Subgroups — C₂×#89783 #11739743

Order 896.

564 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×#1136	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#89783	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#6377	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₄	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₄×C₇×D₄	C₄	1	4	yes	—
13	C₄×C₇×D₄	C₂²	1	4	yes	—
14	C₂×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
15	C₂×C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic
16	C₂×C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic
17	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
18	C₂×#1136	—	2	4	no	—
19	C₂×#1136	C₂²	1	4	yes	complemented
20	C₂²×C₇⋊C₈	—	2	4	no	—
21	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
24	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
25	C₂²×C₇⋊C₈	C₂²	1	4	yes	complemented
26	C₂²⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
27	C₂²⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
28	C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
29	#6377	C₂²	1	4	yes	complemented
30	C₂×#435	—	7	1	no	maximal, Sylow
31	C₂⁴×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂⁴×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₇×D₄	—	2	4	no	—
34	C₂×C₇×D₄	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₇	—	2	2	no	—
36	C₂²×C₄×C₇	—	2	2	no	—
37	C₂²×C₄×C₇	—	2	2	no	—
38	C₂²×C₄×C₇	—	2	2	no	—
39	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
45	C₂²×C₄×C₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
49	C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
50	C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
51	C₄²×C₇	C₂×C₄	1	2	yes	—
52	C₄²×C₇	C₂³	1	2	yes	—
53	#1136	—	2	8	no	—
54	C₂×C₇⋊C₈	—	2	2	no	—
55	C₂×C₇⋊C₈	—	2	2	no	—
56	C₂×C₇⋊C₈	—	2	2	no	—
57	C₂×C₇⋊C₈	—	2	8	no	—
58	C₂×C₇⋊C₈	D₄	1	4	yes	complemented
59	C₂×C₇⋊C₈	C₂³	1	2	yes	—
60	C₂×C₇⋊C₈	C₂³	1	2	yes	—
61	C₂×C₇⋊C₈	C₂³	1	2	yes	—
62	C₂×C₇⋊C₈	C₂³	1	2	yes	—
63	C₂²×M₁₆	—	7	1	no	—
64	C₂³×C₈	—	7	1	no	—
65	C₂×C₄×D₄	D₇	1	1	yes	characteristic
66	#435	—	7	8	no	—
67	C₂×C₄⋊C₈	—	7	1	no	—
68	C₂×C₂²⋊C₈	—	7	1	no	—
69	C₂×C₂²⋊C₈	—	7	1	no	—
70	C₂×#97	—	7	1	no	—
71	C₂³×C₇	—	2	2	no	—
72	C₂³×C₇	—	2	2	no	—
73	C₂³×C₇	—	2	2	no	—
74	C₂³×C₇	—	2	2	no	—
75	C₂³×C₇	—	2	2	no	—
76	C₂³×C₇	M₁₆	1	4	yes	complemented
77	C₂³×C₇	C₂²×C₄	1	2	yes	—
78	C₂³×C₇	C₂²×C₄	1	2	yes	—
79	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
80	C₇×D₄	—	2	8	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	4	no	—
86	C₂×C₄×C₇	—	2	4	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₇	#41	1	2	yes	—
98	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
99	C₇⋊C₈	—	2	2	no	—
100	C₇⋊C₈	—	2	2	no	—
101	C₇⋊C₈	—	2	2	no	—
102	C₇⋊C₈	C₂×D₄	1	4	yes	complemented
103	C₂²×D₄	Dic₇	1	1	yes	characteristic
104	C₂³×C₄	C₂×D₇	1	1	yes	characteristic
105	C₂³×C₄	C₂×D₇	1	1	yes	characteristic
106	C₂×M₁₆	—	7	4	no	—
107	C₂×M₁₆	—	14	4	no	—
108	C₂²×C₈	—	7	1	no	—
109	C₂²×C₈	—	7	4	no	—
110	C₂²×C₈	—	7	1	no	—
111	C₂²×C₈	—	7	1	no	—
112	C₂²×C₈	—	7	1	no	—
113	C₂²×C₈	—	14	4	no	—
114	C₄×D₄	Dic₇	1	4	yes	—
115	C₄×D₄	C₂×D₇	1	4	yes	—
116	C₂×C₄⋊C₄	Dic₇	1	1	yes	characteristic
117	C₂×C₂²⋊C₄	Dic₇	1	1	yes	characteristic
118	C₂×C₂²⋊C₄	Dic₇	1	1	yes	characteristic
119	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
120	C₄⋊C₈	—	7	4	no	—
121	C₂²⋊C₈	—	7	4	no	—
122	C₂²⋊C₈	—	7	4	no	—
123	#97	—	7	4	no	—
124	C₂²×C₇	—	2	2	no	—
125	C₂²×C₇	—	2	2	no	—
126	C₂²×C₇	—	2	8	no	—
127	C₂²×C₇	—	2	2	no	—
128	C₂²×C₇	—	2	2	no	—
129	C₂²×C₇	—	2	2	no	—
130	C₂²×C₇	—	4	4	no	—
131	C₂²×C₇	C₄×D₄	1	2	yes	—
132	C₂²×C₇	C₂×M₁₆	1	2	yes	—
133	C₂²×C₇	C₂×M₁₆	1	4	yes	complemented
134	C₂²×C₇	#131	1	2	yes	—
