Subgroups — C₇×#12871 #13190087

Order 1792.

442 conjugacy classes of subgroups.

The subgroup lattice has 254 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	C₇×#2112	C₂	1	1	yes	maximal, characteristic
3	C₇×#1810	C₂	1	1	yes	maximal, characteristic
4	C₇×#1387	C₂	1	4	yes	maximal, complemented
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₇×C₄⋊C₄	—	2	1	no	—
7	C₂×C₇×M₃₂	C₂²	1	2	yes	—
8	C₂×C₇×#108	C₂²	1	2	yes	—
9	C₂×C₇×#107	—	2	1	no	—
10	C₂×C₇×#107	—	2	1	no	—
11	C₂×C₇×#107	—	2	1	no	—
12	C₂×C₇×#107	C₄	1	2	yes	—
13	C₂×C₇×#107	C₄	1	2	yes	—
14	C₂×C₇×#107	C₂²	1	2	yes	—
15	C₇×#407	—	2	1	no	—
16	C₇×C₂²⋊C₁₆	—	2	1	no	—
17	C₇×C₂²⋊C₁₆	—	2	1	no	—
18	C₇×#341	—	2	1	no	—
19	#12871	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₇	—	4	1	no	—
21	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
22	C₂×C₇×M₁₆	—	2	2	no	—
23	C₂²×C₇×C₈	—	2	4	no	—
24	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
30	C₂×C₇×C₄⋊C₄	—	2	2	no	—
31	C₂×C₇×C₄⋊C₄	—	2	2	no	—
32	C₂×C₇×C₄⋊C₄	—	2	2	no	—
33	C₂×C₇×C₄⋊C₄	—	4	1	no	—
34	C₂×C₇×C₄⋊C₄	—	4	1	no	—
35	C₂×C₇×C₄⋊C₄	—	4	1	no	—
36	C₇×M₃₂	—	2	4	no	—
37	C₂×C₇×C₁₆	—	2	2	no	—
38	C₇×#108	—	2	4	no	—
39	C₇×#107	—	2	2	no	—
40	C₇×#107	—	2	2	no	—
41	C₇×#107	—	2	2	no	—
42	C₇×#107	C₂×C₄	1	4	yes	—
43	C₇×C₂²⋊C₈	—	4	1	no	—
44	C₇×C₂²⋊C₈	—	4	1	no	—
45	C₂²×#107	C₂×C₇	1	1	yes	characteristic
46	#2112	C₂×C₇	1	1	yes	characteristic
47	#1810	C₂×C₇	1	1	yes	characteristic
48	#1387	C₂×C₇	1	4	yes	complemented
49	C₂⁴×C₇	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₇	—	2	2	no	—
51	C₂²×C₄×C₇	—	2	2	no	—
52	C₂²×C₄×C₇	—	4	2	no	—
53	C₂²×C₄×C₇	—	4	2	no	—
54	C₂²×C₄×C₇	—	4	2	no	—
55	C₂²×C₄×C₇	—	8	4	no	—
56	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₇×M₁₆	—	4	2	no	—
63	C₇×M₁₆	—	4	2	no	—
64	C₂×C₇×C₈	—	2	2	no	—
65	C₂×C₇×C₈	—	2	2	no	—
66	C₂×C₇×C₈	—	2	2	no	—
67	C₂×C₇×C₈	—	2	2	no	—
68	C₂×C₇×C₈	—	2	2	no	—
69	C₂×C₇×C₈	—	4	2	no	—
70	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₇×C₈	#41	1	4	yes	—
75	C₇×C₄⋊C₄	—	2	4	no	—
76	C₇×C₄⋊C₄	—	4	2	no	—
77	C₇×C₄⋊C₄	—	4	2	no	—
78	C₇×C₄⋊C₄	—	4	2	no	—
79	C₇×C₁₆	—	4	2	no	—
80	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
81	C₂²×C₄⋊C₄	—	2	1	no	—
82	C₂×M₃₂	C₂²×C₇	1	2	yes	—
83	C₂×#108	C₂²×C₇	1	2	yes	—
84	C₂×#107	—	2	1	no	—
85	C₂×#107	—	2	1	no	—
86	C₂×#107	—	2	1	no	—
87	C₂×#107	C₄×C₇	1	2	yes	—
88	C₂×#107	C₄×C₇	1	2	yes	—
89	C₂×#107	C₂²×C₇	1	2	yes	—
90	#407	—	2	1	no	—
91	C₂²⋊C₁₆	—	2	1	no	—
92	C₂²⋊C₁₆	—	2	1	no	—
93	#341	—	2	1	no	—
94	C₂³×C₇	—	2	2	no	—
95	C₂³×C₇	—	2	2	no	—
96	C₂³×C₇	—	2	2	no	—
97	C₂³×C₇	#102	1	1	yes	characteristic
98	C₂³×C₇	#102	1	1	yes	characteristic
99	C₂³×C₇	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	2	1	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	4	4	no	—
108	C₂×C₄×C₇	—	8	4	no	—
109	C₂×C₄×C₇	—	8	4	no	—
110	C₂×C₄×C₇	—	8	4	no	—
111	C₂×C₄×C₇	#102	1	1	yes	characteristic
112	C₂×C₄×C₇	#102	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇	#123	1	2	yes	—
115	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₇	#137	1	2	yes	—
117	C₇×C₈	—	2	1	no	—
118	C₇×C₈	—	2	1	no	—
119	C₇×C₈	—	8	2	no	—
120	C₇×C₈	#117	1	2	yes	—
121	C₇×C₈	#123	1	2	yes	—
122	C₂³×C₄	—	4	1	no	—
123	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
124	C₂×M₁₆	—	2	2	no	—
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
127	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
128	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
