Subgroups — C₇×#23380 #13200596

Order 1792.

612 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2401	C₂	1	1	yes	maximal, characteristic
2	C₇×#1940	C₂	1	3	yes	maximal, complemented
3	C₇×#1924	C₂	1	3	yes	maximal
4	C₇×#1917	C₂	1	3	yes	maximal, complemented
5	C₇×#1850	C₂	1	3	yes	maximal
6	C₇×#1843	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1840	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
11	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
12	C₂×C₇×C₂²⋊C₈	C₂²	1	3	yes	—
13	C₂×C₇×#97	C₂²	1	3	yes	—
14	C₇×#401	C₄	1	1	yes	characteristic
15	C₇×#400	C₄	1	3	yes	—
16	C₇×#397	C₄	1	3	yes	—
17	C₇×#396	C₄	1	1	yes	characteristic
18	C₄×C₇×C₄⋊C₄	C₂²	1	3	yes	—
19	C₄×C₇×C₂²⋊C₄	C₂²	1	3	yes	—
20	C₇×#376	C₂²	1	1	yes	characteristic
21	C₇×#336	C₂²	1	3	yes	—
22	C₇×#336	C₂²	1	3	yes	—
23	C₇×#336	C₂²	1	3	yes	—
24	C₇×#336	C₂²	1	3	yes	—
25	#23380	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂²×C₇×C₈	—	4	3	no	—
28	C₂²×C₇×C₈	C₂³	1	3	yes	—
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂³	1	3	yes	—
32	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
33	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
34	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	3	yes	—
35	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	3	yes	—
36	C₂×C₄²×C₇	C₂³	1	3	yes	—
37	C₂×C₄²×C₇	C₂³	1	3	yes	—
38	C₇×C₄⋊C₈	—	4	3	no	—
39	C₇×C₄⋊C₈	—	4	3	no	—
40	C₇×C₄⋊C₈	—	4	3	no	—
41	C₇×C₂²⋊C₈	—	4	3	no	—
42	C₇×#97	—	4	3	no	—
43	C₇×#95	C₂×C₄	1	1	yes	characteristic
44	C₇×#95	C₂×C₄	1	3	yes	—
45	C₇×#95	C₂×C₄	1	3	yes	—
46	C₇×#95	C₂×C₄	1	3	yes	—
47	C₇×#95	C₂×C₄	1	3	yes	—
48	C₇×#95	C₂×C₄	1	3	yes	—
49	#2401	C₂×C₇	1	1	yes	characteristic
50	#1940	C₂×C₇	1	3	yes	complemented
51	#1924	C₂×C₇	1	3	yes	—
52	#1917	C₂×C₇	1	3	yes	complemented
53	#1850	C₂×C₇	1	3	yes	—
54	#1843	C₂×C₇	1	1	yes	characteristic, complemented
55	#1840	C₂×C₇	1	1	yes	characteristic, complemented
56	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	—	2	3	no	—
58	C₂²×C₄×C₇	—	2	3	no	—
59	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
61	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
62	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
65	C₂×C₇×C₈	—	2	3	no	—
66	C₂×C₇×C₈	—	2	3	no	—
67	C₂×C₇×C₈	—	2	3	no	—
68	C₂×C₇×C₈	—	2	3	no	—
69	C₂×C₇×C₈	—	2	3	no	—
70	C₂×C₇×C₈	—	2	3	no	—
71	C₂×C₇×C₈	—	2	6	no	—
72	C₂×C₇×C₈	—	4	4	no	—
73	C₇×C₄⋊C₄	—	4	3	no	—
74	C₇×C₄⋊C₄	—	4	3	no	—
