Subgroups — #11731782

Order 896.

452 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6452	C₂	1	2	yes	maximal
2	C₇×#384	C₂	1	1	yes	maximal, characteristic
3	#89757	C₂	1	1	yes	maximal, characteristic
4	#89756	C₂	1	2	yes	maximal
5	#89752	C₂	1	2	yes	maximal
6	#89748	C₂	1	1	yes	maximal, characteristic
7	#89748	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
9	#89744	C₂	1	2	yes	maximal
10	C₄×Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic
11	#89700	C₂	1	1	yes	maximal, characteristic
12	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
22	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
23	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
24	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
25	C₂×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₇	C₂²	1	2	yes	—
28	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
30	#6452	—	2	8	no	—
31	C₇⋊#95	C₂²	1	2	yes	—
32	C₇⋊#95	C₂²	1	1	yes	characteristic
33	C₇⋊#95	C₂²	1	1	yes	characteristic
34	C₇⋊#95	C₂²	1	2	yes	—
35	C₇⋊#95	C₂²	1	2	yes	—
36	#2552	—	7	1	no	maximal, Sylow
37	C₂²×C₄×C₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂²×Dic₇	C₂³	1	2	yes	—
43	C₂²×Dic₇	C₂³	1	2	yes	—
44	C₂²×Dic₇	C₂³	1	1	yes	characteristic
45	C₂²×Dic₇	C₂³	1	1	yes	characteristic
46	C₂²×Dic₇	C₂³	1	1	yes	characteristic
47	C₂²×Dic₇	C₂³	1	1	yes	characteristic
48	C₇×C₄⋊C₄	—	2	4	no	—
49	C₇×C₄⋊C₄	—	2	2	no	—
50	C₇×C₄⋊C₄	—	2	2	no	—
51	C₇×C₄⋊C₄	—	4	2	no	—
52	C₄²×C₇	—	2	2	no	—
53	C₄⋊Dic₇	—	2	2	no	—
54	C₄⋊Dic₇	—	2	2	no	—
55	C₄⋊Dic₇	—	2	4	no	—
56	Dic₇⋊C₄	—	2	4	no	—
57	Dic₇⋊C₄	—	2	4	no	—
58	Dic₇⋊C₄	—	2	4	no	—
59	Dic₇⋊C₄	—	4	2	no	—
60	Dic₇⋊C₄	—	4	2	no	—
61	Dic₇⋊C₄	D₄	1	4	yes	—
62	Dic₇⋊C₄	Q₈	1	4	yes	—
63	C₄×Dic₇	—	2	2	no	—
64	C₄×Dic₇	—	2	4	no	—
65	C₄×Dic₇	Q₈	1	4	yes	—
66	C₂×#125	—	7	2	no	—
67	#398	—	7	2	no	—
68	#398	—	7	2	no	—
69	#389	—	7	2	no	—
70	#389	—	7	1	no	—
71	#389	—	7	1	no	—
72	#384	—	7	1	no	—
73	#384	—	7	1	no	—
74	#384	D₇	1	1	yes	characteristic
75	C₄×C₄⋊C₄	—	7	1	no	—
76	C₄×C₄⋊C₄	—	7	1	no	—
77	C₂³×C₇	C₂⁴	1	1	yes	characteristic
78	C₂×C₄×C₇	—	2	1	no	—
79	C₂×C₄×C₇	—	2	1	no	—
80	C₂×C₄×C₇	—	2	1	no	—
81	C₂×C₄×C₇	—	2	1	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	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
90	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₇	#41	1	2	yes	—
92	C₂×C₄×C₇	#41	1	2	yes	—
93	C₂×Dic₇	—	2	1	no	—
94	C₂×Dic₇	—	2	1	no	—
95	C₂×Dic₇	—	2	1	no	—
96	C₂×Dic₇	—	2	1	no	—
97	C₂×Dic₇	—	2	1	no	—
98	C₂×Dic₇	—	2	1	no	—
99	C₂×Dic₇	—	2	2	no	—
100	C₂×Dic₇	—	2	2	no	—
101	C₂×Dic₇	—	2	2	no	—
102	C₂×Dic₇	—	2	2	no	—
103	C₂×Dic₇	—	2	2	no	—
104	C₂×Dic₇	—	2	2	no	—
105	C₂×Dic₇	C₂×D₄	1	2	yes	—
106	C₂×Dic₇	C₂×D₄	1	2	yes	—
107	C₂×Dic₇	C₂×Q₈	1	2	yes	—
108	C₂×Dic₇	C₂×Q₈	1	2	yes	—
109	C₂×Dic₇	C₂×Q₈	1	2	yes	—
110	C₂×Dic₇	C₂×Q₈	1	2	yes	—
111	#125	—	14	8	no	—
112	C₂×C₄⋊C₄	—	7	1	no	—
113	C₂×C₄⋊C₄	—	7	1	no	—
114	C₂×C₄⋊C₄	—	7	2	no	—
115	C₂×C₄⋊C₄	—	7	2	no	—
116	C₂×C₄⋊C₄	—	7	2	no	—
117	C₂×C₄⋊C₄	—	7	2	no	—
118	C₂×C₄⋊C₄	—	7	1	no	—
119	C₂×C₄⋊C₄	—	7	1	no	—
120	C₂×C₄⋊C₄	—	7	2	no	—
121	C₂×C₄⋊C₄	—	7	2	no	—
122	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
123	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
126	C₂×C₄²	—	7	1	no	—
127	C₂×C₄²	—	7	2	no	—
128	C₂×C₄²	—	7	1	no	—
129	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
