Subgroups — C₇×#14684 #13191900

Order 1792.

2286 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#381	C₂	1	2	yes	maximal
3	C₇×#1434	C₂	1	8	yes	maximal, complemented
4	C₇×C₂²⋊#95	C₂	1	4	yes	maximal
5	C₂⁴×C₄×C₇	C₂²	1	2	yes	—
6	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
8	C₂²×C₄²×C₇	—	2	8	no	—
9	C₂²×C₄²×C₇	C₄	1	8	yes	complemented
10	C₂²×C₄²×C₇	C₂²	1	4	yes	—
11	C₇×#381	—	2	8	no	—
12	C₂×C₇×#95	C₂²	1	8	yes	—
13	C₂×C₇×#95	C₂²	1	16	yes	—
14	#14684	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂⁵×C₇	C₂³	1	1	yes	characteristic
16	C₂³×C₄×C₇	—	2	8	no	—
17	C₂³×C₄×C₇	—	2	8	no	—
18	C₂³×C₄×C₇	—	2	16	no	—
19	C₂³×C₄×C₇	C₂×C₄	1	8	yes	—
20	C₂³×C₄×C₇	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₇	D₄	1	8	yes	—
22	C₂³×C₄×C₇	C₂³	1	8	yes	—
23	C₂³×C₄×C₇	C₂³	1	2	yes	—
24	C₂³×C₄×C₇	C₂³	1	4	yes	—
25	C₂×C₇×C₂²⋊C₄	—	2	8	no	—
26	C₂×C₇×C₂²⋊C₄	—	2	16	no	—
27	C₂×C₄²×C₇	—	2	32	no	—
28	C₂×C₄²×C₇	—	2	16	no	—
29	C₂×C₄²×C₇	C₂×C₄	1	16	yes	—
30	C₇×#95	—	2	16	no	—
31	C₇×#95	—	2	32	no	—
32	C₂³×C₄²	C₂×C₇	1	1	yes	characteristic
33	C₂×#381	C₂×C₇	1	2	yes	—
34	#1434	C₂×C₇	1	8	yes	complemented
35	C₂²⋊#95	C₂×C₇	1	4	yes	—
36	C₂⁴×C₇	—	2	8	no	—
37	C₂⁴×C₇	C₂²×C₄	1	2	yes	—
38	C₂⁴×C₇	C₂×D₄	1	4	yes	—
39	C₂⁴×C₇	#41	1	8	yes	—
40	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
41	C₂²×C₄×C₇	—	2	64	no	—
42	C₂²×C₄×C₇	—	2	16	no	—
43	C₂²×C₄×C₇	—	2	16	no	—
44	C₂²×C₄×C₇	—	2	32	no	—
45	C₂²×C₄×C₇	—	2	8	no	—
46	C₂²×C₄×C₇	—	2	8	no	—
47	C₂²×C₄×C₇	—	2	8	no	—
48	C₂²×C₄×C₇	—	2	32	no	—
49	C₂²×C₄×C₇	—	2	8	no	—
50	C₂²×C₄×C₇	—	2	32	no	—
51	C₂²×C₄×C₇	—	2	16	no	—
52	C₂²×C₄×C₇	C₂²⋊C₄	1	16	yes	—
53	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
55	C₂²×C₄×C₇	C₂×D₄	1	8	yes	—
56	C₂²×C₄×C₇	#41	1	16	yes	—
57	C₇×C₂²⋊C₄	—	4	16	no	—
58	C₄²×C₇	—	2	32	no	—
59	C₂⁴×C₄	C₂²×C₇	1	2	yes	—
60	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
61	C₂²×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
62	C₂²×C₄²	—	2	8	no	—
63	C₂²×C₄²	C₄×C₇	1	8	yes	complemented
64	C₂²×C₄²	C₂²×C₇	1	4	yes	—
65	#381	—	2	8	no	—
66	C₂×#95	C₂²×C₇	1	16	yes	—
67	C₂×#95	C₂²×C₇	1	8	yes	—
68	C₂³×C₇	—	2	8	no	—
69	C₂³×C₇	—	2	16	no	—
70	C₂³×C₇	—	2	32	no	—
71	C₂³×C₇	C₂×C₂²⋊C₄	1	4	yes	—
72	C₂³×C₇	#117	1	8	yes	—
73	C₂³×C₇	#123	1	16	yes	—
74	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
75	C₂³×C₇	C₂²×D₄	1	2	yes	—
76	C₂³×C₇	C₂×#41	1	4	yes	—
77	C₂³×C₇	C₂×#41	1	8	yes	—
78	C₂×C₄×C₇	—	2	64	no	—
79	C₂×C₄×C₇	—	2	8	no	—
80	C₂×C₄×C₇	—	2	8	no	—
81	C₂×C₄×C₇	—	2	8	no	—
82	C₂×C₄×C₇	—	2	16	no	—
83	C₂×C₄×C₇	—	2	32	no	—
84	C₂×C₄×C₇	—	2	32	no	—
85	C₂×C₄×C₇	—	4	8	no	—
86	C₂×C₄×C₇	—	4	16	no	—
87	C₂×C₄×C₇	—	4	32	no	—
88	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	8	yes	—
89	C₂×C₄×C₇	#117	1	16	yes	—
90	C₂⁵	C₂³×C₇	1	1	yes	characteristic
91	C₂³×C₄	—	2	8	no	—
92	C₂³×C₄	—	2	8	no	—
93	C₂³×C₄	—	2	16	no	—
94	C₂³×C₄	C₂×C₄×C₇	1	8	yes	—
95	C₂³×C₄	C₂×C₄×C₇	1	4	yes	—
96	C₂³×C₄	C₇×D₄	1	8	yes	—
97	C₂³×C₄	C₂³×C₇	1	8	yes	—
98	C₂³×C₄	C₂³×C₇	1	2	yes	—
99	C₂³×C₄	C₂³×C₇	1	4	yes	—
100	C₂×C₂²⋊C₄	—	2	8	no	—
