Subgroups — #117×C₇⋊C₈ #13352001

Order 1792.

1174 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#89697	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×C₇⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂²⋊C₄×C₇⋊C₈	C₂	1	4	yes	maximal
6	#11727142	C₂	1	2	yes	maximal, complemented
7	C₄²×C₇⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₂×C₇×#117	C₄	1	2	yes	—
9	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂³×C₇⋊C₈	C₂²	1	2	yes	—
11	C₄×C₇×C₄⋊C₄	C₄	1	2	yes	—
12	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₄×C₇×C₂²⋊C₄	C₄	1	2	yes	—
14	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
15	C₇×#376	C₄	1	2	yes	—
16	C₄³×C₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊(C₇⋊C₈)	C₂²	1	2	yes	—
18	#89697	C₄	1	8	yes	complemented
19	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
20	C₂×C₄×C₇⋊C₈	C₄	1	8	yes	complemented
21	C₂×C₄×C₇⋊C₈	C₂²	1	8	yes	—
22	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
24	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
26	#89319	C₂²	1	8	yes	—
27	C₈×#117	—	7	1	no	maximal, Sylow
28	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₇×#117	C₈	1	16	yes	complemented
32	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₇	C₂³	1	4	yes	—
41	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
43	C₂²×C₇⋊C₈	—	2	8	no	—
44	C₂²×C₇⋊C₈	C₂×C₄	1	8	yes	—
45	C₂²×C₇⋊C₈	C₂³	1	4	yes	—
46	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
47	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
48	C₇×#95	C₂×C₄	1	4	yes	—
49	C₇×#95	C₂×C₄	1	4	yes	—
50	C₂²⋊(C₇⋊C₈)	C₂×C₄	1	8	yes	—
51	C₄⋊(C₇⋊C₈)	C₂×C₄	1	8	yes	—
52	C₄×C₇⋊C₈	—	2	16	no	—
53	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
54	C₄×C₇⋊C₈	C₂×C₄	1	8	yes	—
55	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
56	C₄×C₇⋊C₈	C₂×C₄	1	8	yes	—
57	C₂×#431	—	7	1	no	—
58	C₂²×C₄×C₈	—	7	1	no	—
59	C₄×#117	D₇	1	1	yes	characteristic
60	C₈×C₄⋊C₄	—	7	4	no	—
61	C₈×C₂²⋊C₄	—	7	4	no	—
62	#1744	—	7	2	no	—
63	C₄²×C₈	—	7	2	no	—
64	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	—	2	4	no	—
66	C₂²×C₄×C₇	—	2	4	no	—
67	C₂²×C₄×C₇	C₄²	1	4	yes	—
68	C₂²×C₄×C₇	C₄²	1	4	yes	—
69	C₂²×C₄×C₇	C₂×C₈	1	8	yes	—
70	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
79	C₇×C₄⋊C₄	C₂×C₈	1	16	yes	complemented
80	C₇×C₂²⋊C₄	C₂×C₈	1	16	yes	—
81	C₄²×C₇	—	2	8	no	—
82	C₄²×C₇	—	2	8	no	—
83	C₄²×C₇	C₄²	1	4	yes	—
84	C₄²×C₇	C₄²	1	4	yes	—
85	C₄²×C₇	C₂×C₈	1	16	yes	complemented
86	C₄²×C₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₇	C₂²×C₄	1	2	yes	—
90	C₂×C₇⋊C₈	—	2	4	no	—
91	C₂×C₇⋊C₈	—	2	8	no	—
92	C₂×C₇⋊C₈	—	2	16	no	—
93	C₂×C₇⋊C₈	C₂²×C₄	1	8	yes	—
94	C₂×C₇⋊C₈	C₂²×C₄	1	4	yes	—
95	C₂×C₇⋊C₈	C₂²×C₄	1	4	yes	—
96	C₂×C₇⋊C₈	#41	1	8	yes	—
97	C₂³×C₈	—	7	2	no	—
98	C₂×#117	Dic₇	1	2	yes	—
99	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
100	#431	—	7	8	no	—
101	C₂×C₄⋊C₈	—	7	2	no	—
102	C₂×C₂²⋊C₈	—	7	2	no	—
103	C₂×C₄×C₈	—	7	8	no	—
104	C₂×C₄×C₈	—	7	2	no	—
105	C₂×C₄×C₈	—	7	1	no	—
106	C₂×C₄×C₈	—	7	1	no	—