130	#95	—	7	1	no	—
131	#95	—	7	1	no	—
132	#95	—	7	2	no	—
133	#95	—	7	2	no	—
134	#95	—	7	2	no	—
135	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
136	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
137	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
138	C₂²×C₇	C₂×#41	1	1	yes	characteristic
139	C₂²×C₇	C₂×#41	1	1	yes	characteristic
140	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
142	C₄×C₇	—	2	2	no	—
143	C₄×C₇	—	2	2	no	—
144	C₄×C₇	—	4	2	no	—
145	C₄×C₇	—	4	2	no	—
146	C₄×C₇	#125	1	4	yes	complemented
147	Dic₇	—	2	2	no	—
148	Dic₇	—	2	2	no	—
149	Dic₇	—	2	2	no	—
150	Dic₇	—	4	2	no	—
151	Dic₇	—	4	2	no	—
152	Dic₇	C₄⋊Q₈	1	4	yes	—
153	C₂²×C₄	—	7	1	no	—
154	C₂²×C₄	—	7	1	no	—
155	C₂²×C₄	—	7	1	no	—
156	C₂²×C₄	—	7	1	no	—
157	C₂²×C₄	—	7	2	no	—
158	C₂²×C₄	—	7	2	no	—
159	C₂²×C₄	C₂²×D₇	1	2	yes	—
160	C₂²×C₄	C₂²×D₇	1	2	yes	—
161	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	7	4	no	—
169	C₄⋊C₄	—	7	4	no	—
170	C₄⋊C₄	—	14	4	no	—
171	C₄⋊C₄	—	14	4	no	—
172	C₄⋊C₄	—	14	4	no	—
173	C₄⋊C₄	—	14	2	no	—
174	C₄⋊C₄	—	14	4	no	—
175	C₄⋊C₄	—	14	2	no	—
176	C₄⋊C₄	—	28	2	no	—
177	C₄⋊C₄	—	28	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	7	4	no	—
180	C₄²	—	14	4	no	—
181	C₄²	—	14	2	no	—
182	C₂×C₇	C₂×#125	1	1	yes	characteristic
183	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
184	C₂×C₇	#544	1	1	yes	characteristic
185	C₂×C₇	#547	1	1	yes	characteristic
186	C₂×C₇	#550	1	1	yes	characteristic
187	C₂×C₇	#554	1	1	yes	characteristic
188	C₂×C₇	#557	1	1	yes	characteristic
189	C₂³	C₂³×D₇	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	7	2	no	—
201	C₂×C₄	—	7	2	no	—
202	C₂×C₄	—	7	2	no	—
203	C₂×C₄	—	7	2	no	—
204	C₂×C₄	—	7	2	no	—
205	C₂×C₄	—	7	2	no	—
206	C₂×C₄	—	14	1	no	—
207	C₂×C₄	—	14	1	no	—
208	C₂×C₄	—	14	1	no	—
209	C₂×C₄	—	14	1	no	—
210	C₂×C₄	—	14	1	no	—
211	C₂×C₄	—	14	1	no	—
212	C₂×C₄	—	14	2	no	—
213	C₂×C₄	—	14	2	no	—
214	C₂×C₄	—	14	2	no	—
215	C₂×C₄	—	14	2	no	—
216	C₂×C₄	—	14	2	no	—
217	C₂×C₄	—	14	2	no	—
218	C₂×C₄	#1157	1	2	yes	—
219	C₂×C₄	D₄×D₇	1	2	yes	—
220	C₂×C₄	#1159	1	2	yes	—
221	C₂×C₄	Q₈×D₇	1	2	yes	—
222	C₂×C₄	Q₈×D₇	1	2	yes	—
223	C₇	#2552	1	1	yes	characteristic, complemented, Sylow
224	C₂²	C₂×#1157	1	1	yes	characteristic
225	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
226	C₂²	C₂×#1159	1	1	yes	characteristic
227	C₂²	#6548	1	1	yes	characteristic
228	C₂²	C₂×Q₈×D₇	1	1	yes	characteristic
229	C₂²	C₂×Q₈×D₇	1	1	yes	characteristic
230	C₂²	#6551	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	7	4	no	—
236	C₄	—	14	2	no	—
237	C₄	—	14	2	no	—
238	C₄	—	14	2	no	—
239	C₄	—	28	2	no	—
240	C₄	—	28	2	no	—
241	C₄	#6452	1	4	yes	complemented
242	C₂	C₂×#6452	1	1	yes	characteristic
243	C₂	#90198	1	1	yes	characteristic
244	C₂	#90202	1	1	yes	characteristic
245	C₂	#90409	1	1	yes	characteristic
246	C₂	#90412	1	1	yes	characteristic
247	C₂	D₇×C₄⋊Q₈	1	1	yes	characteristic
248	C₂	#90420	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#384	C₂
hypercenter	#384	D₇
hypocenter	C₇	#2552
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
146. C₄×C₇	#125	semidirect
223. C₇	#2552	semidirect
241. C₄	#6452	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6452	2
C₇×#384	2
#89757	2
#89756	2
#89752	2
#89748	2
C₄⋊C₄×Dic₇	2
#89744	2
C₄×Dic₇⋊C₄	2
#89700	2
#2552	7