Subgroups — C₇×#23200 #13200416

Order 1792.

878 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#494	C₂	1	1	yes	maximal, characteristic
2	C₇×#2950	C₂	1	4	yes	maximal
3	C₇×#1979	C₂	1	4	yes	maximal
4	C₇×#1886	C₂	1	4	yes	maximal
5	C₇×#1772	C₂	1	2	yes	maximal
6	C₂²×C₇×Q₁₆	C₂²	1	4	yes	—
7	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₇×#519	C₂²	1	4	yes	—
9	C₇×#494	C₄	1	8	yes	complemented
10	C₇×#441	—	2	8	no	—
11	C₄×C₇×Q₁₆	—	2	8	no	—
12	C₂×C₇×#108	C₂²	1	2	yes	—
13	C₇×#427	C₂²	1	2	yes	—
14	C₇×#419	C₂²	1	8	yes	—
15	C₂×C₇×#101	C₂²	1	4	yes	—
16	C₇×#405	C₂²	1	4	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₇×#339	C₂²	1	4	yes	—
19	#23200	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×Q₈	C₂³	1	4	yes	—
21	C₂×C₇×Q₁₆	—	2	4	no	—
22	C₂×C₇×Q₁₆	—	2	4	no	—
23	C₂×C₇×Q₁₆	—	2	8	no	—
24	C₂×C₇×Q₁₆	C₂×C₄	1	16	yes	—
25	C₂×C₇×M₁₆	C₂³	1	4	yes	—
26	C₂²×C₇×C₈	C₂³	1	2	yes	—
27	C₇×C₄⋊Q₈	—	4	2	no	—
28	C₇×C₄⋊Q₈	C₂×C₄	1	8	yes	—
29	C₄×C₇×Q₈	—	2	16	no	—
30	C₇×#117	—	4	4	no	—
31	C₇×#117	C₂³	1	4	yes	—
32	C₂×C₇×C₄⋊C₄	—	4	2	no	—
33	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
34	C₇×#108	—	2	4	no	—
35	C₇×#107	D₄	1	4	yes	—
36	C₇×#106	D₄	1	4	yes	—
37	C₇×#103	—	2	16	no	—
38	C₇×#101	—	4	4	no	—
39	C₇×#97	—	2	4	no	—
40	C₄×C₇×C₈	—	2	4	no	—
41	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
42	C₂×#494	C₂×C₇	1	1	yes	characteristic
43	#2950	C₂×C₇	1	4	yes	—
44	#1979	C₂×C₇	1	4	yes	—
45	#1886	C₂×C₇	1	4	yes	—
46	#1772	C₂×C₇	1	2	yes	—
47	C₂×C₇×Q₈	—	2	4	no	—
48	C₂×C₇×Q₈	—	2	8	no	—
49	C₂×C₇×Q₈	—	2	8	no	—
50	C₂×C₇×Q₈	C₂²×C₄	1	8	yes	—
51	C₂²×C₄×C₇	—	2	1	no	—
52	C₂²×C₄×C₇	—	2	4	no	—
53	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
54	C₇×Q₁₆	—	2	16	no	—
55	C₇×Q₁₆	—	2	16	no	—
56	C₇×M₁₆	—	2	4	no	—
57	C₇×M₁₆	—	2	4	no	—
58	C₂×C₇×C₈	—	2	4	no	—
59	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
60	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
61	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
62	C₂×C₇×C₈	#41	1	4	yes	—
63	C₇×C₄⋊C₄	—	4	8	no	—
64	C₇×C₄⋊C₄	—	4	8	no	—
65	C₇×C₄⋊C₄	C₂×D₄	1	8	yes	—
66	C₇×C₂²⋊C₄	—	2	4	no	—
67	C₄²×C₇	—	2	4	no	—
68	C₄²×C₇	—	2	1	no	—
69	C₄²×C₇	—	4	8	no	—
70	C₄²×C₇	C₂²×C₄	1	2	yes	—
71	C₂²×Q₁₆	C₂²×C₇	1	4	yes	—
72	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
73	#519	C₂²×C₇	1	4	yes	—
74	#494	C₄×C₇	1	8	yes	complemented
75	#441	—	2	8	no	—
76	C₄×Q₁₆	—	2	8	no	—
77	C₂×#108	C₂²×C₇	1	2	yes	—
78	#427	C₂²×C₇	1	2	yes	—
79	#419	C₂²×C₇	1	8	yes	—
80	C₂×#101	C₂²×C₇	1	4	yes	—
81	#405	C₂²×C₇	1	4	yes	—
82	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
83	#339	C₂²×C₇	1	4	yes	—
84	C₂³×C₇	H(5,𝔽₂)	1	1	yes	characteristic
85	C₇×Q₈	—	2	16	no	—
86	C₇×Q₈	—	4	8	no	—
87	C₂×C₄×C₇	—	2	4	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	8	no	—
90	C₂×C₄×C₇	—	2	8	no	—
91	C₂×C₄×C₇	—	4	4	no	—
92	C₂×C₄×C₇	—	4	4	no	—
93	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
95	C₂×C₄×C₇	C₂×#41	1	2	yes	—
96	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
