Subgroups — C₇×#12005 #13189221

Order 1792.

242 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×#1450	C₂	1	1	yes	maximal, characteristic
4	C₇×#1370	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×M₃₂	—	2	1	no	—
6	C₂²×C₇×C₁₆	—	2	1	no	—
7	C₇×#431	C₄	1	1	yes	characteristic
8	C₇×#431	C₄	1	1	yes	characteristic
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₇×C₄⋊C₁₆	—	2	1	no	—
11	C₇×C₄⋊C₁₆	—	2	1	no	—
12	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
13	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
14	C₇×#322	C₄	1	2	yes	—
15	C₇×C₈²	C₂²	1	2	yes	—
16	#12005	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
20	C₇×M₃₂	—	4	2	no	—
21	C₂×C₇×C₁₆	—	2	2	no	—
22	C₂×C₇×C₁₆	—	2	2	no	—
23	C₂×C₇×C₁₆	—	4	2	no	—
24	C₇×C₄⋊C₈	—	2	1	no	—
25	C₇×C₄⋊C₈	C₈	1	2	yes	—
26	C₇×C₂²⋊C₈	—	2	1	no	—
27	C₇×C₂²⋊C₈	C₈	1	2	yes	—
28	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
31	C₄×C₇×C₈	D₄	1	1	yes	characteristic
32	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
33	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
34	#2150	C₂×C₇	1	1	yes	characteristic
35	C₂×C₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic
36	#1450	C₂×C₇	1	1	yes	characteristic
37	#1370	C₂×C₇	1	4	yes	complemented
38	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
41	C₂×C₇×C₈	—	2	2	no	—
42	C₂×C₇×C₈	—	2	2	no	—
43	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
46	C₂×C₇×C₈	D₈	1	2	yes	—
47	C₂×C₇×C₈	Q₁₆	1	2	yes	—
48	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₇	M₁₆	1	1	yes	characteristic
53	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
54	C₇×C₁₆	—	4	2	no	—
55	C₇×C₁₆	—	4	2	no	—
56	C₂×M₃₂	—	2	1	no	—
57	C₂²×C₁₆	—	2	1	no	—
58	#431	C₄×C₇	1	1	yes	characteristic
59	#431	C₄×C₇	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
61	C₄⋊C₁₆	—	2	1	no	—
62	C₄⋊C₁₆	—	2	1	no	—
63	C₄⋊C₁₆	C₂²×C₇	1	2	yes	—
64	C₄⋊C₁₆	C₂²×C₇	1	2	yes	—
65	#322	C₄×C₇	1	2	yes	—
66	C₈²	C₂²×C₇	1	2	yes	—
67	C₂³×C₇	C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₇	—	2	2	no	—
69	C₂×C₄×C₇	—	4	1	no	—
70	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
73	C₂×C₄×C₇	#107	1	2	yes	—
74	C₂×C₄×C₇	#108	1	2	yes	—
75	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
78	C₇×C₈	—	2	1	no	—
79	C₇×C₈	—	2	1	no	—
80	C₇×C₈	—	2	4	no	—
81	C₇×C₈	—	4	2	no	—
82	C₇×C₈	#107	1	2	yes	—
83	C₇×C₈	C₂×D₈	1	1	yes	characteristic
84	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
85	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
86	C₂²×C₈	C₇×Q₈	1	1	yes	characteristic
87	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
88	M₃₂	—	4	2	no	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	4	2	no	—
92	C₄⋊C₈	—	2	1	no	—
93	C₄⋊C₈	C₇×C₈	1	2	yes	—
94	C₂²⋊C₈	—	2	1	no	—
95	C₂²⋊C₈	C₇×C₈	1	2	yes	—
96	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
97	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
98	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
99	C₄×C₈	C₇×D₄	1	1	yes	characteristic
100	C₄×C₈	C₇×Q₈	1	1	yes	characteristic
101	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
102	C₂²×C₇	—	2	1	no	—
103	C₂²×C₇	—	2	1	no	—
104	C₂²×C₇	#335	1	2	yes	—
105	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₇	—	2	1	no	—
107	C₄×C₇	—	2	1	no	—
108	C₄×C₇	—	2	1	no	—
109	C₄×C₇	#335	1	2	yes	—
110	C₄×C₇	C₂×#107	1	1	yes	characteristic
111	C₄×C₇	C₂×#108	1	1	yes	characteristic
112	C₄×C₇	#504	1	2	yes	—
113	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
115	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₈	C₂×C₇×C₈	1	4	yes	—
121	C₂×C₈	C₇×D₈	1	2	yes	—
122	C₂×C₈	C₇×Q₁₆	1	2	yes	—
123	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
124	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
125	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
126	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
127	C₄²	C₇×M₁₆	1	1	yes	characteristic
128	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
129	C₁₆	—	4	2	no	—
130	C₁₆	—	4	2	no	—
131	C₂×C₇	—	2	2	no	—
132	C₂×C₇	C₂×#335	1	1	yes	characteristic
133	C₂×C₇	#2151	1	1	yes	characteristic
134	C₂×C₇	#2153	1	1	yes	characteristic
135	C₂³	C₇×C₄⋊C₈	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	C₇×C₄⋊C₈	1	1	yes	characteristic
139	C₂×C₄	C₇×C₄⋊C₈	1	1	yes	characteristic
140	C₂×C₄	C₇×C₄⋊C₈	1	1	yes	characteristic
141	C₂×C₄	C₇×#107	1	2	yes	—
142	C₂×C₄	C₇×#108	1	2	yes	—
143	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
145	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
146	C₈	—	2	4	no	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	4	2	no	—
150	C₈	C₇×#107	1	2	yes	—
151	C₈	C₂×C₇×D₈	1	1	yes	characteristic
152	C₈	C₂×C₇×Q₁₆	1	1	yes	characteristic
153	C₇	#12005	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	C₇×#335	1	2	yes	—
157	C₂²	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	C₇×#335	1	2	yes	—
162	C₄	C₂×C₇×#107	1	1	yes	characteristic
163	C₄	C₂×C₇×#108	1	1	yes	characteristic
164	C₄	C₇×#504	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	C₂×C₇×#335	1	1	yes	characteristic
167	C₂	C₇×#2151	1	1	yes	characteristic
168	C₂	C₇×#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1370	C₂	semidirect
16. #12005	C₇	direct (N × H)
37. #1370	C₂×C₇	semidirect
153. C₇	#12005	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×#2150	2
C₂×C₇×C₄⋊C₁₆	2
C₇×#1450	2
C₇×#1370	2
#12005	7

Subgroups — C7×#12005 #13189221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12005 #13189221