Subgroups — C₇×#13288 #13190504

Order 1792.

354 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3160	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₇×#1846	C₂	1	1	yes	maximal, characteristic
4	C₇×#1386	C₂	1	4	yes	maximal
5	C₇×#533	—	2	1	no	—
6	C₂×C₇×M₃₂	C₂²	1	2	yes	—
7	C₇×#500	C₄	1	2	yes	complemented
8	C₇×#499	—	2	1	no	—
9	C₇×#499	C₄	1	2	yes	complemented
10	C₇×#498	—	2	1	no	—
11	C₇×#427	—	2	1	no	—
12	C₇×#427	C₂²	1	2	yes	—
13	C₂×C₇×#107	—	2	1	no	—
14	C₂×C₇×#106	C₂²	1	2	yes	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₇×#382	—	2	1	no	—
17	C₇×#346	—	2	1	no	—
18	C₄×C₇×C₁₆	—	2	1	no	—
19	#13288	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₇×C₄⋊Q₈	—	2	2	no	—
23	C₇×#125	—	2	2	no	—
24	C₇×C₂²⋊Q₈	—	4	2	no	—
25	C₄×C₇×Q₈	—	4	2	no	—
26	C₇×#117	—	2	2	no	—
27	C₂×C₇×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
29	C₇×M₃₂	—	2	2	no	—
30	C₇×M₃₂	Q₈	1	4	yes	complemented
31	C₂×C₇×C₁₆	—	2	2	no	—
32	C₇×#107	—	2	2	no	—
33	C₇×#107	—	4	2	no	—
34	C₇×#107	C₂×C₄	1	4	yes	—
35	C₇×#106	—	2	2	no	—
36	C₇×#106	—	2	2	no	—
37	C₇×#106	—	2	4	no	—
38	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
39	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₇×C₈	D₄	1	1	yes	characteristic
41	C₄×C₇×C₈	D₄	1	1	yes	characteristic
42	C₇×#95	—	4	2	no	—
43	#3160	C₂×C₇	1	1	yes	characteristic
44	C₄×M₃₂	C₂×C₇	1	1	yes	characteristic
45	#1846	C₂×C₇	1	1	yes	characteristic
46	#1386	C₂×C₇	1	4	yes	—
47	C₂×C₇×Q₈	—	4	2	no	—
48	C₂²×C₄×C₇	—	4	2	no	—
49	C₂²×C₄×C₇	D₈	1	2	yes	—
50	C₂²×C₄×C₇	C₂×D₄	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	C₂×C₇×C₈	SD₁₆	1	2	yes	—
54	C₂×C₇×C₈	Q₁₆	1	2	yes	—
55	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
58	C₂×C₇×C₈	#41	1	2	yes	—
59	C₇×C₄⋊C₄	—	2	4	no	—
60	C₇×C₄⋊C₄	—	4	4	no	—
61	C₇×C₄⋊C₄	—	4	2	no	—
62	C₇×C₄⋊C₄	—	4	2	no	—
63	C₇×C₄⋊C₄	—	4	2	no	—
64	C₇×C₂²⋊C₄	—	4	2	no	—
65	C₄²×C₇	—	4	2	no	—
66	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₇	SD₁₆	1	2	yes	—
69	C₇×C₁₆	—	2	4	no	—
70	#533	—	2	1	no	—
71	C₂×M₃₂	C₂²×C₇	1	2	yes	—
72	#500	C₄×C₇	1	2	yes	complemented
73	#499	—	2	1	no	—
74	#499	C₄×C₇	1	2	yes	complemented
75	#498	—	2	1	no	—
76	#427	—	2	1	no	—
77	#427	C₂²×C₇	1	2	yes	—
78	C₂×#107	—	2	1	no	—
79	C₂×#106	C₂²×C₇	1	2	yes	—
80	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
81	#382	—	2	1	no	—
82	#346	—	2	1	no	—
83	C₄×C₁₆	—	2	1	no	—
84	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
85	C₇×Q₈	—	8	4	no	—
86	C₂×C₄×C₇	—	4	2	no	—
87	C₂×C₄×C₇	—	4	4	no	—
88	C₂×C₄×C₇	—	8	4	no	—
89	C₂×C₄×C₇	—	8	2	no	—
90	C₂×C₄×C₇	#102	1	2	yes	—
91	C₂×C₄×C₇	#102	1	2	yes	—
92	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇	#124	1	1	yes	characteristic
94	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
98	C₇×C₈	#103	1	2	yes	—
99	C₇×C₈	#103	1	2	yes	—
100	C₇×C₈	C₄×Q₈	1	2	yes	—
101	C₇×C₈	C₂²⋊Q₈	1	2	yes	—
102	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
103	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
104	C₄⋊Q₈	—	2	2	no	—
105	#125	—	2	2	no	—
106	C₂²⋊Q₈	—	4	2	no	—
107	C₄×Q₈	—	4	2	no	—
108	#117	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
111	M₃₂	—	2	2	no	—
112	M₃₂	C₇×Q₈	1	4	yes	complemented
113	C₂×C₁₆	—	2	2	no	—
114	#107	—	2	2	no	—
115	#107	—	4	2	no	—