97	C₇×C₈	—	4	4	no	—
98	C₇×C₈	C₄×D₄	1	4	yes	—
99	C₇×C₈	C₄×D₄	1	4	yes	—
100	C₂²×Q₈	C₂³×C₇	1	4	yes	—
101	C₂×Q₁₆	—	2	4	no	—
102	C₂×Q₁₆	—	2	4	no	—
103	C₂×Q₁₆	—	2	8	no	—
104	C₂×Q₁₆	C₂×C₄×C₇	1	16	yes	—
105	C₂×M₁₆	C₂³×C₇	1	4	yes	—
106	C₂²×C₈	C₂³×C₇	1	2	yes	—
107	C₄⋊Q₈	—	4	2	no	—
108	C₄⋊Q₈	C₂×C₄×C₇	1	8	yes	—
109	C₄×Q₈	—	2	16	no	—
110	#117	—	4	4	no	—
111	#117	C₂³×C₇	1	4	yes	—
112	C₂×C₄⋊C₄	—	4	2	no	—
113	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
114	#108	—	2	4	no	—
115	#107	C₇×D₄	1	4	yes	—
116	#106	C₇×D₄	1	4	yes	—
117	#103	—	2	16	no	—
118	#101	—	4	4	no	—
119	#97	—	2	4	no	—
120	C₄×C₈	—	2	4	no	—
121	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
122	C₂²×C₇	—	2	2	no	—
123	C₂²×C₇	#518	1	1	yes	characteristic
124	C₂²×C₇	D₄²	1	1	yes	characteristic
125	C₂²×C₇	#559	1	1	yes	characteristic
126	C₄×C₇	—	2	4	no	—
127	C₄×C₇	—	4	8	no	—
128	C₄×C₇	—	4	8	no	—
129	C₄×C₇	C₂×C₄×D₄	1	2	yes	—
130	C₄×C₇	#550	1	2	yes	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	8	no	—
133	C₂×Q₈	—	2	8	no	—
134	C₂×Q₈	C₂²×C₄×C₇	1	8	yes	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
138	Q₁₆	—	2	16	no	—
139	Q₁₆	—	2	16	no	—
140	M₁₆	—	2	4	no	—
141	M₁₆	—	2	4	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
144	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
145	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
146	C₂×C₈	C₇×#41	1	4	yes	—
147	C₄⋊C₄	—	4	8	no	—
148	C₄⋊C₄	—	4	8	no	—
149	C₄⋊C₄	C₂×C₇×D₄	1	8	yes	—
150	C₂²⋊C₄	—	2	4	no	—
151	C₄²	—	2	4	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	4	8	no	—
154	C₄²	C₂²×C₄×C₇	1	2	yes	—
155	C₂×C₇	—	2	1	no	—
156	C₂×C₇	—	2	1	no	—
157	C₂×C₇	#2366	1	1	yes	characteristic
158	C₂×C₇	#3293	1	2	yes	—
159	C₂³	C₇×H(5,𝔽₂)	1	1	yes	characteristic
160	Q₈	—	2	16	no	—
161	Q₈	—	4	8	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	8	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
169	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
170	C₂×C₄	C₂×C₇×#41	1	2	yes	—
171	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
172	C₈	—	4	4	no	—
173	C₈	C₄×C₇×D₄	1	4	yes	—
174	C₈	C₄×C₇×D₄	1	4	yes	—
175	C₇	#23200	1	1	yes	characteristic, direct product factor, Sylow
176	C₂²	—	2	2	no	—
177	C₂²	C₇×#518	1	1	yes	characteristic
178	C₂²	C₇×D₄²	1	1	yes	characteristic
179	C₂²	C₇×#559	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	4	8	no	—
182	C₄	—	4	8	no	—
183	C₄	C₂×C₄×C₇×D₄	1	2	yes	—
184	C₄	C₇×#550	1	2	yes	—
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	C₇×#2366	1	1	yes	characteristic
188	C₂	C₇×#3293	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂³×C₇	H(5,𝔽₂)
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₇×#494	C₄	semidirect
19. #23200	C₇	direct (N × H)
74. #494	C₄×C₇	semidirect
175. C₇	#23200	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#494	2
C₇×#2950	2
C₇×#1979	2
C₇×#1886	2
C₇×#1772	2
#23200	7

Subgroups — C7×#23200 #13200416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23200 #13200416