Subgroups — #13263515

Order 1792.

394 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2931	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#6386	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725567	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×#131	—	2	1	no	—
6	C₇×#532	C₄	1	1	yes	characteristic
7	C₇×#436	—	2	1	no	—
8	C₇×#436	C₂²	1	2	yes	—
9	C₇×C₈×D₄	—	2	1	no	—
10	C₇×C₈×D₄	C₄	1	2	yes	—
11	C₇×#423	C₄	1	1	yes	characteristic
12	C₇×#408	—	2	1	no	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6386	—	2	1	no	—
15	C₂²×C₇⋊C₁₆	—	2	1	no	—
16	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
17	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
18	#89256	C₂²	1	2	yes	—
19	C₄×C₇⋊C₁₆	—	2	1	no	—
20	#11586	—	7	1	no	maximal, Sylow
21	C₂×C₇×#41	—	2	1	no	—
22	C₇×#131	—	4	2	no	—
23	C₇×#131	—	4	2	no	—
24	C₂×C₇×M₁₆	—	2	1	no	—
25	C₂×C₇×M₁₆	—	2	2	no	—
26	C₂²×C₇×C₈	—	2	2	no	—
27	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
30	C₇×#127	C₈	1	1	yes	characteristic
31	C₇×#124	—	2	1	no	—
32	C₇×#121	—	2	1	no	—
33	C₇×#121	C₈	1	2	yes	—
34	C₄×C₇×D₄	—	2	1	no	—
35	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₇×C₄⋊C₈	—	2	1	no	—
38	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₇×C₂²⋊C₈	—	2	2	no	—
40	C₇×C₂²⋊C₈	—	2	2	no	—
41	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₇×C₈	D₄	1	1	yes	characteristic
43	C₄×C₇×C₈	D₄	1	1	yes	characteristic
44	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
45	#6386	—	2	4	no	—
46	C₂×C₇⋊C₁₆	—	2	2	no	—
47	C₂×C₇⋊C₁₆	—	2	2	no	—
48	C₂×C₇⋊C₁₆	—	4	2	no	—
49	#2931	D₇	1	1	yes	characteristic
50	C₂×C₄⋊C₁₆	—	7	1	no	—
51	C₄×M₃₂	—	7	1	no	—
52	#1330	—	7	4	no	—
53	C₇×#41	—	4	2	no	—
54	C₇×#41	—	4	2	no	—
55	C₂×C₇×Q₈	—	2	1	no	—
56	C₂×C₇×D₄	—	2	2	no	—
57	C₂×C₇×D₄	—	2	1	no	—
58	C₂×C₇×D₄	—	2	1	no	—
59	C₂×C₇×D₄	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₇	—	2	2	no	—
61	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
64	C₇×M₁₆	—	4	2	no	—
65	C₇×M₁₆	—	4	2	no	—
66	C₇×M₁₆	—	4	2	no	—
67	C₂×C₇×C₈	—	2	2	no	—
68	C₂×C₇×C₈	—	2	2	no	—
69	C₂×C₇×C₈	—	4	4	no	—
70	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₇×C₈	D₈	1	2	yes	—
73	C₂×C₇×C₈	SD₁₆	1	2	yes	—
74	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
76	C₇×C₄⋊C₄	—	2	1	no	—
77	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
78	C₇×C₂²⋊C₄	—	2	2	no	—
79	C₇×C₂²⋊C₄	—	2	2	no	—
80	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₇	M₁₆	1	1	yes	characteristic
83	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₇⋊C₁₆	—	2	4	no	—
85	C₇⋊C₁₆	—	4	2	no	—
86	C₂×#131	—	2	1	no	—
87	#532	Dic₇	1	1	yes	characteristic
88	C₂×M₃₂	—	14	1	no	—
89	C₂²×C₁₆	—	14	1	no	—
90	#436	—	2	1	no	—
91	#436	C₂×D₇	1	2	yes	—
92	C₈×D₄	—	2	1	no	—
93	C₈×D₄	Dic₇	1	2	yes	—
94	#423	Dic₇	1	1	yes	characteristic
95	#408	—	2	1	no	—
96	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
97	C₄⋊C₁₆	—	7	2	no	—
98	C₄⋊C₁₆	—	14	1	no	—
99	#346	—	7	2	no	—
100	C₄×C₁₆	—	14	1	no	—
101	C₂³×C₇	—	2	2	no	—
102	C₂³×C₇	C₂²⋊C₈	1	1	yes	characteristic
103	C₇×Q₈	—	4	2	no	—
104	C₇×D₄	—	4	2	no	—
105	C₇×D₄	—	4	4	no	—
106	C₇×D₄	—	4	2	no	—
107	C₇×D₄	—	4	2	no	—
108	C₂×C₄×C₇	—	2	2	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	4	1	no	—
111	C₂×C₄×C₇	—	4	2	no	—
112	C₂×C₄×C₇	—	4	2	no	—
113	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
116	C₂×C₄×C₇	#102	1	2	yes	—
117	C₂×C₄×C₇	#104	1	2	yes	—
118	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
121	C₇×C₈	—	2	1	no	—
122	C₇×C₈	—	2	1	no	—
123	C₇×C₈	—	4	2	no	—
124	C₇×C₈	#102	1	2	yes	—
125	C₇×C₈	C₂×D₈	1	1	yes	characteristic
126	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×#41	—	2	1	no	—
128	#131	—	4	2	no	—
129	#131	—	4	2	no	—
130	C₂×M₁₆	—	2	1	no	—
131	C₂×M₁₆	—	2	2	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
