Subgroups — C₇×#3207 #11742516

Order 896.

438 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×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#463	C₂	1	4	yes	maximal, complemented
8	C₇×#449	C₂	1	4	yes	maximal, complemented
9	C₇×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×#41	—	2	1	no	—
11	C₂×C₇×#41	C₂²	1	2	yes	—
12	C₂×C₇×SD₁₆	C₂²	1	4	yes	—
13	C₂×C₇×D₈	C₂²	1	4	yes	—
14	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₇×#127	C₂²	1	1	yes	characteristic
17	C₇×#127	C₂²	1	1	yes	characteristic
18	C₇×#124	—	2	1	no	—
19	C₇×#124	C₂²	1	2	yes	—
20	C₇×#121	—	2	1	no	—
21	C₇×#121	—	2	1	no	—
22	C₇×#121	—	2	1	no	—
23	C₇×#121	C₂²	1	2	yes	—
24	C₄×C₇×D₄	—	2	1	no	—
25	C₄×C₇×D₄	—	2	1	no	—
26	C₄×C₇×D₄	—	2	1	no	—
27	C₄×C₇×D₄	C₂²	1	2	yes	—
28	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
29	C₇×C₄⋊C₈	C₂²	1	2	yes	—
30	C₇×#103	C₂²	1	4	yes	—
31	C₇×#102	C₂²	1	4	yes	—
32	C₇×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
34	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
35	#3207	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₇×#41	—	4	4	no	—
37	C₇×#41	—	4	4	no	—
38	C₂×C₇×Q₈	—	2	1	no	—
39	C₂×C₇×Q₈	C₂³	1	2	yes	—
40	C₂×C₇×D₄	—	2	2	no	—
41	C₂×C₇×D₄	—	2	2	no	—
42	C₂×C₇×D₄	—	2	1	no	—
43	C₂×C₇×D₄	—	2	1	no	—
44	C₂×C₇×D₄	—	2	1	no	—
45	C₂×C₇×D₄	—	2	1	no	—
46	C₂×C₇×D₄	C₂³	1	2	yes	—
47	C₂×C₇×D₄	C₂³	1	2	yes	—
48	C₂²×C₄×C₇	—	2	2	no	—
49	C₂²×C₄×C₇	—	2	2	no	—
50	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₇×SD₁₆	—	4	4	no	—
54	C₇×D₈	—	4	4	no	—
55	C₂×C₇×C₈	C₂³	1	2	yes	—
56	C₂×C₇×C₈	C₂³	1	2	yes	—
57	C₇×C₄⋊C₄	—	2	1	no	—
58	C₇×C₄⋊C₄	—	2	1	no	—
59	C₇×C₄⋊C₄	—	2	1	no	—
60	C₇×C₄⋊C₄	C₂³	1	2	yes	—
61	C₇×C₂²⋊C₄	—	2	2	no	—
62	C₇×C₂²⋊C₄	—	2	2	no	—
63	C₇×C₂²⋊C₄	—	2	2	no	—
64	C₇×C₂²⋊C₄	—	2	2	no	—
65	C₄²×C₇	D₄	1	1	yes	characteristic
66	C₄²×C₇	D₄	1	1	yes	characteristic
67	C₄²×C₇	C₂³	1	1	yes	characteristic
68	C₄²×C₇	C₂³	1	1	yes	characteristic
69	#532	C₂×C₇	1	1	yes	characteristic, complemented
70	#532	C₂×C₇	1	1	yes	characteristic, complemented
71	#493	C₂×C₇	1	1	yes	characteristic, complemented
72	#492	C₂×C₇	1	1	yes	characteristic, complemented
73	#487	C₂×C₇	1	1	yes	characteristic, complemented
74	#486	C₂×C₇	1	1	yes	characteristic, complemented
75	#463	C₂×C₇	1	4	yes	complemented
76	#449	C₂×C₇	1	4	yes	complemented
77	#431	C₂×C₇	1	1	yes	characteristic, complemented
78	C₂³×C₇	—	2	2	no	—
79	C₂³×C₇	—	2	2	no	—
80	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
81	C₇×Q₈	—	2	4	no	—
82	C₇×Q₈	—	4	2	no	—
83	C₇×D₄	—	2	4	no	—
84	C₇×D₄	—	2	4	no	—
85	C₇×D₄	—	4	4	no	—
86	C₇×D₄	—	4	4	no	—
87	C₇×D₄	—	4	2	no	—
88	C₇×D₄	—	4	2	no	—
89	C₇×D₄	—	4	2	no	—
90	C₇×D₄	—	4	2	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	4	4	no	—
97	C₂×C₄×C₇	—	4	4	no	—
98	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
104	C₇×C₈	—	2	2	no	—
105	C₇×C₈	—	2	2	no	—
106	C₂×#41	—	2	1	no	—
107	C₂×#41	C₂²×C₇	1	2	yes	—
108	C₂×SD₁₆	C₂²×C₇	1	4	yes	—
109	C₂×D₈	C₂²×C₇	1	4	yes	—
110	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
111	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
112	#127	C₂²×C₇	1	1	yes	characteristic
113	#127	C₂²×C₇	1	1	yes	characteristic
114	#124	—	2	1	no	—
115	#124	C₂²×C₇	1	2	yes	—
116	#121	—	2	1	no	—
117	#121	—	2	1	no	—
118	#121	—	2	1	no	—
119	#121	C₂²×C₇	1	2	yes	—
120	C₄×D₄	—	2	1	no	—
121	C₄×D₄	—	2	1	no	—
122	C₄×D₄	—	2	1	no	—
123	C₄×D₄	C₂²×C₇	1	2	yes	—
124	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
