Subgroups — C₇×#10279 #13187495

Order 1792.

510 conjugacy classes of subgroups.

The subgroup lattice has 238 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×#477	C₂	1	2	yes	maximal
3	C₇×#1984	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	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₇×#500	—	2	4	no	—
9	C₇×#477	—	2	4	no	—
10	C₇×#477	—	2	4	no	—
11	C₂×C₇×#107	—	2	2	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×#333	—	2	4	no	—
19	#10279	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	C₂²×C₇×C₈	—	2	2	no	—
23	C₂²×C₇×C₈	—	2	1	no	—
24	C₂²×C₇×C₈	D₄	1	2	yes	—
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	C₄×C₇×Q₈	—	2	2	no	—
30	C₄×C₇×Q₈	—	2	2	no	—
31	C₂×C₇×C₄⋊C₄	—	2	2	no	—
32	C₂×C₇×C₄⋊C₄	—	2	2	no	—
33	C₂×C₇×C₄⋊C₄	—	2	1	no	—
34	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	—	2	1	no	—
36	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
37	C₇×#107	—	2	8	no	—
38	C₇×#107	—	2	8	no	—
39	C₇×C₄⋊C₈	—	2	2	no	—
40	C₇×C₄⋊C₈	—	2	2	no	—
41	C₇×C₄⋊C₈	D₄	1	4	yes	—
42	C₇×#103	—	2	8	no	—
43	C₄×C₇×C₈	—	2	2	no	—
44	C₂×#500	C₂×C₇	1	1	yes	characteristic
45	C₂×#477	C₂×C₇	1	2	yes	—
46	#1984	C₂×C₇	1	1	yes	characteristic
47	C₂×#333	C₂×C₇	1	2	yes	—
48	#1277	C₂×C₇	1	1	yes	characteristic
49	C₂×C₇×Q₈	—	2	1	no	—
50	C₂×C₇×Q₈	—	2	2	no	—
51	C₂×C₇×Q₈	—	2	4	no	—
52	C₂×C₇×Q₈	—	2	2	no	—
53	C₂×C₇×Q₈	—	4	2	no	—
54	C₂×C₇×Q₈	—	8	4	no	—
55	C₂×C₇×Q₈	Q₁₆	1	2	yes	—
56	C₂²×C₄×C₇	—	2	1	no	—
57	C₂²×C₄×C₇	—	2	1	no	—
58	C₂²×C₄×C₇	—	4	2	no	—
59	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	#41	1	1	yes	characteristic
62	C₂×C₇×C₈	—	2	2	no	—
63	C₂×C₇×C₈	—	2	2	no	—
64	C₂×C₇×C₈	—	2	4	no	—
65	C₂×C₇×C₈	—	4	2	no	—
66	C₂×C₇×C₈	—	4	2	no	—
67	C₂×C₇×C₈	—	4	1	no	—
68	C₂×C₇×C₈	—	4	2	no	—
69	C₂×C₇×C₈	Q₁₆	1	4	yes	—
70	C₇×C₄⋊C₄	—	2	4	no	—
71	C₇×C₄⋊C₄	—	2	4	no	—
72	C₇×C₄⋊C₄	—	2	2	no	—
73	C₇×C₄⋊C₄	—	4	2	no	—
74	C₇×C₄⋊C₄	—	4	2	no	—
75	C₇×C₄⋊C₄	—	4	2	no	—
76	C₄²×C₇	—	4	1	no	—
77	C₄²×C₇	—	4	1	no	—
78	C₄²×C₇	C₂×D₄	1	2	yes	—
79	C₄²×C₇	C₂×D₄	1	2	yes	—
80	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
81	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
82	#500	—	2	4	no	—
83	#477	—	2	4	no	—
84	#477	—	2	4	no	—
85	C₂×#107	—	2	2	no	—
86	C₂×#107	—	2	2	no	—
87	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
88	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
89	C₂×#103	—	2	2	no	—
90	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
91	#336	—	2	2	no	—
92	#333	—	2	4	no	—
93	C₂³×C₇	#121	1	1	yes	characteristic
94	C₇×Q₈	—	2	4	no	—
95	C₇×Q₈	—	4	2	no	—
96	C₇×Q₈	—	8	4	no	—
97	C₇×Q₈	—	8	4	no	—
98	C₂×C₄×C₇	—	2	2	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	1	no	—
103	C₂×C₄×C₇	—	4	4	no	—
104	C₂×C₄×C₇	—	4	2	no	—
105	C₂×C₄×C₇	—	8	4	no	—
106	C₂×C₄×C₇	#120	1	1	yes	characteristic
107	C₂×C₄×C₇	#120	1	2	yes	—
108	C₂×C₄×C₇	#121	1	2	yes	—
109	C₂×C₄×C₇	#127	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
112	C₂×C₄×C₇	#135	1	1	yes	characteristic
113	C₂×C₄×C₇	#137	1	2	yes	—
114	C₂×C₄×C₇	#137	1	2	yes	—
115	C₇×C₈	—	2	4	no	—
