Subgroups — C₂×#89461 #11735896

Order 896.

592 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#1131	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#6435	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#6389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#6377	C₂	1	1	yes	maximal, characteristic, complemented
8	#89461	C₂	1	8	yes	maximal, direct product factor
9	C₂²×C₄×D₇	C₂²	1	1	yes	characteristic
10	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₂×(C₂×D₇)⋊C₄	C₄	1	2	yes	—
14	C₂×Dic₇⋊C₄	C₄	1	2	yes	—
15	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
16	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×#1131	C₄	1	8	yes	complemented
19	C₂×#1131	C₂²	1	4	yes	—
20	C₂×C₈×D₇	C₄	1	8	yes	complemented
21	C₂×C₈×D₇	C₂²	1	4	yes	—
22	#6435	C₄	1	4	yes	—
23	#6435	C₂²	1	4	yes	—
24	C₄×C₇×C₈	C₂²	1	4	yes	complemented
25	#6389	C₂²	1	4	yes	complemented
26	C₈×Dic₇	C₂²	1	4	yes	complemented
27	#6377	C₂²	1	4	yes	complemented
28	C₂×#405	—	7	1	no	maximal, Sylow
29	C₂³×D₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₂²×Dic₇	C₂×C₄	1	2	yes	—
33	C₂²×Dic₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₇	C₂×C₄	1	4	yes	—
36	C₂×C₄×D₇	C₂×C₄	1	4	yes	—
37	C₂×C₄×D₇	C₂³	1	2	yes	—
38	C₂×C₇×C₈	C₂×C₄	1	4	yes	—
39	C₂×C₇×C₈	C₂×C₄	1	4	yes	—
40	C₂×C₇×C₈	C₂³	1	2	yes	—
41	C₂×C₇×C₈	C₂³	1	2	yes	—
42	C₄²×C₇	C₂×C₄	1	2	yes	—
43	C₄²×C₇	C₂³	1	2	yes	—
44	(C₂×D₇)⋊C₄	C₂×C₄	1	8	yes	—
45	Dic₇⋊C₄	C₂×C₄	1	8	yes	—
46	C₄×Dic₇	C₂×C₄	1	2	yes	—
47	C₄×Dic₇	C₂³	1	2	yes	—
48	C₂×C₇⋊C₈	C₂×C₄	1	4	yes	—
49	C₂×C₇⋊C₈	C₂×C₄	1	4	yes	—
50	C₂×C₇⋊C₈	C₂³	1	2	yes	—
51	C₂×C₇⋊C₈	C₂³	1	2	yes	—
52	#1131	C₂×C₄	1	16	yes	complemented
53	C₈×D₇	C₂×C₄	1	16	yes	complemented
54	C₂²×M₁₆	—	7	1	no	—
55	C₂³×C₈	—	7	1	no	—
56	C₂×#117	—	7	1	no	—
57	#405	—	7	8	no	—
58	C₂×#97	—	7	1	no	—
59	C₂×#97	—	7	1	no	—
60	C₂×C₄×C₈	—	7	1	no	—
61	C₂×C₄×C₈	D₇	1	1	yes	characteristic, complemented
62	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×D₇	—	2	4	no	—
64	C₂²×D₇	C₄²	1	4	yes	—
65	C₂²×D₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₇	—	2	4	no	—
67	C₂×C₄×C₇	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₇	—	2	4	no	—
73	C₂×Dic₇	C₄²	1	4	yes	—
74	C₂×Dic₇	C₂²×C₄	1	4	yes	—
75	C₂×Dic₇	C₂²×C₄	1	2	yes	—
76	C₄×D₇	C₄²	1	8	yes	—
77	C₄×D₇	C₂²×C₄	1	4	yes	—
78	C₄×D₇	C₂²×C₄	1	4	yes	—
79	C₇×C₈	C₂²×C₄	1	4	yes	—
80	C₇×C₈	C₂²×C₄	1	4	yes	—
81	C₇⋊C₈	C₂²×C₄	1	4	yes	—
82	C₇⋊C₈	C₂²×C₄	1	4	yes	—
83	C₂³×C₄	—	7	1	no	—
84	C₂×M₁₆	—	7	8	no	—
85	C₂×M₁₆	—	7	4	no	—
86	C₂²×C₈	—	7	1	no	—
87	C₂²×C₈	—	7	1	no	—
88	C₂²×C₈	—	7	8	no	—
89	C₂²×C₈	—	7	4	no	—
90	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
91	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
92	#117	—	7	4	no	—
93	#117	—	7	4	no	—
94	C₂×C₄⋊C₄	—	7	2	no	—
95	C₂×C₂²⋊C₄	—	7	2	no	—
96	C₂×C₄²	—	7	1	no	—
97	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
98	#97	—	7	4	no	—
99	#97	—	7	4	no	—
100	C₄×C₈	—	7	4	no	—
101	C₄×C₈	C₂×D₇	1	4	yes	complemented
102	C₂²×C₇	C₂×C₄²	1	2	yes	—
103	C₂²×C₇	#131	1	4	yes	—
104	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×D₇	—	2	4	no	—
