Subgroups — C₇×#20752 #13197968

Order 1792.

1158 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#423	C₂	1	2	yes	maximal
2	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic
3	C₇×#1831	C₂	1	8	yes	maximal, complemented
4	C₇×#1728	C₂	1	4	yes	maximal
5	C₂²×C₇×M₁₆	C₂²	1	4	yes	—
6	C₂²×C₄²×C₇	C₄	1	4	yes	—
7	C₂²×C₄²×C₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₇×#423	—	2	8	no	—
10	C₂×C₇×C₄⋊C₈	C₂²	1	8	yes	—
11	C₄³×C₇	C₄	1	2	yes	—
12	C₄³×C₇	C₄	1	2	yes	—
13	C₄³×C₇	C₂²	1	4	yes	—
14	C₇×#336	C₂²	1	16	yes	—
15	#20752	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₄×C₇	C₂×C₄	1	4	yes	—
17	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂×C₇×M₁₆	—	2	8	no	—
20	C₂×C₇×M₁₆	—	2	16	no	—
21	C₂²×C₇×C₈	C₂³	1	8	yes	—
22	C₂×C₄²×C₇	—	2	8	no	—
23	C₂×C₄²×C₇	—	2	16	no	—
24	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₇	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₇	D₄	1	4	yes	—
31	C₂×C₄²×C₇	Q₈	1	4	yes	—
32	C₂×C₄²×C₇	C₂³	1	4	yes	—
33	C₂×C₄²×C₇	C₂³	1	2	yes	—
34	C₇×C₄⋊C₈	—	2	16	no	—
35	C₂×#423	C₂×C₇	1	2	yes	—
36	C₂×C₄³	C₂×C₇	1	1	yes	characteristic
37	#1831	C₂×C₇	1	8	yes	complemented
38	#1728	C₂×C₇	1	4	yes	—
39	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇	—	2	8	no	—
41	C₂²×C₄×C₇	—	2	16	no	—
42	C₂²×C₄×C₇	—	2	8	no	—
43	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
44	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
46	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₇	#41	1	8	yes	—
51	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
52	C₇×M₁₆	—	4	16	no	—
53	C₂×C₇×C₈	—	2	8	no	—
54	C₂×C₇×C₈	—	2	16	no	—
55	C₄²×C₇	—	2	8	no	—
56	C₄²×C₇	—	2	16	no	—
57	C₄²×C₇	—	2	8	no	—
58	C₄²×C₇	—	2	8	no	—
59	C₄²×C₇	C₄⋊C₄	1	4	yes	—
60	C₄²×C₇	C₄⋊C₄	1	4	yes	—
61	C₄²×C₇	M₁₆	1	16	yes	complemented
62	C₄²×C₇	C₂×D₄	1	4	yes	—
63	C₄²×C₇	C₂×Q₈	1	4	yes	—
64	C₂²×M₁₆	C₂²×C₇	1	4	yes	—
65	C₂²×C₄²	C₄×C₇	1	4	yes	—
66	C₂²×C₄²	C₂²×C₇	1	2	yes	—
67	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
68	#423	—	2	8	no	—
69	C₂×C₄⋊C₈	C₂²×C₇	1	8	yes	—
70	C₄³	C₄×C₇	1	2	yes	—
71	C₄³	C₄×C₇	1	2	yes	—
72	C₄³	C₂²×C₇	1	4	yes	—
73	#336	C₂²×C₇	1	16	yes	—
74	C₂³×C₇	—	2	4	no	—
75	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
76	C₂³×C₇	#117	1	4	yes	—
77	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₇	—	2	32	no	—
79	C₂×C₄×C₇	—	2	8	no	—
80	C₂×C₄×C₇	—	2	4	no	—
81	C₂×C₄×C₇	—	2	8	no	—
82	C₂×C₄×C₇	—	2	16	no	—
83	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₇	#117	1	4	yes	—
87	C₂×C₄×C₇	#124	1	8	yes	—
88	C₂×C₄×C₇	#125	1	8	yes	—
89	C₂×C₄×C₇	C₂×M₁₆	1	16	yes	—
90	C₂×C₄×C₇	C₂×M₁₆	1	8	yes	—
91	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂×#41	1	4	yes	—
94	C₇×C₈	—	4	8	no	—
95	C₂³×C₄	C₂×C₄×C₇	1	4	yes	—
96	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
97	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
98	C₂×M₁₆	—	2	8	no	—
99	C₂×M₁₆	—	2	16	no	—
100	C₂²×C₈	C₂³×C₇	1	8	yes	—
101	C₂×C₄²	—	2	8	no	—
102	C₂×C₄²	—	2	16	no	—
103	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
104	C₂×C₄²	C₂×C₄×C₇	1	8	yes	—
105	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
106	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
107	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
