Subgroups — C₇×#20799 #13198015

Order 1792.

1086 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×#427	C₂	1	4	yes	maximal
2	C₂²×C₇×#97	C₂	1	1	yes	maximal, characteristic
3	C₇×#2285	C₂	1	2	yes	maximal
4	C₇×#1847	C₂	1	8	yes	maximal, complemented
5	C₂³×C₇×C₈	C₂²	1	2	yes	—
6	C₂×C₇×#117	C₂²	1	4	yes	—
7	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₇×#427	—	2	16	no	—
9	C₂×C₇×#107	C₂²	1	8	yes	—
10	C₂×C₇×#106	C₂²	1	8	yes	—
11	C₂×C₇×#97	C₄	1	8	yes	complemented
12	C₂×C₇×#97	C₂²	1	4	yes	—
13	C₇×#384	C₂²	1	4	yes	—
14	C₇×#382	C₂²	1	4	yes	—
15	C₇×#380	—	2	4	no	—
16	C₄×C₇×C₄⋊C₄	—	2	4	no	—
17	#20799	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₇	D₄	1	2	yes	—
19	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
21	C₂²×C₇×C₈	D₄	1	8	yes	—
22	C₂²×C₇×C₈	Q₈	1	8	yes	—
23	C₂²×C₇×C₈	C₂³	1	4	yes	—
24	C₇×#117	—	2	16	no	—
25	C₂×C₇×C₄⋊C₄	—	2	8	no	—
26	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
27	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
28	C₂×C₄²×C₇	—	2	4	no	—
29	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₇	D₄	1	2	yes	—
31	C₂×C₄²×C₇	Q₈	1	4	yes	—
32	C₂×C₄²×C₇	C₂³	1	2	yes	—
33	C₇×#107	—	2	16	no	—
34	C₇×#106	—	2	16	no	—
35	C₇×#97	C₂×C₄	1	16	yes	—
36	C₇×#95	—	2	8	no	—
37	C₂×#427	C₂×C₇	1	4	yes	—
38	C₂²×#97	C₂×C₇	1	1	yes	characteristic
39	#2285	C₂×C₇	1	2	yes	—
40	#1847	C₂×C₇	1	8	yes	complemented
41	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₇	—	2	8	no	—
43	C₂²×C₄×C₇	—	4	4	no	—
44	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
45	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
48	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
52	C₂×C₇×C₈	C₄⋊C₄	1	32	yes	complemented
53	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
54	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
55	C₂×C₇×C₈	C₂×Q₈	1	8	yes	—
56	C₇×C₄⋊C₄	—	2	16	no	—
57	C₇×C₄⋊C₄	—	4	16	no	—
58	C₇×C₂²⋊C₄	—	4	8	no	—
59	C₄²×C₇	—	4	8	no	—
60	C₄²×C₇	C₄⋊C₄	1	8	yes	—
61	C₄²×C₇	C₂²×C₄	1	4	yes	—
62	C₄²×C₇	C₂×Q₈	1	4	yes	—
63	C₂³×C₈	C₂²×C₇	1	2	yes	—
64	C₂×#117	C₂²×C₇	1	4	yes	—
65	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
66	#427	—	2	16	no	—
67	C₂×#107	C₂²×C₇	1	8	yes	—
68	C₂×#106	C₂²×C₇	1	8	yes	—
69	C₂×#97	C₄×C₇	1	8	yes	complemented
70	C₂×#97	C₂²×C₇	1	4	yes	—
71	#384	C₂²×C₇	1	4	yes	—
72	#382	C₂²×C₇	1	4	yes	—
73	#380	—	2	4	no	—
74	C₄×C₄⋊C₄	—	2	4	no	—
75	C₂³×C₇	—	2	4	no	—
76	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
77	C₂³×C₇	#127	1	2	yes	—
78	C₂³×C₇	C₄⋊Q₈	1	2	yes	—
79	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₇	—	2	8	no	—
81	C₂×C₄×C₇	—	4	8	no	—
82	C₂×C₄×C₇	—	4	8	no	—
83	C₂×C₄×C₇	—	4	16	no	—
84	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
85	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
86	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₇	#127	1	2	yes	—
88	C₂×C₄×C₇	C₄⋊Q₈	1	8	yes	—
89	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
90	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
93	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
94	C₇×C₈	C₂×C₄⋊C₄	1	16	yes	—
95	C₂³×C₄	C₇×D₄	1	2	yes	—
96	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
97	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
98	C₂²×C₈	C₇×D₄	1	8	yes	—
99	C₂²×C₈	C₇×Q₈	1	8	yes	—
100	C₂²×C₈	C₂³×C₇	1	4	yes	—
101	#117	—	2	16	no	—
102	C₂×C₄⋊C₄	—	2	8	no	—
103	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
104	C₂×C₂²⋊C₄	—	2	4	no	—
105	C₂×C₄²	—	2	4	no	—
106	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
