Subgroups — C₇×#11968 #13189184

Order 1792.

296 conjugacy classes of subgroups.

The subgroup lattice has 218 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	C₇×#2162	C₂	1	1	yes	maximal, characteristic
3	C₇×#1566	C₂	1	1	yes	maximal, characteristic
4	C₇×#1363	C₂	1	4	yes	maximal, complemented
5	C₇×#533	—	2	1	no	—
6	C₇×#500	C₄	1	1	yes	characteristic
7	C₇×#500	C₄	1	1	yes	characteristic
8	C₇×#499	—	2	1	no	—
9	C₇×#499	C₂²	1	2	yes	—
10	C₇×#498	—	2	1	no	—
11	C₇×#427	—	2	1	no	—
12	C₇×#427	C₄	1	2	yes	—
13	C₇×#423	—	2	1	no	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₇×C₂²⋊C₁₆	—	2	1	no	—
17	C₄×C₇×C₁₆	—	2	1	no	—
18	C₇×#334	—	2	1	no	—
19	C₇×#334	C₂²	1	2	yes	—
20	#11968	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₇×M₁₆	—	4	1	no	—
22	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
24	C₇×C₄⋊Q₈	—	2	1	no	—
25	C₇×C₄⋊Q₈	—	2	1	no	—
26	C₇×#125	—	2	2	no	—
27	C₇×C₂²⋊Q₈	—	4	1	no	—
28	C₇×C₂²⋊Q₈	—	4	1	no	—
29	C₄×C₇×Q₈	—	4	1	no	—
30	C₄×C₇×Q₈	—	4	1	no	—
31	C₇×#117	—	2	2	no	—
32	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
33	C₂×C₇×C₁₆	—	2	2	no	—
34	C₇×#107	—	2	1	no	—
35	C₇×#107	—	2	1	no	—
36	C₇×#107	C₂×C₄	1	2	yes	—
37	C₇×#107	C₂×C₄	1	2	yes	—
38	C₇×#106	—	2	2	no	—
39	C₇×#106	—	2	2	no	—
40	C₇×C₄⋊C₈	—	2	2	no	—
41	C₇×C₄⋊C₈	—	4	1	no	—
42	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₇×C₈	D₄	1	1	yes	characteristic
44	C₄×C₇×C₈	D₄	1	1	yes	characteristic
45	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
46	#3160	C₂×C₇	1	1	yes	characteristic
47	#2162	C₂×C₇	1	1	yes	characteristic
48	#1566	C₂×C₇	1	1	yes	characteristic
49	#1363	C₂×C₇	1	4	yes	complemented
50	C₂×C₇×Q₈	—	4	1	no	—
51	C₂×C₇×Q₈	—	4	1	no	—
52	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
55	C₇×M₁₆	—	8	2	no	—
56	C₂×C₇×C₈	—	2	2	no	—
57	C₂×C₇×C₈	—	2	2	no	—
58	C₂×C₇×C₈	—	4	2	no	—
59	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
63	C₇×C₄⋊C₄	—	2	2	no	—
64	C₇×C₄⋊C₄	—	2	2	no	—
65	C₇×C₄⋊C₄	—	4	1	no	—
66	C₇×C₄⋊C₄	—	4	1	no	—
67	C₇×C₄⋊C₄	—	4	2	no	—
68	C₇×C₄⋊C₄	—	4	2	no	—
69	C₇×C₂²⋊C₄	—	4	2	no	—
70	C₄²×C₇	—	4	2	no	—
71	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₇	D₈	1	1	yes	characteristic
73	C₄²×C₇	SD₁₆	1	1	yes	characteristic
74	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
75	C₇×C₁₆	—	4	2	no	—
76	#533	—	2	1	no	—
77	#500	C₄×C₇	1	1	yes	characteristic
78	#500	C₄×C₇	1	1	yes	characteristic
79	#499	—	2	1	no	—
80	#499	C₂²×C₇	1	2	yes	—
81	#498	—	2	1	no	—
82	#427	—	2	1	no	—
83	#427	C₄×C₇	1	2	yes	—
84	#423	—	2	1	no	—
85	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
86	C₄⋊C₁₆	C₂²×C₇	1	2	yes	—
87	C₂²⋊C₁₆	—	2	1	no	—
88	C₄×C₁₆	—	2	1	no	—
89	#334	—	2	1	no	—
90	#334	C₂²×C₇	1	2	yes	—
91	C₂³×C₇	#102	1	1	yes	characteristic
92	C₇×Q₈	—	8	2	no	—
93	C₇×Q₈	—	8	2	no	—
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	4	2	no	—
99	C₂×C₄×C₇	—	4	2	no	—
100	C₂×C₄×C₇	#102	1	1	yes	characteristic
101	C₂×C₄×C₇	#102	1	1	yes	characteristic
102	C₂×C₄×C₇	#102	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₇×C₈	—	2	1	no	—
107	C₇×C₈	—	2	1	no	—
108	C₇×C₈	—	8	2	no	—
109	C₇×C₈	#101	1	2	yes	—
110	C₇×C₈	#137	1	2	yes	—
111	C₂×M₁₆	—	4	1	no	—