101	C₂×C₂²⋊C₄	—	2	16	no	—
102	C₂×C₄²	—	2	32	no	—
103	C₂×C₄²	—	2	16	no	—
104	C₂×C₄²	C₂×C₄×C₇	1	16	yes	—
105	#95	—	2	16	no	—
106	#95	—	2	32	no	—
107	C₂²×C₇	—	2	32	no	—
108	C₂²×C₇	—	2	8	no	—
109	C₂²×C₇	—	2	16	no	—
110	C₂²×C₇	#381	1	8	yes	—
111	C₂²×C₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×C₇	C₂×#117	1	2	yes	—
113	C₂²×C₇	C₂×#117	1	4	yes	—
114	C₂²×C₇	C₂×#123	1	4	yes	—
115	C₂²×C₇	#525	1	8	yes	—
116	C₂²×C₇	#529	1	16	yes	—
117	C₄×C₇	—	2	8	no	—
118	C₄×C₇	—	2	16	no	—
119	C₄×C₇	—	8	8	no	—
120	C₂⁴	—	2	8	no	—
121	C₂⁴	C₂²×C₄×C₇	1	2	yes	—
122	C₂⁴	C₂×C₇×D₄	1	4	yes	—
123	C₂⁴	C₇×#41	1	8	yes	—
124	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
125	C₂²×C₄	—	2	64	no	—
126	C₂²×C₄	—	2	16	no	—
127	C₂²×C₄	—	2	16	no	—
128	C₂²×C₄	—	2	32	no	—
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	—	2	8	no	—
131	C₂²×C₄	—	2	8	no	—
132	C₂²×C₄	—	2	32	no	—
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	2	32	no	—
135	C₂²×C₄	—	2	16	no	—
136	C₂²×C₄	C₇×C₂²⋊C₄	1	16	yes	—
137	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
138	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
139	C₂²×C₄	C₂×C₇×D₄	1	8	yes	—
140	C₂²×C₄	C₇×#41	1	16	yes	—
141	C₂²⋊C₄	—	4	16	no	—
142	C₄²	—	2	32	no	—
143	C₂×C₇	—	2	8	no	—
144	C₂×C₇	C₂×#381	1	1	yes	characteristic
145	C₂×C₇	#2280	1	2	yes	—
146	C₂×C₇	#2283	1	4	yes	—
147	C₂×C₇	#2853	1	8	yes	—
148	C₂³	—	2	8	no	—
149	C₂³	—	2	16	no	—
150	C₂³	—	2	32	no	—
151	C₂³	C₂×C₇×C₂²⋊C₄	1	4	yes	—
152	C₂³	C₇×#117	1	8	yes	—
153	C₂³	C₇×#123	1	16	yes	—
154	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
155	C₂³	C₂²×C₇×D₄	1	2	yes	—
156	C₂³	C₂×C₇×#41	1	4	yes	—
157	C₂³	C₂×C₇×#41	1	8	yes	—
158	C₂×C₄	—	2	64	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	32	no	—
163	C₂×C₄	—	2	16	no	—
164	C₂×C₄	—	2	32	no	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	4	16	no	—
167	C₂×C₄	—	4	32	no	—
168	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	8	yes	—
169	C₂×C₄	C₇×#117	1	16	yes	—
170	C₇	#14684	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²	—	2	8	no	—
172	C₂²	—	2	16	no	—
173	C₂²	—	2	32	no	—
174	C₂²	C₇×#381	1	8	yes	—
175	C₂²	C₂²×C₇×C₂²⋊C₄	1	1	yes	characteristic
176	C₂²	C₂×C₇×#117	1	2	yes	—
177	C₂²	C₂×C₇×#117	1	4	yes	—
178	C₂²	C₂×C₇×#123	1	4	yes	—
179	C₂²	C₇×#525	1	8	yes	—
180	C₂²	C₇×#529	1	16	yes	—
181	C₄	—	2	8	no	—
182	C₄	—	2	16	no	—
183	C₄	—	8	8	no	—
184	C₂	—	2	8	no	—
185	C₂	C₂×C₇×#381	1	1	yes	characteristic
186	C₂	C₇×#2280	1	2	yes	—
187	C₂	C₇×#2283	1	4	yes	—
188	C₂	C₇×#2853	1	8	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂⁴	C₂⁴×C₇
prime radical	C₂⁵×C₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1434	C₂	semidirect
9. C₂²×C₄²×C₇	C₄	semidirect
14. #14684	C₇	direct (N × H)
34. #1434	C₂×C₇	semidirect
63. C₂²×C₄²	C₄×C₇	semidirect
170. C₇	#14684	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₄²×C₇	2
C₂×C₇×#381	2
C₇×#1434	2
C₇×C₂²⋊#95	2
#14684	7

Subgroups — C7×#14684 #13191900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#14684 #13191900