116	#107	C₂×C₄×C₇	1	4	yes	—
117	#106	—	2	2	no	—
118	#106	—	2	2	no	—
119	#106	—	2	4	no	—
120	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
121	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
122	C₄×C₈	C₇×D₄	1	1	yes	characteristic
123	C₄×C₈	C₇×D₄	1	1	yes	characteristic
124	#95	—	4	2	no	—
125	C₂²×C₇	—	2	2	no	—
126	C₂²×C₇	C₂×#102	1	1	yes	characteristic
127	C₂²×C₇	#486	1	1	yes	characteristic
128	C₂²×C₇	#500	1	1	yes	characteristic
129	C₄×C₇	—	2	2	no	—
130	C₄×C₇	—	2	2	no	—
131	C₄×C₇	—	8	4	no	—
132	C₄×C₇	—	16	2	no	—
133	C₄×C₇	#391	1	1	yes	characteristic
134	C₄×C₇	C₂×#103	1	1	yes	characteristic
135	C₄×C₇	#487	1	1	yes	characteristic
136	C₄×C₇	#498	1	1	yes	characteristic
137	C₂×Q₈	—	4	2	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	C₇×D₈	1	2	yes	—
140	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
141	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₈	C₇×SD₁₆	1	2	yes	—
144	C₂×C₈	C₇×Q₁₆	1	2	yes	—
145	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
146	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
147	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
148	C₂×C₈	C₇×#41	1	2	yes	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₂²⋊C₄	—	4	2	no	—
155	C₄²	—	4	2	no	—
156	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
157	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
158	C₄²	C₇×SD₁₆	1	2	yes	—
159	C₁₆	—	2	4	no	—
160	C₂×C₇	—	2	1	no	—
161	C₂×C₇	—	2	1	no	—
162	C₂×C₇	#1990	1	1	yes	characteristic
163	C₂×C₇	#2145	1	2	yes	—
164	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
165	Q₈	—	8	4	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	8	4	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	C₇×#102	1	2	yes	—
171	C₂×C₄	C₇×#102	1	2	yes	—
172	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₄	C₇×#124	1	1	yes	characteristic
174	C₂×C₄	C₇×C₄⋊Q₈	1	1	yes	characteristic
175	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
176	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
177	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
178	C₈	C₇×#103	1	2	yes	—
179	C₈	C₇×#103	1	2	yes	—
180	C₈	C₄×C₇×Q₈	1	2	yes	—
181	C₈	C₇×C₂²⋊Q₈	1	2	yes	—
182	C₇	#13288	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²	—	2	2	no	—
184	C₂²	C₂×C₇×#102	1	1	yes	characteristic
185	C₂²	C₇×#486	1	1	yes	characteristic
186	C₂²	C₇×#500	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	8	4	no	—
190	C₄	—	16	2	no	—
191	C₄	C₇×#391	1	1	yes	characteristic
192	C₄	C₂×C₇×#103	1	1	yes	characteristic
193	C₄	C₇×#487	1	1	yes	characteristic
194	C₄	C₇×#498	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	C₇×#1990	1	1	yes	characteristic
198	C₂	C₇×#2145	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂³×C₇	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₇×#500	C₄	semidirect
9. C₇×#499	C₄	semidirect
19. #13288	C₇	direct (N × H)
30. C₇×M₃₂	Q₈	semidirect
72. #500	C₄×C₇	semidirect
74. #499	C₄×C₇	semidirect
112. M₃₂	C₇×Q₈	semidirect
182. C₇	#13288	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×#3160	2
C₄×C₇×M₃₂	2
C₇×#1846	2
C₇×#1386	2
#13288	7

Subgroups — C7×#13288 #13190504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13288 #13190504