Subgroups — C₇×#10796 #13188012

Order 1792.

838 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#376	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#336	C₂	1	2	yes	maximal
3	C₂×C₇×#336	C₂	1	2	yes	maximal
4	C₂×C₇×#336	C₂	1	2	yes	maximal
5	C₂³×C₇×C₈	C₂²	1	4	yes	—
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₇×#376	—	2	4	no	—
10	C₂×C₇×#95	C₄	1	4	yes	—
11	C₇×#336	—	2	4	no	—
12	C₇×#336	—	2	4	no	—
13	C₇×#336	C₄	1	8	yes	complemented
14	#10796	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
16	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂²×C₇×C₈	—	2	4	no	—
20	C₂²×C₇×C₈	—	2	4	no	—
21	C₂²×C₇×C₈	—	2	4	no	—
22	C₂²×C₇×C₈	—	2	4	no	—
23	C₂²×C₇×C₈	—	2	4	no	—
24	C₂²×C₇×C₈	—	2	4	no	—
25	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₇	—	2	2	no	—
27	C₂×C₄²×C₇	—	2	2	no	—
28	C₂×C₄²×C₇	—	2	2	no	—
29	C₂×C₄²×C₇	—	2	2	no	—
30	C₂×C₄²×C₇	—	2	2	no	—
31	C₂×C₄²×C₇	—	2	2	no	—
32	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₇	D₄	1	4	yes	—
34	C₂×C₄²×C₇	D₄	1	2	yes	—
35	C₂×C₄²×C₇	Q₈	1	2	yes	—
36	C₇×#95	—	2	8	no	—
37	C₂×#376	C₂×C₇	1	1	yes	characteristic
38	C₂×#336	C₂×C₇	1	2	yes	—
39	C₂×#336	C₂×C₇	1	2	yes	—
40	C₂×#336	C₂×C₇	1	2	yes	—
41	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇	—	2	2	no	—
43	C₂²×C₄×C₇	—	2	2	no	—
44	C₂²×C₄×C₇	—	2	2	no	—
45	C₂²×C₄×C₇	—	2	2	no	—
46	C₂²×C₄×C₇	—	2	2	no	—
47	C₂²×C₄×C₇	—	2	2	no	—
48	C₂²×C₄×C₇	—	2	8	no	—
49	C₂²×C₄×C₇	—	2	8	no	—
50	C₂²×C₄×C₇	—	2	8	no	—
51	C₂²×C₄×C₇	C₄²	1	4	yes	—
52	C₂²×C₄×C₇	C₂²⋊C₄	1	4	yes	—
53	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₇	#41	1	2	yes	—
59	C₂²×C₄×C₇	#41	1	2	yes	—
60	C₂²×C₄×C₇	#41	1	2	yes	—
61	C₂²×C₄×C₇	#41	1	2	yes	—
62	C₂×C₇×C₈	—	2	8	no	—
63	C₂×C₇×C₈	—	2	8	no	—
64	C₂×C₇×C₈	—	2	8	no	—
65	C₂×C₇×C₈	—	4	4	no	—
66	C₂×C₇×C₈	—	4	4	no	—
67	C₂×C₇×C₈	—	4	4	no	—
68	C₂×C₇×C₈	—	4	4	no	—
69	C₄²×C₇	—	2	8	no	—
70	C₄²×C₇	—	2	4	no	—
71	C₄²×C₇	—	2	4	no	—
72	C₄²×C₇	—	2	8	no	—
73	C₂³×C₈	C₂²×C₇	1	4	yes	—
74	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
75	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
76	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
77	#376	—	2	4	no	—
78	C₂×#95	C₄×C₇	1	4	yes	—
79	#336	—	2	4	no	—
80	#336	—	2	4	no	—
81	#336	C₄×C₇	1	8	yes	complemented
82	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
83	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂³×C₇	#117	1	1	yes	characteristic
86	C₂³×C₇	#117	1	1	yes	characteristic
87	C₂³×C₇	#117	1	1	yes	characteristic
88	C₂³×C₇	#117	1	1	yes	characteristic
89	C₂³×C₇	#131	1	8	yes	—
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	—	2	4	no	—
92	C₂×C₄×C₇	—	2	4	no	—
93	C₂×C₄×C₇	—	2	8	no	—
94	C₂×C₄×C₇	—	2	8	no	—
95	C₂×C₄×C₇	—	2	8	no	—
96	C₂×C₄×C₇	—	4	4	no	—
97	C₂×C₄×C₇	—	4	4	no	—
98	C₂×C₄×C₇	—	4	4	no	—
99	C₂×C₄×C₇	—	4	4	no	—
100	C₂×C₄×C₇	—	4	4	no	—
101	C₂×C₄×C₇	—	4	4	no	—
102	C₂×C₄×C₇	#117	1	4	yes	—
103	C₂×C₄×C₇	C₄×D₄	1	4	yes	—
104	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₇	#121	1	2	yes	—
107	C₂×C₄×C₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₇	#123	1	4	yes	—
109	C₂×C₄×C₇	#124	1	2	yes	—
110	C₂×C₄×C₇	#125	1	2	yes	—
111	C₂×C₄×C₇	#126	1	4	yes	—
112	C₇×C₈	—	4	8	no	—
113	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
114	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
115	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
116	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
117	C₂²×C₈	—	2	4	no	—
118	C₂²×C₈	—	2	4	no	—
119	C₂²×C₈	—	2	4	no	—
120	C₂²×C₈	—	2	4	no	—
121	C₂²×C₈	—	2	4	no	—
122	C₂²×C₈	—	2	4	no	—
123	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
124	C₂×C₄²	—	2	2	no	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	—	2	2	no	—
127	C₂×C₄²	—	2	2	no	—
