Subgroups — C₂×#22577427 #31789344

Order 17536.

576 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555494	C₂	1	2	yes	maximal
2	#22577427	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#11764040	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14555494	—	2	8	no	—
6	C₂×#14555494	C₂²	1	8	yes	—
7	C₂²×#11764040	C₂²	1	4	yes	—
8	C₄⋊#11764040	C₂²	1	16	yes	complemented
9	C₂³×C₄×C₁₃₇	C₄	1	2	yes	—
10	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic
12	C₂×C₄²×C₁₃₇	C₄	1	4	yes	—
13	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₃₇	C₂²	1	4	yes	—
16	#14555494	—	2	16	no	—
17	C₂×#11764040	—	2	8	no	—
18	C₂×#11764040	C₂³	1	8	yes	—
19	C₂⁴×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₃₇	—	2	8	no	—
21	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₃₇	C₂×C₄	1	8	yes	—
23	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₃₇	D₄	1	4	yes	—
27	C₂²×C₄×C₁₃₇	Q₈	1	4	yes	—
28	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
29	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
31	C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
32	C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
33	C₄²×C₁₃₇	C₂³	1	4	yes	—
34	#11764040	—	2	8	no	—
35	C₂³×C₁₃₇	—	2	4	no	—
36	C₂³×C₁₃₇	C₄⋊C₄	1	4	yes	—
37	C₂³×C₁₃₇	C₂²×C₄	1	2	yes	—
38	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₇	—	2	4	no	—
40	C₂×C₄×C₁₃₇	—	2	16	no	—
41	C₂×C₄×C₁₃₇	C₄⋊C₄	1	8	yes	—
42	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
43	C₂×C₄×C₁₃₇	M₁₆	1	8	yes	complemented
44	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
51	C₂×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
53	C₂²×C₁₃₇	—	2	4	no	—
54	C₂²×C₁₃₇	—	2	8	no	—
55	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
56	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
57	C₂²×C₁₃₇	C₂×M₁₆	1	4	yes	—
58	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₁₃₇	—	2	4	no	—
60	C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
61	C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
62	C₄×C₁₃₇	C₂×M₁₆	1	8	yes	complemented
63	C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₁₃₇	—	2	4	no	—
66	C₂×C₁₃₇	#423	1	4	yes	complemented
67	C₂×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₁₃₇	C₂²×M₁₆	1	2	yes	—
69	C₁₃₇	C₂×#423	1	1	yes	characteristic, complemented, Sylow
70	C₂×#423	—	137	1	no	maximal, Sylow
71	C₂²×M₁₆	—	137	2	no	—
72	C₂²×C₄²	D₁₃₇	1	1	yes	characteristic
73	#423	—	137	8	no	—
74	C₂×C₄⋊C₈	—	137	4	no	—
75	C₂³×C₄	Dic₁₃₇	1	2	yes	—
76	C₂³×C₄	C₂×D₁₃₇	1	1	yes	characteristic
77	C₂×M₁₆	—	274	8	no	—
78	C₂×M₁₆	—	137	8	no	—
79	C₂²×C₈	—	137	4	no	—
80	C₂×C₄²	Dic₁₃₇	1	1	yes	characteristic
81	C₂×C₄²	Dic₁₃₇	1	4	yes	—
82	C₂×C₄²	Dic₁₃₇	1	1	yes	characteristic
83	C₂×C₄²	C₂×D₁₃₇	1	4	yes	—
