Subgroups — C₇×#10264 #13187480

Order 1792.

626 conjugacy classes of subgroups.

The subgroup lattice has 282 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, complemented
2	C₂×C₇×#461	C₂	1	2	yes	maximal
3	C₇×#1954	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#332	C₂	1	2	yes	maximal, complemented
5	C₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×SD₁₆	—	2	2	no	—
7	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
9	C₇×#500	—	2	4	no	—
10	C₇×#461	—	2	4	no	—
11	C₇×#461	—	2	4	no	—
12	C₂×C₇×#107	—	2	2	no	—
13	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₇×#103	—	2	2	no	—
16	C₂×C₇×C₂²⋊C₈	—	2	2	no	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₇×#332	—	2	4	no	—
19	#10264	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×Q₈	—	2	2	no	—
21	C₂²×C₇×D₄	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₇	—	2	1	no	—
23	C₂×C₇×SD₁₆	—	2	8	no	—
24	C₂×C₇×SD₁₆	—	4	4	no	—
25	C₂×C₇×SD₁₆	—	4	4	no	—
26	C₂²×C₇×C₈	—	2	2	no	—
27	C₂²×C₇×C₈	—	2	1	no	—
28	C₂²×C₇×C₈	D₄	1	2	yes	—
29	C₇×C₄⋊Q₈	—	2	2	no	—
30	C₇×C₄⋊Q₈	—	2	2	no	—
31	C₇×C₄⋊Q₈	—	2	2	no	—
32	C₇×C₄⋊Q₈	D₄	1	4	yes	—
33	C₄×C₇×D₄	—	2	2	no	—
34	C₄×C₇×D₄	—	2	2	no	—
35	C₂×C₇×C₄⋊C₄	—	2	2	no	—
36	C₂×C₇×C₄⋊C₄	—	2	2	no	—
37	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
39	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
40	C₇×#107	—	2	8	no	—
41	C₇×C₄⋊C₈	—	2	2	no	—
42	C₇×C₄⋊C₈	—	2	2	no	—
43	C₇×C₄⋊C₈	D₄	1	4	yes	—
44	C₇×#103	—	2	8	no	—
45	C₇×C₂²⋊C₈	—	4	4	no	—
46	C₄×C₇×C₈	—	2	2	no	—
47	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
48	C₂×#461	C₂×C₇	1	2	yes	—
49	#1954	C₂×C₇	1	1	yes	characteristic
50	C₂×#332	C₂×C₇	1	2	yes	complemented
51	#1277	C₂×C₇	1	1	yes	characteristic, complemented
52	C₂⁴×C₇	—	2	1	no	—
53	C₂×C₇×Q₈	—	2	4	no	—
54	C₂×C₇×Q₈	—	4	2	no	—
55	C₂×C₇×Q₈	—	4	2	no	—
56	C₂×C₇×Q₈	—	4	2	no	—
57	C₂×C₇×Q₈	—	4	2	no	—
58	C₂×C₇×Q₈	—	4	2	no	—
59	C₂×C₇×D₄	—	2	2	no	—
60	C₂×C₇×D₄	—	2	1	no	—
61	C₂×C₇×D₄	—	2	2	no	—
62	C₂×C₇×D₄	D₈	1	2	yes	—
63	C₂²×C₄×C₇	—	2	1	no	—
64	C₂²×C₄×C₇	—	4	2	no	—
65	C₂²×C₄×C₇	—	4	1	no	—
66	C₂²×C₄×C₇	—	4	1	no	—
67	C₂²×C₄×C₇	—	4	2	no	—
68	C₂²×C₄×C₇	—	4	2	no	—
69	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	#41	1	1	yes	characteristic
72	C₇×SD₁₆	—	4	16	no	—
73	C₂×C₇×C₈	—	2	2	no	—
74	C₂×C₇×C₈	—	2	2	no	—
75	C₂×C₇×C₈	—	2	4	no	—
76	C₂×C₇×C₈	—	4	2	no	—
77	C₂×C₇×C₈	—	4	2	no	—
78	C₂×C₇×C₈	—	4	1	no	—
79	C₂×C₇×C₈	—	4	2	no	—
80	C₂×C₇×C₈	SD₁₆	1	4	yes	complemented
81	C₇×C₄⋊C₄	—	2	2	no	—
82	C₇×C₄⋊C₄	—	2	4	no	—
83	C₇×C₄⋊C₄	—	2	4	no	—
84	C₇×C₄⋊C₄	—	4	2	no	—
85	C₇×C₄⋊C₄	—	4	2	no	—
86	C₇×C₂²⋊C₄	—	4	2	no	—
87	C₄²×C₇	C₂×D₄	1	2	yes	—
88	C₄²×C₇	C₂×D₄	1	2	yes	—
