Subgroups — C₇×#12866 #13190082

Order 1792.

470 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×#1808	C₂	1	1	yes	maximal, characteristic
4	C₇×#1380	C₂	1	4	yes	maximal, complemented
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
7	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
8	C₂×C₇×#107	—	2	1	no	—
9	C₂×C₇×#107	—	2	2	no	—
10	C₂×C₇×#107	C₄	1	4	yes	complemented
11	C₂×C₇×#107	C₂²	1	2	yes	—
12	C₂×C₇×#107	C₂²	1	2	yes	—
13	C₇×#380	—	2	1	no	—
14	C₇×C₂²⋊C₁₆	—	2	2	no	—
15	C₇×#340	—	2	1	no	—
16	#12866	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₄×C₇	—	4	1	no	—
18	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	—	2	4	no	—
20	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	D₄	1	2	yes	—
23	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₇×C₄⋊C₄	—	2	4	no	—
25	C₂×C₇×C₄⋊C₄	—	2	2	no	—
26	C₂×C₇×C₄⋊C₄	—	2	2	no	—
27	C₂×C₇×C₄⋊C₄	—	4	1	no	—
28	C₂×C₇×C₄⋊C₄	—	4	2	no	—
29	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
30	C₂×C₇×C₁₆	—	2	2	no	—
31	C₂×C₇×C₁₆	—	2	4	no	—
32	C₇×#107	—	2	4	no	—
33	C₇×#107	—	2	2	no	—
34	C₇×#107	—	2	4	no	—
35	C₇×#107	C₂×C₄	1	4	yes	—
36	C₇×#95	—	4	1	no	—
37	C₂²×#107	C₂×C₇	1	1	yes	characteristic
38	C₂×C₂²⋊C₁₆	C₂×C₇	1	1	yes	characteristic
39	#1808	C₂×C₇	1	1	yes	characteristic
40	#1380	C₂×C₇	1	4	yes	complemented
41	C₂⁴×C₇	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₇	—	2	2	no	—
43	C₂²×C₄×C₇	—	2	2	no	—
44	C₂²×C₄×C₇	—	4	4	no	—
45	C₂²×C₄×C₇	—	4	2	no	—
46	C₂²×C₄×C₇	—	4	2	no	—
47	C₂²×C₄×C₇	—	8	2	no	—
48	C₂²×C₄×C₇	—	8	2	no	—
49	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
52	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₇×C₈	—	2	2	no	—
54	C₂×C₇×C₈	—	2	4	no	—
55	C₂×C₇×C₈	—	2	4	no	—
56	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
57	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₇×C₈	#41	1	4	yes	—
60	C₇×C₄⋊C₄	—	2	4	no	—
61	C₇×C₄⋊C₄	—	4	4	no	—
62	C₇×C₄⋊C₄	—	4	2	no	—
63	C₇×C₄⋊C₄	—	4	2	no	—
64	C₇×C₄⋊C₄	—	4	2	no	—
65	C₇×C₂²⋊C₄	—	8	2	no	—
66	C₇×C₁₆	—	4	2	no	—
67	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
68	C₂²×C₄⋊C₄	—	2	1	no	—
69	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
70	C₂×#107	—	2	1	no	—
71	C₂×#107	—	2	2	no	—
72	C₂×#107	C₄×C₇	1	4	yes	complemented
73	C₂×#107	C₂²×C₇	1	2	yes	—
74	C₂×#107	C₂²×C₇	1	2	yes	—
75	#380	—	2	1	no	—
76	C₂²⋊C₁₆	—	2	2	no	—
77	#340	—	2	1	no	—
78	C₂³×C₇	—	2	2	no	—
79	C₂³×C₇	—	2	2	no	—
80	C₂³×C₇	#102	1	2	yes	—
81	C₂³×C₇	D₁₆	1	2	yes	—
82	C₂³×C₇	SD₃₂	1	2	yes	—
83	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	4	no	—
88	C₂×C₄×C₇	—	4	4	no	—
89	C₂×C₄×C₇	—	8	8	no	—
90	C₂×C₄×C₇	—	8	2	no	—
91	C₂×C₄×C₇	—	8	2	no	—
92	C₂×C₄×C₇	—	8	2	no	—
93	C₂×C₄×C₇	—	8	2	no	—
94	C₂×C₄×C₇	—	8	2	no	—
95	C₂×C₄×C₇	#102	1	2	yes	—
96	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇	#123	1	2	yes	—
98	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₇	#137	1	2	yes	—
100	C₇×C₈	—	2	2	no	—
101	C₇×C₈	#117	1	2	yes	—
102	C₇×C₈	#123	1	2	yes	—
103	C₂³×C₄	—	4	1	no	—
104	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
105	C₂²×C₈	—	2	4	no	—
106	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
107	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
108	C₂²×C₈	C₇×D₄	1	2	yes	—
109	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
