Subgroups — C₇×#21281 #13198497

Order 1792.

766 conjugacy classes of subgroups.

The subgroup lattice has 286 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×#1969	C₂	1	2	yes	maximal
4	C₇×#1942	C₂	1	2	yes	maximal
5	C₇×#1935	C₂	1	4	yes	maximal
6	C₄×C₇×#107	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	2	yes	—
13	C₂×C₇×#107	C₂²	1	1	yes	characteristic
14	C₂×C₇×#103	C₂²	1	4	yes	—
15	C₂×C₇×#103	C₂²	1	2	yes	—
16	C₂×C₇×#97	C₂²	1	1	yes	characteristic
17	C₂×C₇×#97	C₂²	1	2	yes	—
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	#21281	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	—	2	4	no	—
47	C₇×#107	C₂×C₄	1	4	yes	—
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	1	yes	characteristic
59	C₇×#95	D₄	1	1	yes	characteristic
60	C₇×#95	D₄	1	2	yes	—
61	C₂×C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic
62	#2320	C₂×C₇	1	2	yes	—
63	#1969	C₂×C₇	1	2	yes	—
64	#1942	C₂×C₇	1	2	yes	—
65	#1935	C₂×C₇	1	4	yes	—
66	C₄×#107	C₂×C₇	1	1	yes	characteristic
67	#1763	C₂×C₇	1	2	yes	—
68	#1726	C₂×C₇	1	1	yes	characteristic
69	C₂×C₇×Q₈	—	2	2	no	—
70	C₂×C₇×Q₈	—	2	4	no	—
71	C₂×C₇×Q₈	—	2	4	no	—
72	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₇	—	2	2	no	—
74	C₂²×C₄×C₇	—	2	2	no	—
75	C₂²×C₄×C₇	—	2	4	no	—
76	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
81	C₇×Q₁₆	—	2	8	no	—
82	C₂×C₇×C₈	—	2	2	no	—
83	C₂×C₇×C₈	—	2	2	no	—
84	C₂×C₇×C₈	—	2	2	no	—
85	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₇×C₈	#41	1	4	yes	—
88	C₂×C₇×C₈	#41	1	4	yes	—
89	C₇×C₄⋊C₄	—	2	8	no	—
90	C₇×C₄⋊C₄	—	2	2	no	—
91	C₇×C₄⋊C₄	—	2	4	no	—
92	C₇×C₄⋊C₄	—	2	4	no	—
93	C₇×C₄⋊C₄	—	8	4	no	—
94	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
95	C₄²×C₇	—	2	8	no	—
96	C₄²×C₇	—	2	2	no	—
97	C₄²×C₇	—	2	2	no	—
98	C₄²×C₇	—	8	2	no	—
99	C₄²×C₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₇	#41	1	2	yes	—
101	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
102	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
103	C₄×Q₁₆	C₄×C₇	1	8	yes	complemented
104	C₂×#107	C₂²×C₇	1	2	yes	—
105	C₂×#107	C₂²×C₇	1	1	yes	characteristic
106	C₂×#103	C₂²×C₇	1	4	yes	—
107	C₂×#103	C₂²×C₇	1	2	yes	—
108	C₂×#97	C₂²×C₇	1	2	yes	—
109	C₂×#97	C₂²×C₇	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
111	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
112	#391	C₂²×C₇	1	4	yes	—
113	#389	C₂²×C₇	1	4	yes	—
114	#387	—	2	4	no	—
115	#383	—	2	2	no	—
116	#382	—	2	2	no	—
117	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
118	#376	C₂²×C₇	1	1	yes	characteristic
119	#340	C₂²×C₇	1	2	yes	—
120	#340	C₂²×C₇	1	4	yes	—
121	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
122	C₇×Q₈	—	2	4	no	—
123	C₇×Q₈	—	2	4	no	—
124	C₇×Q₈	—	4	4	no	—
125	C₂×C₄×C₇	—	2	1	no	—
126	C₂×C₄×C₇	—	2	2	no	—
127	C₂×C₄×C₇	—	2	1	no	—
128	C₂×C₄×C₇	—	2	4	no	—
129	C₂×C₄×C₇	—	2	4	no	—
130	C₂×C₄×C₇	—	2	2	no	—
131	C₂×C₄×C₇	—	2	2	no	—
132	C₂×C₄×C₇	—	2	2	no	—
133	C₂×C₄×C₇	—	2	2	no	—
134	C₂×C₄×C₇	—	2	2	no	—
135	C₂×C₄×C₇	—	2	2	no	—
136	C₂×C₄×C₇	—	4	4	no	—
137	C₂×C₄×C₇	—	4	2	no	—
138	C₂×C₄×C₇	—	4	2	no	—
139	C₂×C₄×C₇	—	4	4	no	—
140	C₂×C₄×C₇	—	4	4	no	—
141	C₂×C₄×C₇	—	4	4	no	—
142	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
143	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂×#41	1	2	yes	—
146	C₂×C₄×C₇	C₂×#41	1	2	yes	—
147	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
148	C₇×C₈	—	2	2	no	—
149	C₇×C₈	—	2	4	no	—
150	C₇×C₈	#117	1	4	yes	—
151	C₂²×Q₈	C₂³×C₇	1	2	yes	—
152	C₂×Q₁₆	—	2	4	no	—
153	C₂×Q₁₆	C₂×C₄×C₇	1	4	yes	—
