Subgroups — C₇×#22746 #13199962

Order 1792.

642 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	C₇×Q₈×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#1950	C₂	1	6	yes	maximal
4	C₇×#1837	C₂	1	3	yes	maximal
5	C₄×C₇×#108	C₂	1	3	yes	maximal
6	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₇×#446	—	2	6	no	—
8	C₇×C₈×Q₈	—	2	2	no	—
9	C₇×C₈×Q₈	C₄	1	6	yes	—
10	C₇×C₈×Q₈	C₄	1	2	yes	—
11	C₂×C₇×#108	C₂²	1	6	yes	—
12	C₇×#423	C₂²	1	6	yes	—
13	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
14	C₄×C₇×M₁₆	C₂²	1	6	yes	—
15	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
16	C₇×#341	C₂²	1	1	yes	characteristic
17	C₇×#341	C₂²	1	3	yes	—
18	C₇×#341	C₂²	1	6	yes	—
19	#22746	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×Q₈	Q₈	1	1	yes	characteristic
21	C₂×C₇×M₁₆	C₂³	1	6	yes	—
22	C₂×C₇×M₁₆	C₂³	1	2	yes	—
23	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂³	1	3	yes	—
25	C₄×C₇×Q₈	C₂×C₄	1	3	yes	—
26	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
28	C₄×C₇×Q₈	D₄	1	3	yes	—
29	C₂×C₇×C₄⋊C₄	Q₈	1	3	yes	—
30	C₂×C₄²×C₇	C₂³	1	3	yes	—
31	C₇×#108	—	2	12	no	—
32	C₇×C₄⋊C₈	—	2	6	no	—
33	C₇×C₄⋊C₈	—	2	6	no	—
34	C₇×C₄⋊C₈	C₂×C₄	1	6	yes	—
35	C₇×C₄⋊C₈	C₂×C₄	1	6	yes	—
36	C₇×#97	—	2	6	no	—
37	C₄×C₇×C₈	—	2	6	no	—
38	C₄×C₇×C₈	C₂×C₄	1	6	yes	—
39	C₄×C₇×C₈	C₂×C₄	1	6	yes	—
40	Q₈×M₁₆	C₂×C₇	1	2	yes	—
41	C₂×C₈×Q₈	C₂×C₇	1	1	yes	characteristic
42	#1950	C₂×C₇	1	6	yes	—
43	#1837	C₂×C₇	1	3	yes	—
44	C₄×#108	C₂×C₇	1	3	yes	—
45	C₂×C₇×Q₈	—	2	4	no	—
46	C₂×C₇×Q₈	C₄⋊C₄	1	3	yes	—
47	C₂×C₇×Q₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×Q₈	1	3	yes	—
49	C₂²×C₄×C₇	C₂×Q₈	1	3	yes	—
50	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
51	C₇×M₁₆	—	2	2	no	—
52	C₇×M₁₆	—	2	6	no	—
53	C₇×M₁₆	—	2	6	no	—
54	C₇×M₁₆	#41	1	4	yes	—
55	C₂×C₇×C₈	—	2	2	no	—
56	C₂×C₇×C₈	—	2	6	no	—
57	C₂×C₇×C₈	—	2	2	no	—
58	C₂×C₇×C₈	—	2	6	no	—
59	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
60	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
61	C₇×C₄⋊C₄	C₄⋊C₄	1	6	yes	—
62	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
63	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
64	C₄²×C₇	C₂²×C₄	1	3	yes	—
65	C₄²×C₇	C₂²×C₄	1	3	yes	—
66	C₄²×C₇	C₂×D₄	1	3	yes	—
67	C₄²×C₇	C₂×D₄	1	3	yes	—
68	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
69	#446	—	2	6	no	—
70	C₈×Q₈	—	2	2	no	—
71	C₈×Q₈	C₄×C₇	1	6	yes	—
72	C₈×Q₈	C₄×C₇	1	2	yes	—
73	C₂×#108	C₂²×C₇	1	6	yes	—
74	#423	C₂²×C₇	1	6	yes	—
75	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
76	C₄×M₁₆	C₂²×C₇	1	6	yes	—
77	C₂×C₄×C₈	C₂²×C₇	1	3	yes	—
78	#341	C₂²×C₇	1	3	yes	—
79	#341	C₂²×C₇	1	6	yes	—
80	#341	C₂²×C₇	1	1	yes	characteristic
81	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
82	C₇×Q₈	—	2	6	no	—
83	C₇×Q₈	#108	1	4	yes	complemented
84	C₂×C₄×C₇	—	2	6	no	—
85	C₂×C₄×C₇	—	4	3	no	—
86	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
87	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
88	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
89	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
90	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂×#41	1	2	yes	—
93	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
94	C₇×C₈	—	2	6	no	—
95	C₇×C₈	—	2	2	no	—
96	C₇×C₈	—	2	4	no	—
97	C₇×C₈	—	4	6	no	—
98	C₂²×Q₈	C₇×Q₈	1	1	yes	characteristic
99	C₂×M₁₆	C₂³×C₇	1	6	yes	—
100	C₂×M₁₆	C₂³×C₇	1	2	yes	—