107	C₂×C₄²	C₇×D₄	1	2	yes	—
108	C₂×C₄²	C₇×Q₈	1	4	yes	—
109	C₂×C₄²	C₂³×C₇	1	2	yes	—
110	#107	—	2	16	no	—
111	#106	—	2	16	no	—
112	#97	C₂×C₄×C₇	1	16	yes	—
113	#95	—	2	8	no	—
114	C₂²×C₇	—	2	4	no	—
115	C₂²×C₇	—	2	4	no	—
116	C₂²×C₇	—	2	4	no	—
117	C₂²×C₇	#384	1	4	yes	—
118	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₇	C₂×#127	1	1	yes	characteristic
120	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂²×C₇	#575	1	4	yes	—
122	C₄×C₇	—	2	8	no	—
123	C₄×C₇	—	8	8	no	—
124	C₄×C₇	#384	1	4	yes	—
125	C₄×C₇	C₂²×C₄⋊C₄	1	2	yes	—
126	C₄×C₇	C₂×C₄⋊Q₈	1	2	yes	—
127	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
128	C₂²×C₄	—	2	8	no	—
129	C₂²×C₄	—	4	4	no	—
130	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
131	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
132	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
133	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
135	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
136	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
137	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
138	C₂×C₈	C₇×C₄⋊C₄	1	32	yes	complemented
139	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
140	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
141	C₂×C₈	C₂×C₇×Q₈	1	8	yes	—
142	C₄⋊C₄	—	2	16	no	—
143	C₄⋊C₄	—	4	16	no	—
144	C₂²⋊C₄	—	4	8	no	—
145	C₄²	—	4	8	no	—
146	C₄²	C₇×C₄⋊C₄	1	8	yes	—
147	C₄²	C₂²×C₄×C₇	1	4	yes	—
148	C₄²	C₂×C₇×Q₈	1	4	yes	—
149	C₂×C₇	—	2	4	no	—
150	C₂×C₇	C₂×#384	1	1	yes	characteristic
151	C₂×C₇	#2911	1	2	yes	—
152	C₂×C₇	#3150	1	2	yes	—
153	C₂×C₇	#3162	1	2	yes	—
154	C₂³	—	2	4	no	—
155	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
156	C₂³	C₇×#127	1	2	yes	—
157	C₂³	C₇×C₄⋊Q₈	1	2	yes	—
158	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	4	8	no	—
161	C₂×C₄	—	4	8	no	—
162	C₂×C₄	—	4	16	no	—
163	C₂×C₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
164	C₂×C₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
165	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
166	C₂×C₄	C₇×#127	1	2	yes	—
167	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
168	C₂×C₄	C₇×C₄⋊Q₈	1	8	yes	—
169	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
170	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
171	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
172	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
173	C₈	C₂×C₇×C₄⋊C₄	1	16	yes	—
174	C₇	#20799	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²	—	2	4	no	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	4	no	—
178	C₂²	C₇×#384	1	4	yes	—
179	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
180	C₂²	C₂×C₇×#127	1	1	yes	characteristic
181	C₂²	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
182	C₂²	C₇×#575	1	4	yes	—
183	C₄	—	2	8	no	—
184	C₄	—	8	8	no	—
185	C₄	C₇×#384	1	4	yes	—
186	C₄	C₂²×C₇×C₄⋊C₄	1	2	yes	—
187	C₄	C₂×C₇×C₄⋊Q₈	1	2	yes	—
188	C₂	—	2	4	no	—
189	C₂	C₂×C₇×#384	1	1	yes	characteristic
190	C₂	C₇×#2911	1	2	yes	—
191	C₂	C₇×#3150	1	2	yes	—
192	C₂	C₇×#3162	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₇×#1847	C₂	semidirect
11. C₂×C₇×#97	C₄	semidirect
17. #20799	C₇	direct (N × H)
40. #1847	C₂×C₇	semidirect
52. C₂×C₇×C₈	C₄⋊C₄	semidirect
69. C₂×#97	C₄×C₇	semidirect
138. C₂×C₈	C₇×C₄⋊C₄	semidirect
174. C₇	#20799	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×#427	2
C₂²×C₇×#97	2
C₇×#2285	2
C₇×#1847	2
#20799	7

Subgroups — C7×#20799 #13198015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20799 #13198015