Subgroups — C₇×#20768 #13197984

Order 1792.

778 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×#2905	C₂	1	2	yes	maximal
2	C₂×C₇×#405	C₂	1	1	yes	maximal, characteristic
3	C₇×#1838	C₂	1	4	yes	maximal, complemented
4	C₇×#1837	C₂	1	4	yes	maximal, complemented
5	C₇×#1738	C₂	1	2	yes	maximal
6	C₂×C₇×#341	C₂	1	2	yes	maximal
7	C₂²×C₇×M₁₆	C₂²	1	4	yes	—
8	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×#117	C₂²	1	1	yes	characteristic
11	C₇×#424	—	2	8	no	—
12	C₇×#423	C₂²	1	8	yes	—
13	C₇×#405	C₄	1	8	yes	—
14	C₂×C₇×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
16	C₇×#341	C₂²	1	4	yes	—
17	C₇×#341	C₂²	1	4	yes	—
18	C₇×#341	C₂²	1	8	yes	—
19	#20768	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂×C₇×M₁₆	—	2	4	no	—
22	C₂×C₇×M₁₆	—	2	4	no	—
23	C₂×C₇×M₁₆	—	2	8	no	—
24	C₂×C₇×M₁₆	—	2	8	no	—
25	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
26	C₂×C₇×M₁₆	C₂³	1	8	yes	—
27	C₂²×C₇×C₈	—	2	4	no	—
28	C₂²×C₇×C₈	—	2	4	no	—
29	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂³	1	2	yes	—
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₇×#117	C₂×C₄	1	4	yes	—
34	C₇×#117	D₄	1	4	yes	—
35	C₂×C₇×C₄⋊C₄	Q₈	1	2	yes	—
36	C₂×C₇×C₂²⋊C₄	Q₈	1	2	yes	—
37	C₂×C₄²×C₇	C₂³	1	2	yes	—
38	C₇×C₄⋊C₈	—	2	16	no	—
39	C₇×#97	C₂×C₄	1	8	yes	—
40	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
41	#2905	C₂×C₇	1	2	yes	—
42	C₂×#405	C₂×C₇	1	1	yes	characteristic
43	#1838	C₂×C₇	1	4	yes	complemented
44	#1837	C₂×C₇	1	4	yes	complemented
45	#1738	C₂×C₇	1	2	yes	—
46	C₂×#341	C₂×C₇	1	2	yes	—
47	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₇	#41	1	4	yes	—
53	C₂²×C₄×C₇	#41	1	4	yes	—
54	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
55	C₇×M₁₆	—	2	8	no	—
56	C₇×M₁₆	—	2	8	no	—
57	C₇×M₁₆	—	2	8	no	—
58	C₇×M₁₆	—	4	8	no	—
59	C₂×C₇×C₈	—	2	8	no	—
60	C₂×C₇×C₈	—	2	2	no	—
61	C₂×C₇×C₈	—	2	2	no	—
62	C₂×C₇×C₈	—	2	4	no	—
63	C₂×C₇×C₈	—	2	8	no	—
64	C₂×C₇×C₈	—	4	8	no	—
65	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
66	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
67	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
68	C₇×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
69	C₇×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
70	C₄²×C₇	C₂²×C₄	1	4	yes	—
71	C₄²×C₇	C₂×D₄	1	4	yes	—
72	C₂²×M₁₆	C₂²×C₇	1	4	yes	—
73	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
74	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
75	C₂×#117	C₂²×C₇	1	1	yes	characteristic
76	#424	—	2	8	no	—
77	#423	C₂²×C₇	1	8	yes	—
78	#405	C₄×C₇	1	8	yes	—
79	C₂×#97	C₂²×C₇	1	2	yes	—
80	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
81	#341	C₂²×C₇	1	4	yes	—
82	#341	C₂²×C₇	1	8	yes	—
83	#341	C₂²×C₇	1	4	yes	—
84	C₂³×C₇	—	4	2	no	—
85	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂³×C₇	#125	1	4	yes	—
87	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₇	—	4	4	no	—
89	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
90	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₇	#117	1	4	yes	—
92	C₂×C₄×C₇	#117	1	4	yes	—
93	C₂×C₄×C₇	#124	1	8	yes	—
94	C₂×C₄×C₇	#125	1	4	yes	—
95	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂×#41	1	2	yes	—
98	C₂×C₄×C₇	C₂×#41	1	2	yes	—
99	C₇×C₈	—	2	2	no	—
100	C₇×C₈	—	2	2	no	—
101	C₇×C₈	—	2	4	no	—
102	C₇×C₈	—	4	8	no	—
103	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
104	C₂×M₁₆	—	2	4	no	—
105	C₂×M₁₆	—	2	4	no	—
106	C₂×M₁₆	—	2	8	no	—