125	C₄⋊C₈	C₂²×C₇	1	2	yes	—
126	#103	C₂²×C₇	1	4	yes	—
127	#102	C₂²×C₇	1	4	yes	—
128	C₂²⋊C₈	C₂²×C₇	1	2	yes	—
129	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
130	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
131	C₂²×C₇	—	2	1	no	—
132	C₂²×C₇	—	2	1	no	—
133	C₂²×C₇	—	2	1	no	—
134	C₂²×C₇	—	4	2	no	—
135	C₂²×C₇	—	4	2	no	—
136	C₂²×C₇	—	4	2	no	—
137	C₂²×C₇	—	4	2	no	—
138	C₂²×C₇	—	4	2	no	—
139	C₂²×C₇	—	4	2	no	—
140	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₇	—	2	1	no	—
142	C₄×C₇	—	2	1	no	—
143	C₄×C₇	—	2	1	no	—
144	C₄×C₇	—	4	2	no	—
145	C₄×C₇	—	4	2	no	—
146	C₄×C₇	#135	1	2	yes	—
147	C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
148	C₄×C₇	H(5,𝔽₂)	1	2	yes	—
149	#41	—	4	4	no	—
150	#41	—	4	4	no	—
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	C₂³×C₇	1	2	yes	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	1	no	—
156	C₂×D₄	—	2	1	no	—
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	C₂³×C₇	1	2	yes	—
160	C₂×D₄	C₂³×C₇	1	2	yes	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
164	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
166	SD₁₆	—	4	4	no	—
167	D₈	—	4	4	no	—
168	C₂×C₈	C₂³×C₇	1	2	yes	—
169	C₂×C₈	C₂³×C₇	1	2	yes	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	C₂³×C₇	1	2	yes	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₄²	C₇×D₄	1	1	yes	characteristic
179	C₄²	C₇×D₄	1	1	yes	characteristic
180	C₄²	C₂³×C₇	1	1	yes	characteristic
181	C₄²	C₂³×C₇	1	1	yes	characteristic
182	C₂×C₇	—	4	1	no	—
183	C₂×C₇	—	8	2	no	—
184	C₂×C₇	—	8	2	no	—
185	C₂×C₇	#535	1	1	yes	characteristic
186	C₂×C₇	C₂×#135	1	1	yes	characteristic
187	C₂×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₂³	—	2	2	no	—
189	C₂³	—	2	2	no	—
190	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
191	Q₈	—	2	4	no	—
192	Q₈	—	4	2	no	—
193	D₄	—	2	4	no	—
194	D₄	—	2	4	no	—
195	D₄	—	4	4	no	—
196	D₄	—	4	4	no	—
197	D₄	—	4	2	no	—
198	D₄	—	4	2	no	—
199	D₄	—	4	2	no	—
200	D₄	—	4	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
209	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
210	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
211	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
212	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
213	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₇	#3207	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	4	2	no	—
232	C₄	C₇×#135	1	2	yes	—
233	C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
234	C₄	C₇×H(5,𝔽₂)	1	2	yes	—
235	C₂	—	4	1	no	—
236	C₂	—	8	2	no	—
237	C₂	—	8	2	no	—
238	C₂	C₇×#535	1	1	yes	characteristic
239	C₂	C₂×C₇×#135	1	1	yes	characteristic
240	C₂	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#532	C₂	semidirect
2. C₇×#532	C₂	semidirect
3. C₇×#493	C₂	semidirect
4. C₇×#492	C₂	semidirect
5. C₇×#487	C₂	semidirect
6. C₇×#486	C₂	semidirect
7. C₇×#463	C₂	semidirect
8. C₇×#449	C₂	semidirect
9. C₇×#431	C₂	semidirect
35. #3207	C₇	direct (N × H)
69. #532	C₂×C₇	semidirect
70. #532	C₂×C₇	semidirect
71. #493	C₂×C₇	semidirect
72. #492	C₂×C₇	semidirect
73. #487	C₂×C₇	semidirect
74. #486	C₂×C₇	semidirect
75. #463	C₂×C₇	semidirect
76. #449	C₂×C₇	semidirect
77. #431	C₂×C₇	semidirect
216. C₇	#3207	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#532	2
C₇×#493	2
C₇×#492	2
C₇×#487	2
C₇×#486	2
C₇×#463	2
C₇×#449	2
C₇×#431	2
#3207	7

Subgroups — C7×#3207 #11742516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3207 #11742516