Subgroups — C₇×#2832 #11742141

Order 896.

416 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#400	C₂	1	2	yes	maximal, complemented
2	C₇×#397	C₂	1	2	yes	maximal, complemented
3	C₇×#395	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#394	C₂	1	2	yes	maximal
5	C₇×#394	C₂	1	1	yes	maximal, characteristic
6	C₇×#393	C₂	1	2	yes	maximal, complemented
7	C₇×#393	C₂	1	2	yes	maximal, complemented
8	C₇×#391	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#388	C₂	1	2	yes	maximal, complemented
10	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
23	C₇×#95	C₂²	1	2	yes	—
24	C₇×#95	C₂²	1	2	yes	—
25	C₇×#95	C₂²	1	2	yes	—
26	C₇×#95	C₂²	1	2	yes	—
27	C₇×#95	C₂²	1	2	yes	—
28	C₇×#95	C₂²	1	2	yes	—
29	C₇×#95	C₂²	1	1	yes	characteristic
30	C₇×#95	C₂²	1	1	yes	characteristic
31	#2832	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₇	C₂³	1	2	yes	—
33	C₂×C₇×Q₈	—	2	2	no	—
34	C₂×C₇×Q₈	—	4	1	no	—
35	C₂×C₇×Q₈	—	4	1	no	—
36	C₂×C₇×Q₈	—	4	1	no	—
37	C₂×C₇×Q₈	—	4	2	no	—
38	C₂×C₇×D₄	—	4	1	no	—
39	C₂×C₇×D₄	—	4	2	no	—
40	C₂²×C₄×C₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₇×C₄⋊C₄	—	4	1	no	—
49	C₇×C₄⋊C₄	—	4	2	no	—
50	C₇×C₄⋊C₄	—	4	2	no	—
51	C₇×C₂²⋊C₄	—	2	4	no	—
52	C₇×C₂²⋊C₄	—	4	2	no	—
53	C₇×C₂²⋊C₄	—	4	2	no	—
54	C₇×C₂²⋊C₄	—	4	2	no	—
55	C₇×C₂²⋊C₄	—	4	2	no	—
56	C₇×C₂²⋊C₄	—	4	2	no	—
57	C₄²×C₇	—	4	1	no	—
58	#400	C₂×C₇	1	2	yes	complemented
59	#397	C₂×C₇	1	2	yes	complemented
60	#395	C₂×C₇	1	1	yes	characteristic, complemented
61	#394	C₂×C₇	1	2	yes	—
62	#394	C₂×C₇	1	1	yes	characteristic
63	#393	C₂×C₇	1	2	yes	complemented
64	#393	C₂×C₇	1	2	yes	complemented
65	#391	C₂×C₇	1	1	yes	characteristic, complemented
66	#388	C₂×C₇	1	2	yes	complemented
67	C₂³×C₇	—	2	2	no	—
68	C₂³×C₇	—	2	2	no	—
69	C₂³×C₇	—	2	2	no	—
70	C₂³×C₇	—	2	2	no	—
71	C₂³×C₇	—	2	2	no	—
72	C₂³×C₇	—	2	2	no	—
73	C₂³×C₇	—	2	2	no	—
74	C₂³×C₇	C₂⁴	1	1	yes	characteristic
75	C₇×Q₈	—	4	4	no	—
76	C₇×Q₈	—	4	4	no	—
77	C₇×D₄	—	4	4	no	—
78	C₂×C₄×C₇	—	2	1	no	—
79	C₂×C₄×C₇	—	2	1	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	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	—	2	2	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	1	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	2	1	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	2	2	no	—
97	C₂×C₄×C₇	—	2	2	no	—
98	C₂×C₄×C₇	—	2	2	no	—
99	C₂×C₄×C₇	#41	1	2	yes	—
100	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
101	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
102	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
103	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
104	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
106	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
107	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
108	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
109	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
110	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
111	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
112	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
