Subgroups — C₇×#23372 #13200588

Order 1792.

584 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	C₄×C₇×#125	C₂	1	1	yes	maximal, characteristic
2	C₇×#1940	C₂	1	4	yes	maximal
3	C₇×#1917	C₂	1	4	yes	maximal
4	C₇×#1917	C₂	1	4	yes	maximal
5	C₇×#1831	C₂	1	2	yes	maximal
6	C₂×C₇×#125	C₄	1	1	yes	characteristic
7	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
10	C₂×C₇×#97	C₂²	1	4	yes	—
11	C₇×#389	C₄	1	2	yes	—
12	C₇×#387	C₄	1	4	yes	—
13	C₇×#382	C₄	1	1	yes	characteristic
14	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
15	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
16	C₄³×C₇	C₂²	1	1	yes	characteristic
17	C₇×#336	C₂²	1	4	yes	—
18	C₇×#336	C₂²	1	4	yes	—
19	C₇×#336	C₂²	1	4	yes	—
20	C₇×#336	C₂²	1	4	yes	—
21	#23372	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₇×C₈	C₂³	1	4	yes	—
23	C₂²×C₇×C₈	C₂³	1	4	yes	—
24	C₇×#125	—	2	4	no	—
25	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
26	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
27	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂³	1	2	yes	—
34	C₂×C₄²×C₇	C₂³	1	4	yes	—
35	C₇×C₄⋊C₈	—	4	4	no	—
36	C₇×C₄⋊C₈	—	4	2	no	—
37	C₇×C₄⋊C₈	—	4	2	no	—
38	C₇×#97	—	4	4	no	—
39	C₇×#95	C₂×C₄	1	4	yes	—
40	C₇×#95	C₂×C₄	1	4	yes	—
41	C₇×#95	C₂×C₄	1	4	yes	—
42	C₄×#125	C₂×C₇	1	1	yes	characteristic
43	#1940	C₂×C₇	1	4	yes	—
44	#1917	C₂×C₇	1	4	yes	—
45	#1917	C₂×C₇	1	4	yes	—
46	#1831	C₂×C₇	1	2	yes	—
47	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
54	C₂×C₇×C₈	—	2	4	no	—
55	C₂×C₇×C₈	—	2	4	no	—
56	C₂×C₇×C₈	—	2	4	no	—
57	C₂×C₇×C₈	—	2	4	no	—
58	C₂×C₇×C₈	—	2	4	no	—
59	C₂×C₇×C₈	—	2	4	no	—
60	C₇×C₄⋊C₄	—	2	8	no	—
61	C₇×C₄⋊C₄	—	2	4	no	—
62	C₇×C₄⋊C₄	—	4	2	no	—
63	C₇×C₄⋊C₄	—	4	4	no	—
64	C₄²×C₇	—	2	4	no	—
65	C₄²×C₇	—	2	2	no	—
66	C₄²×C₇	—	2	1	no	—
67	C₄²×C₇	—	2	1	no	—
68	C₄²×C₇	—	2	2	no	—
69	C₄²×C₇	—	2	4	no	—
70	C₄²×C₇	—	4	1	no	—
71	C₄²×C₇	—	4	2	no	—
72	C₄²×C₇	M₁₆	1	4	yes	—
73	C₂×#125	C₄×C₇	1	1	yes	characteristic
74	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
75	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
76	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
77	C₂×#97	C₂²×C₇	1	4	yes	—
78	#389	C₄×C₇	1	2	yes	—
79	#387	C₄×C₇	1	4	yes	—
80	#382	C₄×C₇	1	1	yes	characteristic
81	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
82	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
83	C₄³	C₂²×C₇	1	1	yes	characteristic
84	#336	C₂²×C₇	1	4	yes	—
85	#336	C₂²×C₇	1	4	yes	—
86	#336	C₂²×C₇	1	4	yes	—
87	#336	C₂²×C₇	1	4	yes	—
88	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	—	2	1	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	1	no	—
94	C₂×C₄×C₇	—	2	4	no	—
95	C₂×C₄×C₇	—	2	4	no	—
96	C₂×C₄×C₇	—	2	2	no	—
97	C₂×C₄×C₇	—	2	4	no	—
98	C₂×C₄×C₇	—	2	2	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	4	no	—
102	C₂×C₄×C₇	—	2	4	no	—
103	C₂×C₄×C₇	—	2	4	no	—
104	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
106	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
107	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
110	C₇×C₈	—	4	4	no	—
111	C₇×C₈	—	4	4	no	—
112	C₂²×C₈	C₂³×C₇	1	4	yes	—
113	C₂²×C₈	C₂³×C₇	1	4	yes	—
114	#125	—	2	4	no	—
115	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
116	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
