Subgroups — C₇×#13854 #13191070

Order 1792.

1422 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×C₄⋊C₄	C₂	1	2	yes	maximal
2	C₂×C₇×#376	C₂	1	1	yes	maximal, characteristic
3	C₇×#1442	C₂	1	8	yes	maximal
4	C₄×C₇×#95	C₂	1	4	yes	maximal
5	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
6	C₂²×C₄²×C₇	C₂²	1	4	yes	—
7	C₂²×C₄²×C₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₄×C₇×C₄⋊C₄	—	2	8	no	—
10	C₇×#376	C₄	1	8	yes	complemented
11	C₂×C₇×#95	C₂²	1	4	yes	—
12	C₂×C₇×#95	C₂²	1	16	yes	—
13	C₂×C₇×#95	C₂²	1	4	yes	—
14	#13854	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₄×C₇	C₂³	1	4	yes	—
16	C₂³×C₄×C₇	C₂³	1	8	yes	—
17	C₂³×C₄×C₇	C₂³	1	2	yes	—
18	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₄	—	2	8	no	—
20	C₂×C₇×C₄⋊C₄	—	2	16	no	—
21	C₂×C₄²×C₇	—	2	8	no	—
22	C₂×C₄²×C₇	—	2	8	no	—
23	C₂×C₄²×C₇	—	2	16	no	—
24	C₂×C₄²×C₇	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₇	D₄	1	4	yes	—
27	C₂×C₄²×C₇	Q₈	1	4	yes	—
28	C₇×#95	—	2	8	no	—
29	C₇×#95	—	2	32	no	—
30	C₇×#95	C₂×C₄	1	16	yes	—
31	C₂×C₄×C₄⋊C₄	C₂×C₇	1	2	yes	—
32	C₂×#376	C₂×C₇	1	1	yes	characteristic
33	#1442	C₂×C₇	1	8	yes	—
34	C₄×#95	C₂×C₇	1	4	yes	—
35	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
36	C₂²×C₄×C₇	—	2	8	no	—
37	C₂²×C₄×C₇	—	2	8	no	—
38	C₂²×C₄×C₇	—	2	16	no	—
39	C₂²×C₄×C₇	—	2	16	no	—
40	C₂²×C₄×C₇	—	2	8	no	—
41	C₂²×C₄×C₇	—	2	16	no	—
42	C₂²×C₄×C₇	—	2	8	no	—
43	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
44	C₂²×C₄×C₇	C₂²×C₄	1	4	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	2	yes	—
48	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₇	#41	1	16	yes	—
51	C₂²×C₄×C₇	#41	1	8	yes	—
52	C₇×C₄⋊C₄	—	2	32	no	—
53	C₄²×C₇	—	2	16	no	—
54	C₄²×C₇	—	4	16	no	—
55	C₄²×C₇	C₄⋊C₄	1	16	yes	—
56	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
57	C₂²×C₄²	C₂²×C₇	1	4	yes	—
58	C₂²×C₄²	C₂²×C₇	1	2	yes	—
59	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
60	C₄×C₄⋊C₄	—	2	8	no	—
61	#376	C₄×C₇	1	8	yes	complemented
62	C₂×#95	C₂²×C₇	1	16	yes	—
63	C₂×#95	C₂²×C₇	1	4	yes	—
64	C₂×#95	C₂²×C₇	1	4	yes	—
65	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
66	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
67	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
68	C₂³×C₇	C₂×#41	1	4	yes	—
69	C₂³×C₇	C₂×#41	1	8	yes	—
70	C₂×C₄×C₇	—	2	8	no	—
71	C₂×C₄×C₇	—	2	8	no	—
72	C₂×C₄×C₇	—	2	8	no	—
73	C₂×C₄×C₇	—	2	8	no	—
74	C₂×C₄×C₇	—	2	16	no	—
75	C₂×C₄×C₇	—	2	16	no	—
76	C₂×C₄×C₇	—	2	32	no	—
77	C₂×C₄×C₇	—	4	8	no	—
78	C₂×C₄×C₇	—	4	8	no	—
79	C₂×C₄×C₇	—	4	16	no	—
80	C₂×C₄×C₇	—	4	16	no	—
81	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
82	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
83	C₂×C₄×C₇	#117	1	8	yes	—
84	C₂×C₄×C₇	#124	1	8	yes	—
85	C₂×C₄×C₇	#125	1	8	yes	—
86	C₂³×C₄	C₂³×C₇	1	4	yes	—
87	C₂³×C₄	C₂³×C₇	1	8	yes	—
88	C₂³×C₄	C₂³×C₇	1	2	yes	—
89	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	2	8	no	—
91	C₂×C₄⋊C₄	—	2	16	no	—
92	C₂×C₄²	—	2	8	no	—
93	C₂×C₄²	—	2	8	no	—
94	C₂×C₄²	—	2	16	no	—
