Subgroups — C₇×#10543 #13187759

Order 1792.

422 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic
2	C₇×#1987	C₂	1	2	yes	maximal
3	C₇×#1939	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#333	C₂	1	2	yes	maximal
5	C₇×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₇×#500	—	2	4	no	—
8	C₂×C₇×#107	—	2	1	no	—
9	C₂×C₇×#107	—	2	1	no	—
10	C₂×C₇×C₄⋊C₈	—	2	1	no	—
11	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₇×#340	—	2	2	no	—
16	C₇×#340	—	2	2	no	—
17	C₇×#336	—	2	1	no	—
18	C₇×#333	—	2	4	no	—
19	#10543	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×Q₈	—	2	2	no	—
21	C₂²×C₇×C₈	—	2	1	no	—
22	C₂²×C₇×C₈	—	2	2	no	—
23	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
25	C₇×C₄⋊Q₈	—	2	2	no	—
26	C₇×C₄⋊Q₈	—	2	2	no	—
27	C₇×C₄⋊Q₈	—	2	2	no	—
28	C₇×C₄⋊Q₈	D₄	1	4	yes	—
29	C₂×C₇×C₄⋊C₄	—	2	2	no	—
30	C₂×C₇×C₄⋊C₄	—	2	2	no	—
31	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
32	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₇	—	2	1	no	—
34	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₇×#107	—	2	4	no	—
36	C₇×#107	—	2	4	no	—
37	C₇×C₄⋊C₈	—	2	2	no	—
38	C₇×C₄⋊C₈	—	2	2	no	—
39	C₇×C₄⋊C₈	—	2	2	no	—
40	C₇×C₄⋊C₈	—	4	2	no	—
41	C₄×C₇×C₈	—	2	2	no	—
42	C₇×#95	—	2	1	no	—
43	C₂×#500	C₂×C₇	1	1	yes	characteristic
44	#1987	C₂×C₇	1	2	yes	—
45	#1939	C₂×C₇	1	1	yes	characteristic
46	C₂×#333	C₂×C₇	1	2	yes	—
47	#1277	C₂×C₇	1	1	yes	characteristic
48	C₂×C₇×Q₈	—	2	4	no	—
49	C₂×C₇×Q₈	—	4	2	no	—
50	C₂×C₇×Q₈	—	8	2	no	—
51	C₂×C₇×Q₈	—	8	2	no	—
52	C₂²×C₄×C₇	—	2	1	no	—
53	C₂²×C₄×C₇	—	2	1	no	—
54	C₂²×C₄×C₇	—	4	2	no	—
55	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	#41	1	1	yes	characteristic
58	C₂×C₇×C₈	—	2	1	no	—
59	C₂×C₇×C₈	—	2	1	no	—
60	C₂×C₇×C₈	—	2	1	no	—
61	C₂×C₇×C₈	—	2	1	no	—
62	C₂×C₇×C₈	—	4	1	no	—
63	C₂×C₇×C₈	—	4	2	no	—
64	C₂×C₇×C₈	—	4	1	no	—
65	C₂×C₇×C₈	—	4	2	no	—
66	C₂×C₇×C₈	—	4	1	no	—
67	C₂×C₇×C₈	—	4	2	no	—
68	C₂×C₇×C₈	SD₁₆	1	2	yes	—
69	C₂×C₇×C₈	Q₁₆	1	2	yes	—
70	C₇×C₄⋊C₄	—	2	1	no	—
71	C₇×C₄⋊C₄	—	2	4	no	—
72	C₇×C₄⋊C₄	—	2	4	no	—
73	C₇×C₄⋊C₄	—	2	2	no	—
74	C₇×C₄⋊C₄	—	4	2	no	—
75	C₇×C₄⋊C₄	—	4	2	no	—
76	C₇×C₄⋊C₄	Q₁₆	1	2	yes	—
77	C₄²×C₇	—	4	1	no	—
78	C₄²×C₇	—	4	1	no	—
79	C₄²×C₇	C₂×D₄	1	2	yes	—
80	C₄²×C₇	#41	1	2	yes	—
81	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
82	#500	—	2	4	no	—
83	C₂×#107	—	2	1	no	—
84	C₂×#107	—	2	1	no	—
85	C₂×C₄⋊C₈	—	2	1	no	—
86	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
87	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
88	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
89	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
90	#340	—	2	2	no	—
91	#340	—	2	2	no	—
92	#336	—	2	1	no	—
93	#333	—	2	4	no	—
94	C₂³×C₇	#121	1	1	yes	characteristic
95	C₇×Q₈	—	8	4	no	—
96	C₇×Q₈	—	8	4	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	4	1	no	—
101	C₂×C₄×C₇	—	4	1	no	—
102	C₂×C₄×C₇	—	4	4	no	—
103	C₂×C₄×C₇	—	4	1	no	—
104	C₂×C₄×C₇	—	4	1	no	—
105	C₂×C₄×C₇	—	4	1	no	—
106	C₂×C₄×C₇	—	8	2	no	—
107	C₂×C₄×C₇	—	8	2	no	—
108	C₂×C₄×C₇	#120	1	1	yes	characteristic
109	C₂×C₄×C₇	#121	1	2	yes	—
110	C₂×C₄×C₇	#123	1	2	yes	—
111	C₂×C₄×C₇	#124	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×C₄×C₇	#135	1	1	yes	characteristic
116	C₂×C₄×C₇	#137	1	1	yes	characteristic
117	C₂×C₄×C₇	#137	1	1	yes	characteristic
118	C₂×C₄×C₇	#137	1	2	yes	—
119	C₇×C₈	—	2	2	no	—
120	C₇×C₈	—	2	2	no	—
121	C₇×C₈	—	8	1	no	—
122	C₇×C₈	—	8	2	no	—
123	C₂²×Q₈	—	2	2	no	—
124	C₂²×C₈	—	2	1	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