112	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
113	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
114	C₄⋊Q₈	—	2	1	no	—
115	C₄⋊Q₈	—	2	1	no	—
116	#125	—	2	2	no	—
117	C₂²⋊Q₈	—	4	1	no	—
118	C₂²⋊Q₈	—	4	1	no	—
119	C₄×Q₈	—	4	1	no	—
120	C₄×Q₈	—	4	1	no	—
121	#117	—	2	2	no	—
122	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	#107	—	2	1	no	—
125	#107	—	2	1	no	—
126	#107	C₂×C₄×C₇	1	2	yes	—
127	#107	C₂×C₄×C₇	1	2	yes	—
128	#106	—	2	2	no	—
129	#106	—	2	2	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	4	1	no	—
132	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
133	C₄×C₈	C₇×D₄	1	1	yes	characteristic
134	C₄×C₈	C₇×D₄	1	1	yes	characteristic
135	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
136	C₂²×C₇	—	2	1	no	—
137	C₂²×C₇	—	2	1	no	—
138	C₂²×C₇	—	2	1	no	—
139	C₂²×C₇	C₂×#102	1	1	yes	characteristic
140	C₄×C₇	—	2	1	no	—
141	C₄×C₇	—	2	1	no	—
142	C₄×C₇	—	2	1	no	—
143	C₄×C₇	—	8	2	no	—
144	C₄×C₇	—	8	2	no	—
145	C₄×C₇	#359	1	2	yes	—
146	C₄×C₇	#360	1	2	yes	—
147	C₄×C₇	C₂×#101	1	1	yes	characteristic
148	C₄×C₇	#419	1	1	yes	characteristic
149	C₂×Q₈	—	4	1	no	—
150	C₂×Q₈	—	4	1	no	—
151	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
153	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
154	M₁₆	—	8	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
161	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄⋊C₄	—	4	1	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₂²⋊C₄	—	4	2	no	—
169	C₄²	—	4	2	no	—
170	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
171	C₄²	C₇×D₈	1	1	yes	characteristic
172	C₄²	C₇×SD₁₆	1	1	yes	characteristic
173	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
174	C₁₆	—	4	2	no	—
175	C₂×C₇	—	4	1	no	—
176	C₂×C₇	#1546	1	1	yes	characteristic
177	C₂×C₇	C₂×#359	1	1	yes	characteristic
178	C₂×C₇	C₂×#360	1	1	yes	characteristic
179	C₂³	C₇×#102	1	1	yes	characteristic
180	Q₈	—	8	2	no	—
181	Q₈	—	8	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₇×#102	1	1	yes	characteristic
189	C₂×C₄	C₇×#102	1	1	yes	characteristic
190	C₂×C₄	C₇×#102	1	1	yes	characteristic
191	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
192	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
193	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	8	2	no	—
197	C₈	C₇×#101	1	2	yes	—
198	C₈	C₇×#137	1	2	yes	—
199	C₇	#11968	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×C₇×#102	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	8	2	no	—
208	C₄	—	8	2	no	—
209	C₄	C₇×#359	1	2	yes	—
210	C₄	C₇×#360	1	2	yes	—
211	C₄	C₂×C₇×#101	1	1	yes	characteristic
212	C₄	C₇×#419	1	1	yes	characteristic
213	C₂	—	4	1	no	—
214	C₂	C₇×#1546	1	1	yes	characteristic
215	C₂	C₂×C₇×#359	1	1	yes	characteristic
216	C₂	C₂×C₇×#360	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂³×C₇	#102
LCS	G ▷ C₂×C₈ ▷ C₂×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
4. C₇×#1363	C₂	semidirect
20. #11968	C₇	direct (N × H)
49. #1363	C₂×C₇	semidirect
199. C₇	#11968	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
C₇×#2162	2
C₇×#1566	2
C₇×#1363	2
#11968	7

Subgroups — C7×#11968 #13189184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11968 #13189184