Subgroups — C₇×#10278 #13187494

Order 1792.

562 conjugacy classes of subgroups.

The subgroup lattice has 264 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×#476	C₂	1	2	yes	maximal
3	C₇×#1954	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#333	C₂	1	2	yes	maximal, complemented
5	C₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×C₄×C₇×D₄	C₂²	1	1	yes	characteristic
8	C₇×#500	—	2	4	no	—
9	C₇×#476	—	2	4	no	—
10	C₇×#476	—	2	4	no	—
11	C₂×C₇×#107	—	2	2	no	—
12	C₂×C₇×#106	—	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×#102	—	2	2	no	—
16	C₂×C₇×C₂²⋊C₈	—	2	2	no	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₇×#333	—	2	4	no	—
19	#10278	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	C₂²×C₇×C₈	—	2	2	no	—
24	C₂²×C₇×C₈	—	2	1	no	—
25	C₂²×C₇×C₈	D₄	1	2	yes	—
26	C₇×C₄⋊Q₈	—	2	2	no	—
27	C₇×C₄⋊Q₈	—	2	2	no	—
28	C₇×C₄⋊Q₈	—	2	2	no	—
29	C₇×C₄⋊Q₈	D₄	1	4	yes	—
30	C₄×C₇×D₄	—	2	2	no	—
31	C₄×C₇×D₄	—	2	2	no	—
32	C₂×C₇×C₄⋊C₄	—	2	2	no	—
33	C₂×C₇×C₄⋊C₄	—	2	2	no	—
34	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
35	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
36	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
37	C₇×#107	—	2	8	no	—
38	C₇×#106	—	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×#102	—	2	8	no	—
43	C₇×C₂²⋊C₈	—	4	4	no	—
44	C₄×C₇×C₈	—	2	2	no	—
45	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
46	C₂×#476	C₂×C₇	1	2	yes	—
47	#1954	C₂×C₇	1	1	yes	characteristic
48	C₂×#333	C₂×C₇	1	2	yes	complemented
49	#1277	C₂×C₇	1	1	yes	characteristic, complemented
50	C₂⁴×C₇	—	2	1	no	—
51	C₂×C₇×Q₈	—	2	4	no	—
52	C₂×C₇×Q₈	—	4	2	no	—
53	C₂×C₇×Q₈	—	8	4	no	—
54	C₂×C₇×D₄	—	2	2	no	—
55	C₂×C₇×D₄	—	2	1	no	—
56	C₂×C₇×D₄	—	2	2	no	—
57	C₂×C₇×D₄	Q₁₆	1	2	yes	—
58	C₂²×C₄×C₇	—	2	1	no	—
59	C₂²×C₄×C₇	—	4	2	no	—
60	C₂²×C₄×C₇	—	4	1	no	—
61	C₂²×C₄×C₇	—	4	1	no	—
62	C₂²×C₄×C₇	—	4	2	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	C₂×C₇×C₈	—	2	2	no	—
68	C₂×C₇×C₈	—	2	2	no	—
69	C₂×C₇×C₈	—	2	4	no	—
70	C₂×C₇×C₈	—	4	2	no	—
71	C₂×C₇×C₈	—	4	2	no	—
72	C₂×C₇×C₈	—	4	1	no	—
73	C₂×C₇×C₈	—	4	2	no	—
74	C₂×C₇×C₈	SD₁₆	1	4	yes	—
75	C₇×C₄⋊C₄	—	2	4	no	—
76	C₇×C₄⋊C₄	—	2	2	no	—
77	C₇×C₄⋊C₄	—	2	4	no	—
78	C₇×C₄⋊C₄	—	4	2	no	—
79	C₇×C₄⋊C₄	—	4	2	no	—
80	C₇×C₂²⋊C₄	—	4	2	no	—
81	C₄²×C₇	C₂×D₄	1	2	yes	—
82	C₄²×C₇	C₂×D₄	1	2	yes	—
83	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
84	C₂×C₄×D₄	C₂²×C₇	1	1	yes	characteristic
85	#500	—	2	4	no	—
86	#476	—	2	4	no	—
87	#476	—	2	4	no	—
88	C₂×#107	—	2	2	no	—
89	C₂×#106	—	2	2	no	—
90	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
91	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
92	C₂×#102	—	2	2	no	—
93	C₂×C₂²⋊C₈	—	2	2	no	—
94	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
95	#333	—	2	4	no	—
96	C₂³×C₇	—	2	2	no	—
97	C₂³×C₇	—	4	1	no	—
98	C₂³×C₇	—	4	2	no	—
99	C₂³×C₇	—	4	1	no	—
100	C₂³×C₇	—	4	2	no	—
101	C₂³×C₇	#121	1	1	yes	characteristic
102	C₇×Q₈	—	8	4	no	—
103	C₇×Q₈	—	8	4	no	—
104	C₇×D₄	—	2	4	no	—
105	C₇×D₄	—	4	2	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	4	1	no	—
108	C₂×C₄×C₇	—	4	4	no	—
109	C₂×C₄×C₇	—	4	2	no	—
110	C₂×C₄×C₇	—	8	4	no	—
111	C₂×C₄×C₇	—	8	2	no	—
112	C₂×C₄×C₇	—	8	1	no	—
113	C₂×C₄×C₇	—	8	1	no	—
114	C₂×C₄×C₇	#120	1	1	yes	characteristic
115	C₂×C₄×C₇	#120	1	2	yes	—
116	C₂×C₄×C₇	#121	1	2	yes	—
117	C₂×C₄×C₇	#127	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇	#135	1	1	yes	characteristic
121	C₂×C₄×C₇	#137	1	2	yes	—
122	C₂×C₄×C₇	#137	1	2	yes	—
123	C₇×C₈	—	2	4	no	—
124	C₇×C₈	—	4	4	no	—
125	C₇×C₈	—	8	1	no	—
126	C₂²×Q₈	—	2	2	no	—