127	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	C₇×D₄	1	4	yes	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
136	C₂×C₄²	—	2	1	no	—
137	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
138	#107	—	2	4	no	—
139	#107	—	2	4	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	4	2	no	—
144	C₄×C₈	—	2	2	no	—
145	#95	—	2	1	no	—
146	C₂²×C₇	#394	1	1	yes	characteristic
147	C₂²×C₇	#461	1	1	yes	characteristic
148	C₂²×C₇	#462	1	1	yes	characteristic
149	C₂²×C₇	#467	1	1	yes	characteristic
150	C₂²×C₇	#470	1	1	yes	characteristic
151	C₂²×C₇	#472	1	1	yes	characteristic
152	C₂²×C₇	#472	1	1	yes	characteristic
153	C₄×C₇	—	2	2	no	—
154	C₄×C₇	—	4	2	no	—
155	C₄×C₇	—	8	1	no	—
156	C₄×C₇	—	8	4	no	—
157	C₄×C₇	#448	1	1	yes	characteristic
158	C₄×C₇	#451	1	1	yes	characteristic
159	C₄×C₇	#462	1	2	yes	—
160	C₄×C₇	#464	1	2	yes	—
161	C₄×C₇	#487	1	1	yes	characteristic
162	C₄×C₇	#491	1	1	yes	characteristic
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂×Q₈	—	8	2	no	—
166	C₂×Q₈	—	8	2	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
172	C₂²×C₄	C₇×#41	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	C₇×SD₁₆	1	2	yes	—
184	C₂×C₈	C₇×Q₁₆	1	2	yes	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	C₇×Q₁₆	1	2	yes	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	C₂×C₇×D₄	1	2	yes	—
195	C₄²	C₇×#41	1	2	yes	—
196	C₂×C₇	#1675	1	1	yes	characteristic
197	C₂×C₇	#1676	1	1	yes	characteristic
198	C₂×C₇	#1695	1	1	yes	characteristic
199	C₂×C₇	#1696	1	1	yes	characteristic
200	C₂×C₇	#2043	1	1	yes	characteristic
201	C₂×C₇	#2045	1	1	yes	characteristic
202	C₂×C₇	#2056	1	1	yes	characteristic
203	C₂³	C₇×#121	1	1	yes	characteristic
204	Q₈	—	8	4	no	—
205	Q₈	—	8	4	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	C₇×#120	1	1	yes	characteristic
218	C₂×C₄	C₇×#121	1	2	yes	—
219	C₂×C₄	C₇×#123	1	2	yes	—
220	C₂×C₄	C₇×#124	1	1	yes	characteristic
221	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
222	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
223	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
224	C₂×C₄	C₇×#135	1	1	yes	characteristic
225	C₂×C₄	C₇×#137	1	1	yes	characteristic
226	C₂×C₄	C₇×#137	1	1	yes	characteristic
227	C₂×C₄	C₇×#137	1	2	yes	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	8	1	no	—
231	C₈	—	8	2	no	—
232	C₇	#10543	1	1	yes	characteristic, direct product factor, Sylow
233	C₂²	C₇×#394	1	1	yes	characteristic
234	C₂²	C₇×#461	1	1	yes	characteristic
235	C₂²	C₇×#462	1	1	yes	characteristic
236	C₂²	C₇×#467	1	1	yes	characteristic
237	C₂²	C₇×#470	1	1	yes	characteristic
238	C₂²	C₇×#472	1	1	yes	characteristic
239	C₂²	C₇×#472	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	8	4	no	—
243	C₄	—	8	1	no	—
244	C₄	C₇×#448	1	1	yes	characteristic
245	C₄	C₇×#451	1	1	yes	characteristic
246	C₄	C₇×#462	1	2	yes	—
247	C₄	C₇×#464	1	2	yes	—
248	C₄	C₇×#487	1	1	yes	characteristic
249	C₄	C₇×#491	1	1	yes	characteristic
250	C₂	C₇×#1675	1	1	yes	characteristic
251	C₂	C₇×#1676	1	1	yes	characteristic
252	C₂	C₇×#1695	1	1	yes	characteristic
253	C₂	C₇×#1696	1	1	yes	characteristic
254	C₂	C₇×#2043	1	1	yes	characteristic
255	C₂	C₇×#2045	1	1	yes	characteristic
256	C₂	C₇×#2056	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
19. #10543	C₇	direct (N × H)
232. C₇	#10543	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×#500	2
C₇×#1987	2
C₇×#1939	2
C₂×C₇×#333	2
C₇×#1277	2
#10543	7

Subgroups — C7×#10543 #13187759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10543 #13187759