Subgroups — C₇×#20811 #13198027

Order 1792.

834 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#432	C₂	1	2	yes	maximal
2	C₄×C₇×#117	C₂	1	1	yes	maximal, characteristic
3	C₇×#1843	C₂	1	2	yes	maximal, complemented
4	C₇×#1832	C₂	1	2	yes	maximal, complemented
5	C₇×#1798	C₂	1	4	yes	maximal
6	C₇×#1767	C₂	1	4	yes	maximal, complemented
7	C₂×C₇×#117	C₂²	1	2	yes	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₇×#432	—	2	8	no	—
10	C₂×C₇×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×C₇×C₄⋊C₈	C₂²	1	8	yes	—
12	C₂×C₇×C₂²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₇×#97	C₂²	1	4	yes	—
14	C₄×C₇×C₄⋊C₄	C₄	1	4	yes	—
15	C₄×C₇×C₂²⋊C₄	C₄	1	4	yes	—
16	C₇×#376	C₂²	1	1	yes	characteristic
17	C₇×#376	C₂²	1	1	yes	characteristic
18	C₄³×C₇	C₂²	1	2	yes	—
19	C₇×#336	C₂²	1	8	yes	—
20	#20811	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂²×C₇×C₈	C₂³	1	8	yes	—
24	C₇×#117	—	2	8	no	—
25	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
26	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₇	C₂×C₄	1	8	yes	—
28	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇	D₄	1	4	yes	—
30	C₂×C₄²×C₇	Q₈	1	4	yes	—
31	C₂×C₄²×C₇	C₂³	1	2	yes	—
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇	C₂³	1	2	yes	—
35	C₇×C₄⋊C₈	—	2	8	no	—
36	C₇×C₄⋊C₈	—	2	16	no	—
37	C₇×C₂²⋊C₈	—	2	8	no	—
38	C₇×#97	—	2	8	no	—
39	C₇×#95	C₂×C₄	1	4	yes	—
40	C₇×#95	C₂×C₄	1	4	yes	—
41	C₂×#432	C₂×C₇	1	2	yes	—
42	C₄×#117	C₂×C₇	1	1	yes	characteristic
43	#1843	C₂×C₇	1	2	yes	complemented
44	#1832	C₂×C₇	1	2	yes	complemented
45	#1798	C₂×C₇	1	4	yes	—
46	#1767	C₂×C₇	1	4	yes	complemented
47	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	—	2	4	no	—
49	C₂²×C₄×C₇	—	4	4	no	—
50	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
51	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
59	C₂×C₇×C₈	—	2	8	no	—
60	C₂×C₇×C₈	—	2	16	no	—
61	C₇×C₄⋊C₄	—	2	8	no	—
62	C₇×C₂²⋊C₄	—	2	8	no	—
63	C₄²×C₇	—	2	4	no	—
64	C₄²×C₇	—	4	8	no	—
65	C₄²×C₇	C₄⋊C₄	1	8	yes	—
66	C₄²×C₇	C₂×D₄	1	2	yes	—
67	C₄²×C₇	C₂×D₄	1	2	yes	—
68	C₄²×C₇	C₂×Q₈	1	2	yes	—
69	C₄²×C₇	C₂×Q₈	1	2	yes	—
70	C₄²×C₇	#41	1	8	yes	—
71	C₂×#117	C₂²×C₇	1	2	yes	—
72	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
73	#432	—	2	8	no	—
74	C₂×C₄⋊C₈	C₂²×C₇	1	8	yes	—
75	C₂×C₄⋊C₈	C₂²×C₇	1	4	yes	—
76	C₂×C₂²⋊C₈	C₂²×C₇	1	4	yes	—
77	C₂×#97	C₂²×C₇	1	4	yes	—
78	C₄×C₄⋊C₄	C₄×C₇	1	4	yes	—
79	C₄×C₂²⋊C₄	C₄×C₇	1	4	yes	—
80	#376	C₂²×C₇	1	1	yes	characteristic
81	#376	C₂²×C₇	1	1	yes	characteristic
82	C₄³	C₂²×C₇	1	2	yes	—
83	#336	C₂²×C₇	1	8	yes	—
84	C₂³×C₇	—	2	2	no	—
85	C₂³×C₇	—	2	4	no	—
86	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	4	no	—
90	C₂×C₄×C₇	—	2	8	no	—
91	C₂×C₄×C₇	—	2	4	no	—
92	C₂×C₄×C₇	—	2	4	no	—
93	C₂×C₄×C₇	—	2	4	no	—
94	C₂×C₄×C₇	—	2	4	no	—
95	C₂×C₄×C₇	—	2	8	no	—
96	C₂×C₄×C₇	—	4	8	no	—
97	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₇	#117	1	8	yes	—
101	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂×#41	1	4	yes	—
104	C₇×C₈	—	4	8	no	—
105	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
106	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
107	C₂²×C₈	C₂³×C₇	1	8	yes	—
108	#117	—	2	8	no	—
109	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
110	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	4	yes	—