135	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
136	C₄×C₇	—	2	2	no	—
137	C₄×C₇	—	2	2	no	—
138	C₄×C₇	—	2	2	no	—
139	C₄×C₇	—	2	2	no	—
140	C₄×C₇	C₄×D₄	1	2	yes	—
141	C₄×C₇	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₇	C₂×#41	1	1	yes	characteristic
143	C₂⁴	C₂×Dic₇	1	1	yes	characteristic
144	C₂⁴	C₂×Dic₇	1	1	yes	characteristic
145	C₂×D₄	—	2	4	no	—
146	C₂×D₄	C₂×Dic₇	1	4	yes	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
152	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
153	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
154	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
155	C₂²×C₄	C₂×Dic₇	1	2	yes	—
156	C₂²×C₄	C₂×Dic₇	1	2	yes	—
157	C₂²×C₄	C₂²×D₇	1	2	yes	—
158	C₂²×C₄	C₂²×D₇	1	2	yes	—
159	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
160	M₁₆	—	14	8	no	—
161	C₂×C₈	—	7	4	no	—
162	C₂×C₈	—	7	2	no	—
163	C₂×C₈	—	7	2	no	—
164	C₂×C₈	—	7	2	no	—
165	C₂×C₈	—	7	2	no	—
166	C₂×C₈	—	14	8	no	—
167	C₂×C₈	—	14	2	no	—
168	C₂×C₈	—	14	2	no	—
169	C₂×C₈	—	14	2	no	—
170	C₄⋊C₄	C₂×Dic₇	1	4	yes	—
171	C₂²⋊C₄	C₂×Dic₇	1	4	yes	—
172	C₂²⋊C₄	C₂×Dic₇	1	4	yes	—
173	C₄²	C₂×Dic₇	1	2	yes	—
174	C₄²	C₂²×D₇	1	2	yes	—
175	C₂×C₇	—	2	4	no	—
176	C₂×C₇	—	4	2	no	—
177	C₂×C₇	#435	1	4	yes	complemented
178	C₂×C₇	C₂×C₄×D₄	1	1	yes	characteristic
179	C₂×C₇	C₂²×M₁₆	1	1	yes	characteristic
180	C₂×C₇	C₂×#131	1	1	yes	characteristic
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	#1136	1	4	yes	complemented
187	C₂³	C₂²×Dic₇	1	2	yes	—
188	C₂³	C₂²×Dic₇	1	2	yes	—
189	C₂³	C₂²×Dic₇	1	1	yes	characteristic
190	D₄	—	2	8	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	D₄×D₇	1	2	yes	—
202	C₂×C₄	#1159	1	2	yes	—
203	C₂×C₄	C₂²×Dic₇	1	2	yes	—
204	C₂×C₄	C₂²×Dic₇	1	2	yes	—
205	C₂×C₄	C₂²×Dic₇	1	2	yes	—
206	C₂×C₄	C₂²×Dic₇	1	2	yes	—
207	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
209	C₈	—	7	4	no	—
210	C₈	—	14	2	no	—
211	C₈	—	14	2	no	—
212	C₈	—	14	2	no	—
213	C₇	C₂×#435	1	1	yes	characteristic, complemented, Sylow
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	C₂×#1136	1	2	yes	—
222	C₂²	C₂×#1136	1	4	yes	complemented
223	C₂²	D₄×Dic₇	1	2	yes	—
224	C₂²	#6511	1	2	yes	—
225	C₂²	C₂³×Dic₇	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	D₄×Dic₇	1	2	yes	—
231	C₄	C₂×D₄×D₇	1	1	yes	characteristic
232	C₄	C₂×#1159	1	1	yes	characteristic
233	C₂	—	2	4	no	—
234	C₂	—	4	2	no	—
235	C₂	#89783	1	4	yes	direct product factor
236	C₂	C₂²×#1136	1	1	yes	characteristic
237	C₂	C₂×D₄×Dic₇	1	1	yes	characteristic
238	C₂	C₂×#6511	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₇
commutator	C₂²×C₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₂×C₄×C₇×D₄	C₂
socle	C₂³×C₇	C₂²×C₄
prime radical	C₂²×C₇×D₄	C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	C₂×C₄×D₄	D₇
hypocenter	C₇	C₂×#435
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#1136	C₂	semidirect
3. C₂³×C₇⋊C₈	C₂	semidirect
4. C₂×C₂²⋊(C₇⋊C₈)	C₂	semidirect
5. C₂×C₂²⋊(C₇⋊C₈)	C₂	semidirect
6. #89783	C₂	direct (N × H)
7. C₂×C₄⋊(C₇⋊C₈)	C₂	semidirect
8. C₂×#6377	C₂	semidirect
19. C₂×#1136	C₂²	semidirect
25. C₂²×C₇⋊C₈	C₂²	semidirect
26. C₂²⋊(C₇⋊C₈)	C₂²	semidirect
27. C₂²⋊(C₇⋊C₈)	C₂²	semidirect
28. C₄⋊(C₇⋊C₈)	C₂²	semidirect
29. #6377	C₂²	semidirect
58. C₂×C₇⋊C₈	D₄	semidirect
76. C₂³×C₇	M₁₆	semidirect
102. C₇⋊C₈	C₂×D₄	semidirect
133. C₂²×C₇	C₂×M₁₆	semidirect
177. C₂×C₇	#435	semidirect
186. C₂³	#1136	semidirect
213. C₇	C₂×#435	semidirect
222. C₂²	C₂×#1136	semidirect
235. C₂	#89783	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×C₄×C₇×D₄	2
C₂²×#1136	2
C₂³×C₇⋊C₈	2
C₂×C₂²⋊(C₇⋊C₈)	2
#89783	2
C₂×C₄⋊(C₇⋊C₈)	2
C₂×#6377	2
C₂×#435	7

Subgroups — C2×#89783 #11739743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#89783 #11739743