Subgroups — C₇×#3228 #11742537

Order 896.

370 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#532	C₂	1	1	yes	maximal, characteristic
3	C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#486	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#482	C₂	1	4	yes	maximal, complemented
8	C₇×#478	C₂	1	4	yes	maximal, complemented
9	C₇×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×#41	—	2	1	no	—
11	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₇×#127	C₂²	1	1	yes	characteristic
15	C₇×#125	C₂²	1	2	yes	—
16	C₇×#124	—	2	1	no	—
17	C₇×C₂²⋊Q₈	—	2	1	no	—
18	C₇×C₂²⋊Q₈	—	2	1	no	—
19	C₇×#121	—	2	1	no	—
20	C₇×#121	C₂²	1	2	yes	—
21	C₄×C₇×Q₈	—	2	1	no	—
22	C₄×C₇×Q₈	—	2	1	no	—
23	C₄×C₇×D₄	—	2	1	no	—
24	C₄×C₇×D₄	C₂²	1	2	yes	—
25	C₇×#117	C₂²	1	2	yes	—
26	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
27	C₇×#107	C₂²	1	4	yes	—
28	C₇×#106	C₂²	1	4	yes	—
29	C₇×C₄⋊C₈	C₂²	1	2	yes	—
30	C₇×#102	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	#3228	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₇×#41	—	4	4	no	—
37	C₂×C₇×Q₈	—	2	1	no	—
38	C₂×C₇×Q₈	—	2	1	no	—
39	C₂×C₇×Q₈	—	2	1	no	—
40	C₂×C₇×D₄	—	2	2	no	—
41	C₂×C₇×D₄	—	2	1	no	—
42	C₂×C₇×D₄	—	2	1	no	—
43	C₂×C₇×D₄	C₂³	1	2	yes	—
44	C₂²×C₄×C₇	—	2	2	no	—
45	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂×C₇×C₈	C₂³	1	2	yes	—
49	C₂×C₇×C₈	C₂³	1	2	yes	—
50	C₇×C₄⋊C₄	—	2	1	no	—
51	C₇×C₄⋊C₄	—	2	2	no	—
52	C₇×C₄⋊C₄	—	2	2	no	—
53	C₇×C₄⋊C₄	—	2	1	no	—
54	C₇×C₄⋊C₄	—	2	1	no	—
55	C₇×C₄⋊C₄	C₂³	1	2	yes	—
56	C₇×C₄⋊C₄	C₂³	1	2	yes	—
57	C₇×C₄⋊C₄	C₂³	1	2	yes	—
58	C₇×C₂²⋊C₄	—	2	2	no	—
59	C₇×C₂²⋊C₄	—	2	2	no	—
60	C₇×C₂²⋊C₄	—	2	2	no	—
61	C₄²×C₇	—	2	2	no	—
62	C₄²×C₇	D₄	1	1	yes	characteristic
63	C₄²×C₇	D₄	1	1	yes	characteristic
64	C₄²×C₇	C₂³	1	1	yes	characteristic
65	C₄²×C₇	C₂³	1	1	yes	characteristic
66	#533	C₂×C₇	1	1	yes	characteristic, complemented
67	#532	C₂×C₇	1	1	yes	characteristic
68	#500	C₂×C₇	1	1	yes	characteristic, complemented
69	#498	C₂×C₇	1	1	yes	characteristic, complemented
70	#486	C₂×C₇	1	1	yes	characteristic, complemented
71	#486	C₂×C₇	1	1	yes	characteristic, complemented
72	#482	C₂×C₇	1	4	yes	complemented
73	#478	C₂×C₇	1	4	yes	complemented
74	#431	C₂×C₇	1	1	yes	characteristic, complemented
75	C₂³×C₇	—	2	2	no	—
76	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
77	C₇×Q₈	—	4	2	no	—
78	C₇×Q₈	—	4	2	no	—
79	C₇×Q₈	—	4	2	no	—
80	C₇×D₄	—	2	4	no	—
81	C₇×D₄	—	4	4	no	—
82	C₇×D₄	—	4	2	no	—
83	C₇×D₄	—	4	2	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	4	4	no	—
91	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
97	C₇×C₈	—	2	2	no	—
98	C₇×C₈	—	2	2	no	—
99	C₂×#41	—	2	1	no	—
100	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
101	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
102	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
103	#127	C₂²×C₇	1	1	yes	characteristic
104	#125	C₂²×C₇	1	2	yes	—
105	#124	—	2	1	no	—
106	C₂²⋊Q₈	—	2	1	no	—
107	C₂²⋊Q₈	—	2	1	no	—
108	#121	—	2	1	no	—
109	#121	C₂²×C₇	1	2	yes	—
110	C₄×Q₈	—	2	1	no	—
111	C₄×Q₈	—	2	1	no	—
112	C₄×D₄	—	2	1	no	—
113	C₄×D₄	C₂²×C₇	1	2	yes	—
114	#117	C₂²×C₇	1	2	yes	—
115	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
116	#107	C₂²×C₇	1	4	yes	—
117	#106	C₂²×C₇	1	4	yes	—
118	C₄⋊C₈	C₂²×C₇	1	2	yes	—
