Subgroups — C₇×#21293 #13198509

Order 1792.

746 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	C₄×C₇×#135	C₂	1	1	yes	maximal, characteristic
2	C₇×#2922	C₂	1	2	yes	maximal, complemented
3	C₇×#1975	C₂	1	2	yes	maximal
4	C₇×#1946	C₂	1	2	yes	maximal, complemented
5	C₇×#1938	C₂	1	4	yes	maximal, complemented
6	C₄×C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₇×#104	C₂	1	2	yes	maximal, complemented
8	C₄×C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×#135	C₂²	1	1	yes	characteristic
10	C₄×C₇×#41	C₂²	1	2	yes	—
11	C₄×C₇×Q₁₆	C₄	1	2	yes	complemented
12	C₄×C₇×SD₁₆	C₄	1	4	yes	complemented
13	C₄×C₇×D₈	C₄	1	2	yes	complemented
14	C₇×#433	—	2	4	no	—
15	C₇×#431	—	2	4	no	—
16	C₂×C₇×#108	C₂²	1	2	yes	—
17	C₇×#427	C₂²	1	1	yes	characteristic
18	C₇×#424	C₂²	1	4	yes	—
19	C₂×C₇×#104	C₂²	1	4	yes	—
20	C₇×#416	C₂²	1	2	yes	—
21	C₇×#408	C₂²	1	4	yes	—
22	C₄×C₇×M₁₆	C₂²	1	2	yes	—
23	C₂×C₇×#97	C₂²	1	2	yes	—
24	C₂×C₇×#97	C₂²	1	2	yes	—
25	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
27	C₄³×C₇	C₂²	1	1	yes	characteristic
28	C₇×#341	C₂²	1	2	yes	—
29	C₇×#339	C₂²	1	4	yes	—
30	#21293	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂×C₇×#41	C₂³	1	2	yes	—
32	C₇×#135	—	2	4	no	—
33	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₇×D₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₇×M₁₆	C₂³	1	4	yes	—
37	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₇×C₈	C₂³	1	2	yes	—
39	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
41	C₄×C₇×D₄	—	2	4	no	—
42	C₄×C₇×D₄	C₂×C₄	1	4	yes	—
43	C₇×#117	—	2	2	no	—
44	C₇×#117	C₂³	1	2	yes	—
45	C₂×C₄²×C₇	—	2	2	no	—
46	C₂×C₄²×C₇	—	2	2	no	—
47	C₂×C₄²×C₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	C₇×#108	—	2	4	no	—
50	C₇×#107	C₂×C₄	1	2	yes	—
51	C₇×#106	C₂×C₄	1	2	yes	—
52	C₇×C₄⋊C₈	—	2	8	no	—
53	C₇×#104	—	2	8	no	—
54	C₇×#103	C₂×C₄	1	4	yes	—
55	C₇×#102	C₂×C₄	1	4	yes	—
56	C₇×C₂²⋊C₈	—	2	8	no	—
57	C₇×#97	—	2	4	no	—
58	C₇×#97	—	2	2	no	—
59	C₇×#97	—	2	4	no	—
60	C₄×C₇×C₈	—	2	2	no	—
61	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
62	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
63	C₄×C₇×C₈	D₄	1	2	yes	—
64	C₄×C₇×C₈	D₄	1	2	yes	—
65	C₄×#135	C₂×C₇	1	1	yes	characteristic
66	#2922	C₂×C₇	1	2	yes	complemented
67	#1975	C₂×C₇	1	2	yes	—
68	#1946	C₂×C₇	1	2	yes	complemented
69	#1938	C₂×C₇	1	4	yes	complemented
70	C₄×#108	C₂×C₇	1	1	yes	characteristic, complemented
71	C₄×#104	C₂×C₇	1	2	yes	complemented
72	C₄×#97	C₂×C₇	1	1	yes	characteristic, complemented
73	C₇×#41	—	2	4	no	—
74	C₇×#41	—	2	4	no	—
75	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
76	C₂×C₇×D₄	—	2	2	no	—
77	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₇	—	2	2	no	—
79	C₂²×C₄×C₇	—	2	4	no	—
80	C₂²×C₄×C₇	—	2	2	no	—
81	C₂²×C₄×C₇	#41	1	2	yes	—
82	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
83	C₇×Q₁₆	—	2	2	no	—
84	C₇×SD₁₆	—	2	4	no	—
85	C₇×D₈	—	2	2	no	—
86	C₇×M₁₆	—	2	4	no	—
87	C₇×M₁₆	—	2	4	no	—
88	C₂×C₇×C₈	—	2	2	no	—
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	—	2	2	no	—
91	C₂×C₇×C₈	—	2	4	no	—
