Subgroups — C₇×#20753 #13197969

Order 1792.

926 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#423	C₂	1	2	yes	maximal
2	C₂×C₇×#376	C₂	1	1	yes	maximal, characteristic
3	C₇×#1832	C₂	1	8	yes	maximal, complemented
4	C₇×#1728	C₂	1	4	yes	maximal
5	C₂²×C₇×M₁₆	C₂²	1	4	yes	—
6	C₂²×C₄²×C₇	C₂²	1	2	yes	—
7	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₇×#423	—	2	8	no	—
9	C₂×C₇×C₄⋊C₈	C₂²	1	8	yes	—
10	C₇×#376	C₄	1	4	yes	—
11	C₇×#376	C₂²	1	4	yes	—
12	C₂×C₇×#95	C₄	1	4	yes	—
13	C₇×#336	C₂²	1	16	yes	—
14	#20753	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₄×C₇	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂×C₇×M₁₆	—	2	8	no	—
19	C₂×C₇×M₁₆	—	2	16	no	—
20	C₂²×C₇×C₈	C₂³	1	8	yes	—
21	C₂×C₄²×C₇	—	2	8	no	—
22	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₇	D₄	1	4	yes	—
25	C₂×C₄²×C₇	Q₈	1	4	yes	—
26	C₂×C₄²×C₇	C₂³	1	4	yes	—
27	C₂×C₄²×C₇	C₂³	1	2	yes	—
28	C₇×C₄⋊C₈	—	2	16	no	—
29	C₇×#95	C₂×C₄	1	16	yes	—
30	C₂×#423	C₂×C₇	1	2	yes	—
31	C₂×#376	C₂×C₇	1	1	yes	characteristic
32	#1832	C₂×C₇	1	8	yes	complemented
33	#1728	C₂×C₇	1	4	yes	—
34	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₇	—	2	8	no	—
36	C₂²×C₄×C₇	—	4	4	no	—
37	C₂²×C₄×C₇	—	4	8	no	—
38	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
39	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
40	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
44	C₂²×C₄×C₇	#41	1	8	yes	—
45	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
46	C₇×M₁₆	—	4	16	no	—
47	C₂×C₇×C₈	—	2	8	no	—
48	C₂×C₇×C₈	—	2	16	no	—
49	C₄²×C₇	—	2	8	no	—
50	C₄²×C₇	—	2	8	no	—
51	C₄²×C₇	C₄⋊C₄	1	8	yes	—
52	C₄²×C₇	C₂×D₄	1	4	yes	—
53	C₄²×C₇	C₂×Q₈	1	4	yes	—
54	C₂²×M₁₆	C₂²×C₇	1	4	yes	—
55	C₂²×C₄²	C₂²×C₇	1	2	yes	—
56	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
57	#423	—	2	8	no	—
58	C₂×C₄⋊C₈	C₂²×C₇	1	8	yes	—
59	#376	C₄×C₇	1	4	yes	—
60	#376	C₂²×C₇	1	4	yes	—
61	C₂×#95	C₄×C₇	1	4	yes	—
62	#336	C₂²×C₇	1	16	yes	—
63	C₂³×C₇	—	2	4	no	—
64	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
65	C₂³×C₇	#117	1	4	yes	—
66	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
67	C₂×C₄×C₇	—	2	8	no	—
68	C₂×C₄×C₇	—	2	8	no	—
69	C₂×C₄×C₇	—	2	16	no	—
70	C₂×C₄×C₇	—	4	8	no	—
71	C₂×C₄×C₇	—	4	16	no	—
72	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
73	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₇	#117	1	4	yes	—
75	C₂×C₄×C₇	#124	1	8	yes	—
76	C₂×C₄×C₇	#125	1	8	yes	—
77	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₇	C₂×#41	1	4	yes	—
80	C₇×C₈	—	4	8	no	—
81	C₂³×C₄	C₂×C₄×C₇	1	4	yes	—
82	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
83	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
84	C₂×M₁₆	—	2	8	no	—
85	C₂×M₁₆	—	2	16	no	—
86	C₂²×C₈	C₂³×C₇	1	8	yes	—
87	C₂×C₄²	—	2	8	no	—
88	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
89	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
