Subgroups — C₇×#10265 #13187481

Order 1792.

574 conjugacy classes of subgroups.

The subgroup lattice has 256 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	C₂×C₇×#462	C₂	1	2	yes	maximal
3	C₇×#1984	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#332	C₂	1	2	yes	maximal
5	C₇×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₁₆	—	2	2	no	—
7	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₇×#500	—	2	4	no	—
10	C₇×#462	—	2	4	no	—
11	C₇×#462	—	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×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	C₇×#336	—	2	2	no	—
18	C₇×#332	—	2	4	no	—
19	#10265	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂²×C₇×Q₈	—	2	2	no	—
21	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
22	C₂×C₇×Q₁₆	—	2	8	no	—
23	C₂×C₇×Q₁₆	—	4	4	no	—
24	C₂×C₇×Q₁₆	—	4	4	no	—
25	C₂²×C₇×C₈	—	2	2	no	—
26	C₂²×C₇×C₈	—	2	1	no	—
27	C₂²×C₇×C₈	D₄	1	2	yes	—
28	C₇×C₄⋊Q₈	—	2	2	no	—
29	C₇×C₄⋊Q₈	—	2	2	no	—
30	C₇×C₄⋊Q₈	—	2	2	no	—
31	C₇×C₄⋊Q₈	D₄	1	4	yes	—
32	C₄×C₇×Q₈	—	2	2	no	—
33	C₄×C₇×Q₈	—	2	2	no	—
34	C₂×C₇×C₄⋊C₄	—	2	2	no	—
35	C₂×C₇×C₄⋊C₄	—	2	2	no	—
36	C₂×C₇×C₄⋊C₄	—	2	1	no	—
37	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	—	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	C₄×C₇×C₈	—	2	2	no	—
46	C₂×#500	C₂×C₇	1	1	yes	characteristic
47	C₂×#462	C₂×C₇	1	2	yes	—
48	#1984	C₂×C₇	1	1	yes	characteristic
49	C₂×#332	C₂×C₇	1	2	yes	—
50	#1277	C₂×C₇	1	1	yes	characteristic
51	C₂×C₇×Q₈	—	2	1	no	—
52	C₂×C₇×Q₈	—	2	2	no	—
53	C₂×C₇×Q₈	—	2	4	no	—
54	C₂×C₇×Q₈	—	2	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×Q₈	—	4	2	no	—
60	C₂×C₇×Q₈	D₈	1	2	yes	—
61	C₂²×C₄×C₇	—	2	1	no	—
62	C₂²×C₄×C₇	—	2	1	no	—
63	C₂²×C₄×C₇	—	4	2	no	—
64	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	#41	1	1	yes	characteristic
67	C₇×Q₁₆	—	4	16	no	—
68	C₂×C₇×C₈	—	2	2	no	—
69	C₂×C₇×C₈	—	2	2	no	—
70	C₂×C₇×C₈	—	2	4	no	—
71	C₂×C₇×C₈	—	4	2	no	—
72	C₂×C₇×C₈	—	4	2	no	—
73	C₂×C₇×C₈	—	4	1	no	—
74	C₂×C₇×C₈	—	4	2	no	—
75	C₂×C₇×C₈	Q₁₆	1	4	yes	complemented
76	C₇×C₄⋊C₄	—	2	2	no	—
77	C₇×C₄⋊C₄	—	2	4	no	—
78	C₇×C₄⋊C₄	—	2	4	no	—
79	C₇×C₄⋊C₄	—	4	2	no	—
80	C₇×C₄⋊C₄	—	4	2	no	—
81	C₇×C₄⋊C₄	—	4	2	no	—
82	C₄²×C₇	—	4	1	no	—
83	C₄²×C₇	—	4	1	no	—
84	C₄²×C₇	C₂×D₄	1	2	yes	—
85	C₄²×C₇	C₂×D₄	1	2	yes	—
86	C₂²×Q₁₆	—	2	2	no	—
87	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
88	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
89	#500	—	2	4	no	—
90	#462	—	2	4	no	—
91	#462	—	2	4	no	—
92	C₂×#107	—	2	2	no	—
93	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
94	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
95	C₂×#103	—	2	2	no	—
96	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
97	#336	—	2	2	no	—
98	#332	—	2	4	no	—
99	C₂³×C₇	#121	1	1	yes	characteristic
100	C₇×Q₈	—	2	4	no	—
101	C₇×Q₈	—	4	2	no	—
102	C₇×Q₈	—	4	4	no	—
103	C₇×Q₈	—	4	4	no	—
104	C₇×Q₈	—	8	2	no	—
105	C₇×Q₈	—	8	2	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	4	1	no	—
109	C₂×C₄×C₇	—	4	1	no	—
110	C₂×C₄×C₇	—	4	1	no	—
111	C₂×C₄×C₇	—	4	4	no	—
112	C₂×C₄×C₇	—	4	2	no	—
113	C₂×C₄×C₇	—	8	2	no	—
114	C₂×C₄×C₇	—	8	2	no	—
115	C₂×C₄×C₇	#120	1	2	yes	—
116	C₂×C₄×C₇	#120	1	1	yes	characteristic
117	C₂×C₄×C₇	#121	1	2	yes	—
118	C₂×C₄×C₇	#127	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
121	C₂×C₄×C₇	#135	1	1	yes	characteristic
