Subgroups — C₂×#22577456 #31789405

Order 17536.

528 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577456	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#11764040	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#11764040	C₂	1	2	yes	maximal
4	C₂×C₄×#11764040	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄²×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11764040	C₂²	1	4	yes	—
7	C₄⋊#11764040	C₂²	1	8	yes	complemented
8	C₂²⋊#11764040	C₂²	1	8	yes	—
9	C₄×#11764040	C₂²	1	8	yes	complemented
10	C₂³×C₄×C₁₃₇	C₄	1	1	yes	characteristic
11	C₂³×C₄×C₁₃₇	C₄	1	1	yes	characteristic
12	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₃₇	C₄	1	4	yes	—
15	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₃₇	C₂²	1	4	yes	—
18	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×#11764040	—	2	8	no	—
20	C₂×#11764040	C₂³	1	8	yes	—
21	C₂⁴×C₁₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₇	—	2	4	no	—
23	C₂²×C₄×C₁₃₇	—	2	4	no	—
24	C₂²×C₄×C₁₃₇	C₈	1	8	yes	complemented
25	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
36	C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
37	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
38	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
39	C₄²×C₁₃₇	C₂³	1	2	yes	—
40	C₄²×C₁₃₇	C₂³	1	2	yes	—
41	#11764040	—	2	8	no	—
42	C₂³×C₁₃₇	—	2	2	no	—
43	C₂³×C₁₃₇	—	2	2	no	—
44	C₂³×C₁₃₇	C₂×C₈	1	4	yes	—
45	C₂³×C₁₃₇	C₂²×C₄	1	2	yes	—
46	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃₇	—	2	2	no	—
48	C₂×C₄×C₁₃₇	—	2	2	no	—
49	C₂×C₄×C₁₃₇	—	2	4	no	—
50	C₂×C₄×C₁₃₇	—	2	8	no	—
51	C₂×C₄×C₁₃₇	—	2	4	no	—
52	C₂×C₄×C₁₃₇	C₂×C₈	1	4	yes	—
53	C₂×C₄×C₁₃₇	C₂×C₈	1	4	yes	—
54	C₂×C₄×C₁₃₇	C₂×C₈	1	16	yes	complemented
55	C₂×C₄×C₁₃₇	M₁₆	1	4	yes	—
56	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₁₃₇	#41	1	4	yes	—
63	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
64	C₂²×C₁₃₇	—	2	2	no	—
65	C₂²×C₁₃₇	—	2	2	no	—
66	C₂²×C₁₃₇	—	2	8	no	—
67	C₂²×C₁₃₇	#117	1	2	yes	—
68	C₂²×C₁₃₇	C₂²×C₈	1	4	yes	—
69	C₂²×C₁₃₇	C₂²×C₈	1	2	yes	—
70	C₂²×C₁₃₇	C₂×M₁₆	1	2	yes	—
71	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
72	C₄×C₁₃₇	—	2	2	no	—
73	C₄×C₁₃₇	—	2	2	no	—
74	C₄×C₁₃₇	#117	1	2	yes	—
75	C₄×C₁₃₇	C₂²×C₈	1	4	yes	—
76	C₄×C₁₃₇	C₂²×C₈	1	4	yes	—
77	C₄×C₁₃₇	C₂×M₁₆	1	4	yes	—
78	C₄×C₁₃₇	C₂×#41	1	2	yes	—
79	C₂×C₁₃₇	—	2	4	no	—
80	C₂×C₁₃₇	#431	1	4	yes	complemented
81	C₂×C₁₃₇	C₂×#117	1	1	yes	characteristic
82	C₂×C₁₃₇	C₂³×C₈	1	1	yes	characteristic
83	C₂×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
84	C₁₃₇	C₂×#431	1	1	yes	characteristic, complemented, Sylow
85	C₂×#431	—	137	1	no	maximal, Sylow
86	C₂²×C₄²	D₁₃₇	1	1	yes	characteristic
87	#431	—	137	8	no	—
88	C₂×C₄⋊C₈	—	137	2	no	—
89	C₂×C₂²⋊C₈	—	137	2	no	—
90	C₂×C₄×C₈	—	137	2	no	—
91	C₂³×C₄	Dic₁₃₇	1	1	yes	characteristic
92	C₂³×C₄	Dic₁₃₇	1	1	yes	characteristic
93	C₂³×C₄	C₂×D₁₃₇	1	1	yes	characteristic
94	C₂²×C₈	—	137	4	no	—
95	C₂×C₄²	Dic₁₃₇	1	1	yes	characteristic
96	C₂×C₄²	Dic₁₃₇	1	1	yes	characteristic
97	C₂×C₄²	Dic₁₃₇	1	4	yes	—
98	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
99	C₂×C₄²	C₂×D₁₃₇	1	4	yes	—
