Subgroups — #11735995

Order 896.

556 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6470	C₂	1	1	yes	maximal, characteristic
2	C₂×#6464	C₂	1	1	yes	maximal, characteristic
3	#90213	C₂	1	1	yes	maximal, characteristic
4	C₇×#405	C₂	1	1	yes	maximal, characteristic, complemented
5	M₁₆×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#6389	C₂	1	1	yes	maximal, characteristic, complemented
7	#89769	C₂	1	1	yes	maximal, characteristic, complemented
8	#89480	C₂	1	2	yes	maximal, complemented
9	D₇×#97	C₂	1	2	yes	maximal, complemented
10	#89461	C₂	1	2	yes	maximal, complemented
11	C₄×#1131	C₂	1	2	yes	maximal, complemented
12	C₂×#1157	C₂²	1	1	yes	characteristic
13	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₇×#117	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
17	C₂×#1136	C₂²	1	1	yes	characteristic
18	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#6470	C₄	1	8	yes	complemented
20	#6464	C₄	1	8	yes	complemented
21	C₂×#1131	C₂²	1	2	yes	—
22	C₂×#1131	C₂²	1	2	yes	—
23	C₂×C₈×D₇	C₂²	1	2	yes	—
24	C₂×C₈×D₇	C₂²	1	2	yes	—
25	#6456	C₄	1	2	yes	—
26	#6450	C₄	1	2	yes	—
27	(C₂²⋊D₇)⋊C₄	C₄	1	4	yes	—
28	#6435	C₂²	1	2	yes	—
29	C₄²×D₇	C₂²	1	2	yes	—
30	C₇×#97	C₂²	1	2	yes	—
31	C₄×C₇×C₈	C₂²	1	2	yes	—
32	#6389	C₂²	1	1	yes	characteristic
33	#6389	C₂²	1	1	yes	characteristic
34	#6389	C₂²	1	1	yes	characteristic
35	#6389	C₂²	1	1	yes	characteristic
36	#6389	C₂²	1	2	yes	—
37	C₈×Dic₇	C₂²	1	2	yes	—
38	#6377	C₂²	1	2	yes	—
39	C₄×C₇⋊C₈	C₂²	1	2	yes	—
40	#2875	—	7	1	no	maximal, Sylow
41	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₂²⋊D₇	C₂×C₄	1	2	yes	—
43	C₂²×Dic₇	C₂×C₄	1	2	yes	—
44	#1157	C₂×C₄	1	4	yes	—
45	#1157	C₂×C₄	1	4	yes	—
46	C₂×D₂₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₇	C₂×C₄	1	4	yes	—
48	C₂×C₄×D₇	C₂³	1	2	yes	—
49	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
50	C₇×M₁₆	C₂×C₄	1	4	yes	complemented
51	C₂×C₇×C₈	C₂×C₄	1	4	yes	complemented
52	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂³	1	2	yes	—
54	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
56	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
57	C₄²×C₇	C₂³	1	2	yes	—
58	(C₂×D₇)⋊C₄	C₂×C₄	1	4	yes	—
59	Dic₇⋊C₄	C₂×C₄	1	4	yes	—
60	C₄×Dic₇	C₂×C₄	1	4	yes	—
61	C₄×Dic₇	C₂×C₄	1	2	yes	—
62	C₄×Dic₇	C₂³	1	1	yes	characteristic
63	C₄×Dic₇	C₂³	1	1	yes	characteristic
64	#1136	C₂×C₄	1	4	yes	complemented
65	C₂×C₇⋊C₈	C₂×C₄	1	4	yes	complemented
66	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₇⋊C₈	C₂³	1	2	yes	—
68	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