75	C₇×C₂²⋊C₄	—	2	6	no	—
76	C₇×C₂²⋊C₄	—	4	3	no	—
77	C₄²×C₇	—	2	3	no	—
78	C₄²×C₇	—	2	3	no	—
79	C₄²×C₇	—	2	3	no	—
80	C₄²×C₇	—	2	3	no	—
81	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
82	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
83	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
84	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
85	C₂×C₂²⋊C₈	C₂²×C₇	1	3	yes	—
86	C₂×#97	C₂²×C₇	1	3	yes	—
87	#401	C₄×C₇	1	1	yes	characteristic
88	#400	C₄×C₇	1	3	yes	—
89	#397	C₄×C₇	1	3	yes	—
90	#396	C₄×C₇	1	1	yes	characteristic
91	C₄×C₄⋊C₄	C₂²×C₇	1	3	yes	—
92	C₄×C₂²⋊C₄	C₂²×C₇	1	3	yes	—
93	#376	C₂²×C₇	1	1	yes	characteristic
94	#336	C₂²×C₇	1	3	yes	—
95	#336	C₂²×C₇	1	3	yes	—
96	#336	C₂²×C₇	1	3	yes	—
97	#336	C₂²×C₇	1	3	yes	—
98	C₂³×C₇	—	2	3	no	—
99	C₂³×C₇	—	2	3	no	—
100	C₂³×C₇	#131	1	2	yes	—
101	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₇	—	2	3	no	—
103	C₂×C₄×C₇	—	2	3	no	—
104	C₂×C₄×C₇	—	2	3	no	—
105	C₂×C₄×C₇	—	2	3	no	—
106	C₂×C₄×C₇	—	2	3	no	—
107	C₂×C₄×C₇	—	2	3	no	—
108	C₂×C₄×C₇	—	2	3	no	—
109	C₂×C₄×C₇	—	2	3	no	—
110	C₂×C₄×C₇	—	2	3	no	—
111	C₂×C₄×C₇	—	2	3	no	—
112	C₂×C₄×C₇	—	2	3	no	—
113	C₂×C₄×C₇	—	2	3	no	—
114	C₂×C₄×C₇	—	2	3	no	—
115	C₂×C₄×C₇	—	4	1	no	—
116	C₂×C₄×C₇	—	4	3	no	—
117	C₂×C₄×C₇	H(5,𝔽₂)	1	3	yes	—
118	C₂×C₄×C₇	2₋¹⁺⁴	1	3	yes	—
119	C₇×C₈	—	4	3	no	—
120	C₇×C₈	—	4	1	no	—
121	C₇×C₈	—	4	1	no	—
122	C₇×C₈	—	4	3	no	—
123	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
124	C₂²×C₈	—	4	3	no	—
125	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
127	C₂²×C₈	C₂³×C₇	1	3	yes	—
128	C₂²×C₈	C₂³×C₇	1	3	yes	—
129	C₂×C₄⋊C₄	C₂×C₄×C₇	1	3	yes	—
130	C₂×C₄⋊C₄	C₂×C₄×C₇	1	3	yes	—
131	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	3	yes	—
132	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	3	yes	—
133	C₂×C₄²	C₂³×C₇	1	3	yes	—
134	C₂×C₄²	C₂³×C₇	1	3	yes	—
135	C₄⋊C₈	—	4	3	no	—
136	C₄⋊C₈	—	4	3	no	—
137	C₄⋊C₈	—	4	3	no	—
138	C₂²⋊C₈	—	4	3	no	—
139	#97	—	4	3	no	—
140	#95	C₂×C₄×C₇	1	1	yes	characteristic
141	#95	C₂×C₄×C₇	1	3	yes	—
142	#95	C₂×C₄×C₇	1	3	yes	—
143	#95	C₂×C₄×C₇	1	3	yes	—
144	#95	C₂×C₄×C₇	1	3	yes	—
145	#95	C₂×C₄×C₇	1	3	yes	—
146	C₂²×C₇	—	2	6	no	—
147	C₂²×C₇	—	4	1	no	—
148	C₂²×C₇	—	4	3	no	—
149	C₂²×C₇	#520	1	3	yes	—
150	C₂²×C₇	C₂×#131	1	1	yes	characteristic
151	C₂²×C₇	#568	1	3	yes	—
152	C₄×C₇	—	4	1	no	—
153	C₄×C₇	—	4	3	no	—
154	C₄×C₇	—	4	3	no	—
155	C₄×C₇	—	4	3	no	—
156	C₄×C₇	—	4	3	no	—