113	#95	C₂²×C₇	1	2	yes	—
114	#95	C₂²×C₇	1	2	yes	—
115	#95	C₂²×C₇	1	2	yes	—
116	#95	C₂²×C₇	1	2	yes	—
117	#95	C₂²×C₇	1	2	yes	—
118	#95	C₂²×C₇	1	2	yes	—
119	#95	C₂²×C₇	1	1	yes	characteristic
120	#95	C₂²×C₇	1	1	yes	characteristic
121	C₂²×C₇	—	4	2	no	—
122	C₂²×C₇	—	4	2	no	—
123	C₂²×C₇	—	4	2	no	—
124	C₂²×C₇	—	4	2	no	—
125	C₂²×C₇	—	4	2	no	—
126	C₂²×C₇	—	4	2	no	—
127	C₂²×C₇	—	4	2	no	—
128	C₂²×C₇	C₂×#41	1	1	yes	characteristic
129	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
132	C₂²×C₇	2₋¹⁺⁴	1	2	yes	—
133	C₄×C₇	—	2	2	no	—
134	C₄×C₇	—	4	2	no	—
135	C₄×C₇	—	4	2	no	—
136	C₄×C₇	—	4	1	no	—
137	C₄×C₇	—	4	2	no	—
138	C₄×C₇	—	4	1	no	—
139	C₄×C₇	—	4	2	no	—
140	C₄×C₇	—	4	2	no	—
141	C₂⁴	C₂³×C₇	1	2	yes	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	1	no	—
145	C₂×Q₈	—	4	1	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×D₄	—	4	1	no	—
148	C₂×D₄	—	4	2	no	—
149	C₂²×C₄	C₂³×C₇	1	2	yes	—
150	C₂²×C₄	C₂³×C₇	1	2	yes	—
151	C₂²×C₄	C₂³×C₇	1	2	yes	—
152	C₂²×C₄	C₂³×C₇	1	2	yes	—
153	C₂²×C₄	C₂³×C₇	1	2	yes	—
154	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
155	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
157	C₄⋊C₄	—	4	1	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	4	2	no	—
164	C₂²⋊C₄	—	4	2	no	—
165	C₂²⋊C₄	—	4	2	no	—
166	C₄²	—	4	1	no	—
167	C₂×C₇	—	8	2	no	—
168	C₂×C₇	#538	1	1	yes	characteristic
169	C₂×C₇	#542	1	2	yes	—
170	C₂×C₇	H(3,𝔽₄)	1	1	yes	characteristic
171	C₂×C₇	#562	1	2	yes	—
172	C₂×C₇	#563	1	1	yes	characteristic
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	C₂⁴×C₇	1	1	yes	characteristic
181	Q₈	—	4	4	no	—
182	Q₈	—	4	4	no	—
183	D₄	—	4	4	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	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	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	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	C₇×#41	1	2	yes	—
206	C₇	#2832	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	C₂×C₇×#41	1	1	yes	characteristic
215	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
216	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
217	C₂²	C₇×H(5,𝔽₂)	1	2	yes	—
218	C₂²	C₇×2₋¹⁺⁴	1	2	yes	—
219	C₄	—	2	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₂	—	8	2	no	—
228	C₂	C₇×#538	1	1	yes	characteristic
229	C₂	C₇×#542	1	2	yes	—
230	C₂	C₇×H(3,𝔽₄)	1	1	yes	characteristic
231	C₂	C₇×#562	1	2	yes	—
232	C₂	C₇×#563	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#400	C₂	semidirect
2. C₇×#397	C₂	semidirect
3. C₇×#395	C₂	semidirect
6. C₇×#393	C₂	semidirect
7. C₇×#393	C₂	semidirect
8. C₇×#391	C₂	semidirect
9. C₇×#388	C₂	semidirect
31. #2832	C₇	direct (N × H)
58. #400	C₂×C₇	semidirect
59. #397	C₂×C₇	semidirect
60. #395	C₂×C₇	semidirect
63. #393	C₂×C₇	semidirect
64. #393	C₂×C₇	semidirect
65. #391	C₂×C₇	semidirect
66. #388	C₂×C₇	semidirect
206. C₇	#2832	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×#400	2
C₇×#397	2
C₇×#395	2
C₇×#394	2
C₇×#393	2
C₇×#391	2
C₇×#388	2
#2832	7

Subgroups — C7×#2832 #11742141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2832 #11742141