69	#1131	C₂×C₄	1	8	yes	complemented
70	#1131	C₂×C₄	1	8	yes	complemented
71	C₈×D₇	C₂×C₄	1	8	yes	complemented
72	C₈×D₇	C₂×C₄	1	8	yes	complemented
73	C₂×#131	—	7	1	no	—
74	C₂×#131	—	7	1	no	—
75	C₄×#41	—	7	1	no	—
76	#405	—	7	1	no	—
77	#405	—	7	2	no	—
78	#405	—	7	2	no	—
79	#405	D₇	1	1	yes	characteristic, complemented
80	C₄×M₁₆	—	7	1	no	—
81	C₄×M₁₆	—	7	2	no	—
82	C₂×#97	—	7	1	no	—
83	C₂×#97	—	7	2	no	—
84	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×D₇	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₇	C₂²×C₄	1	4	yes	—
87	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
91	C₂²⋊D₇	C₄²	1	8	yes	—
92	C₂×Dic₇	—	4	2	no	—
93	C₂×Dic₇	C₂²×C₄	1	2	yes	—
94	C₂×Dic₇	C₂²×C₄	1	2	yes	—
95	C₂×Dic₇	C₂²×C₄	1	2	yes	—
96	D₂₈	C₄²	1	4	yes	—
97	C₄×D₇	—	4	4	no	—
98	C₄×D₇	C₂²×C₄	1	4	yes	—
99	C₄×D₇	C₂²×C₄	1	4	yes	—
100	Dic₁₄	C₄²	1	4	yes	—
101	C₇×C₈	C₂²×C₄	1	2	yes	—
102	C₇×C₈	C₂²×C₄	1	4	yes	—
103	C₇×C₈	C₂²×C₄	1	2	yes	—
104	C₇⋊C₈	C₂²×C₄	1	2	yes	—
105	C₇⋊C₈	C₂²×C₄	1	4	yes	—
106	C₇⋊C₈	C₂²×C₄	1	2	yes	—
107	C₂×#41	—	7	1	no	—
108	#131	—	7	8	no	—
109	#131	—	7	8	no	—
110	C₂×M₁₆	—	7	1	no	—
111	C₂×M₁₆	—	7	2	no	—
112	C₂×M₁₆	—	7	2	no	—
113	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
114	C₂²×C₈	—	7	1	no	—
115	C₂²×C₈	—	7	2	no	—
116	C₂²×C₈	—	7	2	no	—
117	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
118	C₄×Q₈	—	7	2	no	—
119	C₄×D₄	—	7	2	no	—
120	C₄×D₄	—	7	4	no	—
121	#117	—	7	2	no	—
122	#117	C₂×D₇	1	1	yes	characteristic
123	C₂×C₄²	—	7	1	no	—
124	C₂×C₄²	—	7	2	no	—
125	#97	—	7	1	no	—
126	#97	—	7	2	no	—
127	#97	—	7	1	no	—
128	#97	—	7	1	no	—
129	#97	—	7	1	no	—
130	#97	—	7	2	no	—
131	#97	C₂×D₇	1	2	yes	—
132	C₄×C₈	—	7	2	no	—
133	C₄×C₈	—	7	2	no	—
134	C₄×C₈	C₂×D₇	1	2	yes	—
135	C₂²×C₇	—	2	2	no	—
136	C₂²×C₇	C₂×C₄²	1	2	yes	—
137	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×D₇	—	2	4	no	—
139	C₂×D₇	C₂×C₄²	1	4	yes	—
140	C₄×C₇	—	2	4	no	—
141	C₄×C₇	C₂×C₄²	1	2	yes	—
142	C₄×C₇	C₂³×C₄	1	1	yes	characteristic
143	C₄×C₇	C₂³×C₄	1	1	yes	characteristic
144	Dic₇	—	2	2	no	—
145	Dic₇	—	2	2	no	—
146	Dic₇	C₂×C₄²	1	4	yes	—
147	#41	—	7	4	no	—
148	#41	—	7	4	no	—
149	C₂×Q₈	—	7	1	no	—
150	C₂×D₄	—	7	2	no	—
151	C₂×D₄	—	7	1	no	—
152	C₂²×C₄	—	7	2	no	—
153	C₂²×C₄	—	7	2	no	—
154	C₂²×C₄	—	7	4	no	—
155	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
156	M₁₆	—	7	4	no	—
157	M₁₆	—	7	8	no	—
158	M₁₆	—	7	8	no	—
159	M₁₆	C₄×D₇	1	4	yes	complemented
160	C₂×C₈	—	7	4	no	—
161	C₂×C₈	—	7	2	no	—
162	C₂×C₈	—	7	1	no	—
163	C₂×C₈	—	7	1	no	—
164	C₂×C₈	—	7	8	no	—
165	C₂×C₈	—	7	8	no	—