89	C₂²×SD₁₆	—	2	2	no	—
90	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
91	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
92	#500	—	2	4	no	—
93	#461	—	2	4	no	—
94	#461	—	2	4	no	—
95	C₂×#107	—	2	2	no	—
96	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
97	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
98	C₂×#103	—	2	2	no	—
99	C₂×C₂²⋊C₈	—	2	2	no	—
100	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
101	#332	—	2	4	no	—
102	C₂³×C₇	—	2	2	no	—
103	C₂³×C₇	—	4	1	no	—
104	C₂³×C₇	—	4	2	no	—
105	C₂³×C₇	—	4	1	no	—
106	C₂³×C₇	—	4	2	no	—
107	C₂³×C₇	#121	1	1	yes	characteristic
108	C₇×Q₈	—	4	4	no	—
109	C₇×Q₈	—	4	4	no	—
110	C₇×Q₈	—	8	2	no	—
111	C₇×Q₈	—	8	2	no	—
112	C₇×D₄	—	2	4	no	—
113	C₇×D₄	—	4	2	no	—
114	C₂×C₄×C₇	—	2	2	no	—
115	C₂×C₄×C₇	—	4	1	no	—
116	C₂×C₄×C₇	—	4	4	no	—
117	C₂×C₄×C₇	—	4	2	no	—
118	C₂×C₄×C₇	—	8	2	no	—
119	C₂×C₄×C₇	—	8	2	no	—
120	C₂×C₄×C₇	—	8	2	no	—
121	C₂×C₄×C₇	—	8	1	no	—
122	C₂×C₄×C₇	—	8	1	no	—
123	C₂×C₄×C₇	#120	1	2	yes	—
124	C₂×C₄×C₇	#120	1	1	yes	characteristic
125	C₂×C₄×C₇	#121	1	2	yes	—
126	C₂×C₄×C₇	#127	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₇	#135	1	1	yes	characteristic
130	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₇	#137	1	2	yes	—
132	C₇×C₈	—	2	4	no	—
133	C₇×C₈	—	4	4	no	—
134	C₇×C₈	—	8	1	no	—
135	C₂²×Q₈	—	2	2	no	—
136	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
137	C₂³×C₄	—	2	1	no	—
138	C₂×SD₁₆	—	2	8	no	—
139	C₂×SD₁₆	—	4	4	no	—
140	C₂×SD₁₆	—	4	4	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	—	2	1	no	—
143	C₂²×C₈	C₇×D₄	1	2	yes	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	C₇×D₄	1	4	yes	—
148	C₄×D₄	—	2	2	no	—
149	C₄×D₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
153	C₂×C₂²⋊C₄	—	2	1	no	—
154	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
155	#107	—	2	8	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	C₇×D₄	1	4	yes	—
159	#103	—	2	8	no	—
160	C₂²⋊C₈	—	4	4	no	—
161	C₄×C₈	—	2	2	no	—
162	C₂²×C₇	—	4	4	no	—
163	C₂²×C₇	—	4	2	no	—
164	C₂²×C₇	—	8	2	no	—
165	C₂²×C₇	—	8	1	no	—
166	C₂²×C₇	—	8	1	no	—
167	C₂²×C₇	#392	1	1	yes	characteristic
168	C₂²×C₇	#461	1	2	yes	—
169	C₂²×C₇	#466	1	1	yes	characteristic
170	C₂²×C₇	#469	1	1	yes	characteristic
171	C₂²×C₇	#473	1	2	yes	—
172	C₄×C₇	—	2	2	no	—
173	C₄×C₇	—	8	4	no	—
174	C₄×C₇	—	8	1	no	—
175	C₄×C₇	#447	1	2	yes	—
176	C₄×C₇	#448	1	2	yes	—
177	C₄×C₇	#449	1	2	yes	—
178	C₄×C₇	#492	1	1	yes	characteristic
179	C₄×C₇	#497	1	1	yes	characteristic
180	C₂⁴	—	2	1	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	C₇×D₈	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₇×#41	1	1	yes	characteristic
