Subgroups — #11733638

Order 896.

400 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#389	C₂	1	1	yes	maximal, characteristic
2	#89757	C₂	1	1	yes	maximal, characteristic
3	#89753	C₂	1	2	yes	maximal
4	#89750	C₂	1	2	yes	maximal
5	#89749	C₂	1	1	yes	maximal, characteristic
6	#89749	C₂	1	1	yes	maximal, characteristic
7	#89745	C₂	1	1	yes	maximal, characteristic
8	C₄²×Dic₇	C₂	1	1	yes	maximal, characteristic
9	#89700	C₂	1	1	yes	maximal, characteristic
10	#89425	C₂	1	2	yes	maximal
11	#89424	C₂	1	2	yes	maximal
12	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
27	C₇×#95	C₂²	1	2	yes	—
28	C₇⋊#95	C₂²	1	2	yes	—
29	C₇⋊#95	C₂²	1	2	yes	—
30	C₇⋊#95	C₂²	1	2	yes	—
31	C₇⋊#95	C₂²	1	2	yes	—
32	C₇⋊#95	C₂²	1	2	yes	—
33	C₇⋊#95	C₂²	1	2	yes	—
34	C₇⋊#95	C₂²	1	2	yes	—
35	#2864	—	7	1	no	maximal, Sylow
36	C₂²×C₄×C₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂²×Dic₇	C₂³	1	2	yes	—
42	C₂²×Dic₇	C₂³	1	2	yes	—
43	C₂²×Dic₇	C₂³	1	1	yes	characteristic
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	C₇×C₄⋊C₄	—	2	2	no	—
48	C₇×C₄⋊C₄	—	2	4	no	—
49	C₇×C₄⋊C₄	—	2	2	no	—
50	C₄²×C₇	—	2	2	no	—
51	C₄⋊Dic₇	—	2	2	no	—
52	C₄⋊Dic₇	—	2	2	no	—
53	C₄⋊Dic₇	—	2	4	no	—
54	Dic₇⋊C₄	—	2	4	no	—
55	Dic₇⋊C₄	—	2	4	no	—
56	C₄×Dic₇	—	2	2	no	—
57	C₄×Dic₇	—	2	2	no	—
58	C₄×Dic₇	—	2	2	no	—
59	C₄×Dic₇	—	2	2	no	—
60	C₄×Dic₇	—	2	2	no	—
61	C₄×Dic₇	Q₈	1	4	yes	—
62	#389	—	7	1	no	—
63	#389	—	7	1	no	—
64	#389	—	7	1	no	—
65	#389	D₇	1	1	yes	characteristic
66	#387	—	7	2	no	—
67	#387	—	7	2	no	—
68	#387	—	7	2	no	—
69	#387	—	7	2	no	—
70	#382	—	7	1	no	—
71	#382	—	7	1	no	—
72	C₄³	—	7	1	no	—
73	C₂³×C₇	C₂⁴	1	1	yes	characteristic
74	C₂×C₄×C₇	—	2	1	no	—
75	C₂×C₄×C₇	—	2	1	no	—
76	C₂×C₄×C₇	—	2	1	no	—
77	C₂×C₄×C₇	—	2	1	no	—
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	2	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	C₂×Q₈	1	2	yes	—
85	C₂×C₄×C₇	#41	1	2	yes	—
86	C₂×C₄×C₇	#41	1	2	yes	—
87	C₂×C₄×C₇	#41	1	2	yes	—
88	C₂×C₄×C₇	#41	1	2	yes	—
89	C₂×Dic₇	—	2	1	no	—
90	C₂×Dic₇	—	2	1	no	—
91	C₂×Dic₇	—	2	1	no	—
92	C₂×Dic₇	—	2	1	no	—
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	2	no	—
98	C₂×Dic₇	—	2	2	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	C₂×Q₈	1	2	yes	—
104	C₂×Dic₇	C₂×Q₈	1	2	yes	—
105	C₂×Dic₇	#41	1	2	yes	—
106	C₂×Dic₇	#41	1	2	yes	—
107	C₂×C₄⋊C₄	—	7	2	no	—
108	C₂×C₄⋊C₄	—	7	2	no	—
109	C₂×C₄⋊C₄	—	7	1	no	—
110	C₂×C₄⋊C₄	—	7	1	no	—
111	C₂×C₄⋊C₄	—	7	2	no	—
112	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
115	C₂×C₄²	—	7	1	no	—
116	C₂×C₄²	—	7	1	no	—
117	C₂×C₄²	—	7	1	no	—
118	C₂×C₄²	—	7	1	no	—
119	C₂×C₄²	—	7	1	no	—
120	C₂×C₄²	—	7	1	no	—
121	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
122	#95	—	7	2	no	—
123	#95	—	7	2	no	—
124	#95	—	7	2	no	—
125	#95	—	7	2	no	—
126	#95	—	7	2	no	—
127	#95	—	7	2	no	—
128	#95	—	7	2	no	—
129	#95	C₂×D₇	1	2	yes	—
130	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
131	C₂²×C₇	C₂×#41	1	1	yes	characteristic
132	C₂²×C₇	C₂×#41	1	1	yes	characteristic