119	#102	C₂²×C₇	1	4	yes	—
120	#102	C₂²×C₇	1	4	yes	—
121	C₂²⋊C₈	C₂²×C₇	1	2	yes	—
122	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
123	C₄×C₈	C₂²×C₇	1	1	yes	characteristic
124	C₂²×C₇	—	2	1	no	—
125	C₂²×C₇	—	2	1	no	—
126	C₂²×C₇	—	2	1	no	—
127	C₂²×C₇	—	4	2	no	—
128	C₂²×C₇	—	4	2	no	—
129	C₂²×C₇	—	4	2	no	—
130	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₇	—	2	1	no	—
132	C₄×C₇	—	2	1	no	—
133	C₄×C₇	—	2	1	no	—
134	C₄×C₇	—	4	2	no	—
135	C₄×C₇	—	4	2	no	—
136	C₄×C₇	—	4	2	no	—
137	C₄×C₇	#135	1	2	yes	—
138	C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
139	C₄×C₇	2₋¹⁺⁴	1	2	yes	—
140	#41	—	4	4	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	2	1	no	—
143	C₂×Q₈	—	2	1	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	2	1	no	—
147	C₂×D₄	C₂³×C₇	1	2	yes	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
150	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
152	C₂×C₈	C₂³×C₇	1	2	yes	—
153	C₂×C₈	C₂³×C₇	1	2	yes	—
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	1	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	C₂³×C₇	1	2	yes	—
160	C₄⋊C₄	C₂³×C₇	1	2	yes	—
161	C₄⋊C₄	C₂³×C₇	1	2	yes	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	C₇×D₄	1	1	yes	characteristic
167	C₄²	C₇×D₄	1	1	yes	characteristic
168	C₄²	C₂³×C₇	1	1	yes	characteristic
169	C₄²	C₂³×C₇	1	1	yes	characteristic
170	C₂×C₇	—	4	1	no	—
171	C₂×C₇	—	8	2	no	—
172	C₂×C₇	#536	1	1	yes	characteristic
173	C₂×C₇	C₂×#135	1	1	yes	characteristic
174	C₂×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂³	—	2	2	no	—
176	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
177	Q₈	—	4	2	no	—
178	Q₈	—	4	2	no	—
179	Q₈	—	4	2	no	—
180	D₄	—	2	4	no	—
181	D₄	—	4	4	no	—
182	D₄	—	4	2	no	—
183	D₄	—	4	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
192	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
193	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
194	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
195	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
196	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	—	2	2	no	—
199	C₇	#3228	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	C₇×#135	1	2	yes	—
214	C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
215	C₄	C₇×2₋¹⁺⁴	1	2	yes	—
216	C₂	—	4	1	no	—
217	C₂	—	8	2	no	—
218	C₂	C₇×#536	1	1	yes	characteristic
219	C₂	C₂×C₇×#135	1	1	yes	characteristic
220	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
prime radical	C₇×#532	C₂
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×#533	C₂	semidirect
3. C₇×#500	C₂	semidirect
4. C₇×#498	C₂	semidirect
5. C₇×#486	C₂	semidirect
6. C₇×#486	C₂	semidirect
7. C₇×#482	C₂	semidirect
8. C₇×#478	C₂	semidirect
9. C₇×#431	C₂	semidirect
35. #3228	C₇	direct (N × H)
66. #533	C₂×C₇	semidirect
68. #500	C₂×C₇	semidirect
69. #498	C₂×C₇	semidirect
70. #486	C₂×C₇	semidirect
71. #486	C₂×C₇	semidirect
72. #482	C₂×C₇	semidirect
73. #478	C₂×C₇	semidirect
74. #431	C₂×C₇	semidirect
199. C₇	#3228	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×#533	2
C₇×#532	2
C₇×#500	2
C₇×#498	2
C₇×#486	2
C₇×#482	2
C₇×#478	2
C₇×#431	2
#3228	7

Subgroups — C7×#3228 #11742537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3228 #11742537