Subgroups — C₇×#21975 #13199191

Order 1792.

798 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#423	C₂	1	1	yes	maximal, characteristic
3	C₇×#2283	C₂	1	1	yes	maximal, characteristic
4	C₇×#1939	C₂	1	4	yes	maximal, complemented
5	C₇×#1851	C₂	1	4	yes	maximal
6	C₇×#1843	C₂	1	2	yes	maximal, complemented
7	C₇×#1832	C₂	1	2	yes	maximal, complemented
8	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₇×#431	—	2	4	no	—
11	C₇×#423	—	2	4	no	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₄⋊C₈	C₂²	1	8	yes	—
17	C₂×C₇×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
19	C₇×#383	C₄	1	2	yes	—
20	C₇×#382	C₄	1	2	yes	—
21	C₇×#381	C₄	1	2	yes	—
22	C₇×#380	C₄	1	2	yes	—
23	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₇×#376	C₂²	1	2	yes	—
26	C₇×#336	C₂²	1	8	yes	—
27	#21975	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₂×C₇×M₁₆	—	2	4	no	—
32	C₂×C₇×M₁₆	—	4	4	no	—
33	C₂²×C₇×C₈	C₂³	1	2	yes	—
34	C₂²×C₇×C₈	C₂³	1	4	yes	—
35	C₂²×C₇×C₈	C₂³	1	2	yes	—
36	C₂×C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇	D₄	1	4	yes	—
41	C₂×C₄²×C₇	Q₈	1	4	yes	—
42	C₂×C₄²×C₇	C₂³	1	4	yes	—
43	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
45	C₇×C₄⋊C₈	—	2	4	no	—
46	C₇×C₄⋊C₈	—	2	4	no	—
47	C₇×C₄⋊C₈	—	2	2	no	—
48	C₇×C₄⋊C₈	—	2	2	no	—
49	C₇×C₄⋊C₈	—	4	8	no	—
50	C₇×C₂²⋊C₈	—	2	4	no	—
51	C₄×C₇×C₈	—	2	4	no	—
52	C₇×#95	C₂×C₄	1	4	yes	—
53	C₇×#95	C₂×C₄	1	4	yes	—
54	C₇×#95	C₂×C₄	1	4	yes	—
55	C₇×#95	C₂×C₄	1	4	yes	—
56	C₂×#431	C₂×C₇	1	1	yes	characteristic
57	C₂×#423	C₂×C₇	1	1	yes	characteristic
58	#2283	C₂×C₇	1	1	yes	characteristic
59	#1939	C₂×C₇	1	4	yes	complemented
60	#1851	C₂×C₇	1	4	yes	—
61	#1843	C₂×C₇	1	2	yes	complemented
62	#1832	C₂×C₇	1	2	yes	complemented
63	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	—	2	4	no	—
65	C₂²×C₄×C₇	—	2	2	no	—
66	C₂²×C₄×C₇	—	2	2	no	—
67	C₂²×C₄×C₇	—	4	2	no	—
68	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
79	C₇×M₁₆	—	4	8	no	—
80	C₂×C₇×C₈	—	2	2	no	—
81	C₂×C₇×C₈	—	2	2	no	—
82	C₂×C₇×C₈	—	2	4	no	—
83	C₂×C₇×C₈	—	2	4	no	—
84	C₂×C₇×C₈	—	2	8	no	—
85	C₂×C₇×C₈	—	2	4	no	—
86	C₇×C₄⋊C₄	—	2	8	no	—
87	C₇×C₂²⋊C₄	—	4	4	no	—
88	C₄²×C₇	—	2	4	no	—
89	C₄²×C₇	—	2	4	no	—
90	C₄²×C₇	C₄⋊C₄	1	4	yes	—
91	C₄²×C₇	C₄⋊C₄	1	4	yes	—
92	C₄²×C₇	C₂×D₄	1	2	yes	—
93	C₄²×C₇	C₂×D₄	1	2	yes	—
94	C₄²×C₇	C₂×Q₈	1	2	yes	—
95	C₄²×C₇	C₂×Q₈	1	2	yes	—
96	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
97	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
98	#431	—	2	4	no	—
99	#423	—	2	4	no	—
100	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
101	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
102	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
103	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
104	C₂×C₄⋊C₈	C₂²×C₇	1	8	yes	—
105	C₂×C₂²⋊C₈	C₂²×C₇	1	2	yes	—
106	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
107	#383	C₄×C₇	1	2	yes	—
108	#382	C₄×C₇	1	2	yes	—
109	#381	C₄×C₇	1	2	yes	—
110	#380	C₄×C₇	1	2	yes	—
111	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
112	C₄×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
113	#376	C₂²×C₇	1	2	yes	—
114	#336	C₂²×C₇	1	8	yes	—
115	C₂³×C₇	—	2	2	no	—
116	C₂³×C₇	—	2	2	no	—
117	C₂³×C₇	—	2	2	no	—
118	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₇	—	2	2	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	2	2	no	—
125	C₂×C₄×C₇	—	2	2	no	—
126	C₂×C₄×C₇	—	2	4	no	—
127	C₂×C₄×C₇	—	2	8	no	—
128	C₂×C₄×C₇	—	2	8	no	—
129	C₂×C₄×C₇	—	2	8	no	—
130	C₂×C₄×C₇	—	2	8	no	—
131	C₂×C₄×C₇	—	4	4	no	—
132	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₇	#131	1	4	yes	—
138	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₇	H(5,𝔽₂)	1	2	yes	—
141	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
142	C₇×C₈	—	4	2	no	—
143	C₇×C₈	—	4	4	no	—
144	C₇×C₈	—	4	2	no	—
145	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