111	C₂×C₄²	C₂×C₄×C₇	1	8	yes	—
112	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
113	C₂×C₄²	C₇×D₄	1	4	yes	—
114	C₂×C₄²	C₇×Q₈	1	4	yes	—
115	C₂×C₄²	C₂³×C₇	1	2	yes	—
116	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
117	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
118	C₂×C₄²	C₂³×C₇	1	2	yes	—
119	C₄⋊C₈	—	2	8	no	—
120	C₄⋊C₈	—	2	16	no	—
121	C₂²⋊C₈	—	2	8	no	—
122	#97	—	2	8	no	—
123	#95	C₂×C₄×C₇	1	4	yes	—
124	#95	C₂×C₄×C₇	1	4	yes	—
125	C₂²×C₇	—	2	4	no	—
126	C₂²×C₇	—	2	2	no	—
127	C₂²×C₇	—	4	4	no	—
128	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₇	C₂×#117	1	2	yes	—
130	C₂²×C₇	#568	1	4	yes	—
131	C₄×C₇	—	2	4	no	—
132	C₄×C₇	—	2	4	no	—
133	C₄×C₇	—	2	8	no	—
134	C₄×C₇	—	2	2	no	—
135	C₄×C₇	—	4	4	no	—
136	C₄×C₇	#532	1	2	yes	—
137	C₄×C₇	#533	1	2	yes	—
138	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	4	4	no	—
141	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
142	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
143	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
144	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
145	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
146	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
147	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
148	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
149	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	2	16	no	—
152	C₄⋊C₄	—	2	8	no	—
153	C₂²⋊C₄	—	2	8	no	—
154	C₄²	—	2	4	no	—
155	C₄²	—	4	8	no	—
156	C₄²	C₇×C₄⋊C₄	1	8	yes	—
157	C₄²	C₂×C₇×D₄	1	2	yes	—
158	C₄²	C₂×C₇×D₄	1	2	yes	—
159	C₄²	C₂×C₇×Q₈	1	2	yes	—
160	C₄²	C₂×C₇×Q₈	1	2	yes	—
161	C₄²	C₇×#41	1	8	yes	—
162	C₂×C₇	—	4	2	no	—
163	C₂×C₇	#2285	1	1	yes	characteristic
164	C₂×C₇	#2905	1	2	yes	—
165	C₂×C₇	#2923	1	4	yes	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	4	no	—
168	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
169	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
180	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
181	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
182	C₂×C₄	C₇×#117	1	8	yes	—
183	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
185	C₂×C₄	C₂×C₇×#41	1	4	yes	—
186	C₈	—	4	8	no	—
187	C₇	#20811	1	1	yes	characteristic, direct product factor, Sylow
188	C₂²	—	2	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	4	no	—
191	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
192	C₂²	C₂×C₇×#117	1	2	yes	—
193	C₂²	C₇×#568	1	4	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	2	4	no	—
196	C₄	—	2	8	no	—
197	C₄	—	2	2	no	—
198	C₄	—	4	4	no	—
199	C₄	C₇×#532	1	2	yes	—
200	C₄	C₇×#533	1	2	yes	—
201	C₂	—	4	2	no	—
202	C₂	C₇×#2285	1	1	yes	characteristic
203	C₂	C₇×#2905	1	2	yes	—
204	C₂	C₇×#2923	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×#1843	C₂	semidirect
4. C₇×#1832	C₂	semidirect
6. C₇×#1767	C₂	semidirect
20. #20811	C₇	direct (N × H)
43. #1843	C₂×C₇	semidirect
44. #1832	C₂×C₇	semidirect
46. #1767	C₂×C₇	semidirect
187. C₇	#20811	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×#432	2
C₄×C₇×#117	2
C₇×#1843	2
C₇×#1832	2
C₇×#1798	2
C₇×#1767	2
#20811	7

Subgroups — C7×#20811 #13198027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20811 #13198027