116	C₇×C₈	—	4	4	no	—
117	C₇×C₈	—	8	1	no	—
118	C₂²×Q₈	—	2	2	no	—
119	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	C₇×D₄	1	2	yes	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	C₇×D₄	1	4	yes	—
127	C₄×Q₈	—	2	2	no	—
128	C₄×Q₈	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
133	C₂×C₄²	—	2	1	no	—
134	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
135	#107	—	2	8	no	—
136	#107	—	2	8	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	C₇×D₄	1	4	yes	—
140	#103	—	2	8	no	—
141	C₄×C₈	—	2	2	no	—
142	C₂²×C₇	#392	1	1	yes	characteristic
143	C₂²×C₇	#462	1	2	yes	—
144	C₂²×C₇	#467	1	1	yes	characteristic
145	C₂²×C₇	#470	1	1	yes	characteristic
146	C₂²×C₇	#472	1	2	yes	—
147	C₄×C₇	—	2	2	no	—
148	C₄×C₇	—	4	2	no	—
149	C₄×C₇	—	8	1	no	—
150	C₄×C₇	—	8	4	no	—
151	C₄×C₇	#451	1	2	yes	—
152	C₄×C₇	#451	1	2	yes	—
153	C₄×C₇	#452	1	2	yes	—
154	C₄×C₇	#494	1	1	yes	characteristic
155	C₄×C₇	#497	1	1	yes	characteristic
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	8	4	no	—
162	C₂×Q₈	C₇×Q₁₆	1	2	yes	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	4	2	no	—
166	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
168	C₂²×C₄	C₇×#41	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	C₇×Q₁₆	1	4	yes	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄²	—	4	1	no	—
184	C₄²	—	4	1	no	—
185	C₄²	C₂×C₇×D₄	1	2	yes	—
186	C₄²	C₂×C₇×D₄	1	2	yes	—
187	C₂×C₇	#1676	1	2	yes	—
188	C₂×C₇	#1696	1	2	yes	—
189	C₂×C₇	#2001	1	1	yes	characteristic
190	C₂×C₇	#2016	1	1	yes	characteristic
191	C₂×C₇	#2018	1	1	yes	characteristic
192	C₂³	C₇×#121	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	8	4	no	—
196	Q₈	—	8	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	C₇×#120	1	1	yes	characteristic
206	C₂×C₄	C₇×#120	1	2	yes	—
207	C₂×C₄	C₇×#121	1	2	yes	—
208	C₂×C₄	C₇×#127	1	1	yes	characteristic
209	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
210	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
211	C₂×C₄	C₇×#135	1	1	yes	characteristic
212	C₂×C₄	C₇×#137	1	2	yes	—
213	C₂×C₄	C₇×#137	1	2	yes	—
214	C₈	—	2	4	no	—
215	C₈	—	4	4	no	—
216	C₈	—	8	1	no	—
217	C₇	#10279	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²	C₇×#392	1	1	yes	characteristic
219	C₂²	C₇×#462	1	2	yes	—
220	C₂²	C₇×#467	1	1	yes	characteristic
221	C₂²	C₇×#470	1	1	yes	characteristic
222	C₂²	C₇×#472	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	8	1	no	—
226	C₄	—	8	4	no	—
227	C₄	C₇×#451	1	2	yes	—
228	C₄	C₇×#451	1	2	yes	—
229	C₄	C₇×#452	1	2	yes	—
230	C₄	C₇×#494	1	1	yes	characteristic
231	C₄	C₇×#497	1	1	yes	characteristic
232	C₂	C₇×#1676	1	2	yes	—
233	C₂	C₇×#1696	1	2	yes	—
234	C₂	C₇×#2001	1	1	yes	characteristic
235	C₂	C₇×#2016	1	1	yes	characteristic
236	C₂	C₇×#2018	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. #10279	C₇	direct (N × H)
217. C₇	#10279	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×#477	2
C₇×#1984	2
C₂×C₇×#333	2
C₇×#1277	2
#10279	7

Subgroups — C7×#10279 #13187495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10279 #13187495