108	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
109	C₂×C₄²	C₇×D₄	1	4	yes	—
110	C₂×C₄²	C₇×Q₈	1	4	yes	—
111	C₂×C₄²	C₂³×C₇	1	4	yes	—
112	C₂×C₄²	C₂³×C₇	1	2	yes	—
113	C₄⋊C₈	—	2	16	no	—
114	C₂²×C₇	—	2	4	no	—
115	C₂²×C₇	—	2	8	no	—
116	C₂²×C₇	#382	1	4	yes	—
117	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₇	C₂×#117	1	2	yes	—
119	C₂²×C₇	C₂²×M₁₆	1	4	yes	—
120	C₄×C₇	—	2	4	no	—
121	C₄×C₇	—	2	16	no	—
122	C₄×C₇	#382	1	4	yes	—
123	C₄×C₇	#423	1	8	yes	—
124	C₄×C₇	C₂×#124	1	2	yes	—
125	C₄×C₇	C₂×#125	1	2	yes	—
126	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
127	C₂²×C₄	—	2	8	no	—
128	C₂²×C₄	—	2	16	no	—
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
131	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
132	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
133	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
134	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
135	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
136	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
137	C₂²×C₄	C₇×#41	1	8	yes	—
138	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
139	M₁₆	—	4	16	no	—
140	C₂×C₈	—	2	8	no	—
141	C₂×C₈	—	2	16	no	—
142	C₄²	—	2	8	no	—
143	C₄²	—	2	16	no	—
144	C₄²	—	2	8	no	—
145	C₄²	—	2	8	no	—
146	C₄²	C₇×C₄⋊C₄	1	4	yes	—
147	C₄²	C₇×C₄⋊C₄	1	4	yes	—
148	C₄²	C₇×M₁₆	1	16	yes	complemented
149	C₄²	C₂×C₇×D₄	1	4	yes	—
150	C₄²	C₂×C₇×Q₈	1	4	yes	—
151	C₂×C₇	—	2	4	no	—
152	C₂×C₇	C₂×#382	1	1	yes	characteristic
153	C₂×C₇	C₂×#423	1	2	yes	—
154	C₂×C₇	#2920	1	4	yes	—
155	C₂³	—	2	4	no	—
156	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
157	C₂³	C₇×#117	1	4	yes	—
158	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
159	C₂×C₄	—	2	32	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	2	16	no	—
164	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
165	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
166	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
167	C₂×C₄	C₇×#117	1	4	yes	—
168	C₂×C₄	C₇×#124	1	8	yes	—
169	C₂×C₄	C₇×#125	1	8	yes	—
170	C₂×C₄	C₂×C₇×M₁₆	1	16	yes	—
171	C₂×C₄	C₂×C₇×M₁₆	1	8	yes	—
172	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
173	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
174	C₂×C₄	C₂×C₇×#41	1	4	yes	—
175	C₈	—	4	8	no	—
176	C₇	#20752	1	1	yes	characteristic, direct product factor, Sylow
177	C₂²	—	2	4	no	—
178	C₂²	—	2	8	no	—
179	C₂²	C₇×#382	1	4	yes	—
180	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
181	C₂²	C₂×C₇×#117	1	2	yes	—
182	C₂²	C₂²×C₇×M₁₆	1	4	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	2	16	no	—
185	C₄	C₇×#382	1	4	yes	—
186	C₄	C₇×#423	1	8	yes	—
187	C₄	C₂×C₇×#124	1	2	yes	—
188	C₄	C₂×C₇×#125	1	2	yes	—
189	C₂	—	2	4	no	—
190	C₂	C₂×C₇×#382	1	1	yes	characteristic
191	C₂	C₂×C₇×#423	1	2	yes	—
192	C₂	C₇×#2920	1	4	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
3. C₇×#1831	C₂	semidirect
15. #20752	C₇	direct (N × H)
37. #1831	C₂×C₇	semidirect
61. C₄²×C₇	M₁₆	semidirect
148. C₄²	C₇×M₁₆	semidirect
176. C₇	#20752	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×#423	2
C₂×C₄³×C₇	2
C₇×#1831	2
C₇×#1728	2
#20752	7

Subgroups — C7×#20752 #13197968

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20752 #13197968