Subgroups — #13271968

Order 1792.

446 conjugacy classes of subgroups.

The subgroup lattice has 260 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×#89356	C₂	1	4	yes	maximal, complemented
3	C₂×C₄⋊(C₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#6386	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×#131	C₂²	1	2	yes	—
6	C₇×#532	C₄	1	1	yes	characteristic
7	C₇×#436	—	2	1	no	—
8	C₇×#436	—	2	1	no	—
9	C₇×C₈×D₄	—	2	1	no	—
10	C₇×C₈×D₄	—	2	1	no	—
11	C₇×#423	C₄	1	1	yes	characteristic
12	C₇×#408	C₄	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6386	C₂²	1	2	yes	—
15	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
16	#89356	—	2	8	no	—
17	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
18	C₄⋊(C₇⋊C₁₆)	—	2	1	no	—
19	#89256	—	2	1	no	—
20	C₄×C₇⋊C₁₆	—	2	1	no	—
21	#13193	—	7	1	no	maximal, Sylow
22	C₂×C₇×#41	C₂×C₄	1	2	yes	—
23	C₇×#131	—	2	4	no	—
24	C₇×#131	—	2	4	no	—
25	C₂×C₇×M₁₆	—	2	2	no	—
26	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
27	C₂²×C₇×C₈	—	2	2	no	—
28	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₇×C₄⋊Q₈	C₈	1	1	yes	characteristic
31	C₇×#127	C₈	1	1	yes	characteristic
32	C₇×#124	C₈	1	2	yes	—
33	C₇×#121	—	2	1	no	—
34	C₇×#121	—	2	1	no	—
35	C₄×C₇×D₄	—	2	1	no	—
36	C₄×C₇×D₄	—	2	1	no	—
37	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
38	C₇×C₄⋊C₈	—	2	1	no	—
39	C₇×C₄⋊C₈	—	2	1	no	—
40	C₇×C₂²⋊C₈	—	2	2	no	—
41	C₇×C₂²⋊C₈	—	2	2	no	—
42	C₄×C₇×C₈	D₄	1	1	yes	characteristic
43	C₄×C₇×C₈	D₄	1	1	yes	characteristic
44	C₄×C₇×C₈	D₄	1	1	yes	characteristic
45	C₄×C₇×C₈	D₄	1	1	yes	characteristic
46	#6386	—	2	2	no	—
47	#6386	D₄	1	4	yes	complemented
48	C₂×C₇⋊C₁₆	—	2	2	no	—
49	C₂×C₇⋊C₁₆	—	2	4	no	—
50	C₂×C₇⋊C₁₆	—	2	2	no	—
51	#2931	D₇	1	1	yes	characteristic
52	C₂×C₄⋊C₁₆	—	7	1	no	—
53	C₂×#350	—	7	4	no	—
54	C₄×M₃₂	—	7	1	no	—
55	C₇×#41	—	2	4	no	—
56	C₇×#41	—	2	4	no	—
57	C₂×C₇×Q₈	C₂×C₈	1	2	yes	—
58	C₂×C₇×D₄	—	2	2	no	—
59	C₂×C₇×D₄	—	2	1	no	—
60	C₂×C₇×D₄	—	2	1	no	—
61	C₂×C₇×D₄	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₇	—	2	2	no	—
63	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
66	C₇×M₁₆	—	2	4	no	—
67	C₇×M₁₆	—	4	2	no	—
68	C₇×M₁₆	—	4	2	no	—
69	C₂×C₇×C₈	—	2	2	no	—
70	C₂×C₇×C₈	—	2	2	no	—
71	C₂×C₇×C₈	—	4	4	no	—
72	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
76	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₇×C₈	#41	1	2	yes	—
78	C₇×C₄⋊C₄	—	2	1	no	—
79	C₇×C₄⋊C₄	—	2	1	no	—
80	C₇×C₂²⋊C₄	—	2	2	no	—
81	C₇×C₂²⋊C₄	—	2	2	no	—
82	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
85	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
86	C₇⋊C₁₆	—	2	4	no	—
87	C₇⋊C₁₆	—	4	2	no	—
88	C₂×#131	C₂×D₇	1	2	yes	—
89	#532	Dic₇	1	1	yes	characteristic
90	C₂×M₃₂	—	7	2	no	—
91	C₂²×C₁₆	—	7	2	no	—
92	#436	—	2	1	no	—
93	#436	—	2	1	no	—
94	C₈×D₄	—	2	1	no	—
95	C₈×D₄	—	2	1	no	—
96	#423	Dic₇	1	1	yes	characteristic
97	#408	Dic₇	1	2	yes	—
98	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
99	C₄⋊C₁₆	—	14	1	no	—
100	C₄⋊C₁₆	—	14	1	no	—
101	#350	—	14	8	no	—
102	#346	—	14	1	no	—
103	C₄×C₁₆	—	14	1	no	—
104	C₂³×C₇	—	2	2	no	—
105	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
106	C₇×Q₈	—	2	4	no	—
107	C₇×D₄	—	2	4	no	—
108	C₇×D₄	—	4	2	no	—
109	C₇×D₄	—	4	4	no	—
110	C₇×D₄	—	4	2	no	—
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	—	2	2	no	—
113	C₂×C₄×C₇	—	4	1	no	—
114	C₂×C₄×C₇	—	4	2	no	—
115	C₂×C₄×C₇	—	4	2	no	—
116	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₇	#131	1	2	yes	—
124	C₇×C₈	—	2	1	no	—
125	C₇×C₈	—	2	1	no	—
126	C₇×C₈	—	4	2	no	—
127	C₇×C₈	#121	1	2	yes	—
128	C₇×C₈	#124	1	1	yes	characteristic
129	C₇×C₈	#127	1	1	yes	characteristic
130	C₂×#41	C₂×Dic₇	1	2	yes	—
131	#131	—	2	4	no	—