157	C₄×C₇	#560	1	1	yes	characteristic
158	C₄×C₇	#563	1	3	yes	—
159	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
160	C₂²×C₄	—	2	3	no	—
161	C₂²×C₄	—	2	3	no	—
162	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
163	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
164	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
165	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
166	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
167	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
168	C₂×C₈	—	2	3	no	—
169	C₂×C₈	—	2	3	no	—
170	C₂×C₈	—	2	3	no	—
171	C₂×C₈	—	2	3	no	—
172	C₂×C₈	—	2	3	no	—
173	C₂×C₈	—	2	3	no	—
174	C₂×C₈	—	2	6	no	—
175	C₂×C₈	—	4	4	no	—
176	C₄⋊C₄	—	4	3	no	—
177	C₄⋊C₄	—	4	3	no	—
178	C₂²⋊C₄	—	2	6	no	—
179	C₂²⋊C₄	—	4	3	no	—
180	C₄²	—	2	3	no	—
181	C₄²	—	2	3	no	—
182	C₄²	—	2	3	no	—
183	C₄²	—	2	3	no	—
184	C₂×C₇	—	4	2	no	—
185	C₂×C₇	#2383	1	1	yes	characteristic
186	C₂×C₇	#2992	1	3	yes	—
187	C₂×C₇	#2995	1	3	yes	—
188	C₂³	—	2	3	no	—
189	C₂³	—	2	3	no	—
190	C₂³	C₇×#131	1	2	yes	—
191	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	—	2	3	no	—
199	C₂×C₄	—	2	3	no	—
200	C₂×C₄	—	2	3	no	—
201	C₂×C₄	—	2	3	no	—
202	C₂×C₄	—	2	3	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	3	no	—
207	C₂×C₄	C₇×H(5,𝔽₂)	1	3	yes	—
208	C₂×C₄	C₇×2₋¹⁺⁴	1	3	yes	—
209	C₈	—	4	3	no	—
210	C₈	—	4	1	no	—
211	C₈	—	4	1	no	—
212	C₈	—	4	3	no	—
213	C₇	#23380	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	—	2	6	no	—
215	C₂²	—	4	3	no	—
216	C₂²	—	4	1	no	—
217	C₂²	C₇×#520	1	3	yes	—
218	C₂²	C₂×C₇×#131	1	1	yes	characteristic
219	C₂²	C₇×#568	1	3	yes	—
220	C₄	—	4	3	no	—
221	C₄	—	4	3	no	—
222	C₄	—	4	3	no	—
223	C₄	—	4	3	no	—
224	C₄	—	4	1	no	—
225	C₄	C₇×#560	1	1	yes	characteristic
226	C₄	C₇×#563	1	3	yes	—
227	C₂	—	4	2	no	—
228	C₂	C₇×#2383	1	1	yes	characteristic
229	C₂	C₇×#2992	1	3	yes	—
230	C₂	C₇×#2995	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#1940	C₂	semidirect
4. C₇×#1917	C₂	semidirect
6. C₇×#1843	C₂	semidirect
7. C₇×#1840	C₂	semidirect
25. #23380	C₇	direct (N × H)
50. #1940	C₂×C₇	semidirect
52. #1917	C₂×C₇	semidirect
54. #1843	C₂×C₇	semidirect
55. #1840	C₂×C₇	semidirect
213. C₇	#23380	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×#2401	2
C₇×#1940	2
C₇×#1924	2
C₇×#1917	2
C₇×#1850	2
C₇×#1843	2
C₇×#1840	2
#23380	7

Subgroups — C7×#23380 #13200596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23380 #13200596