Subgroups — C₇×#11583 #13188799

Order 1792.

450 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×D₄⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂³×C₇×C₈	—	2	1	no	—
6	C₂×C₄×C₇×D₄	C₄	1	1	yes	characteristic
7	C₂×C₇×M₃₂	—	2	1	no	—
8	C₂×C₇×M₃₂	—	2	1	no	—
9	C₇×C₈×D₄	—	2	1	no	—
10	C₇×C₈×D₄	—	2	1	no	—
11	C₇×C₈×D₄	C₄	1	2	yes	—
12	C₇×C₈×D₄	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₇×C₂²⋊C₈	—	2	1	no	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₇×C₄⋊C₁₆	—	2	1	no	—
17	C₇×C₄⋊C₁₆	C₂²	1	2	yes	complemented
18	C₇×#346	—	2	1	no	—
19	C₄×C₇×C₁₆	C₂²	1	2	yes	complemented
20	#11583	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×D₄	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₇	—	2	1	no	—
23	C₂²×C₇×C₈	—	2	1	no	—
24	C₂²×C₇×C₈	—	2	2	no	—
25	C₂²×C₇×C₈	—	2	2	no	—
26	C₂²×C₇×C₈	—	4	2	no	—
27	C₂²×C₇×C₈	—	4	2	no	—
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
30	C₄×C₇×D₄	—	2	1	no	—
31	C₄×C₇×D₄	—	2	1	no	—
32	C₄×C₇×D₄	C₈	1	2	yes	—
33	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
34	C₂×C₇×C₄⋊C₄	C₈	1	1	yes	characteristic
35	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₇×M₃₂	—	2	4	no	—
38	C₇×M₃₂	—	4	2	no	—
39	C₂×C₇×C₁₆	—	2	2	no	—
40	C₂×C₇×C₁₆	—	2	2	no	—
41	C₇×C₄⋊C₈	—	2	1	no	—
42	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₇×C₂²⋊C₈	—	2	2	no	—
44	C₇×C₂²⋊C₈	—	2	2	no	—
45	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₇×C₈	D₄	1	1	yes	characteristic
47	C₄×C₇×C₈	D₄	1	1	yes	characteristic
48	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₈×D₄	C₂×C₇	1	1	yes	characteristic
50	#2150	C₂×C₇	1	1	yes	characteristic, complemented
51	C₄×M₃₂	C₂×C₇	1	1	yes	characteristic, complemented
52	D₄⋊C₁₆	C₂×C₇	1	4	yes	complemented
53	C₂⁴×C₇	—	2	1	no	—
54	C₂×C₇×D₄	—	2	4	no	—
55	C₂×C₇×D₄	—	2	1	no	—
56	C₂×C₇×D₄	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₇	—	2	1	no	—
58	C₂²×C₄×C₇	—	2	2	no	—
59	C₂²×C₄×C₇	—	2	2	no	—
60	C₂²×C₄×C₇	—	4	2	no	—
61	C₂²×C₄×C₇	—	4	2	no	—
62	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
65	C₂×C₇×C₈	—	2	2	no	—
66	C₂×C₇×C₈	—	2	2	no	—
67	C₂×C₇×C₈	—	4	2	no	—
68	C₂×C₇×C₈	—	4	4	no	—
69	C₂×C₇×C₈	—	4	2	no	—
70	C₂×C₇×C₈	—	4	2	no	—
71	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₇×C₈	D₈	1	2	yes	—
74	C₂×C₇×C₈	SD₁₆	1	2	yes	—
75	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
77	C₇×C₄⋊C₄	—	2	1	no	—
78	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
79	C₇×C₂²⋊C₄	—	2	2	no	—
80	C₇×C₂²⋊C₄	—	2	2	no	—
81	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₇	M₁₆	1	1	yes	characteristic
84	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
85	C₇×C₁₆	—	2	4	no	—
86	C₇×C₁₆	—	4	2	no	—
87	C₂³×C₈	—	2	1	no	—
88	C₂×C₄×D₄	C₄×C₇	1	1	yes	characteristic
89	C₂×M₃₂	—	2	1	no	—
90	C₂×M₃₂	—	2	1	no	—
91	C₈×D₄	—	2	1	no	—
92	C₈×D₄	—	2	1	no	—
93	C₈×D₄	C₄×C₇	1	2	yes	—
94	C₈×D₄	C₂²×C₇	1	2	yes	—
95	C₂×C₄⋊C₈	C₄×C₇	1	1	yes	characteristic
96	C₂×C₂²⋊C₈	—	2	1	no	—
97	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
98	C₄⋊C₁₆	—	2	1	no	—
99	C₄⋊C₁₆	C₂²×C₇	1	2	yes	complemented
100	#346	—	2	1	no	—
101	C₄×C₁₆	C₂²×C₇	1	2	yes	complemented
102	C₂³×C₇	—	2	4	no	—
103	C₂³×C₇	—	2	2	no	—
104	C₂³×C₇	—	4	2	no	—
105	C₂³×C₇	—	4	2	no	—
106	C₂³×C₇	C₂²⋊C₈	1	1	yes	characteristic
107	C₇×D₄	—	2	4	no	—
108	C₇×D₄	—	2	2	no	—
109	C₇×D₄	M₃₂	1	4	yes	complemented
110	C₂×C₄×C₇	—	2	2	no	—
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	—	4	1	no	—
113	C₂×C₄×C₇	—	4	2	no	—
114	C₂×C₄×C₇	—	4	4	no	—
115	C₂×C₄×C₇	—	4	2	no	—
116	C₂×C₄×C₇	—	4	2	no	—
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	#102	1	2	yes	—
121	C₂×C₄×C₇	#104	1	2	yes	—
122	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
125	C₇×C₈	—	2	1	no	—
126	C₇×C₈	—	2	1	no	—
127	C₇×C₈	—	4	2	no	—
128	C₇×C₈	#102	1	2	yes	—
129	C₇×C₈	C₂×D₈	1	1	yes	characteristic
130	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
131	C₂²×D₄	C₇×C₈	1	1	yes	characteristic
132	C₂³×C₄	—	2	1	no	—
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	4	2	no	—
