Subgroups — C₇×#21283 #13198499

Order 1792.

766 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₇×#2320	C₂	1	2	yes	maximal
3	C₇×#1971	C₂	1	2	yes	maximal
4	C₇×#1943	C₂	1	2	yes	maximal
5	C₇×#1933	C₂	1	4	yes	maximal
6	C₄×C₇×#106	C₂	1	1	yes	maximal, characteristic
7	C₇×#1763	C₂	1	2	yes	maximal
8	C₇×#1726	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
11	C₄×C₇×Q₁₆	C₄	1	8	yes	complemented
12	C₂×C₇×#107	C₂²	1	1	yes	characteristic
13	C₂×C₇×#106	C₂²	1	2	yes	—
14	C₂×C₇×#103	C₂²	1	4	yes	—
15	C₂×C₇×#103	C₂²	1	2	yes	—
16	C₂×C₇×#97	C₂²	1	2	yes	—
17	C₂×C₇×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₇×#391	C₂²	1	4	yes	—
21	C₇×#389	C₂²	1	4	yes	—
22	C₇×#387	—	2	4	no	—
23	C₇×#383	—	2	2	no	—
24	C₇×#382	—	2	2	no	—
25	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
26	C₇×#376	C₂²	1	1	yes	characteristic
27	C₇×#340	C₂²	1	4	yes	—
28	C₇×#340	C₂²	1	2	yes	—
29	#21283	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×Q₈	C₂³	1	2	yes	—
31	C₂×C₇×Q₁₆	—	2	4	no	—
32	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
33	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₇×C₈	C₂³	1	2	yes	—
35	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₄×C₇×Q₈	—	2	4	no	—
37	C₄×C₇×Q₈	C₂×C₄	1	8	yes	—
38	C₂×C₇×C₄⋊C₄	—	2	2	no	—
39	C₂×C₇×C₄⋊C₄	—	2	4	no	—
40	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₄²×C₇	—	2	2	no	—
43	C₂×C₄²×C₇	—	2	2	no	—
44	C₂×C₄²×C₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
46	C₇×#107	C₂×C₄	1	4	yes	—
47	C₇×#106	—	2	4	no	—
48	C₇×#103	—	2	8	no	—
49	C₇×#103	C₂×C₄	1	8	yes	—
50	C₇×#97	—	2	4	no	—
51	C₇×#97	—	2	2	no	—
52	C₄×C₇×C₈	—	2	4	no	—
53	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
54	C₇×#95	—	2	2	no	—
55	C₇×#95	—	2	4	no	—
56	C₇×#95	—	2	4	no	—
57	C₇×#95	—	2	4	no	—
58	C₇×#95	D₄	1	2	yes	—
59	C₇×#95	D₄	1	2	yes	—
60	C₂×C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic
61	#2320	C₂×C₇	1	2	yes	—
62	#1971	C₂×C₇	1	2	yes	—
63	#1943	C₂×C₇	1	2	yes	—
64	#1933	C₂×C₇	1	4	yes	—
65	C₄×#106	C₂×C₇	1	1	yes	characteristic
66	#1763	C₂×C₇	1	2	yes	—
67	#1726	C₂×C₇	1	1	yes	characteristic
68	C₂×C₇×Q₈	—	2	2	no	—
69	C₂×C₇×Q₈	—	2	4	no	—
70	C₂×C₇×Q₈	—	2	4	no	—
71	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₇	—	2	2	no	—
73	C₂²×C₄×C₇	—	2	2	no	—
74	C₂²×C₄×C₇	—	2	4	no	—
75	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
79	C₇×Q₁₆	—	2	8	no	—
80	C₂×C₇×C₈	—	2	2	no	—
81	C₂×C₇×C₈	—	2	2	no	—
82	C₂×C₇×C₈	—	2	2	no	—
83	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₇×C₈	#41	1	4	yes	—
86	C₂×C₇×C₈	#41	1	4	yes	—
87	C₇×C₄⋊C₄	—	2	8	no	—
88	C₇×C₄⋊C₄	—	2	2	no	—
89	C₇×C₄⋊C₄	—	2	4	no	—
90	C₇×C₄⋊C₄	—	2	4	no	—
91	C₇×C₄⋊C₄	—	8	4	no	—
92	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
93	C₄²×C₇	—	2	8	no	—
94	C₄²×C₇	—	2	2	no	—
95	C₄²×C₇	—	2	2	no	—
96	C₄²×C₇	—	8	2	no	—
97	C₄²×C₇	C₂²×C₄	1	2	yes	—
98	C₄²×C₇	#41	1	2	yes	—
99	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
100	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
101	C₄×Q₁₆	C₄×C₇	1	8	yes	complemented
102	C₂×#107	C₂²×C₇	1	1	yes	characteristic
103	C₂×#106	C₂²×C₇	1	2	yes	—
104	C₂×#103	C₂²×C₇	1	4	yes	—
105	C₂×#103	C₂²×C₇	1	2	yes	—
106	C₂×#97	C₂²×C₇	1	2	yes	—
107	C₂×#97	C₂²×C₇	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
109	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
110	#391	C₂²×C₇	1	4	yes	—
111	#389	C₂²×C₇	1	4	yes	—
112	#387	—	2	4	no	—
113	#383	—	2	2	no	—
114	#382	—	2	2	no	—
115	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
116	#376	C₂²×C₇	1	1	yes	characteristic
117	#340	C₂²×C₇	1	2	yes	—
118	#340	C₂²×C₇	1	4	yes	—
119	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
120	C₇×Q₈	—	2	4	no	—
121	C₇×Q₈	—	2	4	no	—
122	C₇×Q₈	—	4	4	no	—
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	2	4	no	—
126	C₂×C₄×C₇	—	2	4	no	—
127	C₂×C₄×C₇	—	2	4	no	—
128	C₂×C₄×C₇	—	2	4	no	—