154	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₇	1	2	yes	—
156	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
157	C₄×Q₈	—	2	4	no	—
158	C₄×Q₈	C₂×C₄×C₇	1	8	yes	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	4	no	—
161	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
162	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	C₂³×C₇	1	2	yes	—
166	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
167	#107	—	2	4	no	—
168	#107	C₂×C₄×C₇	1	4	yes	—
169	#103	—	2	8	no	—
170	#103	C₂×C₄×C₇	1	8	yes	—
171	#97	—	2	4	no	—
172	#97	—	2	2	no	—
173	C₄×C₈	—	2	4	no	—
174	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
175	#95	—	2	2	no	—
176	#95	—	2	4	no	—
177	#95	—	2	4	no	—
178	#95	—	2	4	no	—
179	#95	C₇×D₄	1	1	yes	characteristic
180	#95	C₇×D₄	1	1	yes	characteristic
181	#95	C₇×D₄	1	2	yes	—
182	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
183	C₂²×C₇	#532	1	2	yes	—
184	C₂²×C₇	#576	1	1	yes	characteristic
185	C₂²×C₇	#578	1	2	yes	—
186	C₂²×C₇	#578	1	1	yes	characteristic
187	C₄×C₇	—	2	2	no	—
188	C₄×C₇	—	2	2	no	—
189	C₄×C₇	—	4	2	no	—
190	C₄×C₇	—	4	2	no	—
191	C₄×C₇	—	4	4	no	—
192	C₄×C₇	—	4	4	no	—
193	C₄×C₇	—	8	4	no	—
194	C₄×C₇	C₂×#117	1	1	yes	characteristic
195	C₄×C₇	C₄×#41	1	1	yes	characteristic
196	C₄×C₇	#529	1	2	yes	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
205	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
206	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
208	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
209	Q₁₆	—	2	8	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
214	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
215	C₂×C₈	C₇×#41	1	4	yes	—
216	C₂×C₈	C₇×#41	1	4	yes	—
217	C₄⋊C₄	—	2	8	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	8	4	no	—
222	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
223	C₄²	—	2	8	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	8	2	no	—
227	C₄²	C₂²×C₄×C₇	1	2	yes	—
228	C₄²	C₇×#41	1	2	yes	—
229	C₂×C₇	#2293	1	1	yes	characteristic
230	C₂×C₇	#2950	1	1	yes	characteristic
231	C₂×C₇	#2951	1	1	yes	characteristic
232	C₂×C₇	#3175	1	2	yes	—
233	C₂×C₇	#3177	1	2	yes	—
234	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
235	Q₈	—	2	4	no	—
236	Q₈	—	2	4	no	—
237	Q₈	—	4	4	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
256	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
257	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
258	C₂×C₄	C₂×C₇×#41	1	2	yes	—
259	C₂×C₄	C₂×C₇×#41	1	2	yes	—
260	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	2	4	no	—
263	C₈	C₇×#117	1	4	yes	—
264	C₇	#21281	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
266	C₂²	C₇×#532	1	2	yes	—
267	C₂²	C₇×#576	1	1	yes	characteristic
268	C₂²	C₇×#578	1	2	yes	—
269	C₂²	C₇×#578	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	4	4	no	—
276	C₄	—	8	4	no	—
277	C₄	C₂×C₇×#117	1	1	yes	characteristic
278	C₄	C₄×C₇×#41	1	1	yes	characteristic
279	C₄	C₇×#529	1	2	yes	—
280	C₂	C₇×#2293	1	1	yes	characteristic
281	C₂	C₇×#2950	1	1	yes	characteristic
282	C₂	C₇×#2951	1	1	yes	characteristic
283	C₂	C₇×#3175	1	2	yes	—
284	C₂	C₇×#3177	1	2	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. #21281	C₇	direct (N × H)
103. C₄×Q₁₆	C₄×C₇	semidirect
264. C₇	#21281	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×#1969	2
C₇×#1942	2
C₇×#1935	2
C₄×C₇×#107	2
C₇×#1763	2
C₇×#1726	2
#21281	7

Subgroups — C7×#21281 #13198497

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21281 #13198497