132	#131	—	2	4	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
138	C₄⋊Q₈	C₇⋊C₈	1	1	yes	characteristic
139	#127	C₇⋊C₈	1	1	yes	characteristic
140	#124	C₇⋊C₈	1	2	yes	—
141	#121	—	2	1	no	—
142	#121	—	2	1	no	—
143	C₄×D₄	—	2	1	no	—
144	C₄×D₄	—	2	1	no	—
145	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
146	M₃₂	—	7	4	no	—
147	M₃₂	—	14	2	no	—
148	C₂×C₁₆	—	14	2	no	—
149	C₂×C₁₆	—	14	4	no	—
150	C₂×C₁₆	—	14	2	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	2	1	no	—
153	C₂²⋊C₈	—	2	2	no	—
154	C₂²⋊C₈	—	2	2	no	—
155	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
159	C₂²×C₇	—	2	1	no	—
160	C₂²×C₇	—	2	1	no	—
161	C₂²×C₇	—	4	2	no	—
162	C₂²×C₇	—	4	2	no	—
163	C₂²×C₇	—	4	2	no	—
164	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
165	C₂²×C₇	#436	1	2	yes	—
166	C₄×C₇	—	2	1	no	—
167	C₄×C₇	—	2	1	no	—
168	C₄×C₇	—	2	1	no	—
169	C₄×C₇	—	4	2	no	—
170	C₄×C₇	#350	1	2	yes	complemented
171	C₄×C₇	#390	1	1	yes	characteristic
172	C₄×C₇	#408	1	1	yes	characteristic
173	C₄×C₇	C₈×D₄	1	2	yes	—
174	#41	—	2	4	no	—
175	#41	—	2	4	no	—
176	C₂×Q₈	C₂×C₇⋊C₈	1	2	yes	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₂×C₇⋊C₈	1	2	yes	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	#1136	1	1	yes	characteristic
183	C₂²×C₄	#1136	1	1	yes	characteristic
184	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
185	M₁₆	—	2	4	no	—
186	M₁₆	—	4	2	no	—
187	M₁₆	—	4	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	4	4	no	—
191	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
193	C₂×C₈	D₄×D₇	1	2	yes	—
194	C₂×C₈	#1159	1	2	yes	—
195	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
202	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
203	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
204	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
205	C₁₆	—	14	4	no	—
206	C₁₆	—	28	2	no	—
207	C₂×C₇	—	2	2	no	—
208	C₂×C₇	—	8	2	no	—
209	C₂×C₇	#1954	1	1	yes	characteristic
210	C₂×C₇	C₂×#350	1	1	yes	characteristic
211	C₂×C₇	#2117	1	1	yes	characteristic
212	C₂³	—	2	2	no	—
213	C₂³	C₂×#1136	1	1	yes	characteristic
214	Q₈	—	2	4	no	—
215	D₄	—	2	4	no	—
216	D₄	—	4	2	no	—
217	D₄	—	4	4	no	—
218	D₄	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
225	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
226	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
227	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
229	C₂×C₄	D₄×Dic₇	1	2	yes	—
230	C₂×C₄	#6511	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	4	2	no	—
235	C₈	#6499	1	1	yes	characteristic
236	C₈	#6501	1	2	yes	—
237	C₈	#6503	1	1	yes	characteristic
238	C₇	#13193	1	1	yes	characteristic, complemented, Sylow
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	#89784	1	2	yes	—
245	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	2	no	—
250	C₄	#89356	1	2	yes	complemented
251	C₄	D₄×C₇⋊C₈	1	2	yes	—
252	C₄	#89993	1	1	yes	characteristic
253	C₄	#90006	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	8	2	no	—
256	C₂	#11729368	1	1	yes	characteristic
257	C₂	C₂×#89356	1	1	yes	characteristic
258	C₂	#11729760	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#2931	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₇×#532	C₄
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#2931	D₇
hypocenter	C₇	#13193
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×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×#89356	C₂	semidirect
3. C₂×C₄⋊(C₇⋊C₁₆)	C₂	semidirect
4. C₄×#6386	C₂	semidirect
47. #6386	D₄	semidirect
170. C₄×C₇	#350	semidirect
238. C₇	#13193	semidirect
250. C₄	#89356	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×#89356	2
C₂×C₄⋊(C₇⋊C₁₆)	2
C₄×#6386	2
#13193	7