Subgroups — C₇×#12839 #13190055

Order 1792.

362 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#504	C₂	1	1	yes	maximal, characteristic
2	C₇×#2158	C₂	1	2	yes	maximal, complemented
3	C₇×#1812	C₂	1	2	yes	maximal
4	C₇×#1384	C₂	1	2	yes	maximal, complemented
5	C₂×C₇×#131	C₂²	1	1	yes	characteristic
6	C₇×#504	—	2	2	no	—
7	C₇×#504	—	2	2	no	—
8	C₂×C₇×M₃₂	C₄	1	2	yes	complemented
9	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₇×C₁₆	C₄	1	2	yes	complemented
11	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
12	C₇×#427	C₂²	1	4	yes	—
13	C₇×#410	—	2	2	no	—
14	C₇×#367	—	2	2	no	—
15	C₇×#366	—	2	2	no	—
16	C₇×H(3,ℤ/4)	—	2	2	no	—
17	#12839	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂×C₇×#41	D₄	1	1	yes	characteristic
19	C₇×#131	—	2	2	no	—
20	C₇×#131	—	2	2	no	—
21	C₂×C₇×M₁₆	D₄	1	1	yes	characteristic
22	C₂×C₇×M₁₆	Q₈	1	2	yes	—
23	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
25	C₇×#117	—	2	4	no	—
26	C₇×M₃₂	—	2	2	no	—
27	C₇×M₃₂	—	2	4	no	—
28	C₂×C₇×C₁₆	—	2	2	no	—
29	C₂×C₇×C₁₆	—	2	2	no	—
30	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×C₇×C₁₆	D₄	1	2	yes	—
34	C₂×C₇×C₁₆	D₄	1	2	yes	—
35	C₇×#107	—	2	4	no	—
36	C₇×#106	—	2	4	no	—
37	C₇×#99	—	4	4	no	—
38	C₂×#504	C₂×C₇	1	1	yes	characteristic
39	#2158	C₂×C₇	1	2	yes	complemented
40	#1812	C₂×C₇	1	2	yes	—
41	#1384	C₂×C₇	1	2	yes	complemented
42	C₇×#41	—	2	2	no	—
43	C₇×#41	—	2	2	no	—
44	C₂×C₇×Q₈	Q₁₆	1	1	yes	characteristic
45	C₂×C₇×D₄	D₈	1	2	yes	—
46	C₂×C₇×D₄	Q₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₇×M₁₆	—	2	2	no	—
50	C₇×M₁₆	—	2	4	no	—
51	C₂×C₇×C₈	—	2	2	no	—
52	C₂×C₇×C₈	—	2	2	no	—
53	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
54	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
56	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₇×C₈	#41	1	2	yes	—
58	C₇×C₄⋊C₄	—	4	4	no	—
59	C₇×C₂²⋊C₄	—	4	4	no	—
60	C₄²×C₇	—	4	4	no	—
61	C₇×C₁₆	—	2	2	no	—
62	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
63	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
64	C₇×C₁₆	C₂²⋊C₄	1	2	yes	—
65	C₂×#131	C₂²×C₇	1	1	yes	characteristic
66	#504	—	2	2	no	—
67	#504	—	2	2	no	—
68	C₂×M₃₂	C₄×C₇	1	2	yes	complemented
69	C₂×M₃₂	C₂²×C₇	1	1	yes	characteristic
70	C₂²×C₁₆	C₄×C₇	1	2	yes	complemented
71	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
72	#427	C₂²×C₇	1	4	yes	—
73	#410	—	2	2	no	—
74	#367	—	2	2	no	—
75	#366	—	2	2	no	—
76	H(3,ℤ/4)	—	2	2	no	—
77	C₂³×C₇	#107	1	2	yes	—
78	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
79	C₇×Q₈	—	2	2	no	—
80	C₇×D₄	—	2	4	no	—
81	C₇×D₄	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	8	4	no	—
85	C₂×C₄×C₇	#107	1	2	yes	—
86	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₇	#121	1	2	yes	—
88	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₇	#135	1	2	yes	—
90	C₇×C₈	—	2	2	no	—
91	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₇×C₈	#117	1	1	yes	characteristic
93	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
94	C₂×#41	C₇×D₄	1	1	yes	characteristic
95	#131	—	2	2	no	—
96	#131	—	2	2	no	—
97	C₂×M₁₆	C₇×D₄	1	1	yes	characteristic
98	C₂×M₁₆	C₇×Q₈	1	2	yes	—
99	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
100	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
