Subgroups — C₇×#20830 #13198046

Order 1792.

770 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2920	C₂	1	2	yes	maximal
2	C₂²×C₇×#97	C₂	1	1	yes	maximal, characteristic
3	C₇×#1849	C₂	1	2	yes	maximal, complemented
4	C₇×#1838	C₂	1	2	yes	maximal, complemented
5	C₇×#1810	C₂	1	4	yes	maximal
6	C₇×#1778	C₂	1	4	yes	maximal, complemented
7	C₂³×C₇×C₈	C₂²	1	2	yes	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₇×#432	—	2	4	no	—
10	C₇×#431	—	2	4	no	—
11	C₂×C₇×#108	C₂²	1	8	yes	—
12	C₇×#423	C₂²	1	4	yes	—
13	C₇×#407	C₂²	1	4	yes	—
14	C₄×C₇×M₁₆	C₂²	1	4	yes	—
15	C₂×C₇×#97	C₄	1	8	yes	—
16	C₂×C₇×#97	C₂²	1	2	yes	—
17	C₂×C₇×#97	C₂²	1	1	yes	characteristic
18	C₂×C₇×#97	C₂²	1	1	yes	characteristic
19	C₇×#341	C₂²	1	8	yes	—
20	#20830	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₇	Q₈	1	2	yes	—
22	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂×C₇×M₁₆	C₂³	1	8	yes	—
24	C₂²×C₇×C₈	—	2	8	no	—
25	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
26	C₂²×C₇×C₈	C₂³	1	2	yes	—
27	C₂²×C₇×C₈	C₂³	1	2	yes	—
28	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₇	D₄	1	4	yes	—
30	C₂×C₄²×C₇	Q₈	1	2	yes	—
31	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	C₇×#108	—	2	16	no	—
34	C₇×C₄⋊C₈	—	2	8	no	—
35	C₇×C₂²⋊C₈	—	2	8	no	—
36	C₇×#97	—	2	4	no	—
37	C₇×#97	C₂×C₄	1	4	yes	—
38	C₇×#97	C₂×C₄	1	8	yes	—
39	C₇×#97	C₂×C₄	1	4	yes	—
40	C₄×C₇×C₈	—	2	4	no	—
41	#2920	C₂×C₇	1	2	yes	—
42	C₂²×#97	C₂×C₇	1	1	yes	characteristic
43	#1849	C₂×C₇	1	2	yes	complemented
44	#1838	C₂×C₇	1	2	yes	complemented
45	#1810	C₂×C₇	1	4	yes	—
46	#1778	C₂×C₇	1	4	yes	complemented
47	C₂⁴×C₇	C₂×Q₈	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	4	yes	—
51	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
58	C₇×M₁₆	—	2	16	no	—
59	C₂×C₇×C₈	—	2	4	no	—
60	C₂×C₇×C₈	—	2	16	no	—
61	C₂×C₇×C₈	—	2	8	no	—
62	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
63	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
64	C₂×C₇×C₈	#41	1	8	yes	—
65	C₄²×C₇	C₄⋊C₄	1	8	yes	—
66	C₄²×C₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₇	C₂×D₄	1	2	yes	—
69	C₄²×C₇	C₂×D₄	1	2	yes	—
70	C₂³×C₈	C₂²×C₇	1	2	yes	—
71	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
72	#432	—	2	4	no	—
73	#431	—	2	4	no	—
74	C₂×#108	C₂²×C₇	1	8	yes	—
75	#423	C₂²×C₇	1	4	yes	—
76	#407	C₂²×C₇	1	4	yes	—
77	C₄×M₁₆	C₂²×C₇	1	4	yes	—
78	C₂×#97	C₄×C₇	1	8	yes	—
79	C₂×#97	C₂²×C₇	1	1	yes	characteristic
80	C₂×#97	C₂²×C₇	1	1	yes	characteristic
81	C₂×#97	C₂²×C₇	1	2	yes	—
82	#341	C₂²×C₇	1	8	yes	—
83	C₂³×C₇	—	2	2	no	—
84	C₂³×C₇	—	2	4	no	—
85	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	8	no	—
89	C₂×C₄×C₇	—	2	4	no	—
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	—	4	4	no	—
92	C₂×C₄×C₇	—	4	2	no	—
93	C₂×C₄×C₇	—	4	2	no	—
94	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×#41	1	4	yes	—
100	C₇×C₈	—	2	4	no	—
101	C₇×C₈	—	4	8	no	—
102	C₇×C₈	#117	1	8	yes	—
103	C₂³×C₄	C₇×Q₈	1	2	yes	—
104	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
105	C₂×M₁₆	C₂³×C₇	1	8	yes	—
106	C₂²×C₈	—	2	8	no	—
107	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
108	C₂²×C₈	C₂³×C₇	1	2	yes	—