107	C₂×C₄×C₈	—	7	2	no	—
108	C₂×C₄×C₈	—	7	8	no	—
109	C₄×C₄⋊C₄	Dic₇	1	2	yes	—
110	C₄×C₄⋊C₄	C₂×D₇	1	2	yes	—
111	C₄×C₂²⋊C₄	Dic₇	1	2	yes	—
112	C₄×C₂²⋊C₄	C₂×D₇	1	2	yes	—
113	#376	Dic₇	1	2	yes	—
114	C₄³	C₂×D₇	1	2	yes	—
115	#336	—	7	8	no	—
116	C₂³×C₇	—	2	2	no	—
117	C₂³×C₇	—	2	2	no	—
118	C₂³×C₇	C₂×C₄²	1	2	yes	—
119	C₂³×C₇	C₂²×C₈	1	4	yes	—
120	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₇	—	2	8	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	2	8	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	2	4	no	—
126	C₂×C₄×C₇	—	2	4	no	—
127	C₂×C₄×C₇	—	2	8	no	—
128	C₂×C₄×C₇	C₄×C₈	1	16	yes	complemented
129	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₇	C₂²×C₈	1	16	yes	—
135	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
136	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
137	C₂×C₄×C₇	#131	1	4	yes	—
138	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂×#41	1	4	yes	—
141	C₇⋊C₈	—	2	4	no	—
142	C₇⋊C₈	—	2	8	no	—
143	C₇⋊C₈	#117	1	8	yes	direct product factor
144	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
145	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
146	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
147	C₂²×C₈	—	7	2	no	—
148	C₂²×C₈	—	7	8	no	—
149	C₂²×C₈	—	7	2	no	—
150	C₂²×C₈	—	7	4	no	—
151	C₂²×C₈	—	14	8	no	—
152	#117	C₇⋊C₈	1	16	yes	direct product factor
153	C₂×C₄⋊C₄	C₂×Dic₇	1	4	yes	—
154	C₂×C₂²⋊C₄	C₂×Dic₇	1	4	yes	—
155	C₂×C₄²	C₄×D₇	1	4	yes	—
156	C₂×C₄²	C₄×D₇	1	4	yes	—
157	C₂×C₄²	C₂×Dic₇	1	4	yes	—
158	C₂×C₄²	C₂×Dic₇	1	4	yes	—
159	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
160	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₇	1	4	yes	—
162	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
163	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
164	C₄⋊C₈	—	7	8	no	—
165	C₂²⋊C₈	—	7	8	no	—
166	C₄×C₈	—	7	8	no	—
167	C₄×C₈	—	7	4	no	—
168	C₄×C₈	—	7	4	no	—
169	C₄×C₈	—	7	8	no	—
170	C₄×C₈	—	14	16	no	—
171	#95	C₂×Dic₇	1	4	yes	—
172	#95	C₂×Dic₇	1	4	yes	—
173	C₂²×C₇	—	2	2	no	—
174	C₂²×C₇	—	2	8	no	—
175	C₂²×C₇	—	2	2	no	—
176	C₂²×C₇	C₂×C₄×C₈	1	4	yes	—
177	C₂²×C₇	C₂²×C₄²	1	1	yes	characteristic
178	C₂²×C₇	C₄×#41	1	2	yes	—
179	C₂²×C₇	C₂³×C₈	1	2	yes	—
180	C₂²×C₇	C₂×#131	1	2	yes	—
181	C₄×C₇	—	2	2	no	—
182	C₄×C₇	—	2	16	no	—
183	C₄×C₇	—	2	2	no	—
184	C₄×C₇	C₂×C₄×C₈	1	4	yes	—
185	C₄×C₇	C₂×C₄×C₈	1	4	yes	—
186	C₄×C₇	#405	1	4	yes	—
187	C₄×C₇	C₂×#117	1	2	yes	—
188	C₄×C₇	C₄×#41	1	2	yes	—
189	C₂⁴	C₂²×Dic₇	1	1	yes	characteristic
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	C₂×C₇⋊C₈	1	8	yes	—
193	C₂²×C₄	C₄×Dic₇	1	4	yes	—
194	C₂²×C₄	C₄×Dic₇	1	4	yes	—
195	C₂²×C₄	C₂×C₄×D₇	1	2	yes	—
196	C₂²×C₄	C₂×C₄×D₇	1	2	yes	—
197	C₂²×C₄	C₂²×Dic₇	1	8	yes	—
198	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
203	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
204	C₂×C₈	—	7	4	no	—
205	C₂×C₈	—	7	8	no	—
206	C₂×C₈	—	7	4	no	—
207	C₂×C₈	—	7	8	no	—
208	C₂×C₈	—	14	16	no	—
209	C₂×C₈	—	14	8	no	—
210	C₂×C₈	—	14	4	no	—
211	C₄⋊C₄	C₂×C₇⋊C₈	1	16	yes	complemented