92	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
96	C₂×C₇×C₈	#41	1	4	yes	—
97	C₇×C₄⋊C₄	—	2	4	no	—
98	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
99	C₇×C₂²⋊C₄	—	2	2	no	—
100	C₇×C₂²⋊C₄	—	2	4	no	—
101	C₄²×C₇	—	2	4	no	—
102	C₄²×C₇	—	2	2	no	—
103	C₄²×C₇	—	2	2	no	—
104	C₄²×C₇	—	4	4	no	—
105	C₄²×C₇	C₂²×C₄	1	2	yes	—
106	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₇	#41	1	4	yes	—
109	C₂×#135	C₂²×C₇	1	1	yes	characteristic
110	C₄×#41	C₂²×C₇	1	2	yes	—
111	C₄×Q₁₆	C₄×C₇	1	2	yes	complemented
112	C₄×SD₁₆	C₄×C₇	1	4	yes	complemented
113	C₄×D₈	C₄×C₇	1	2	yes	complemented
114	#433	—	2	4	no	—
115	#431	—	2	4	no	—
116	C₂×#108	C₂²×C₇	1	2	yes	—
117	#427	C₂²×C₇	1	1	yes	characteristic
118	#424	C₂²×C₇	1	4	yes	—
119	C₂×#104	C₂²×C₇	1	4	yes	—
120	#416	C₂²×C₇	1	2	yes	—
121	#408	C₂²×C₇	1	4	yes	—
122	C₄×M₁₆	C₂²×C₇	1	2	yes	—
123	C₂×#97	C₂²×C₇	1	2	yes	—
124	C₂×#97	C₂²×C₇	1	2	yes	—
125	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
127	C₄³	C₂²×C₇	1	1	yes	characteristic
128	#341	C₂²×C₇	1	2	yes	—
129	#339	C₂²×C₇	1	4	yes	—
130	C₂³×C₇	—	2	2	no	—
131	C₂³×C₇	C₂×#41	1	1	yes	characteristic
132	C₇×Q₈	—	2	2	no	—
133	C₇×Q₈	—	2	2	no	—
134	C₇×D₄	—	2	2	no	—
135	C₇×D₄	—	2	2	no	—
136	C₇×D₄	—	4	4	no	—
137	C₂×C₄×C₇	—	2	4	no	—
138	C₂×C₄×C₇	—	2	4	no	—
139	C₂×C₄×C₇	—	2	2	no	—
140	C₂×C₄×C₇	—	2	4	no	—
141	C₂×C₄×C₇	—	4	4	no	—
142	C₂×C₄×C₇	—	4	4	no	—
143	C₂×C₄×C₇	—	4	8	no	—
144	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
145	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
146	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂×#41	1	2	yes	—
149	C₂×C₄×C₇	C₂×#41	1	2	yes	—
150	C₇×C₈	—	2	2	no	—
151	C₇×C₈	—	2	4	no	—
152	C₇×C₈	—	4	4	no	—
153	C₇×C₈	#117	1	2	yes	—
154	C₇×C₈	#117	1	2	yes	—
155	C₂×#41	C₂³×C₇	1	2	yes	—
156	#135	—	2	4	no	—
157	C₂×Q₁₆	C₂×C₄×C₇	1	1	yes	characteristic
158	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
159	C₂×D₈	C₂×C₄×C₇	1	1	yes	characteristic
160	C₂×M₁₆	C₂³×C₇	1	4	yes	—
161	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
162	C₂²×C₈	C₂³×C₇	1	2	yes	—
163	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
164	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
165	C₄×D₄	—	2	4	no	—
166	C₄×D₄	C₂×C₄×C₇	1	4	yes	—
167	#117	—	2	2	no	—
168	#117	C₂³×C₇	1	2	yes	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	—	2	2	no	—
171	C₂×C₄²	C₂³×C₇	1	2	yes	—
172	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
173	#108	—	2	4	no	—
174	#107	C₂×C₄×C₇	1	2	yes	—
175	#106	C₂×C₄×C₇	1	2	yes	—
176	C₄⋊C₈	—	2	8	no	—
177	#104	—	2	8	no	—
178	#103	C₂×C₄×C₇	1	4	yes	—
179	#102	C₂×C₄×C₇	1	4	yes	—
180	C₂²⋊C₈	—	2	8	no	—
181	#97	—	2	4	no	—
182	#97	—	2	2	no	—
183	#97	—	2	4	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
186	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
187	C₄×C₈	C₇×D₄	1	2	yes	—
188	C₄×C₈	C₇×D₄	1	2	yes	—
189	C₂²×C₇	—	2	2	no	—
190	C₂²×C₇	—	4	4	no	—
191	C₂²×C₇	—	4	2	no	—
192	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
193	C₂²×C₇	#529	1	2	yes	—
194	C₄×C₇	—	2	2	no	—
195	C₄×C₇	—	4	4	no	—
196	C₄×C₇	—	4	4	no	—