84	C₂×C₄²	C₂×D₁₃₇	1	2	yes	—
85	C₄⋊C₈	—	137	16	no	—
86	C₂⁴	C₂×Dic₁₃₇	1	1	yes	characteristic
87	C₂²×C₄	—	2	8	no	—
88	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
89	C₂²×C₄	C₂×Dic₁₃₇	1	8	yes	—
90	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
91	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
92	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
93	C₂²×C₄	D₅₄₈	1	4	yes	—
94	C₂²×C₄	Dic₂₇₄	1	4	yes	—
95	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
97	M₁₆	—	274	16	no	—
98	C₂×C₈	—	274	8	no	—
99	C₂×C₈	—	137	8	no	—
100	C₄²	C₂×Dic₁₃₇	1	4	yes	—
101	C₄²	C₂×Dic₁₃₇	1	4	yes	—
102	C₄²	C₂×Dic₁₃₇	1	4	yes	—
103	C₄²	C₂²×D₁₃₇	1	4	yes	—
104	C₂³	—	2	4	no	—
105	C₂³	C₄⋊Dic₁₃₇	1	4	yes	—
106	C₂³	C₂²×Dic₁₃₇	1	2	yes	—
107	C₂³	C₂²×Dic₁₃₇	1	1	yes	characteristic
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	16	no	—
110	C₂×C₄	C₄⋊Dic₁₃₇	1	8	yes	—
111	C₂×C₄	C₄⋊Dic₁₃₇	1	4	yes	—
112	C₂×C₄	#14555494	1	8	yes	complemented
113	C₂×C₄	C₂²×Dic₁₃₇	1	4	yes	—
114	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
115	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
116	C₂×C₄	C₂²×Dic₁₃₇	1	4	yes	—
117	C₂×C₄	C₂×D₅₄₈	1	2	yes	—
118	C₂×C₄	C₂×D₅₄₈	1	4	yes	—
119	C₂×C₄	C₂×Dic₂₇₄	1	4	yes	—
120	C₂×C₄	C₂×Dic₂₇₄	1	2	yes	—
121	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
122	C₈	—	274	8	no	—
123	C₂²	—	2	8	no	—
124	C₂²	—	2	4	no	—
125	C₂²	C₂×C₄⋊Dic₁₃₇	1	4	yes	—
126	C₂²	C₂×C₄⋊Dic₁₃₇	1	2	yes	—
127	C₂²	C₂×#14555494	1	4	yes	—
128	C₂²	C₂³×Dic₁₃₇	1	1	yes	characteristic
129	C₄	—	2	4	no	—
130	C₄	C₂×C₄⋊Dic₁₃₇	1	4	yes	—
131	C₄	C₂×C₄⋊Dic₁₃₇	1	2	yes	—
132	C₄	C₂×#14555494	1	8	yes	complemented
133	C₄	C₂²×D₅₄₈	1	1	yes	characteristic
134	C₄	C₂²×Dic₂₇₄	1	1	yes	characteristic
135	C₂	—	2	4	no	—
136	C₂	#22577427	1	4	yes	direct product factor
137	C₂	C₂²×C₄⋊Dic₁₃₇	1	1	yes	characteristic
138	C₂	C₂²×#14555494	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₇
commutator	C₂²×C₁₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₇
Fitting	C₂²×C₄²×C₁₃₇	C₂
socle	C₂³×C₁₃₇	C₂²×C₄
prime radical	C₂⁴×C₁₃₇	C₂×C₄
cosocle	C₂×C₄×C₁₃₇	C₂⁴
hypercenter	C₂²×C₄²	D₁₃₇
hypocenter	C₁₃₇	C₂×#423
LCS	G ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. #22577427	C₂	direct (N × H)
3. C₂×C₄⋊#11764040	C₂	semidirect
8. C₄⋊#11764040	C₂²	semidirect
43. C₂×C₄×C₁₃₇	M₁₆	semidirect
62. C₄×C₁₃₇	C₂×M₁₆	semidirect
66. C₂×C₁₃₇	#423	semidirect
69. C₁₃₇	C₂×#423	semidirect
112. C₂×C₄	#14555494	semidirect
132. C₄	C₂×#14555494	semidirect
136. C₂	#22577427	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×#14555494	2
#22577427	2
C₂×C₄⋊#11764040	2
C₂²×C₄²×C₁₃₇	2
C₂×#423	137

Subgroups — C2×#22577427 #31789344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577427 #31789344