106	C₂×D₇	—	2	8	no	—
107	C₂×D₇	C₂×C₄²	1	4	yes	—
108	C₄×C₇	—	2	4	no	—
109	C₄×C₇	C₂×C₄²	1	2	yes	—
110	C₄×C₇	C₂³×C₄	1	1	yes	characteristic
111	C₄×C₇	C₂³×C₄	1	1	yes	characteristic
112	Dic₇	—	2	4	no	—
113	Dic₇	C₂×C₄²	1	4	yes	—
114	C₂⁴	—	7	1	no	—
115	C₂²×C₄	—	7	1	no	—
116	C₂²×C₄	—	7	2	no	—
117	C₂²×C₄	—	7	4	no	—
118	C₂²×C₄	—	7	4	no	—
119	C₂²×C₄	—	7	2	no	—
120	C₂²×C₄	—	7	2	no	—
121	C₂²×C₄	C₄×D₇	1	2	yes	—
122	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
123	M₁₆	—	7	16	no	—
124	C₂×C₈	—	7	4	no	—
125	C₂×C₈	—	7	4	no	—
126	C₂×C₈	—	7	2	no	—
127	C₂×C₈	—	7	2	no	—
128	C₂×C₈	—	7	16	no	—
129	C₂×C₈	C₄×D₇	1	4	yes	—
130	C₂×C₈	C₄×D₇	1	4	yes	—
131	C₂×C₈	C₂²×D₇	1	2	yes	—
132	C₂×C₈	C₂²×D₇	1	2	yes	—
133	C₄⋊C₄	—	7	8	no	—
134	C₂²⋊C₄	—	7	8	no	—
135	C₄²	—	7	2	no	—
136	C₄²	—	7	2	no	—
137	C₄²	C₄×D₇	1	2	yes	—
138	C₄²	C₂²×D₇	1	2	yes	—
139	C₂×C₇	#405	1	4	yes	complemented
140	C₂×C₇	C₂²×C₄²	1	1	yes	characteristic
141	C₂×C₇	C₂×#131	1	2	yes	—
142	D₇	—	2	4	no	—
143	C₂³	—	7	4	no	—
144	C₂³	—	7	2	no	—
145	C₂³	—	14	4	no	—
146	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	7	4	no	—
149	C₂×C₄	—	7	2	no	—
150	C₂×C₄	—	7	8	no	—
151	C₂×C₄	—	7	4	no	—
152	C₂×C₄	—	7	4	no	—
153	C₂×C₄	—	7	4	no	—
154	C₂×C₄	—	14	4	no	—
155	C₂×C₄	C₂×C₄×D₇	1	4	yes	—
156	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
158	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
159	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
160	C₈	—	7	4	no	—
161	C₈	—	7	4	no	—
162	C₈	C₂×C₄×D₇	1	4	yes	—
163	C₈	C₂×C₄×D₇	1	4	yes	—
164	C₇	C₂×#405	1	1	yes	characteristic, complemented, Sylow
165	C₂²	—	7	4	no	—
166	C₂²	—	14	4	no	—
167	C₂²	—	14	8	no	—
168	C₂²	C₄²×D₇	1	2	yes	—
169	C₂²	#6464	1	4	yes	—
170	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
171	C₄	—	2	4	no	—
172	C₄	—	7	4	no	—
173	C₄	—	14	4	no	—
174	C₄	C₄²×D₇	1	2	yes	—
175	C₄	C₂²×C₄×D₇	1	1	yes	characteristic
176	C₄	C₂²×C₄×D₇	1	1	yes	characteristic
177	C₂	—	14	4	no	—
178	C₂	#89461	1	4	yes	direct product factor
179	C₂	C₂×C₄²×D₇	1	1	yes	characteristic
180	C₂	C₂×#6464	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×D₇
commutator	C₂×C₇	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₂×C₄×C₇×C₈	C₂
socle	C₂³×C₇	C₂²×C₄
prime radical	C₂³×D₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₇
hypocenter	C₇	C₂×#405
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₂×C₄×C₇×C₈	C₂	semidirect
5. C₂×#6389	C₂	semidirect
6. C₂×C₈×Dic₇	C₂	semidirect
7. C₂×#6377	C₂	semidirect
8. #89461	C₂	direct (N × H)
18. C₂×#1131	C₄	semidirect
20. C₂×C₈×D₇	C₄	semidirect
24. C₄×C₇×C₈	C₂²	semidirect
25. #6389	C₂²	semidirect
26. C₈×Dic₇	C₂²	semidirect
27. #6377	C₂²	semidirect
52. #1131	C₂×C₄	semidirect
53. C₈×D₇	C₂×C₄	semidirect
61. C₂×C₄×C₈	D₇	semidirect
101. C₄×C₈	C₂×D₇	semidirect
139. C₂×C₇	#405	semidirect
164. C₇	C₂×#405	semidirect
178. C₂	#89461	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×#1131	2
C₂²×C₈×D₇	2
C₂×#6435	2
C₂×C₄×C₇×C₈	2
C₂×#6389	2
C₂×C₈×Dic₇	2
C₂×#6377	2
#89461	2
C₂×#405	7

Subgroups — C2×#89461 #11735896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#89461 #11735896