122	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₇	#137	1	2	yes	—
124	C₇×C₈	—	2	4	no	—
125	C₇×C₈	—	4	4	no	—
126	C₇×C₈	—	8	1	no	—
127	C₂²×Q₈	—	2	2	no	—
128	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
129	C₂×Q₁₆	—	2	8	no	—
130	C₂×Q₁₆	—	4	4	no	—
131	C₂×Q₁₆	—	4	4	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	C₇×D₄	1	2	yes	—
135	C₄⋊Q₈	—	2	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	C₇×D₄	1	4	yes	—
139	C₄×Q₈	—	2	2	no	—
140	C₄×Q₈	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	2	1	no	—
144	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
147	#107	—	2	8	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	C₇×D₄	1	4	yes	—
151	#103	—	2	8	no	—
152	C₄×C₈	—	2	2	no	—
153	C₂²×C₇	#392	1	1	yes	characteristic
154	C₂²×C₇	#462	1	2	yes	—
155	C₂²×C₇	#466	1	1	yes	characteristic
156	C₂²×C₇	#469	1	1	yes	characteristic
157	C₂²×C₇	#473	1	2	yes	—
158	C₄×C₇	—	2	2	no	—
159	C₄×C₇	—	4	2	no	—
160	C₄×C₇	—	8	4	no	—
161	C₄×C₇	—	8	1	no	—
162	C₄×C₇	#447	1	2	yes	—
163	C₄×C₇	#449	1	2	yes	—
164	C₄×C₇	#451	1	2	yes	—
165	C₄×C₇	#494	1	1	yes	characteristic
166	C₄×C₇	#497	1	1	yes	characteristic
167	C₂×Q₈	—	2	1	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂×Q₈	C₇×D₈	1	2	yes	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
182	C₂²×C₄	C₇×#41	1	1	yes	characteristic
183	Q₁₆	—	4	16	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	C₇×Q₁₆	1	4	yes	complemented
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	C₂×C₇×D₄	1	2	yes	—
201	C₄²	C₂×C₇×D₄	1	2	yes	—
202	C₂×C₇	#1674	1	2	yes	—
203	C₂×C₇	#1694	1	2	yes	—
204	C₂×C₇	#1999	1	1	yes	characteristic
205	C₂×C₇	#2016	1	1	yes	characteristic
206	C₂×C₇	#2020	1	1	yes	characteristic
207	C₂³	C₇×#121	1	1	yes	characteristic
208	Q₈	—	2	4	no	—
209	Q₈	—	4	2	no	—
210	Q₈	—	4	4	no	—
211	Q₈	—	4	4	no	—
212	Q₈	—	8	2	no	—
213	Q₈	—	8	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	C₇×#120	1	2	yes	—
224	C₂×C₄	C₇×#120	1	1	yes	characteristic
225	C₂×C₄	C₇×#121	1	2	yes	—
226	C₂×C₄	C₇×#127	1	1	yes	characteristic
227	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
228	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
229	C₂×C₄	C₇×#135	1	1	yes	characteristic
230	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
231	C₂×C₄	C₇×#137	1	2	yes	—
232	C₈	—	2	4	no	—
233	C₈	—	4	4	no	—
234	C₈	—	8	1	no	—
235	C₇	#10265	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²	C₇×#392	1	1	yes	characteristic
237	C₂²	C₇×#462	1	2	yes	—
238	C₂²	C₇×#466	1	1	yes	characteristic
239	C₂²	C₇×#469	1	1	yes	characteristic
240	C₂²	C₇×#473	1	2	yes	—
241	C₄	—	2	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	8	4	no	—
244	C₄	—	8	1	no	—
245	C₄	C₇×#447	1	2	yes	—
246	C₄	C₇×#449	1	2	yes	—
247	C₄	C₇×#451	1	2	yes	—
248	C₄	C₇×#494	1	1	yes	characteristic
249	C₄	C₇×#497	1	1	yes	characteristic
250	C₂	C₇×#1674	1	2	yes	—
251	C₂	C₇×#1694	1	2	yes	—
252	C₂	C₇×#1999	1	1	yes	characteristic
253	C₂	C₇×#2016	1	1	yes	characteristic
254	C₂	C₇×#2020	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. #10265	C₇	direct (N × H)
75. C₂×C₇×C₈	Q₁₆	semidirect
191. C₂×C₈	C₇×Q₁₆	semidirect
235. C₇	#10265	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×#462	2
C₇×#1984	2
C₂×C₇×#332	2
C₇×#1277	2
#10265	7

Subgroups — C7×#10265 #13187481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10265 #13187481