129	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
130	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
131	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	4	1	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄⋊C₄	—	4	1	no	—
138	M₃₂	—	2	4	no	—
139	C₂×C₁₆	—	2	2	no	—
140	#108	—	2	4	no	—
141	#107	—	2	2	no	—
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#107	C₂×C₄×C₇	1	4	yes	—
145	C₂²⋊C₈	—	4	1	no	—
146	C₂²⋊C₈	—	4	1	no	—
147	C₂²×C₇	—	2	8	no	—
148	C₂²×C₇	—	2	1	no	—
149	C₂²×C₇	—	2	1	no	—
150	C₂²×C₇	—	2	1	no	—
151	C₂²×C₇	—	2	1	no	—
152	C₂²×C₇	—	2	1	no	—
153	C₂²×C₇	—	2	1	no	—
154	C₂²×C₇	#361	1	2	yes	—
155	C₂²×C₇	#362	1	2	yes	—
156	C₂²×C₇	C₂×#102	1	1	yes	characteristic
157	C₂²×C₇	#483	1	2	yes	—
158	C₄×C₇	—	2	1	no	—
159	C₄×C₇	—	2	1	no	—
160	C₄×C₇	—	8	4	no	—
161	C₄×C₇	#381	1	1	yes	characteristic
162	C₄×C₇	#419	1	1	yes	characteristic
163	C₄×C₇	#480	1	2	yes	—
164	C₂⁴	C₇×SD₁₆	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	8	4	no	—
171	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
174	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
175	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
177	M₁₆	—	4	2	no	—
178	M₁₆	—	4	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
186	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
187	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
188	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
189	C₂×C₈	C₇×#41	1	4	yes	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₁₆	—	4	2	no	—
195	C₂×C₇	—	2	2	no	—
196	C₂×C₇	—	2	2	no	—
197	C₂×C₇	—	2	2	no	—
198	C₂×C₇	#1823	1	1	yes	characteristic
199	C₂×C₇	C₂×#361	1	1	yes	characteristic
200	C₂×C₇	C₂×#362	1	1	yes	characteristic
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	C₇×#102	1	1	yes	characteristic
205	C₂³	C₇×#102	1	1	yes	characteristic
206	C₂³	C₂×C₇×SD₁₆	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	—	8	4	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	C₇×#102	1	1	yes	characteristic
219	C₂×C₄	C₇×#102	1	1	yes	characteristic
220	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
221	C₂×C₄	C₇×#123	1	2	yes	—
222	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
223	C₂×C₄	C₇×#137	1	2	yes	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	8	2	no	—
227	C₈	C₇×#117	1	2	yes	—
228	C₈	C₇×#123	1	2	yes	—
229	C₇	#12871	1	1	yes	characteristic, direct product factor, Sylow
230	C₂²	—	2	8	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	C₇×#361	1	2	yes	—
238	C₂²	C₇×#362	1	2	yes	—
239	C₂²	C₂×C₇×#102	1	1	yes	characteristic
240	C₂²	C₇×#483	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	8	4	no	—
244	C₄	C₇×#381	1	1	yes	characteristic
245	C₄	C₇×#419	1	1	yes	characteristic
246	C₄	C₇×#480	1	2	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	2	2	no	—
249	C₂	—	2	2	no	—
250	C₂	C₇×#1823	1	1	yes	characteristic
251	C₂	C₂×C₇×#361	1	1	yes	characteristic
252	C₂	C₂×C₇×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂⁴×C₇	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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×#1387	C₂	semidirect
19. #12871	C₇	direct (N × H)
48. #1387	C₂×C₇	semidirect
229. C₇	#12871	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
C₇×#2112	2
C₇×#1810	2
C₇×#1387	2
#12871	7

Subgroups — C7×#12871 #13190087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12871 #13190087