Subgroups — C₂×C₄×#14555522 #31797941

Order 17536.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555522	C₂	1	2	yes	maximal
2	C₄×#14555522	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×#11764047	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555522	C₄	1	8	yes	direct product factor
5	C₂×#14555522	C₂²	1	4	yes	—
6	C₂²×#11764047	C₄	1	1	yes	characteristic
7	C₂²×#11764047	C₂²	1	1	yes	characteristic
8	C₄×#11764047	C₄	1	2	yes	—
9	C₄×#11764047	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₃₇	C₄	1	1	yes	characteristic
11	#14555522	C₂×C₄	1	8	yes	direct product factor
12	C₂×#11764047	C₈	1	4	yes	—
13	C₂×#11764047	C₂×C₄	1	4	yes	—
14	C₂×#11764047	C₂×C₄	1	1	yes	characteristic
15	C₂×#11764047	C₂×C₄	1	2	yes	—
16	C₂×#11764047	C₂³	1	1	yes	characteristic
17	C₂²×Dic₁₃₇	C₈	1	1	yes	characteristic
18	C₂²×Dic₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₁₃₇	C₈	1	2	yes	—
20	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₃₇	C₈	1	1	yes	characteristic
22	#11764047	C₄²	1	2	yes	—
23	#11764047	C₂×C₈	1	4	yes	—
24	#11764047	C₂²×C₄	1	2	yes	—
25	C₂×Dic₁₃₇	C₁₆	1	4	yes	complemented
26	C₂×Dic₁₃₇	C₄²	1	2	yes	—
27	C₂×Dic₁₃₇	C₂×C₈	1	1	yes	characteristic
28	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
29	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
30	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
31	C₂³×C₁₃₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	C₁₆	1	4	yes	complemented
33	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
34	C₂²×C₁₃₇	C₄×C₈	1	2	yes	—
35	C₂²×C₁₃₇	C₂×C₁₆	1	2	yes	—
36	C₂²×C₁₃₇	C₂×C₁₆	1	2	yes	—
37	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
38	C₄×C₁₃₇	C₂×C₁₆	1	4	yes	complemented
39	Dic₁₃₇	C₄×C₈	1	2	yes	—
40	Dic₁₃₇	C₂×C₁₆	1	4	yes	complemented
41	Dic₁₃₇	C₂×C₄²	1	1	yes	characteristic
42	Dic₁₃₇	C₂²×C₈	1	1	yes	characteristic
43	C₂×C₁₃₇	C₄×C₁₆	1	4	yes	complemented
44	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
45	C₂×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
46	C₂×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
47	C₁₃₇	C₂×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₄×C₁₆	—	137	1	no	maximal, Sylow
49	C₂²×C₁₆	—	137	2	no	—
50	C₂×C₄×C₈	—	137	1	no	—
51	C₄×C₁₆	—	137	4	no	—
52	C₂²×C₈	—	137	1	no	—
53	C₂²×C₈	—	137	1	no	—
54	C₂×C₄²	—	137	1	no	—
55	C₂×C₁₆	—	137	4	no	—
56	C₂×C₁₆	—	137	8	no	—
57	C₄×C₈	—	137	2	no	—
58	C₄×C₈	—	137	2	no	—
59	C₂²×C₄	—	137	1	no	—
60	C₂²×C₄	—	137	1	no	—
61	C₂²×C₄	#11764049	1	1	yes	characteristic
62	C₂×C₈	—	137	1	no	—
63	C₂×C₈	—	137	1	no	—
64	C₂×C₈	—	137	2	no	—
65	C₂×C₈	—	137	4	no	—
66	C₂×C₈	—	137	4	no	—
67	C₄²	—	137	2	no	—
68	C₄²	—	137	2	no	—
69	C₁₆	—	137	8	no	—
70	C₂³	C₂×#11764049	1	1	yes	characteristic
71	C₂×C₄	—	137	2	no	—
72	C₂×C₄	—	137	2	no	—
73	C₂×C₄	—	137	4	no	—
74	C₂×C₄	—	137	1	no	—
75	C₂×C₄	—	137	1	no	—
76	C₂×C₄	—	137	2	no	—
77	C₂×C₄	#14555522	1	4	yes	direct product factor
78	C₂×C₄	C₂×#11764049	1	2	yes	—
79	C₈	—	137	2	no	—
80	C₈	—	137	2	no	—
81	C₈	—	137	4	no	—
82	C₂²	C₄×#11764049	1	2	yes	—
83	C₂²	C₂×#14555522	1	2	yes	—
84	C₂²	#17984183	1	2	yes	—
85	C₂²	C₂²×#11764049	1	1	yes	characteristic
86	C₄	—	137	1	no	—
87	C₄	—	137	1	no	—
88	C₄	—	137	2	no	—
89	C₄	—	137	4	no	—
90	C₄	C₂×#14555522	1	4	yes	direct product factor
91	C₂	C₄×#14555522	1	4	yes	direct product factor
92	C₂	C₂×C₄×#11764049	1	1	yes	characteristic
93	C₂	C₂²×#14555522	1	1	yes	characteristic
94	C₂	C₂×#17984183	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	#11764049
commutator, hypocenter	C₁₃₇	C₂×C₄×C₁₆
Frattini	C₂²	C₂²×#11764049
Fitting	C₂²×C₄×C₁₃₇	C₈
socle, prime radical	C₂³×C₁₃₇	C₂×C₈
cosocle	C₂×#11764047	C₂³
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₄×#14555522	C₂	direct (N × H)
4. C₂×#14555522	C₄	direct (N × H)
11. #14555522	C₂×C₄	direct (N × H)
25. C₂×Dic₁₃₇	C₁₆	semidirect
32. C₂×C₄×C₁₃₇	C₁₆	semidirect
38. C₄×C₁₃₇	C₂×C₁₆	semidirect
40. Dic₁₃₇	C₂×C₁₆	semidirect
43. C₂×C₁₃₇	C₄×C₁₆	semidirect
47. C₁₃₇	C₂×C₄×C₁₆	semidirect
77. C₂×C₄	#14555522	direct (N × H)
90. C₄	C₂×#14555522	direct (N × H)
91. C₂	C₄×#14555522	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14555522	2
C₄×#14555522	2
C₂×C₄×#11764047	2
C₂×C₄×C₁₆	137

Subgroups — C2×C4×#14555522 #31797941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14555522 #31797941