	—	2	4	no	—
16	C₂²×C₄×C₁₃₇	Q₈	1	2	yes	—
17	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₁₃₇	—	2	2	no	—
19	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
20	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
22	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
24	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
25	#11764040	—	2	8	no	—
26	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₄×C₁₃₇	—	2	4	no	—
28	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
31	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	#41	1	2	yes	—
34	C₈×C₁₃₇	—	2	2	no	—
35	C₈×C₁₃₇	C₄²	1	4	yes	—
36	C₂²×C₁₃₇	—	2	2	no	—
37	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₁₃₇	C₄×Q₈	1	2	yes	—
39	C₄×C₁₃₇	—	2	2	no	—
40	C₄×C₁₃₇	#108	1	4	yes	complemented
41	C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
42	C₄×C₁₃₇	#117	1	1	yes	characteristic
43	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
44	C₂×C₁₃₇	—	2	2	no	—
45	C₂×C₁₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₁₃₇	C₂×#108	1	2	yes	—
47	C₁₃₇	C₄×#108	1	1	yes	characteristic, complemented, Sylow
48	C₄×#108	—	137	1	no	maximal, Sylow
49	C₂×#108	—	137	2	no	—
50	C₄×M₁₆	—	137	2	no	—
51	C₂×C₄×C₈	D₁₃₇	1	1	yes	characteristic
52	#341	—	137	2	no	—
53	C₂×M₁₆	—	137	4	no	—
54	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
55	C₂²×C₈	C₂×D₁₃₇	1	1	yes	characteristic
56	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
57	#108	—	137	8	no	—
58	#97	—	274	2	no	—
59	C₄×C₈	—	274	2	no	—
60	C₄×C₈	Dic₁₃₇	1	4	yes	—
61	C₂²×C₄	Dic₂₇₄	1	2	yes	—
62	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
63	M₁₆	—	274	4	no	—
64	M₁₆	—	137	8	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	274	4	no	—
67	C₂×C₈	C₄×D₁₃₇	1	4	yes	—
68	C₂×C₈	C₂×Dic₁₃₇	1	2	yes	—
69	C₂×C₈	C₂×Dic₁₃₇	1	2	yes	—
70	C₄²	C₂×Dic₁₃₇	1	2	yes	—
71	C₄²	D₅₄₈	1	1	yes	characteristic
72	C₄²	D₅₄₈	1	1	yes	characteristic
73	C₂³	C₂×Dic₂₇₄	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	C₄⋊Dic₁₃₇	1	2	yes	—
76	C₂×C₄	C₄⋊Dic₁₃₇	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₁₃₇	1	2	yes	—
78	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
79	C₂×C₄	C₂×D₅₄₈	1	1	yes	characteristic
80	C₂×C₄	#14555510	1	2	yes	—
81	C₈	—	2	2	no	—
82	C₈	—	274	8	no	—
83	C₈	C₄×Dic₁₃₇	1	4	yes	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×C₄⋊Dic₁₃₇	1	1	yes	characteristic
86	C₂²	C₄×Dic₂₇₄	1	2	yes	—
87	C₄	—	2	2	no	—
88	C₄	#17984024	1	4	yes	direct product factor
89	C₄	C₂×C₄×Dic₁₃₇	1	1	yes	characteristic
90	C₄	#17984047	1	1	yes	characteristic
91	C₄	C₄×D₅₄₈	1	2	yes	—
92	C₂	—	2	2	no	—
93	C₂	C₄×C₄⋊Dic₁₃₇	1	1	yes	characteristic
94	C₂	C₂×#17984024	1	2	yes	—

	subgroup	G/H
center	C₄²	D₅₄₈
commutator	C₄×C₁₃₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	C₂×C₄×C₈×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₃₇
hypocenter	C₁₃₇	C₄×#108
LCS	G ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #17984024	C₄	direct (N × H)
40. C₄×C₁₃₇	#108	semidirect
47. C₁₃₇	C₄×#108	semidirect
88. C₄	#17984024	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17984024	2
C₄×#14555494	2
#22577168	2
C₂×C₄×C₈×C₁₃₇	2
C₄×#108	137

Subgroups — C4×#17984024 #31780691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17984024 #31780691