146	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
147	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
148	C₂×M₁₆	—	2	4	no	—
149	C₂×M₁₆	—	4	4	no	—
150	C₂²×C₈	C₂³×C₇	1	4	yes	—
151	C₂²×C₈	C₂³×C₇	1	2	yes	—
152	C₂²×C₈	C₂³×C₇	1	2	yes	—
153	C₂×C₄⋊C₄	C₂×C₄×C₇	1	4	yes	—
154	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	4	yes	—
155	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
156	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
157	C₂×C₄²	C₇×D₄	1	4	yes	—
158	C₂×C₄²	C₇×Q₈	1	4	yes	—
159	C₂×C₄²	C₂³×C₇	1	4	yes	—
160	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
162	C₄⋊C₈	—	2	4	no	—
163	C₄⋊C₈	—	2	4	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	4	8	no	—
167	C₂²⋊C₈	—	2	4	no	—
168	C₄×C₈	—	2	4	no	—
169	#95	C₂×C₄×C₇	1	4	yes	—
170	#95	C₂×C₄×C₇	1	4	yes	—
171	#95	C₂×C₄×C₇	1	4	yes	—
172	#95	C₂×C₄×C₇	1	4	yes	—
173	C₂²×C₇	—	2	2	no	—
174	C₂²×C₇	—	2	2	no	—
175	C₂²×C₇	—	2	2	no	—
176	C₂²×C₇	—	4	4	no	—
177	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₇	#520	1	2	yes	—
179	C₂²×C₇	C₂×#131	1	2	yes	—
180	C₂²×C₇	#568	1	2	yes	—
181	C₄×C₇	—	2	2	no	—
182	C₄×C₇	—	2	2	no	—
183	C₄×C₇	—	2	2	no	—
184	C₄×C₇	—	2	2	no	—
185	C₄×C₇	—	4	8	no	—
186	C₄×C₇	#424	1	4	yes	—
187	C₄×C₇	#535	1	1	yes	characteristic
188	C₄×C₇	#536	1	1	yes	characteristic
189	C₄×C₇	#544	1	2	yes	—
190	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
196	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₇	1	8	yes	—
198	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
204	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
205	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
206	M₁₆	—	4	8	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	8	no	—
212	C₂×C₈	—	2	4	no	—
213	C₄⋊C₄	—	2	8	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	2	4	no	—
217	C₄²	C₇×C₄⋊C₄	1	4	yes	—
218	C₄²	C₇×C₄⋊C₄	1	4	yes	—
219	C₄²	C₂×C₇×D₄	1	2	yes	—
220	C₄²	C₂×C₇×D₄	1	2	yes	—
221	C₄²	C₂×C₇×Q₈	1	2	yes	—
222	C₄²	C₂×C₇×Q₈	1	2	yes	—
223	C₂×C₇	—	4	2	no	—
224	C₂×C₇	#2317	1	1	yes	characteristic
225	C₂×C₇	C₂×#424	1	1	yes	characteristic
226	C₂×C₇	#2905	1	1	yes	characteristic
227	C₂×C₇	#2993	1	4	yes	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
232	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
246	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
247	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
248	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
249	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
250	C₂×C₄	C₇×#131	1	4	yes	—
251	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
253	C₂×C₄	C₇×H(5,𝔽₂)	1	2	yes	—
254	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
255	C₈	—	4	2	no	—
256	C₈	—	4	4	no	—
257	C₈	—	4	2	no	—
258	C₇	#21975	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	4	4	no	—
263	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
264	C₂²	C₇×#520	1	2	yes	—
265	C₂²	C₂×C₇×#131	1	2	yes	—
266	C₂²	C₇×#568	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	8	no	—
272	C₄	C₇×#424	1	4	yes	—
273	C₄	C₇×#535	1	1	yes	characteristic
274	C₄	C₇×#536	1	1	yes	characteristic
275	C₄	C₇×#544	1	2	yes	—
276	C₂	—	4	2	no	—
277	C₂	C₇×#2317	1	1	yes	characteristic
278	C₂	C₂×C₇×#424	1	1	yes	characteristic
279	C₂	C₇×#2905	1	1	yes	characteristic
280	C₂	C₇×#2993	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₇×#1939	C₂	semidirect
6. C₇×#1843	C₂	semidirect
7. C₇×#1832	C₂	semidirect
27. #21975	C₇	direct (N × H)
59. #1939	C₂×C₇	semidirect
61. #1843	C₂×C₇	semidirect
62. #1832	C₂×C₇	semidirect
258. C₇	#21975	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×#431	2
C₂×C₇×#423	2
C₇×#2283	2
C₇×#1939	2
C₇×#1851	2
C₇×#1843	2
C₇×#1832	2
#21975	7

Subgroups — C7×#21975 #13199191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21975 #13199191