117	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
118	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
119	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
120	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
121	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
122	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
123	C₂×C₄²	C₂³×C₇	1	2	yes	—
124	C₂×C₄²	C₂³×C₇	1	4	yes	—
125	C₄⋊C₈	—	4	4	no	—
126	C₄⋊C₈	—	4	2	no	—
127	C₄⋊C₈	—	4	2	no	—
128	#97	—	4	4	no	—
129	#95	C₂×C₄×C₇	1	4	yes	—
130	#95	C₂×C₄×C₇	1	4	yes	—
131	#95	C₂×C₄×C₇	1	4	yes	—
132	C₂²×C₇	#519	1	1	yes	characteristic
133	C₂²×C₇	#520	1	2	yes	—
134	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
135	C₂²×C₇	#568	1	1	yes	characteristic
136	C₂²×C₇	#568	1	2	yes	—
137	C₄×C₇	—	2	2	no	—
138	C₄×C₇	—	2	4	no	—
139	C₄×C₇	—	4	1	no	—
140	C₄×C₇	—	4	4	no	—
141	C₄×C₇	—	4	2	no	—
142	C₄×C₇	—	4	4	no	—
143	C₄×C₇	#562	1	1	yes	characteristic
144	C₄×C₇	#563	1	2	yes	—
145	C₄×C₇	#564	1	1	yes	characteristic
146	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
147	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
148	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
149	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
150	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
151	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
152	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₄⋊C₄	—	2	8	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄²	—	2	4	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	4	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	2	no	—
171	C₄²	C₇×M₁₆	1	4	yes	—
172	C₂×C₇	#2382	1	1	yes	characteristic
173	C₂×C₇	#2983	1	1	yes	characteristic
174	C₂×C₇	#2987	1	1	yes	characteristic
175	C₂×C₇	#2989	1	2	yes	—
176	C₂×C₇	#2995	1	2	yes	—
177	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
194	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
195	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
196	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
197	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
198	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
199	C₈	—	4	4	no	—
200	C₈	—	4	4	no	—
201	C₇	#23372	1	1	yes	characteristic, direct product factor, Sylow
202	C₂²	C₇×#519	1	1	yes	characteristic
203	C₂²	C₇×#520	1	2	yes	—
204	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
205	C₂²	C₇×#568	1	1	yes	characteristic
206	C₂²	C₇×#568	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	4	1	no	—
210	C₄	—	4	4	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	4	no	—
213	C₄	C₇×#562	1	1	yes	characteristic
214	C₄	C₇×#563	1	2	yes	—
215	C₄	C₇×#564	1	1	yes	characteristic
216	C₂	C₇×#2382	1	1	yes	characteristic
217	C₂	C₇×#2983	1	1	yes	characteristic
218	C₂	C₇×#2987	1	1	yes	characteristic
219	C₂	C₇×#2989	1	2	yes	—
220	C₂	C₇×#2995	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂³×C₇	C₂³×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
21. #23372	C₇	direct (N × H)
201. C₇	#23372	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×#125	2
C₇×#1940	2
C₇×#1917	2
C₇×#1831	2
#23372	7

Subgroups — C7×#23372 #13200588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23372 #13200588