Subgroups — C₇×#20827 #13198043

Order 1792.

758 conjugacy classes of subgroups.

The subgroup lattice has 208 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	C₂×C₄×C₇×M₁₆	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×#1811	C₂	1	4	yes	maximal
6	C₄×C₇×#108	C₂	1	4	yes	maximal, complemented
7	C₂²×C₇×M₁₆	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	C₄	1	8	yes	—
15	C₄×C₇×M₁₆	C₂²	1	4	yes	—
16	C₂×C₇×#97	C₂²	1	1	yes	characteristic
17	C₂×C₇×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
19	C₇×#341	C₂²	1	8	yes	—
20	#20827	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	—	2	8	no	—
24	C₂×C₇×M₁₆	C₂×C₄	1	8	yes	—
25	C₂×C₇×M₁₆	C₂³	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	2	yes	—
30	C₂×C₄²×C₇	D₄	1	2	yes	—
31	C₂×C₄²×C₇	Q₈	1	2	yes	—
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
34	C₇×#108	—	2	16	no	—
35	C₇×C₄⋊C₈	—	2	8	no	—
36	C₇×C₂²⋊C₈	—	2	8	no	—
37	C₇×#97	—	2	4	no	—
38	C₇×#97	C₂×C₄	1	4	yes	—
39	C₇×#97	C₂×C₄	1	4	yes	—
40	C₄×C₇×C₈	—	2	4	no	—
41	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
42	#2920	C₂×C₇	1	2	yes	—
43	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic
44	#1849	C₂×C₇	1	2	yes	complemented
45	#1838	C₂×C₇	1	2	yes	complemented
46	#1811	C₂×C₇	1	4	yes	—
47	C₄×#108	C₂×C₇	1	4	yes	complemented
48	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₇	—	2	4	no	—
50	C₂²×C₄×C₇	—	4	4	no	—
51	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
59	C₇×M₁₆	—	2	16	no	—
60	C₇×M₁₆	—	2	16	no	—
61	C₂×C₇×C₈	—	2	4	no	—
62	C₂×C₇×C₈	—	2	8	no	—
63	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
64	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
65	C₂×C₇×C₈	#41	1	8	yes	—
66	C₄²×C₇	C₄⋊C₄	1	4	yes	—
67	C₄²×C₇	C₄⋊C₄	1	4	yes	—
68	C₄²×C₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₇	C₂×D₄	1	2	yes	—
72	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
74	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
75	#432	—	2	4	no	—
76	#431	—	2	4	no	—
77	C₂×#108	C₂²×C₇	1	8	yes	—
78	#423	C₂²×C₇	1	4	yes	—
79	#407	C₂²×C₇	1	4	yes	—
80	C₄×M₁₆	C₄×C₇	1	8	yes	—
81	C₄×M₁₆	C₂²×C₇	1	4	yes	—
82	C₂×#97	C₂²×C₇	1	1	yes	characteristic
83	C₂×#97	C₂²×C₇	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
85	#341	C₂²×C₇	1	8	yes	—
86	C₂³×C₇	—	2	2	no	—
87	C₂³×C₇	—	4	2	no	—
88	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	—	2	4	no	—
92	C₂×C₄×C₇	—	2	4	no	—
93	C₂×C₄×C₇	—	2	8	no	—
94	C₂×C₄×C₇	—	2	4	no	—
95	C₂×C₄×C₇	—	4	4	no	—
96	C₂×C₄×C₇	—	4	2	no	—
97	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂×#41	1	4	yes	—
104	C₇×C₈	—	2	4	no	—
105	C₇×C₈	—	4	8	no	—
106	C₇×C₈	#117	1	8	yes	—
107	C₂³×C₄	C₇×Q₈	1	2	yes	—
108	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
109	C₂×M₁₆	—	2	8	no	—
110	C₂×M₁₆	C₂×C₄×C₇	1	8	yes	—
111	C₂×M₁₆	C₂³×C₇	1	8	yes	—
112	C₂²×C₈	C₂³×C₇	1	2	yes	—
113	C₂²×C₈	C₂³×C₇	1	2	yes	—
114	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
115	C₂×C₄²	C₇×D₄	1	2	yes	—
116	C₂×C₄²	C₇×D₄	1	2	yes	—
117	C₂×C₄²	C₇×Q₈	1	2	yes	—
118	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
119	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
120	#108	—	2	16	no	—
121	C₄⋊C₈	—	2	8	no	—
122	C₂²⋊C₈	—	2	8	no	—
123	#97	—	2	4	no	—
124	#97	C₂×C₄×C₇	1	4	yes	—
125	#97	C₂×C₄×C₇	1	4	yes	—
126	C₄×C₈	—	2	4	no	—
127	C₄×C₈	C₂×C₄×C₇	1	8	yes	—
128	C₂²×C₇	—	2	2	no	—
129	C₂²×C₇	—	2	4	no	—
130	C₂²×C₇	—	2	2	no	—
131	C₂²×C₇	—	4	4	no	—
132	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₇	#533	1	2	yes	—
134	C₄×C₇	—	2	4	no	—
135	C₄×C₇	—	2	2	no	—
136	C₄×C₇	—	2	2	no	—
137	C₄×C₇	#430	1	4	yes	—
138	C₄×C₇	C₂×#117	1	2	yes	—
139	C₄×C₇	#532	1	2	yes	—
140	C₂⁴	C₂×C₇×Q₈	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
144	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
145	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
146	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
147	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
149	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
150	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
151	M₁₆	—	2	16	no	—
152	M₁₆	—	2	16	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
156	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
157	C₂×C₈	C₇×#41	1	8	yes	—
158	C₄²	C₇×C₄⋊C₄	1	4	yes	—
159	C₄²	C₇×C₄⋊C₄	1	4	yes	—
160	C₄²	C₂²×C₄×C₇	1	2	yes	—
161	C₄²	C₂²×C₄×C₇	1	2	yes	—
162	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
163	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
164	C₄²	C₂×C₇×D₄	1	2	yes	—
165	C₂×C₇	—	2	2	no	—
166	C₂×C₇	—	4	2	no	—
167	C₂×C₇	#2285	1	1	yes	characteristic
168	C₂×C₇	C₂×#430	1	2	yes	—
169	C₂³	—	2	2	no	—
170	C₂³	—	4	2	no	—
171	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
172	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
181	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
182	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
183	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
184	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
186	C₂×C₄	C₂×C₇×#41	1	4	yes	—
187	C₈	—	2	4	no	—
188	C₈	—	4	8	no	—
189	C₈	C₇×#117	1	8	yes	—
190	C₇	#20827	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²	—	2	2	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	4	4	no	—
195	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
196	C₂²	C₇×#533	1	2	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	C₇×#430	1	4	yes	—
201	C₄	C₂×C₇×#117	1	2	yes	—
202	C₄	C₇×#532	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	4	2	no	—
205	C₂	C₇×#2285	1	1	yes	characteristic
206	C₂	C₂×C₇×#430	1	2	yes	—

	subgroup	G/H
center	C₄²×C₇	C₂×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 ◁ C₄²×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. C₄×C₇×#108	C₂	semidirect
20. #20827	C₇	direct (N × H)
44. #1849	C₂×C₇	semidirect
45. #1838	C₂×C₇	semidirect
47. C₄×#108	C₂×C₇	semidirect
190. C₇	#20827	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
C₂×C₄×C₇×M₁₆	2
C₇×#1849	2
C₇×#1838	2
C₇×#1811	2
C₄×C₇×#108	2
#20827	7

Subgroups — C7×#20827 #13198043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20827 #13198043