100	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
101	C₄⋊C₈	—	137	8	no	—
102	C₂²⋊C₈	—	137	8	no	—
103	C₄×C₈	—	137	8	no	—
104	C₂⁴	C₂×Dic₁₃₇	1	1	yes	characteristic
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	#11764040	1	8	yes	complemented
108	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
109	C₂²×C₄	C₂×Dic₁₃₇	1	2	yes	—
110	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
111	C₂²×C₄	C₂×Dic₁₃₇	1	4	yes	—
112	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
114	C₂²×C₄	C₂×Dic₁₃₇	1	4	yes	—
115	C₂²×C₄	C₂×Dic₁₃₇	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₁₃₇	1	2	yes	—
117	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
118	C₂×C₈	—	274	8	no	—
119	C₂×C₈	—	137	8	no	—
120	C₄²	C₂×Dic₁₃₇	1	4	yes	—
121	C₄²	C₂×Dic₁₃₇	1	4	yes	—
122	C₄²	C₂×Dic₁₃₇	1	2	yes	—
123	C₄²	C₂×Dic₁₃₇	1	2	yes	—
124	C₄²	C₂²×D₁₃₇	1	2	yes	—
125	C₄²	C₂²×D₁₃₇	1	2	yes	—
126	C₂³	—	2	2	no	—
127	C₂³	—	2	2	no	—
128	C₂³	C₂×#11764040	1	4	yes	—
129	C₂³	C₂²×Dic₁₃₇	1	2	yes	—
130	C₂³	C₂²×Dic₁₃₇	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	C₂×#11764040	1	4	yes	—
137	C₂×C₄	C₂×#11764040	1	4	yes	—
138	C₂×C₄	C₂×#11764040	1	16	yes	complemented
139	C₂×C₄	#14555494	1	4	yes	—
140	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
141	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
142	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
143	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
144	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
145	C₂×C₄	C₂²×Dic₁₃₇	1	2	yes	—
146	C₂×C₄	#14555510	1	4	yes	—
147	C₂×C₄	C₂³×D₁₃₇	1	1	yes	characteristic
148	C₈	—	274	8	no	—
149	C₂²	—	2	8	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	#17984047	1	2	yes	—
153	C₂²	C₂²×#11764040	1	4	yes	—
154	C₂²	C₂²×#11764040	1	2	yes	—
155	C₂²	C₂×#14555494	1	2	yes	—
156	C₂²	C₂³×Dic₁₃₇	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	#17984047	1	2	yes	—
160	C₄	C₂²×#11764040	1	4	yes	—
161	C₄	C₂²×#11764040	1	4	yes	—
162	C₄	C₂×#14555494	1	4	yes	—
163	C₄	C₂×#14555510	1	2	yes	—
164	C₂	—	2	4	no	—
165	C₂	#22577456	1	4	yes	direct product factor
166	C₂	C₂×#17984047	1	1	yes	characteristic
167	C₂	C₂³×#11764040	1	1	yes	characteristic
168	C₂	C₂²×#14555494	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₃₇
commutator	C₂×C₁₃₇	C₂³×C₈
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×#431
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #22577456	C₂	direct (N × H)
2. C₂×C₄⋊#11764040	C₂	semidirect
4. C₂×C₄×#11764040	C₂	semidirect
7. C₄⋊#11764040	C₂²	semidirect
9. C₄×#11764040	C₂²	semidirect
24. C₂²×C₄×C₁₃₇	C₈	semidirect
54. C₂×C₄×C₁₃₇	C₂×C₈	semidirect
80. C₂×C₁₃₇	#431	semidirect
84. C₁₃₇	C₂×#431	semidirect
107. C₂²×C₄	#11764040	semidirect
138. C₂×C₄	C₂×#11764040	semidirect
165. C₂	#22577456	direct (N × H)

Maximal subgroups

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

subgroup	index
#22577456	2
C₂×C₄⋊#11764040	2
C₂×C₂²⋊#11764040	2
C₂×C₄×#11764040	2
C₂²×C₄²×C₁₃₇	2
C₂×#431	137

Subgroups — C2×#22577456 #31789405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577456 #31789405