197	C₄×C₇	—	4	2	no	—
198	C₄×C₇	#444	1	4	yes	—
199	C₄×C₇	C₂×#117	1	1	yes	characteristic
200	C₄×C₇	C₄×#41	1	1	yes	characteristic
201	C₄×C₇	#532	1	2	yes	—
202	#41	—	2	4	no	—
203	#41	—	2	4	no	—
204	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	C₇×#41	1	2	yes	—
211	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
212	Q₁₆	—	2	2	no	—
213	SD₁₆	—	2	4	no	—
214	D₈	—	2	2	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
222	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
223	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
224	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
225	C₂×C₈	C₇×#41	1	4	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₄²	—	2	4	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₂²×C₄×C₇	1	2	yes	—
235	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
236	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
237	C₄²	C₇×#41	1	4	yes	—
238	C₂×C₇	—	2	2	no	—
239	C₂×C₇	—	8	2	no	—
240	C₂×C₇	#2293	1	1	yes	characteristic
241	C₂×C₇	C₂×#444	1	2	yes	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₇×#41	1	1	yes	characteristic
244	Q₈	—	2	2	no	—
245	Q₈	—	2	2	no	—
246	D₄	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	4	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
257	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
258	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
260	C₂×C₄	C₂×C₇×#41	1	2	yes	—
261	C₂×C₄	C₂×C₇×#41	1	2	yes	—
262	C₈	—	2	2	no	—
263	C₈	—	2	4	no	—
264	C₈	—	4	4	no	—
265	C₈	C₇×#117	1	2	yes	—
266	C₈	C₇×#117	1	2	yes	—
267	C₇	#21293	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	—	2	2	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	2	no	—
271	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
272	C₂²	C₇×#529	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	4	4	no	—
276	C₄	—	4	2	no	—
277	C₄	C₇×#444	1	4	yes	—
278	C₄	C₂×C₇×#117	1	1	yes	characteristic
279	C₄	C₄×C₇×#41	1	1	yes	characteristic
280	C₄	C₇×#532	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	8	2	no	—
283	C₂	C₇×#2293	1	1	yes	characteristic
284	C₂	C₂×C₇×#444	1	2	yes	—

	subgroup	G/H
center	C₄²×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂×C₇×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2922	C₂	semidirect
4. C₇×#1946	C₂	semidirect
5. C₇×#1938	C₂	semidirect
6. C₄×C₇×#108	C₂	semidirect
7. C₄×C₇×#104	C₂	semidirect
8. C₄×C₇×#97	C₂	semidirect
11. C₄×C₇×Q₁₆	C₄	semidirect
12. C₄×C₇×SD₁₆	C₄	semidirect
13. C₄×C₇×D₈	C₄	semidirect
30. #21293	C₇	direct (N × H)
66. #2922	C₂×C₇	semidirect
68. #1946	C₂×C₇	semidirect
69. #1938	C₂×C₇	semidirect
70. C₄×#108	C₂×C₇	semidirect
71. C₄×#104	C₂×C₇	semidirect
72. C₄×#97	C₂×C₇	semidirect
111. C₄×Q₁₆	C₄×C₇	semidirect
112. C₄×SD₁₆	C₄×C₇	semidirect
113. C₄×D₈	C₄×C₇	semidirect
267. C₇	#21293	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₇×#135	2
C₇×#2922	2
C₇×#1975	2
C₇×#1946	2
C₇×#1938	2
C₄×C₇×#108	2
C₄×C₇×#104	2
C₄×C₇×#97	2
#21293	7

Subgroups — C7×#21293 #13198509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#21293 #13198509