137	C₂²×C₈	—	4	2	no	—
138	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
139	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
140	C₄×D₄	—	2	1	no	—
141	C₄×D₄	—	2	1	no	—
142	C₄×D₄	C₇×C₈	1	2	yes	—
143	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
144	C₂×C₄⋊C₄	C₇×C₈	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	—	2	1	no	—
146	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
147	M₃₂	—	2	4	no	—
148	M₃₂	—	4	2	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	C₂×C₄×C₇	1	2	yes	—
153	C₂²⋊C₈	—	2	2	no	—
154	C₂²⋊C₈	—	2	2	no	—
155	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
156	C₄×C₈	C₇×D₄	1	1	yes	characteristic
157	C₄×C₈	C₇×D₄	1	1	yes	characteristic
158	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
159	C₂²×C₇	—	2	1	no	—
160	C₂²×C₇	—	2	1	no	—
161	C₂²×C₇	—	2	4	no	—
162	C₂²×C₇	—	4	8	no	—
163	C₂²×C₇	—	4	2	no	—
164	C₂²×C₇	—	4	2	no	—
165	C₂²×C₇	D₄⋊C₈	1	2	yes	—
166	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
167	C₄×C₇	—	2	1	no	—
168	C₄×C₇	—	2	1	no	—
169	C₄×C₇	—	2	1	no	—
170	C₄×C₇	—	4	2	no	—
171	C₄×C₇	D₄⋊C₈	1	2	yes	—
172	C₄×C₇	C₂×#102	1	1	yes	characteristic
173	C₄×C₇	C₂×#104	1	1	yes	characteristic
174	C₄×C₇	C₂×M₃₂	1	2	yes	—
175	C₂⁴	—	2	1	no	—
176	C₂×D₄	—	2	4	no	—
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	C₂×C₇×C₈	1	2	yes	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
186	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	4	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₈	C₇×D₈	1	2	yes	—
196	C₂×C₈	C₇×SD₁₆	1	2	yes	—
197	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
198	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	C₂×C₇×C₈	1	2	yes	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
204	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
205	C₄²	C₇×M₁₆	1	1	yes	characteristic
206	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
207	C₁₆	—	2	4	no	—
208	C₁₆	—	4	2	no	—
209	C₂×C₇	—	2	2	no	—
210	C₂×C₇	—	4	4	no	—
211	C₂×C₇	C₂×D₄⋊C₈	1	1	yes	characteristic
212	C₂×C₇	#2112	1	1	yes	characteristic
213	C₂×C₇	#2117	1	1	yes	characteristic
214	C₂³	—	2	4	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	4	2	no	—
217	C₂³	—	4	2	no	—
218	C₂³	C₇×C₂²⋊C₈	1	1	yes	characteristic
219	D₄	—	2	4	no	—
220	D₄	—	2	2	no	—
221	D₄	C₇×M₃₂	1	4	yes	complemented
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
230	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
231	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
232	C₂×C₄	C₇×#102	1	2	yes	—
233	C₂×C₄	C₇×#104	1	2	yes	—
234	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
236	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	4	2	no	—
240	C₈	C₇×#102	1	2	yes	—
241	C₈	C₂×C₇×D₈	1	1	yes	characteristic
242	C₈	C₂×C₇×SD₁₆	1	1	yes	characteristic
243	C₇	#11583	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	4	8	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	C₇×D₄⋊C₈	1	2	yes	—
251	C₂²	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	2	no	—
256	C₄	C₇×D₄⋊C₈	1	2	yes	—
257	C₄	C₂×C₇×#102	1	1	yes	characteristic
258	C₄	C₂×C₇×#104	1	1	yes	characteristic
259	C₄	C₂×C₇×M₃₂	1	2	yes	—
260	C₂	—	2	2	no	—
261	C₂	—	4	4	no	—
262	C₂	C₂×C₇×D₄⋊C₈	1	1	yes	characteristic
263	C₂	C₇×#2112	1	1	yes	characteristic
264	C₂	C₇×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₂²×C₇×D₄	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2150	C₂	semidirect
3. C₄×C₇×M₃₂	C₂	semidirect
4. C₇×D₄⋊C₁₆	C₂	semidirect
17. C₇×C₄⋊C₁₆	C₂²	semidirect
19. C₄×C₇×C₁₆	C₂²	semidirect
20. #11583	C₇	direct (N × H)
50. #2150	C₂×C₇	semidirect
51. C₄×M₃₂	C₂×C₇	semidirect
52. D₄⋊C₁₆	C₂×C₇	semidirect
99. C₄⋊C₁₆	C₂²×C₇	semidirect
101. C₄×C₁₆	C₂²×C₇	semidirect
109. C₇×D₄	M₃₂	semidirect
221. D₄	C₇×M₃₂	semidirect
243. C₇	#11583	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×C₈×D₄	2
C₇×#2150	2
C₄×C₇×M₃₂	2
C₇×D₄⋊C₁₆	2
#11583	7

Subgroups — C7×#11583 #13188799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11583 #13188799