107	C₂×M₁₆	—	2	8	no	—
108	C₂×M₁₆	C₂×C₄×C₇	1	4	yes	—
109	C₂×M₁₆	C₂³×C₇	1	8	yes	—
110	C₂²×C₈	—	2	4	no	—
111	C₂²×C₈	—	2	4	no	—
112	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
113	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
114	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
115	C₂²×C₈	C₂³×C₇	1	2	yes	—
116	#117	C₂×C₄×C₇	1	4	yes	—
117	#117	C₇×D₄	1	4	yes	—
118	C₂×C₄⋊C₄	C₇×Q₈	1	2	yes	—
119	C₂×C₂²⋊C₄	C₇×Q₈	1	2	yes	—
120	C₂×C₄²	C₂³×C₇	1	2	yes	—
121	C₄⋊C₈	—	2	16	no	—
122	#97	C₂×C₄×C₇	1	8	yes	—
123	C₄×C₈	C₂×C₄×C₇	1	8	yes	—
124	C₂²×C₇	—	2	2	no	—
125	C₂²×C₇	—	2	4	no	—
126	C₂²×C₇	—	4	4	no	—
127	C₂²×C₇	#382	1	4	yes	—
128	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₇	C₂×#125	1	2	yes	—
130	C₄×C₇	—	2	8	no	—
131	C₄×C₇	#382	1	4	yes	—
132	C₄×C₇	C₂×#117	1	1	yes	characteristic
133	C₄×C₇	C₂×#117	1	1	yes	characteristic
134	C₄×C₇	C₂×#124	1	2	yes	—
135	C₂⁴	C₂×C₇×Q₈	1	1	yes	characteristic
136	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
137	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
138	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
139	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
140	C₂²×C₄	C₇×#41	1	4	yes	—
141	C₂²×C₄	C₇×#41	1	4	yes	—
142	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
143	M₁₆	—	2	8	no	—
144	M₁₆	—	2	8	no	—
145	M₁₆	—	2	8	no	—
146	M₁₆	—	4	8	no	—
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	4	8	no	—
153	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
154	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
155	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
156	C₄⋊C₄	C₇×C₄⋊C₄	1	8	yes	—
157	C₂²⋊C₄	C₇×C₄⋊C₄	1	8	yes	—
158	C₄²	C₂²×C₄×C₇	1	4	yes	—
159	C₄²	C₂×C₇×D₄	1	4	yes	—
160	C₂×C₇	—	2	2	no	—
161	C₂×C₇	—	2	4	no	—
162	C₂×C₇	C₂×#382	1	1	yes	characteristic
163	C₂×C₇	#2916	1	2	yes	—
164	C₂³	—	4	2	no	—
165	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
166	C₂³	C₇×#125	1	4	yes	—
167	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
170	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
171	C₂×C₄	C₇×#117	1	4	yes	—
172	C₂×C₄	C₇×#117	1	4	yes	—
173	C₂×C₄	C₇×#124	1	8	yes	—
174	C₂×C₄	C₇×#125	1	4	yes	—
175	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
176	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
177	C₂×C₄	C₂×C₇×#41	1	2	yes	—
178	C₂×C₄	C₂×C₇×#41	1	2	yes	—
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₈	—	2	4	no	—
182	C₈	—	4	8	no	—
183	C₇	#20768	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	C₇×#382	1	4	yes	—
188	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
189	C₂²	C₂×C₇×#125	1	2	yes	—
190	C₄	—	2	8	no	—
191	C₄	C₇×#382	1	4	yes	—
192	C₄	C₂×C₇×#117	1	1	yes	characteristic
193	C₄	C₂×C₇×#117	1	1	yes	characteristic
194	C₄	C₂×C₇×#124	1	2	yes	—
195	C₂	—	2	4	no	—
196	C₂	—	2	2	no	—
197	C₂	C₂×C₇×#382	1	1	yes	characteristic
198	C₂	C₇×#2916	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 ◁ C₂×C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1838	C₂	semidirect
4. C₇×#1837	C₂	semidirect
19. #20768	C₇	direct (N × H)
43. #1838	C₂×C₇	semidirect
44. #1837	C₂×C₇	semidirect
183. C₇	#20768	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×#2905	2
C₂×C₇×#405	2
C₇×#1838	2
C₇×#1837	2
C₇×#1738	2
C₂×C₇×#341	2
#20768	7

Subgroups — C7×#20768 #13197984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20768 #13197984