101	#117	—	2	4	no	—
102	M₃₂	—	2	2	no	—
103	M₃₂	—	2	4	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	C₂×C₄×C₇	1	2	yes	—
107	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
108	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
109	C₂×C₁₆	C₇×D₄	1	2	yes	—
110	C₂×C₁₆	C₇×D₄	1	2	yes	—
111	#107	—	2	4	no	—
112	#106	—	2	4	no	—
113	#99	—	4	4	no	—
114	C₂²×C₇	—	2	2	no	—
115	C₂²×C₇	—	4	1	no	—
116	C₂²×C₇	—	4	2	no	—
117	C₂²×C₇	C₂×#107	1	1	yes	characteristic
118	C₂²×C₇	#466	1	2	yes	—
119	C₄×C₇	—	2	2	no	—
120	C₄×C₇	—	16	4	no	—
121	C₄×C₇	#380	1	1	yes	characteristic
122	C₄×C₇	#427	1	1	yes	characteristic
123	C₄×C₇	#467	1	2	yes	—
124	#41	—	2	2	no	—
125	#41	—	2	2	no	—
126	C₂×Q₈	C₇×Q₁₆	1	1	yes	characteristic
127	C₂×D₄	C₇×D₈	1	2	yes	—
128	C₂×D₄	C₇×Q₁₆	1	1	yes	characteristic
129	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
130	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
131	M₁₆	—	2	2	no	—
132	M₁₆	—	2	4	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
136	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
137	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
138	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
139	C₂×C₈	C₇×#41	1	2	yes	—
140	C₄⋊C₄	—	4	4	no	—
141	C₂²⋊C₄	—	4	4	no	—
142	C₄²	—	4	4	no	—
143	C₁₆	—	2	2	no	—
144	C₁₆	C₇×C₂²⋊C₄	1	1	yes	characteristic
145	C₁₆	C₇×C₂²⋊C₄	1	1	yes	characteristic
146	C₁₆	C₇×C₂²⋊C₄	1	2	yes	—
147	C₂×C₇	—	2	2	no	—
148	C₂×C₇	—	2	1	no	—
149	C₂×C₇	—	4	2	no	—
150	C₂×C₇	#1808	1	1	yes	characteristic
151	C₂³	C₇×#107	1	2	yes	—
152	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
153	Q₈	—	2	2	no	—
154	D₄	—	2	4	no	—
155	D₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	8	4	no	—
159	C₂×C₄	C₇×#107	1	2	yes	—
160	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄	C₇×#121	1	2	yes	—
162	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
163	C₂×C₄	C₇×#135	1	2	yes	—
164	C₈	—	2	2	no	—
165	C₈	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
166	C₈	C₇×#117	1	1	yes	characteristic
167	C₈	C₇×C₂²⋊Q₈	1	2	yes	—
168	C₇	#12839	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²	—	2	2	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	2	no	—
172	C₂²	C₂×C₇×#107	1	1	yes	characteristic
173	C₂²	C₇×#466	1	2	yes	—
174	C₄	—	2	2	no	—
175	C₄	—	16	4	no	—
176	C₄	C₇×#380	1	1	yes	characteristic
177	C₄	C₇×#427	1	1	yes	characteristic
178	C₄	C₇×#467	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	—	2	1	no	—
181	C₂	—	4	2	no	—
182	C₂	C₇×#1808	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	#427
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1808
prime radical	C₂×C₇×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₇×M₁₆ ◁ C₂×C₇×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2158	C₂	semidirect
4. C₇×#1384	C₂	semidirect
8. C₂×C₇×M₃₂	C₄	semidirect
10. C₂²×C₇×C₁₆	C₄	semidirect
17. #12839	C₇	direct (N × H)
39. #2158	C₂×C₇	semidirect
41. #1384	C₂×C₇	semidirect
68. C₂×M₃₂	C₄×C₇	semidirect
70. C₂²×C₁₆	C₄×C₇	semidirect
168. C₇	#12839	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×C₇×#504	2
C₇×#2158	2
C₇×#1812	2
C₇×#1384	2
#12839	7

Subgroups — C7×#12839 #13190055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12839 #13190055