129	C₂×C₄×C₇	—	2	4	no	—
130	C₂×C₄×C₇	—	4	4	no	—
131	C₂×C₄×C₇	—	4	2	no	—
132	C₂×C₄×C₇	—	4	2	no	—
133	C₂×C₄×C₇	—	4	4	no	—
134	C₂×C₄×C₇	—	4	4	no	—
135	C₂×C₄×C₇	—	4	4	no	—
136	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂×#41	1	2	yes	—
140	C₂×C₄×C₇	C₂×#41	1	2	yes	—
141	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
142	C₇×C₈	—	2	2	no	—
143	C₇×C₈	—	2	4	no	—
144	C₇×C₈	#117	1	4	yes	—
145	C₂²×Q₈	C₂³×C₇	1	2	yes	—
146	C₂×Q₁₆	—	2	4	no	—
147	C₂×Q₁₆	C₂×C₄×C₇	1	4	yes	—
148	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₇	1	2	yes	—
150	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
151	C₄×Q₈	—	2	4	no	—
152	C₄×Q₈	C₂×C₄×C₇	1	8	yes	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
156	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	C₂³×C₇	1	2	yes	—
160	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
161	#107	C₂×C₄×C₇	1	4	yes	—
162	#106	—	2	4	no	—
163	#103	—	2	8	no	—
164	#103	C₂×C₄×C₇	1	8	yes	—
165	#97	—	2	4	no	—
166	#97	—	2	2	no	—
167	C₄×C₈	—	2	4	no	—
168	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
169	#95	—	2	2	no	—
170	#95	—	2	4	no	—
171	#95	—	2	4	no	—
172	#95	—	2	4	no	—
173	#95	C₇×D₄	1	2	yes	—
174	#95	C₇×D₄	1	2	yes	—
175	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂²×C₇	#532	1	2	yes	—
177	C₂²×C₇	#577	1	2	yes	—
178	C₂²×C₇	#578	1	2	yes	—
179	C₄×C₇	—	2	2	no	—
180	C₄×C₇	—	2	2	no	—
181	C₄×C₇	—	4	4	no	—
182	C₄×C₇	—	4	4	no	—
183	C₄×C₇	—	4	4	no	—
184	C₄×C₇	—	8	4	no	—
185	C₄×C₇	C₂×#117	1	1	yes	characteristic
186	C₄×C₇	C₄×#41	1	1	yes	characteristic
187	C₄×C₇	#529	1	2	yes	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	4	no	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
196	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
199	Q₁₆	—	2	8	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
204	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
205	C₂×C₈	C₇×#41	1	4	yes	—
206	C₂×C₈	C₇×#41	1	4	yes	—
207	C₄⋊C₄	—	2	8	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	8	4	no	—
212	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
213	C₄²	—	2	8	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	8	2	no	—
217	C₄²	C₂²×C₄×C₇	1	2	yes	—
218	C₄²	C₇×#41	1	2	yes	—
219	C₂×C₇	#2293	1	1	yes	characteristic
220	C₂×C₇	#2950	1	1	yes	characteristic
221	C₂×C₇	#2952	1	1	yes	characteristic
222	C₂×C₇	#3178	1	4	yes	—
223	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	2	4	no	—
226	Q₈	—	4	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
241	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
242	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
243	C₂×C₄	C₂×C₇×#41	1	2	yes	—
244	C₂×C₄	C₂×C₇×#41	1	2	yes	—
245	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	2	4	no	—
248	C₈	C₇×#117	1	4	yes	—
249	C₇	#21283	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
251	C₂²	C₇×#532	1	2	yes	—
252	C₂²	C₇×#577	1	2	yes	—
253	C₂²	C₇×#578	1	2	yes	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	4	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	4	no	—
259	C₄	—	8	4	no	—
260	C₄	C₂×C₇×#117	1	1	yes	characteristic
261	C₄	C₄×C₇×#41	1	1	yes	characteristic
262	C₄	C₇×#529	1	2	yes	—
263	C₂	C₇×#2293	1	1	yes	characteristic
264	C₂	C₇×#2950	1	1	yes	characteristic
265	C₂	C₇×#2952	1	1	yes	characteristic
266	C₂	C₇×#3178	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₇×#376 ◁ G

Complements

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

N	H ≅ G/N	type
11. C₄×C₇×Q₁₆	C₄	semidirect
29. #21283	C₇	direct (N × H)
101. C₄×Q₁₆	C₄×C₇	semidirect
249. C₇	#21283	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×C₄×C₇×Q₁₆	2
C₇×#2320	2
C₇×#1971	2
C₇×#1943	2
C₇×#1933	2
C₄×C₇×#106	2
C₇×#1763	2
C₇×#1726	2
#21283	7

Subgroups — C7×#21283 #13198499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21283 #13198499