133	C₂²×C₇	C₂×#41	1	1	yes	characteristic
134	C₂²×C₇	C₂×#41	1	1	yes	characteristic
135	C₂²×C₇	C₂×#41	1	1	yes	characteristic
136	C₂²×C₇	C₂×#41	1	1	yes	characteristic
137	C₄×C₇	—	2	2	no	—
138	C₄×C₇	—	2	2	no	—
139	C₄×C₇	—	4	2	no	—
140	C₄×C₇	—	4	2	no	—
141	C₄×C₇	#125	1	4	yes	—
142	Dic₇	—	2	2	no	—
143	Dic₇	—	2	2	no	—
144	Dic₇	—	2	2	no	—
145	Dic₇	—	2	2	no	—
146	Dic₇	—	4	2	no	—
147	Dic₇	—	4	2	no	—
148	C₂²×C₄	—	7	1	no	—
149	C₂²×C₄	—	7	1	no	—
150	C₂²×C₄	—	7	1	no	—
151	C₂²×C₄	—	7	1	no	—
152	C₂²×C₄	—	7	2	no	—
153	C₂²×C₄	—	7	2	no	—
154	C₂²×C₄	C₂²×D₇	1	2	yes	—
155	C₂²×C₄	C₂²×D₇	1	2	yes	—
156	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	14	4	no	—
163	C₄⋊C₄	—	14	4	no	—
164	C₄⋊C₄	—	14	2	no	—
165	C₄⋊C₄	—	14	2	no	—
166	C₄⋊C₄	—	14	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	7	4	no	—
169	C₄²	—	14	2	no	—
170	C₄²	—	14	2	no	—
171	C₄²	—	14	2	no	—
172	C₄²	—	14	2	no	—
173	C₄²	—	14	2	no	—
174	C₂×C₇	C₂×#125	1	1	yes	characteristic
175	C₂×C₇	#529	1	1	yes	characteristic
176	C₂×C₇	#529	1	1	yes	characteristic
177	C₂×C₇	#529	1	1	yes	characteristic
178	C₂×C₇	#529	1	1	yes	characteristic
179	C₂×C₇	#533	1	1	yes	characteristic
180	C₂×C₇	#533	1	1	yes	characteristic
181	C₂³	C₂³×D₇	1	1	yes	characteristic
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	7	2	no	—
193	C₂×C₄	—	7	2	no	—
194	C₂×C₄	—	7	2	no	—
195	C₂×C₄	—	7	2	no	—
196	C₂×C₄	—	14	1	no	—
197	C₂×C₄	—	14	1	no	—
198	C₂×C₄	—	14	1	no	—
199	C₂×C₄	—	14	1	no	—
200	C₂×C₄	—	14	1	no	—
201	C₂×C₄	—	14	1	no	—
202	C₂×C₄	—	14	1	no	—
203	C₂×C₄	—	14	1	no	—
204	C₂×C₄	—	14	2	no	—
205	C₂×C₄	—	14	2	no	—
206	C₂×C₄	—	14	2	no	—
207	C₂×C₄	—	14	2	no	—
208	C₂×C₄	—	14	2	no	—
209	C₂×C₄	—	14	2	no	—
210	C₂×C₄	#1157	1	2	yes	—
211	C₂×C₄	#1159	1	2	yes	—
212	C₂×C₄	#1159	1	2	yes	—
213	C₂×C₄	Q₈×D₇	1	2	yes	—
214	C₂×C₄	#1161	1	2	yes	—
215	C₇	#2864	1	1	yes	characteristic, complemented, Sylow
216	C₂²	C₂×#1157	1	1	yes	characteristic
217	C₂²	C₂×#1159	1	1	yes	characteristic
218	C₂²	C₂×#1159	1	1	yes	characteristic
219	C₂²	C₂×Q₈×D₇	1	1	yes	characteristic
220	C₂²	C₂×#1161	1	1	yes	characteristic
221	C₂²	D₇×#41	1	1	yes	characteristic
222	C₂²	D₇×#41	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	14	2	no	—
228	C₄	—	14	2	no	—
229	C₄	—	14	2	no	—
230	C₄	—	14	2	no	—
231	C₄	—	28	2	no	—
232	C₄	—	28	2	no	—
233	C₄	#6452	1	4	yes	—
234	C₂	C₂×#6452	1	1	yes	characteristic
235	C₂	#90248	1	1	yes	characteristic
236	C₂	#90271	1	1	yes	characteristic
237	C₂	#90291	1	1	yes	characteristic
238	C₂	#90315	1	1	yes	characteristic
239	C₂	#90379	1	1	yes	characteristic
240	C₂	#90419	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#389	C₂
hypercenter	#389	D₇
hypocenter	C₇	#2864
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
215. C₇	#2864	semidirect

Maximal subgroups

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

subgroup	index
C₇×#389	2
#89757	2
#89753	2
#89750	2
#89749	2
#89745	2
C₄²×Dic₇	2
#89700	2
#89425	2
#89424	2
#2864	7