200	SD₁₆	—	4	16	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	1	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	C₇×SD₁₆	1	4	yes	complemented
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₄²	C₂×C₇×D₄	1	2	yes	—
216	C₄²	C₂×C₇×D₄	1	2	yes	—
217	C₂×C₇	—	8	2	no	—
218	C₂×C₇	#1673	1	2	yes	—
219	C₂×C₇	#1693	1	2	yes	—
220	C₂×C₇	#1999	1	1	yes	characteristic
221	C₂×C₇	#2013	1	1	yes	characteristic
222	C₂×C₇	#2020	1	1	yes	characteristic
223	C₂³	—	2	2	no	—
224	C₂³	—	4	1	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	4	1	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₇×#121	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	Q₈	—	4	4	no	—
231	Q₈	—	8	2	no	—
232	Q₈	—	8	2	no	—
233	D₄	—	2	4	no	—
234	D₄	—	4	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	—	8	1	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	C₇×#120	1	2	yes	—
245	C₂×C₄	C₇×#120	1	1	yes	characteristic
246	C₂×C₄	C₇×#121	1	2	yes	—
247	C₂×C₄	C₇×#127	1	1	yes	characteristic
248	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
249	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
250	C₂×C₄	C₇×#135	1	1	yes	characteristic
251	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
252	C₂×C₄	C₇×#137	1	2	yes	—
253	C₈	—	2	4	no	—
254	C₈	—	4	4	no	—
255	C₈	—	8	1	no	—
256	C₇	#10264	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²	—	4	2	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	8	1	no	—
260	C₂²	—	8	1	no	—
261	C₂²	—	8	2	no	—
262	C₂²	C₇×#392	1	1	yes	characteristic
263	C₂²	C₇×#461	1	2	yes	—
264	C₂²	C₇×#466	1	1	yes	characteristic
265	C₂²	C₇×#469	1	1	yes	characteristic
266	C₂²	C₇×#473	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	8	4	no	—
269	C₄	—	8	1	no	—
270	C₄	C₇×#447	1	2	yes	—
271	C₄	C₇×#448	1	2	yes	—
272	C₄	C₇×#449	1	2	yes	—
273	C₄	C₇×#492	1	1	yes	characteristic
274	C₄	C₇×#497	1	1	yes	characteristic
275	C₂	—	8	2	no	—
276	C₂	C₇×#1673	1	2	yes	—
277	C₂	C₇×#1693	1	2	yes	—
278	C₂	C₇×#1999	1	1	yes	characteristic
279	C₂	C₇×#2013	1	1	yes	characteristic
280	C₂	C₇×#2020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂²×C₇×D₄	D₄
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
1. C₂×C₇×#500	C₂	semidirect
4. C₂×C₇×#332	C₂	semidirect
5. C₇×#1277	C₂	semidirect
19. #10264	C₇	direct (N × H)
47. C₂×#500	C₂×C₇	semidirect
50. C₂×#332	C₂×C₇	semidirect
51. #1277	C₂×C₇	semidirect
80. C₂×C₇×C₈	SD₁₆	semidirect
208. C₂×C₈	C₇×SD₁₆	semidirect
256. C₇	#10264	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
C₂×C₇×#461	2
C₇×#1954	2
C₂×C₇×#332	2
C₇×#1277	2
#10264	7

Subgroups — C7×#10264 #13187480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10264 #13187480