109	C₂²×C₈	C₂³×C₇	1	2	yes	—
110	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
111	C₂×C₄²	C₇×D₄	1	4	yes	—
112	C₂×C₄²	C₇×Q₈	1	2	yes	—
113	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
114	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
115	#108	—	2	16	no	—
116	C₄⋊C₈	—	2	8	no	—
117	C₂²⋊C₈	—	2	8	no	—
118	#97	—	2	4	no	—
119	#97	C₂×C₄×C₇	1	4	yes	—
120	#97	C₂×C₄×C₇	1	8	yes	—
121	#97	C₂×C₄×C₇	1	4	yes	—
122	C₄×C₈	—	2	4	no	—
123	C₂²×C₇	—	2	8	no	—
124	C₂²×C₇	—	2	2	no	—
125	C₂²×C₇	—	2	2	no	—
126	C₂²×C₇	—	2	2	no	—
127	C₂²×C₇	#430	1	4	yes	—
128	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₇	#533	1	2	yes	—
130	C₄×C₇	—	2	4	no	—
131	C₄×C₇	—	2	4	no	—
132	C₄×C₇	—	2	2	no	—
133	C₄×C₇	C₂×#117	1	2	yes	—
134	C₄×C₇	#532	1	2	yes	—
135	C₂⁴	C₂×C₇×Q₈	1	1	yes	characteristic
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	4	4	no	—
138	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
139	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
140	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
141	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
142	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
144	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
145	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
146	M₁₆	—	2	16	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	16	no	—
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
151	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
152	C₂×C₈	C₇×#41	1	8	yes	—
153	C₄²	C₇×C₄⋊C₄	1	8	yes	—
154	C₄²	C₂²×C₄×C₇	1	2	yes	—
155	C₄²	C₂²×C₄×C₇	1	2	yes	—
156	C₄²	C₂×C₇×D₄	1	2	yes	—
157	C₄²	C₂×C₇×D₄	1	2	yes	—
158	C₂×C₇	—	2	2	no	—
159	C₂×C₇	—	2	4	no	—
160	C₂×C₇	#2285	1	1	yes	characteristic
161	C₂×C₇	C₂×#430	1	2	yes	—
162	C₂³	—	2	2	no	—
163	C₂³	—	2	4	no	—
164	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
165	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
174	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
175	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
176	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
178	C₂×C₄	C₂×C₇×#41	1	4	yes	—
179	C₈	—	2	4	no	—
180	C₈	—	4	8	no	—
181	C₈	C₇×#117	1	8	yes	—
182	C₇	#20830	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²	—	2	8	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	C₇×#430	1	4	yes	—
188	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
189	C₂²	C₇×#533	1	2	yes	—
190	C₄	—	2	4	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	C₂×C₇×#117	1	2	yes	—
194	C₄	C₇×#532	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₇×#2285	1	1	yes	characteristic
198	C₂	C₂×C₇×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1849	C₂	semidirect
4. C₇×#1838	C₂	semidirect
6. C₇×#1778	C₂	semidirect
20. #20830	C₇	direct (N × H)
43. #1849	C₂×C₇	semidirect
44. #1838	C₂×C₇	semidirect
46. #1778	C₂×C₇	semidirect
182. C₇	#20830	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×#2920	2
C₂²×C₇×#97	2
C₇×#1849	2
C₇×#1838	2
C₇×#1810	2
C₇×#1778	2
#20830	7

Subgroups — C7×#20830 #13198046

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20830 #13198046