Subgroups — C₇×#11414 #13188630

Order 1792.

248 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2163	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₇×#1451	C₂	1	1	yes	maximal, characteristic
4	C₇×#1313	C₂	1	2	yes	maximal, complemented
5	C₇×#1313	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×M₃₂	C₄	1	2	yes	—
7	C₇×#431	C₄	1	1	yes	characteristic
8	C₄×C₇×M₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₇×C₄⋊C₁₆	C₄	1	2	yes	—
11	C₇×C₂²⋊C₁₆	C₄	1	2	yes	—
12	C₇×#346	C₂²	1	1	yes	characteristic
13	C₇×#346	C₂²	1	1	yes	characteristic
14	C₇×#346	C₂²	1	2	yes	—
15	C₄×C₇×C₁₆	C₄	1	2	yes	—
16	C₇×#322	C₄	1	2	yes	—
17	C₇×#322	C₂²	1	2	yes	—
18	#11414	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₇×M₁₆	—	2	1	no	—
20	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
23	C₇×M₃₂	—	2	4	no	—
24	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
26	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
27	C₇×C₄⋊C₈	—	2	1	no	—
28	C₇×C₂²⋊C₈	—	2	1	no	—
29	C₇×#97	—	2	1	no	—
30	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
33	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
34	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
36	#2163	C₂×C₇	1	1	yes	characteristic
37	C₄×M₃₂	C₂×C₇	1	1	yes	characteristic
38	#1451	C₂×C₇	1	1	yes	characteristic
39	#1313	C₂×C₇	1	2	yes	complemented
40	#1313	C₂×C₇	1	2	yes	complemented
41	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
44	C₇×M₁₆	—	4	2	no	—
45	C₂×C₇×C₈	—	2	2	no	—
46	C₂×C₇×C₈	—	2	2	no	—
47	C₂×C₇×C₈	—	2	2	no	—
48	C₂×C₇×C₈	—	2	2	no	—
49	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
50	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₇	C₄²	1	1	yes	characteristic
54	C₄²×C₇	M₁₆	1	1	yes	characteristic
55	C₄²×C₇	M₁₆	1	1	yes	characteristic
56	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₇×C₁₆	—	2	2	no	—
58	C₇×C₁₆	—	2	4	no	—
59	C₇×C₁₆	M₁₆	1	4	yes	complemented
60	C₂×M₃₂	C₄×C₇	1	2	yes	—
61	#431	C₄×C₇	1	1	yes	characteristic
62	C₄×M₁₆	C₄×C₇	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
64	C₄⋊C₁₆	C₄×C₇	1	2	yes	—
65	C₂²⋊C₁₆	C₄×C₇	1	2	yes	—
66	#346	C₂²×C₇	1	1	yes	characteristic
67	#346	C₂²×C₇	1	1	yes	characteristic
68	#346	C₂²×C₇	1	2	yes	—
69	C₄×C₁₆	C₄×C₇	1	2	yes	—
70	#322	C₄×C₇	1	2	yes	—
71	#322	C₂²×C₇	1	2	yes	—
72	C₂³×C₇	#97	1	1	yes	characteristic
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	—	2	1	no	—
75	C₂×C₄×C₇	—	2	1	no	—
76	C₂×C₄×C₇	—	4	1	no	—
77	C₂×C₄×C₇	#97	1	1	yes	characteristic
78	C₂×C₄×C₇	#97	1	1	yes	characteristic
79	C₂×C₄×C₇	#97	1	1	yes	characteristic
80	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
83	C₇×C₈	—	2	1	no	—
84	C₇×C₈	—	2	1	no	—
85	C₇×C₈	—	4	2	no	—
86	C₇×C₈	—	4	2	no	—
87	C₇×C₈	C₂×M₁₆	1	2	yes	—
88	C₇×C₈	#131	1	2	yes	—
89	C₂×M₁₆	—	2	1	no	—
90	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
91	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
92	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
93	M₃₂	—	2	4	no	—
94	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
95	C₂×C₁₆	C₂×C₄×C₇	1	4	yes	—
96	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
97	C₄⋊C₈	—	2	1	no	—
98	C₂²⋊C₈	—	2	1	no	—
99	#97	—	2	1	no	—
100	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
101	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
102	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
103	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
104	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
105	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
106	C₂²×C₇	—	2	1	no	—
107	C₂²×C₇	—	2	1	no	—
108	C₂²×C₇	—	2	1	no	—
109	C₂²×C₇	C₂×#97	1	1	yes	characteristic
110	C₄×C₇	—	2	1	no	—
111	C₄×C₇	—	2	1	no	—
112	C₄×C₇	—	2	1	no	—
113	C₄×C₇	—	2	1	no	—
114	C₄×C₇	#347	1	2	yes	—
115	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
116	C₄×C₇	#405	1	1	yes	characteristic
117	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
118	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
119	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
120	M₁₆	—	4	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	C₄²×C₇	1	1	yes	characteristic
126	C₂×C₈	C₄²×C₇	1	1	yes	characteristic
127	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
128	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
129	C₄²	C₄²×C₇	1	1	yes	characteristic
130	C₄²	C₇×M₁₆	1	1	yes	characteristic
131	C₄²	C₇×M₁₆	1	1	yes	characteristic
132	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
133	C₁₆	—	2	2	no	—
134	C₁₆	—	2	4	no	—
135	C₁₆	C₇×M₁₆	1	4	yes	complemented
136	C₂×C₇	—	4	1	no	—
137	C₂×C₇	#1451	1	1	yes	characteristic
138	C₂×C₇	C₂×#347	1	1	yes	characteristic
139	C₂×C₇	#2110	1	1	yes	characteristic
140	C₂³	C₇×#97	1	1	yes	characteristic
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	C₇×#97	1	1	yes	characteristic
146	C₂×C₄	C₇×#97	1	1	yes	characteristic
147	C₂×C₄	C₇×#97	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
149	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
150	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	4	2	no	—
154	C₈	—	4	2	no	—
155	C₈	C₂×C₇×M₁₆	1	2	yes	—
156	C₈	C₇×#131	1	2	yes	—
157	C₇	#11414	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	C₂×C₇×#97	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	C₇×#347	1	2	yes	—
167	C₄	C₄×C₇×M₁₆	1	1	yes	characteristic
168	C₄	C₇×#405	1	1	yes	characteristic
169	C₂	—	4	1	no	—
170	C₂	C₇×#1451	1	1	yes	characteristic
171	C₂	C₂×C₇×#347	1	1	yes	characteristic
172	C₂	C₇×#2110	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×#97
prime radical	C₂³×C₇	#97
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1313	C₂	semidirect
5. C₇×#1313	C₂	semidirect
18. #11414	C₇	direct (N × H)
39. #1313	C₂×C₇	semidirect
40. #1313	C₂×C₇	semidirect
59. C₇×C₁₆	M₁₆	semidirect
135. C₁₆	C₇×M₁₆	semidirect
157. C₇	#11414	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×#2163	2
C₄×C₇×M₃₂	2
C₇×#1451	2
C₇×#1313	2
#11414	7

Subgroups — C7×#11414 #13188630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11414 #13188630