110	C₂×C₄⋊C₄	—	2	4	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	4	1	no	—
114	C₂×C₄⋊C₄	—	4	2	no	—
115	C₂×C₂²⋊C₄	—	4	1	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	4	no	—
118	#107	—	2	4	no	—
119	#107	—	2	2	no	—
120	#107	—	2	4	no	—
121	#107	C₂×C₄×C₇	1	4	yes	—
122	#95	—	4	1	no	—
123	C₂²×C₇	—	2	2	no	—
124	C₂²×C₇	—	2	2	no	—
125	C₂²×C₇	—	2	4	no	—
126	C₂²×C₇	—	2	4	no	—
127	C₂²×C₇	#357	1	4	yes	—
128	C₂²×C₇	C₂×#102	1	1	yes	characteristic
129	C₂²×C₇	#480	1	2	yes	—
130	C₂²×C₇	C₂×D₁₆	1	1	yes	characteristic
131	C₂²×C₇	C₂×SD₃₂	1	1	yes	characteristic
132	C₂²×C₇	#510	1	2	yes	—
133	C₄×C₇	—	2	2	no	—
134	C₄×C₇	—	8	4	no	—
135	C₄×C₇	—	16	2	no	—
136	C₄×C₇	#381	1	1	yes	characteristic
137	C₄×C₇	#419	1	1	yes	characteristic
138	C₄×C₇	#483	1	2	yes	—
139	C₂⁴	C₇×D₈	1	1	yes	characteristic
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	—	8	2	no	—
146	C₂²×C₄	—	8	2	no	—
147	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
148	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
149	C₂²×C₄	C₇×SD₁₆	1	2	yes	—
150	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
155	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
156	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
157	C₂×C₈	C₇×#41	1	4	yes	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	8	2	no	—
164	C₁₆	—	4	2	no	—
165	C₂×C₇	—	2	4	no	—
166	C₂×C₇	#1823	1	1	yes	characteristic
167	C₂×C₇	C₂×#357	1	1	yes	characteristic
168	C₂×C₇	#2141	1	1	yes	characteristic
169	C₂×C₇	#2232	1	2	yes	—
170	C₂×C₇	#2235	1	2	yes	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	C₇×#102	1	2	yes	—
174	C₂³	C₇×D₁₆	1	2	yes	—
175	C₂³	C₇×SD₃₂	1	2	yes	—
176	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	8	no	—
185	C₂×C₄	—	8	2	no	—
186	C₂×C₄	—	8	2	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	C₇×#102	1	2	yes	—
189	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
190	C₂×C₄	C₇×#123	1	2	yes	—
191	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
192	C₂×C₄	C₇×#137	1	2	yes	—
193	C₈	—	2	2	no	—
194	C₈	C₇×#117	1	2	yes	—
195	C₈	C₇×#123	1	2	yes	—
196	C₇	#12866	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²	—	2	4	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	2	no	—
201	C₂²	C₇×#357	1	4	yes	—
202	C₂²	C₂×C₇×#102	1	1	yes	characteristic
203	C₂²	C₇×#480	1	2	yes	—
204	C₂²	C₂×C₇×D₁₆	1	1	yes	characteristic
205	C₂²	C₂×C₇×SD₃₂	1	1	yes	characteristic
206	C₂²	C₇×#510	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	8	4	no	—
209	C₄	—	16	2	no	—
210	C₄	C₇×#381	1	1	yes	characteristic
211	C₄	C₇×#419	1	1	yes	characteristic
212	C₄	C₇×#483	1	2	yes	—
213	C₂	—	2	4	no	—
214	C₂	C₇×#1823	1	1	yes	characteristic
215	C₂	C₂×C₇×#357	1	1	yes	characteristic
216	C₂	C₇×#2141	1	1	yes	characteristic
217	C₂	C₇×#2232	1	2	yes	—
218	C₂	C₇×#2235	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂⁴×C₇	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1380	C₂	semidirect
10. C₂×C₇×#107	C₄	semidirect
16. #12866	C₇	direct (N × H)
40. #1380	C₂×C₇	semidirect
72. C₂×#107	C₄×C₇	semidirect
196. C₇	#12866	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₇×#107	2
C₂×C₇×C₂²⋊C₁₆	2
C₇×#1808	2
C₇×#1380	2
#12866	7

Subgroups — C7×#12866 #13190082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12866 #13190082