134	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
135	C₄⋊Q₈	C₇⋊C₈	1	1	yes	characteristic
136	#127	C₇⋊C₈	1	1	yes	characteristic
137	#124	—	2	1	no	—
138	#121	—	2	1	no	—
139	#121	C₇⋊C₈	1	2	yes	—
140	C₄×D₄	—	2	1	no	—
141	C₄×D₄	C₂×Dic₇	1	2	yes	—
142	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
143	M₃₂	—	14	4	no	—
144	C₂×C₁₆	—	14	2	no	—
145	C₂×C₁₆	—	14	2	no	—
146	C₂×C₁₆	—	28	2	no	—
147	C₄⋊C₈	—	2	1	no	—
148	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
149	C₂²⋊C₈	—	2	2	no	—
150	C₂²⋊C₈	—	2	2	no	—
151	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
154	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
155	C₂²×C₇	—	2	1	no	—
156	C₂²×C₇	—	2	1	no	—
157	C₂²×C₇	—	4	2	no	—
158	C₂²×C₇	—	4	2	no	—
159	C₂²×C₇	—	4	2	no	—
160	C₂²×C₇	D₄⋊C₈	1	2	yes	—
161	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₄×C₇	—	2	1	no	—
163	C₄×C₇	—	2	1	no	—
164	C₄×C₇	—	2	1	no	—
165	C₄×C₇	—	4	2	no	—
166	C₄×C₇	D₄⋊C₈	1	2	yes	—
167	C₄×C₇	#350	1	2	yes	—
168	C₄×C₇	C₂×#102	1	1	yes	characteristic
169	C₄×C₇	C₂×#104	1	1	yes	characteristic
170	#41	—	4	2	no	—
171	#41	—	4	2	no	—
172	C₂×Q₈	—	2	1	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	2	1	no	—
176	C₂×D₄	C₂×C₇⋊C₈	1	2	yes	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
179	C₂²×C₄	#1136	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
181	M₁₆	—	4	2	no	—
182	M₁₆	—	4	2	no	—
183	M₁₆	—	4	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	4	no	—
187	C₂×C₈	#1141	1	2	yes	—
188	C₂×C₈	#1142	1	2	yes	—
189	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
191	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	C₂×C₇⋊C₈	1	2	yes	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
198	C₄²	#1136	1	1	yes	characteristic
199	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
200	C₄²	C₂²×Dic₇	1	1	yes	characteristic
201	C₁₆	—	14	4	no	—
202	C₁₆	—	28	2	no	—
203	C₂×C₇	—	2	2	no	—
204	C₂×C₇	—	8	2	no	—
205	C₂×C₇	C₂×D₄⋊C₈	1	1	yes	characteristic
206	C₂×C₇	#2115	1	1	yes	characteristic
207	C₂×C₇	C₂×#350	1	1	yes	characteristic
208	C₂³	—	2	2	no	—
209	C₂³	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
210	Q₈	—	4	2	no	—
211	D₄	—	4	2	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	2	no	—
214	D₄	—	4	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
221	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
222	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
223	C₂×C₄	#6406	1	2	yes	—
224	C₂×C₄	#6411	1	2	yes	—
225	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
226	C₂×C₄	C₂×#1136	1	1	yes	characteristic
227	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
228	C₈	—	2	1	no	—
229	C₈	—	2	1	no	—
230	C₈	—	4	2	no	—
231	C₈	#6406	1	2	yes	—
232	C₈	C₂×#1141	1	1	yes	characteristic
233	C₈	C₂×#1142	1	1	yes	characteristic
234	C₇	#11586	1	1	yes	characteristic, complemented, Sylow
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	D₄⋊(C₇⋊C₈)	1	2	yes	—
241	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	2	no	—
246	C₄	D₄⋊(C₇⋊C₈)	1	2	yes	—
247	C₄	#89356	1	2	yes	—
248	C₄	C₂×#6406	1	1	yes	characteristic
249	C₄	C₂×#6411	1	1	yes	characteristic
250	C₂	—	2	2	no	—
251	C₂	—	8	2	no	—
252	C₂	C₂×D₄⋊(C₇⋊C₈)	1	1	yes	characteristic
253	C₂	C₂×#89356	1	1	yes	characteristic
254	C₂	#11729761	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#2931	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₇×#532	C₄
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#2931	D₇
hypocenter	C₇	#11586
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊(C₇⋊C₁₆)	C₂	semidirect
3. C₄×#6386	C₂	semidirect
4. #11725567	C₂	semidirect
234. C₇	#11586	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2931	2
C₂×C₄⋊(C₇⋊C₁₆)	2
C₄×#6386	2
#11725567	2
#11586	7