128	C₂×C₄²	—	2	2	no	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
131	C₂×C₄²	C₇×D₄	1	4	yes	—
132	C₂×C₄²	C₇×D₄	1	2	yes	—
133	C₂×C₄²	C₇×Q₈	1	2	yes	—
134	#95	—	2	8	no	—
135	C₂²×C₇	#376	1	1	yes	characteristic
136	C₂²×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₇	#380	1	1	yes	characteristic
139	C₂²×C₇	#381	1	1	yes	characteristic
140	C₂²×C₇	#382	1	1	yes	characteristic
141	C₂²×C₇	#383	1	1	yes	characteristic
142	C₂²×C₇	#405	1	4	yes	—
143	C₂²×C₇	#408	1	4	yes	—
144	C₂²×C₇	#424	1	4	yes	—
145	C₂²×C₇	#433	1	4	yes	—
146	C₂²×C₇	#433	1	4	yes	—
147	C₂²×C₇	#433	1	4	yes	—
148	C₂²×C₇	#433	1	4	yes	—
149	C₄×C₇	—	4	4	no	—
150	C₄×C₇	—	4	4	no	—
151	C₄×C₇	—	4	4	no	—
152	C₄×C₇	#387	1	4	yes	—
153	C₄×C₇	#388	1	4	yes	—
154	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	C₄²×C₇	1	4	yes	—
165	C₂²×C₄	C₇×C₂²⋊C₄	1	4	yes	—
166	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
167	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
168	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
170	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
171	C₂²×C₄	C₇×#41	1	2	yes	—
172	C₂²×C₄	C₇×#41	1	2	yes	—
173	C₂²×C₄	C₇×#41	1	2	yes	—
174	C₂²×C₄	C₇×#41	1	2	yes	—
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	—	2	8	no	—
178	C₂×C₈	—	4	4	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	—	4	4	no	—
182	C₄²	—	2	8	no	—
183	C₄²	—	2	4	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	8	no	—
186	C₂×C₇	#1434	1	1	yes	characteristic
187	C₂×C₇	#1746	1	2	yes	—
188	C₂×C₇	#1750	1	2	yes	—
189	C₂×C₇	#1768	1	2	yes	—
190	C₂×C₇	#1798	1	2	yes	—
191	C₂×C₇	#1818	1	2	yes	—
192	C₂×C₇	#1832	1	2	yes	—
193	C₂×C₇	#1840	1	2	yes	—
194	C₂³	C₂×C₄²×C₇	1	1	yes	characteristic
195	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
196	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
197	C₂³	C₇×#117	1	1	yes	characteristic
198	C₂³	C₇×#117	1	1	yes	characteristic
199	C₂³	C₇×#117	1	1	yes	characteristic
200	C₂³	C₇×#117	1	1	yes	characteristic
201	C₂³	C₇×#131	1	8	yes	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	C₇×#117	1	4	yes	—
215	C₂×C₄	C₄×C₇×D₄	1	4	yes	—
216	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
217	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
218	C₂×C₄	C₇×#121	1	2	yes	—
219	C₂×C₄	C₇×C₂²⋊Q₈	1	2	yes	—
220	C₂×C₄	C₇×#123	1	4	yes	—
221	C₂×C₄	C₇×#124	1	2	yes	—
222	C₂×C₄	C₇×#125	1	2	yes	—
223	C₂×C₄	C₇×#126	1	4	yes	—
224	C₈	—	4	8	no	—
225	C₇	#10796	1	1	yes	characteristic, direct product factor, Sylow
226	C₂²	C₇×#376	1	1	yes	characteristic
227	C₂²	C₄×C₇×C₂²⋊C₄	1	1	yes	characteristic
228	C₂²	C₄×C₇×C₄⋊C₄	1	1	yes	characteristic
229	C₂²	C₇×#380	1	1	yes	characteristic
230	C₂²	C₇×#381	1	1	yes	characteristic
231	C₂²	C₇×#382	1	1	yes	characteristic
232	C₂²	C₇×#383	1	1	yes	characteristic
233	C₂²	C₇×#405	1	4	yes	—
234	C₂²	C₇×#408	1	4	yes	—
235	C₂²	C₇×#424	1	4	yes	—
236	C₂²	C₇×#433	1	4	yes	—
237	C₂²	C₇×#433	1	4	yes	—
238	C₂²	C₇×#433	1	4	yes	—
239	C₂²	C₇×#433	1	4	yes	—
240	C₄	—	4	4	no	—
241	C₄	—	4	4	no	—
242	C₄	—	4	4	no	—
243	C₄	C₇×#387	1	4	yes	—
244	C₄	C₇×#388	1	4	yes	—
245	C₂	C₇×#1434	1	1	yes	characteristic
246	C₂	C₇×#1746	1	2	yes	—
247	C₂	C₇×#1750	1	2	yes	—
248	C₂	C₇×#1768	1	2	yes	—
249	C₂	C₇×#1798	1	2	yes	—
250	C₂	C₇×#1818	1	2	yes	—
251	C₂	C₇×#1832	1	2	yes	—
252	C₂	C₇×#1840	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₇	C₂³
commutator	C₂³	C₂×C₄²×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
socle, prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₇×#336	C₄	semidirect
14. #10796	C₇	direct (N × H)
81. #336	C₄×C₇	semidirect
225. C₇	#10796	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×#376	2
C₂×C₇×#336	2
#10796	7

Subgroups — C7×#10796 #13188012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10796 #13188012