127	C₂²×D₄	C₇×D₄	1	1	yes	characteristic
128	C₂³×C₄	—	2	1	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	2	1	no	—
131	C₂²×C₈	C₇×D₄	1	2	yes	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₄⋊Q₈	C₇×D₄	1	4	yes	—
136	C₄×D₄	—	2	2	no	—
137	C₄×D₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	—	2	1	no	—
142	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
143	#107	—	2	8	no	—
144	#106	—	2	8	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	C₇×D₄	1	4	yes	—
148	#102	—	2	8	no	—
149	C₂²⋊C₈	—	4	4	no	—
150	C₄×C₈	—	2	2	no	—
151	C₂²×C₇	—	4	4	no	—
152	C₂²×C₇	—	4	2	no	—
153	C₂²×C₇	—	8	2	no	—
154	C₂²×C₇	—	8	1	no	—
155	C₂²×C₇	—	8	1	no	—
156	C₂²×C₇	#392	1	1	yes	characteristic
157	C₂²×C₇	#461	1	2	yes	—
158	C₂²×C₇	#467	1	1	yes	characteristic
159	C₂²×C₇	#470	1	1	yes	characteristic
160	C₂²×C₇	#472	1	2	yes	—
161	C₄×C₇	—	2	2	no	—
162	C₄×C₇	—	8	1	no	—
163	C₄×C₇	—	8	4	no	—
164	C₄×C₇	#448	1	2	yes	—
165	C₄×C₇	#451	1	2	yes	—
166	C₄×C₇	#452	1	2	yes	—
167	C₄×C₇	#492	1	1	yes	characteristic
168	C₄×C₇	#497	1	1	yes	characteristic
169	C₂⁴	—	2	1	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	8	4	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	C₇×Q₁₆	1	2	yes	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₇×#41	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	C₇×SD₁₆	1	4	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	4	2	no	—
200	C₄²	C₂×C₇×D₄	1	2	yes	—
201	C₄²	C₂×C₇×D₄	1	2	yes	—
202	C₂×C₇	—	8	2	no	—
203	C₂×C₇	#1675	1	2	yes	—
204	C₂×C₇	#1695	1	2	yes	—
205	C₂×C₇	#2001	1	1	yes	characteristic
206	C₂×C₇	#2013	1	1	yes	characteristic
207	C₂×C₇	#2018	1	1	yes	characteristic
208	C₂³	—	2	2	no	—
209	C₂³	—	4	1	no	—
210	C₂³	—	4	2	no	—
211	C₂³	—	4	1	no	—
212	C₂³	—	4	2	no	—
213	C₂³	C₇×#121	1	1	yes	characteristic
214	Q₈	—	8	4	no	—
215	Q₈	—	8	4	no	—
216	D₄	—	2	4	no	—
217	D₄	—	4	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	4	no	—
223	C₂×C₄	—	8	1	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₇×#120	1	1	yes	characteristic
227	C₂×C₄	C₇×#120	1	2	yes	—
228	C₂×C₄	C₇×#121	1	2	yes	—
229	C₂×C₄	C₇×#127	1	1	yes	characteristic
230	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
231	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
232	C₂×C₄	C₇×#135	1	1	yes	characteristic
233	C₂×C₄	C₇×#137	1	2	yes	—
234	C₂×C₄	C₇×#137	1	2	yes	—
235	C₈	—	2	4	no	—
236	C₈	—	4	4	no	—
237	C₈	—	8	1	no	—
238	C₇	#10278	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	4	2	no	—
240	C₂²	—	4	4	no	—
241	C₂²	—	8	1	no	—
242	C₂²	—	8	1	no	—
243	C₂²	—	8	2	no	—
244	C₂²	C₇×#392	1	1	yes	characteristic
245	C₂²	C₇×#461	1	2	yes	—
246	C₂²	C₇×#467	1	1	yes	characteristic
247	C₂²	C₇×#470	1	1	yes	characteristic
248	C₂²	C₇×#472	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	8	1	no	—
251	C₄	—	8	4	no	—
252	C₄	C₇×#448	1	2	yes	—
253	C₄	C₇×#451	1	2	yes	—
254	C₄	C₇×#452	1	2	yes	—
255	C₄	C₇×#492	1	1	yes	characteristic
256	C₄	C₇×#497	1	1	yes	characteristic
257	C₂	—	8	2	no	—
258	C₂	C₇×#1675	1	2	yes	—
259	C₂	C₇×#1695	1	2	yes	—
260	C₂	C₇×#2001	1	1	yes	characteristic
261	C₂	C₇×#2013	1	1	yes	characteristic
262	C₂	C₇×#2018	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×#333	C₂	semidirect
5. C₇×#1277	C₂	semidirect
19. #10278	C₇	direct (N × H)
45. C₂×#500	C₂×C₇	semidirect
48. C₂×#333	C₂×C₇	semidirect
49. #1277	C₂×C₇	semidirect
238. C₇	#10278	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×#476	2
C₇×#1954	2
C₂×C₇×#333	2
C₇×#1277	2
#10278	7

Subgroups — C7×#10278 #13187494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10278 #13187494