90	C₂×C₄²	C₇×D₄	1	4	yes	—
91	C₂×C₄²	C₇×Q₈	1	4	yes	—
92	C₂×C₄²	C₂³×C₇	1	4	yes	—
93	C₂×C₄²	C₂³×C₇	1	2	yes	—
94	C₄⋊C₈	—	2	16	no	—
95	#95	C₂×C₄×C₇	1	16	yes	—
96	C₂²×C₇	—	2	4	no	—
97	C₂²×C₇	—	2	8	no	—
98	C₂²×C₇	#382	1	4	yes	—
99	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₇	C₂×#117	1	2	yes	—
101	C₂²×C₇	#568	1	4	yes	—
102	C₄×C₇	—	2	8	no	—
103	C₄×C₇	—	4	8	no	—
104	C₄×C₇	#382	1	4	yes	—
105	C₄×C₇	C₂×#124	1	2	yes	—
106	C₄×C₇	C₂×#125	1	2	yes	—
107	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
108	C₂²×C₄	—	2	8	no	—
109	C₂²×C₄	—	4	4	no	—
110	C₂²×C₄	—	4	8	no	—
111	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
112	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
113	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
114	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
115	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
116	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
117	C₂²×C₄	C₇×#41	1	8	yes	—
118	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
119	M₁₆	—	4	16	no	—
120	C₂×C₈	—	2	8	no	—
121	C₂×C₈	—	2	16	no	—
122	C₄²	—	2	8	no	—
123	C₄²	—	2	8	no	—
124	C₄²	C₇×C₄⋊C₄	1	8	yes	—
125	C₄²	C₂×C₇×D₄	1	4	yes	—
126	C₄²	C₂×C₇×Q₈	1	4	yes	—
127	C₂×C₇	—	2	4	no	—
128	C₂×C₇	C₂×#382	1	1	yes	characteristic
129	C₂×C₇	#2905	1	2	yes	—
130	C₂×C₇	#2921	1	4	yes	—
131	C₂³	—	2	4	no	—
132	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
133	C₂³	C₇×#117	1	4	yes	—
134	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
135	C₂×C₄	—	2	8	no	—
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	16	no	—
138	C₂×C₄	—	4	8	no	—
139	C₂×C₄	—	4	16	no	—
140	C₂×C₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
141	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
142	C₂×C₄	C₇×#117	1	4	yes	—
143	C₂×C₄	C₇×#124	1	8	yes	—
144	C₂×C₄	C₇×#125	1	8	yes	—
145	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
146	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
147	C₂×C₄	C₂×C₇×#41	1	4	yes	—
148	C₈	—	4	8	no	—
149	C₇	#20753	1	1	yes	characteristic, direct product factor, Sylow
150	C₂²	—	2	4	no	—
151	C₂²	—	2	8	no	—
152	C₂²	C₇×#382	1	4	yes	—
153	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
154	C₂²	C₂×C₇×#117	1	2	yes	—
155	C₂²	C₇×#568	1	4	yes	—
156	C₄	—	2	8	no	—
157	C₄	—	4	8	no	—
158	C₄	C₇×#382	1	4	yes	—
159	C₄	C₂×C₇×#124	1	2	yes	—
160	C₄	C₂×C₇×#125	1	2	yes	—
161	C₂	—	2	4	no	—
162	C₂	C₂×C₇×#382	1	1	yes	characteristic
163	C₂	C₇×#2905	1	2	yes	—
164	C₂	C₇×#2921	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×#1832	C₂	semidirect
14. #20753	C₇	direct (N × H)
32. #1832	C₂×C₇	semidirect
149. C₇	#20753	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×#423	2
C₂×C₇×#376	2
C₇×#1832	2
C₇×#1728	2
#20753	7

Subgroups — C7×#20753 #13197969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20753 #13197969