166	C₂×C₈	C₄×D₇	1	4	yes	complemented
167	C₂×C₈	C₂²×D₇	1	2	yes	—
168	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
169	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
170	C₄⋊C₄	—	7	4	no	—
171	C₄⋊C₄	C₄×D₇	1	2	yes	—
172	C₂²⋊C₄	—	7	4	no	—
173	C₂²⋊C₄	C₄×D₇	1	2	yes	—
174	C₄²	—	7	4	no	—
175	C₄²	—	7	2	no	—
176	C₄²	—	7	1	no	—
177	C₄²	—	7	1	no	—
178	C₄²	C₂²×D₇	1	2	yes	—
179	C₂×C₇	—	2	2	no	—
180	C₂×C₇	C₂²×C₄²	1	1	yes	characteristic
181	C₂×C₇	#568	1	2	yes	—
182	D₇	—	2	4	no	—
183	C₂³	—	7	2	no	—
184	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
185	Q₈	—	7	4	no	—
186	D₄	—	7	8	no	—
187	D₄	—	7	4	no	—
188	C₂×C₄	—	7	2	no	—
189	C₂×C₄	—	7	4	no	—
190	C₂×C₄	—	7	4	no	—
191	C₂×C₄	—	7	2	no	—
192	C₂×C₄	—	7	2	no	—
193	C₂×C₄	—	28	4	no	—
194	C₂×C₄	—	28	2	no	—
195	C₂×C₄	C₂×C₄×D₇	1	4	yes	—
196	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
199	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
200	C₈	—	7	2	no	—
201	C₈	—	7	4	no	—
202	C₈	—	7	2	no	—
203	C₈	C₂×C₄×D₇	1	2	yes	—
204	C₈	C₂×C₄×D₇	1	4	yes	—
205	C₈	C₂×C₄×D₇	1	2	yes	—
206	C₇	#2875	1	1	yes	characteristic, complemented, Sylow
207	C₂²	—	2	2	no	—
208	C₂²	—	7	4	no	—
209	C₂²	—	14	4	no	—
210	C₂²	C₄²×D₇	1	2	yes	—
211	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	7	4	no	—
214	C₄	—	14	2	no	—
215	C₄	—	14	2	no	—
216	C₄	C₄²×D₇	1	2	yes	—
217	C₄	C₂²×C₄×D₇	1	1	yes	characteristic
218	C₄	C₂²×C₄×D₇	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	—	14	4	no	—
221	C₂	C₂×C₄²×D₇	1	1	yes	characteristic
222	C₂	#90441	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator	C₂×C₇	C₂²×C₄²
Fitting	C₇×#405	C₂
socle	C₂²×C₇	C₂³×C₄
prime radical	C₂×#1157	C₂²
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#405	D₇
hypocenter	C₇	#2875
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
4. C₇×#405	C₂	semidirect
5. M₁₆×Dic₇	C₂	semidirect
6. C₂×#6389	C₂	semidirect
7. #89769	C₂	semidirect
8. #89480	C₂	semidirect
9. D₇×#97	C₂	semidirect
10. #89461	C₂	semidirect
11. C₄×#1131	C₂	semidirect
19. #6470	C₄	semidirect
20. #6464	C₄	semidirect
50. C₇×M₁₆	C₂×C₄	semidirect
51. C₂×C₇×C₈	C₂×C₄	semidirect
64. #1136	C₂×C₄	semidirect
65. C₂×C₇⋊C₈	C₂×C₄	semidirect
69. #1131	C₂×C₄	semidirect
70. #1131	C₂×C₄	semidirect
71. C₈×D₇	C₂×C₄	semidirect
72. C₈×D₇	C₂×C₄	semidirect
79. #405	D₇	semidirect
159. M₁₆	C₄×D₇	semidirect
166. C₂×C₈	C₄×D₇	semidirect
206. C₇	#2875	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6470	2
C₂×#6464	2
#90213	2
C₇×#405	2
M₁₆×Dic₇	2
C₂×#6389	2
#89769	2
#89480	2
D₇×#97	2
#89461	2
C₄×#1131	2
#2875	7