212	C₂²⋊C₄	C₂×C₇⋊C₈	1	16	yes	—
213	C₄²	—	2	8	no	—
214	C₄²	—	2	8	no	—
215	C₄²	C₂×C₇⋊C₈	1	16	yes	complemented
216	C₄²	C₄×Dic₇	1	4	yes	—
217	C₄²	C₄×Dic₇	1	4	yes	—
218	C₄²	C₂×C₄×D₇	1	2	yes	—
219	C₄²	C₂×C₄×D₇	1	2	yes	—
220	C₄²	C₂×C₄×D₇	1	2	yes	—
221	C₄²	C₂×C₄×D₇	1	2	yes	—
222	C₂×C₇	—	2	4	no	—
223	C₂×C₇	C₄×#117	1	1	yes	characteristic
224	C₂×C₇	C₂²×C₄×C₈	1	1	yes	characteristic
225	C₂×C₇	C₂×#405	1	1	yes	characteristic
226	C₂×C₇	C₈×#41	1	4	yes	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	C₂²×C₇⋊C₈	1	4	yes	—
230	C₂³	C₂×C₄×Dic₇	1	2	yes	—
231	C₂³	C₂³×Dic₇	1	1	yes	characteristic
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	C₄×C₇⋊C₈	1	16	yes	complemented
240	C₂×C₄	C₂²×C₇⋊C₈	1	16	yes	—
241	C₂×C₄	C₂²×C₇⋊C₈	1	4	yes	—
242	C₂×C₄	C₂²×C₇⋊C₈	1	4	yes	—
243	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
244	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
246	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
247	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
248	C₂×C₄	#6511	1	4	yes	—
249	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
250	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
251	C₂×C₄	D₇×#41	1	4	yes	—
252	C₈	—	7	8	no	—
253	C₈	—	14	4	no	—
254	C₈	—	14	8	no	—
255	C₇	C₈×#117	1	1	yes	characteristic, complemented, Sylow
256	C₂²	—	2	2	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	2	2	no	—
259	C₂²	C₂×C₄×C₇⋊C₈	1	4	yes	—
260	C₂²	C₂³×C₇⋊C₈	1	2	yes	—
261	C₂²	C₂²×C₄×Dic₇	1	1	yes	characteristic
262	C₂²	C₂×#6511	1	2	yes	—
263	C₂²	#41×Dic₇	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	2	16	no	—
266	C₄	—	2	2	no	—
267	C₄	C₂×C₄×C₇⋊C₈	1	4	yes	—
268	C₄	C₂×C₄×C₇⋊C₈	1	4	yes	—
269	C₄	#89769	1	4	yes	—
270	C₄	D₇×#117	1	2	yes	—
271	C₄	#90213	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	C₂²×C₄×C₇⋊C₈	1	1	yes	characteristic
274	C₂	C₂×#89769	1	1	yes	characteristic
275	C₂	Dic₇×#117	1	1	yes	characteristic
276	C₂	#41×C₇⋊C₈	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₇
commutator	C₂×C₇	C₂²×C₄×C₈
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₄×C₇×#117	C₂
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂⁴×C₇	C₂²×C₄
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	C₄×#117	D₇
hypocenter	C₇	C₈×#117
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
4. C₄⋊C₄×C₇⋊C₈	C₂	semidirect
6. #11727142	C₂	semidirect
7. C₄²×C₇⋊C₈	C₂	semidirect
18. #89697	C₄	semidirect
20. C₂×C₄×C₇⋊C₈	C₄	semidirect
31. C₇×#117	C₈	semidirect
79. C₇×C₄⋊C₄	C₂×C₈	semidirect
85. C₄²×C₇	C₂×C₈	semidirect
128. C₂×C₄×C₇	C₄×C₈	semidirect
143. C₇⋊C₈	#117	direct (N × H)
152. #117	C₇⋊C₈	direct (N × H)
211. C₄⋊C₄	C₂×C₇⋊C₈	semidirect
215. C₄²	C₂×C₇⋊C₈	semidirect
239. C₂×C₄	C₄×C₇⋊C₈	semidirect
255. C₇	C₈×#117	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×#117	2
C₂×#89697	2
C₂²×C₄×C₇⋊C₈	2
C₄⋊C₄×C₇⋊C₈	2
C₂²⋊C₄×C₇⋊C₈	2
#11727142	2
C₄²×C₇⋊C₈	2
C₈×#117	7

Subgroups — #117×C7⋊C8 #13352001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #117×C₇⋊C₈ #13352001