101	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
102	C₂²×C₈	C₂³×C₇	1	3	yes	—
103	C₄×Q₈	C₂×C₄×C₇	1	3	yes	—
104	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
105	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
106	C₄×Q₈	C₇×D₄	1	3	yes	—
107	C₂×C₄⋊C₄	C₇×Q₈	1	3	yes	—
108	C₂×C₄²	C₂³×C₇	1	3	yes	—
109	#108	—	2	12	no	—
110	C₄⋊C₈	—	2	6	no	—
111	C₄⋊C₈	—	2	6	no	—
112	C₄⋊C₈	C₂×C₄×C₇	1	6	yes	—
113	C₄⋊C₈	C₂×C₄×C₇	1	6	yes	—
114	#97	—	2	6	no	—
115	C₄×C₈	—	2	6	no	—
116	C₄×C₈	C₂×C₄×C₇	1	6	yes	—
117	C₄×C₈	C₂×C₄×C₇	1	6	yes	—
118	C₂²×C₇	—	2	1	no	—
119	C₂²×C₇	—	2	1	no	—
120	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₇	#557	1	2	yes	—
122	C₄×C₇	—	2	3	no	—
123	C₄×C₇	—	2	3	no	—
124	C₄×C₇	—	2	3	no	—
125	C₄×C₇	C₂×#108	1	6	yes	—
126	C₄×C₇	C₄×#41	1	1	yes	characteristic
127	C₄×C₇	#519	1	1	yes	characteristic
128	C₄×C₇	#548	1	2	yes	—
129	C₂×Q₈	—	2	4	no	—
130	C₂×Q₈	C₇×C₄⋊C₄	1	3	yes	—
131	C₂×Q₈	C₇×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₇×Q₈	1	3	yes	—
133	C₂²×C₄	C₂×C₇×Q₈	1	3	yes	—
134	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
135	M₁₆	—	2	2	no	—
136	M₁₆	—	2	6	no	—
137	M₁₆	—	2	6	no	—
138	M₁₆	C₇×#41	1	4	yes	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	6	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	6	no	—
143	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
144	C₂×C₈	C₂²×C₄×C₇	1	6	yes	—
145	C₄⋊C₄	C₇×C₄⋊C₄	1	6	yes	—
146	C₄⋊C₄	C₇×C₄⋊C₄	1	3	yes	—
147	C₄⋊C₄	C₇×C₄⋊C₄	1	3	yes	—
148	C₄²	C₂²×C₄×C₇	1	3	yes	—
149	C₄²	C₂²×C₄×C₇	1	3	yes	—
150	C₄²	C₂×C₇×D₄	1	3	yes	—
151	C₄²	C₂×C₇×D₄	1	3	yes	—
152	C₂×C₇	—	2	2	no	—
153	C₂×C₇	#2351	1	1	yes	characteristic
154	C₂×C₇	C₂²×#108	1	1	yes	characteristic
155	C₂×C₇	#2916	1	1	yes	characteristic
156	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
157	Q₈	—	2	6	no	—
158	Q₈	C₇×#108	1	4	yes	complemented
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	4	3	no	—
161	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
162	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
163	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
164	C₂×C₄	C₂×C₇×C₄⋊C₄	1	3	yes	—
165	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
166	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
167	C₂×C₄	C₂×C₇×#41	1	2	yes	—
168	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
169	C₈	—	2	6	no	—
170	C₈	—	2	2	no	—
171	C₈	—	2	4	no	—
172	C₈	—	4	6	no	—
173	C₇	#22746	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
177	C₂²	C₇×#557	1	2	yes	—
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	C₂×C₇×#108	1	6	yes	—
182	C₄	C₄×C₇×#41	1	1	yes	characteristic
183	C₄	C₇×#519	1	1	yes	characteristic
184	C₄	C₇×#548	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	C₇×#2351	1	1	yes	characteristic
187	C₂	C₂²×C₇×#108	1	1	yes	characteristic
188	C₂	C₇×#2916	1	1	yes	characteristic

	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	C₂³×C₇	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄×C₇×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
19. #22746	C₇	direct (N × H)
83. C₇×Q₈	#108	semidirect
158. Q₈	C₇×#108	semidirect
173. C₇	#22746	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×Q₈×M₁₆	2
C₂×C₇×C₈×Q₈	2
C₇×#1950	2
C₇×#1837	2
C₄×C₇×#108	2
#22746	7

Subgroups — C7×#22746 #13199962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22746 #13199962