95	C₂×C₄²	C₂×C₄×C₇	1	8	yes	—
96	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
97	C₂×C₄²	C₇×D₄	1	4	yes	—
98	C₂×C₄²	C₇×Q₈	1	4	yes	—
99	#95	—	2	8	no	—
100	#95	—	2	32	no	—
101	#95	C₂×C₄×C₇	1	16	yes	—
102	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₇	C₂×#117	1	2	yes	—
104	C₂²×C₇	C₄×#41	1	4	yes	—
105	C₂²×C₇	#525	1	4	yes	—
106	C₂²×C₇	C₂×#124	1	2	yes	—
107	C₂²×C₇	C₂×#125	1	2	yes	—
108	C₂²×C₇	#529	1	16	yes	—
109	C₂²×C₇	#533	1	4	yes	—
110	C₄×C₇	—	2	8	no	—
111	C₄×C₇	—	4	16	no	—
112	C₄×C₇	—	4	8	no	—
113	C₄×C₇	#382	1	8	yes	—
114	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
115	C₂²×C₄	—	2	8	no	—
116	C₂²×C₄	—	2	8	no	—
117	C₂²×C₄	—	2	16	no	—
118	C₂²×C₄	—	2	16	no	—
119	C₂²×C₄	—	2	8	no	—
120	C₂²×C₄	—	2	16	no	—
121	C₂²×C₄	—	2	8	no	—
122	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
123	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
124	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
125	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
126	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
127	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
128	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
129	C₂²×C₄	C₇×#41	1	16	yes	—
130	C₂²×C₄	C₇×#41	1	8	yes	—
131	C₄⋊C₄	—	2	32	no	—
132	C₄²	—	2	16	no	—
133	C₄²	—	4	16	no	—
134	C₄²	C₇×C₄⋊C₄	1	16	yes	—
135	C₂×C₇	C₂×#382	1	1	yes	characteristic
136	C₂×C₇	#2296	1	2	yes	—
137	C₂×C₇	#2298	1	4	yes	—
138	C₂×C₇	#2398	1	4	yes	—
139	C₂×C₇	#2402	1	4	yes	—
140	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
141	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
142	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
143	C₂³	C₂×C₇×#41	1	4	yes	—
144	C₂³	C₂×C₇×#41	1	8	yes	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	—	2	16	no	—
150	C₂×C₄	—	2	16	no	—
151	C₂×C₄	—	2	32	no	—
152	C₂×C₄	—	4	8	no	—
153	C₂×C₄	—	4	8	no	—
154	C₂×C₄	—	4	16	no	—
155	C₂×C₄	—	4	16	no	—
156	C₂×C₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
157	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
158	C₂×C₄	C₇×#117	1	8	yes	—
159	C₂×C₄	C₇×#124	1	8	yes	—
160	C₂×C₄	C₇×#125	1	8	yes	—
161	C₇	#13854	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
163	C₂²	C₂×C₇×#117	1	2	yes	—
164	C₂²	C₄×C₇×#41	1	4	yes	—
165	C₂²	C₇×#525	1	4	yes	—
166	C₂²	C₂×C₇×#124	1	2	yes	—
167	C₂²	C₂×C₇×#125	1	2	yes	—
168	C₂²	C₇×#529	1	16	yes	—
169	C₂²	C₇×#533	1	4	yes	—
170	C₄	—	2	8	no	—
171	C₄	—	4	16	no	—
172	C₄	—	4	8	no	—
173	C₄	C₇×#382	1	8	yes	—
174	C₂	C₂×C₇×#382	1	1	yes	characteristic
175	C₂	C₇×#2296	1	2	yes	—
176	C₂	C₇×#2298	1	4	yes	—
177	C₂	C₇×#2398	1	4	yes	—
178	C₂	C₇×#2402	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
10. C₇×#376	C₄	semidirect
14. #13854	C₇	direct (N × H)
61. #376	C₄×C₇	semidirect
161. C₇	#13854	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×C₄×C₇×C₄⋊C₄	2
C₂×C₇×#376	2
C₇×#1442	2
C₄×C₇×#95	2
#13854	7

Subgroups — C7×#13854 #13191070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13854 #13191070