Subgroups — C₇×#17970 #13195186

Order 1792.

640 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3169	C₂	1	1	yes	maximal, characteristic
2	C₇×#2981	C₂	1	2	yes	maximal, complemented
3	C₇×#1594	C₂	1	2	yes	maximal, complemented
4	C₇×#1589	C₂	1	2	yes	maximal, complemented
5	C₇×#1580	C₂	1	2	yes	maximal, complemented
6	C₇×C₈×D₈	C₂	1	2	yes	maximal, complemented
7	C₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1496	C₂	1	2	yes	maximal, complemented
9	C₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#532	C₂²	1	2	yes	—
11	C₇×#500	C₄	1	1	yes	characteristic
12	C₇×#493	C₄	1	1	yes	characteristic
13	C₇×#486	C₄	1	2	yes	—
14	C₇×#466	C₄	1	2	yes	—
15	C₇×#466	C₄	1	2	yes	—
16	C₄×C₇×D₈	C₂²	1	2	yes	—
17	C₄×C₇×D₈	C₂²	1	2	yes	—
18	C₇×#436	C₂²	1	4	yes	—
19	C₇×#435	—	2	4	no	—
20	C₇×C₈×D₄	C₂²	1	4	yes	—
21	C₇×#432	C₂²	1	2	yes	—
22	C₇×#431	C₂²	1	1	yes	characteristic
23	C₇×#431	C₂²	1	1	yes	characteristic
24	C₇×#408	—	2	4	no	—
25	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
26	C₇×#335	C₂²	1	2	yes	—
27	C₇×#334	C₂²	1	2	yes	—
28	C₇×D₄⋊C₈	C₂²	1	4	yes	—
29	C₇×D₄⋊C₈	C₂²	1	4	yes	—
30	C₇×#322	C₂²	1	1	yes	characteristic
31	C₇×#322	C₂²	1	1	yes	characteristic
32	C₇×C₈²	C₂²	1	1	yes	characteristic
33	C₇×C₈²	C₂²	1	1	yes	characteristic
34	#17970	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂×C₇×#41	—	2	2	no	—
36	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
37	C₂×C₇×D₈	C₂×C₄	1	2	yes	—
38	C₂×C₇×M₁₆	—	2	4	no	—
39	C₂²×C₇×C₈	—	2	4	no	—
40	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
42	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
43	C₇×#127	C₂×C₄	1	2	yes	—
44	C₇×#124	—	2	2	no	—
45	C₇×#121	—	2	2	no	—
46	C₇×#121	C₂×C₄	1	4	yes	—
47	C₄×C₇×D₄	—	2	2	no	—
48	C₄×C₇×D₄	C₂³	1	4	yes	—
49	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
50	C₇×#107	C₂×C₄	1	2	yes	—
51	C₇×#107	C₂×C₄	1	2	yes	—
52	C₇×C₄⋊C₈	—	2	1	no	—
53	C₇×C₄⋊C₈	D₄	1	2	yes	—
54	C₇×C₄⋊C₈	C₂³	1	4	yes	—
55	C₇×#102	C₂×C₄	1	4	yes	—
56	C₇×#102	C₂×C₄	1	4	yes	—
57	C₇×C₂²⋊C₈	—	2	2	no	—
58	C₇×C₂²⋊C₈	—	2	1	no	—
59	C₇×C₂²⋊C₈	—	2	4	no	—
60	C₇×C₂²⋊C₈	—	2	4	no	—
61	C₇×C₂²⋊C₈	D₄	1	2	yes	—
62	C₇×#97	—	2	2	no	—
63	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
70	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
71	#3169	C₂×C₇	1	1	yes	characteristic
72	#2981	C₂×C₇	1	2	yes	complemented
73	#1594	C₂×C₇	1	2	yes	complemented
74	#1589	C₂×C₇	1	2	yes	complemented
75	#1580	C₂×C₇	1	2	yes	complemented
76	C₈×D₈	C₂×C₇	1	2	yes	complemented
77	#1568	C₂×C₇	1	1	yes	characteristic, complemented
78	#1496	C₂×C₇	1	2	yes	complemented
79	#1450	C₂×C₇	1	1	yes	characteristic, complemented
80	C₇×#41	—	8	4	no	—
81	C₂×C₇×Q₈	—	2	2	no	—
82	C₂×C₇×D₄	—	2	4	no	—
83	C₂×C₇×D₄	—	2	2	no	—
84	C₂×C₇×D₄	—	2	2	no	—
85	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
86	C₂²×C₄×C₇	—	2	4	no	—
87	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇	#41	1	1	yes	characteristic
90	C₇×D₈	—	2	4	no	—
91	C₇×D₈	—	2	4	no	—
92	C₇×M₁₆	—	8	4	no	—
93	C₂×C₇×C₈	—	2	2	no	—
94	C₂×C₇×C₈	—	2	2	no	—
95	C₂×C₇×C₈	—	2	4	no	—
96	C₂×C₇×C₈	—	2	1	no	—
97	C₂×C₇×C₈	—	2	1	no	—
98	C₂×C₇×C₈	—	8	4	no	—
99	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
104	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
105	C₇×C₄⋊C₄	—	2	2	no	—
106	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
107	C₇×C₂²⋊C₄	—	2	4	no	—
108	C₇×C₂²⋊C₄	—	2	4	no	—
109	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₇	#41	1	1	yes	characteristic
112	C₄²×C₇	C₂⁴	1	1	yes	characteristic
113	#532	C₂²×C₇	1	2	yes	—
114	#500	C₄×C₇	1	1	yes	characteristic
115	#493	C₄×C₇	1	1	yes	characteristic
116	#486	C₄×C₇	1	2	yes	—
117	#466	C₄×C₇	1	2	yes	—
118	#466	C₄×C₇	1	2	yes	—
119	C₄×D₈	C₂²×C₇	1	2	yes	—
120	C₄×D₈	C₂²×C₇	1	2	yes	—
121	#436	C₂²×C₇	1	4	yes	—
122	#435	—	2	4	no	—
123	C₈×D₄	C₂²×C₇	1	4	yes	—
124	#432	C₂²×C₇	1	2	yes	—
125	#431	C₂²×C₇	1	1	yes	characteristic
126	#431	C₂²×C₇	1	1	yes	characteristic
127	#408	—	2	4	no	—
128	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
129	#335	C₂²×C₇	1	2	yes	—
130	#334	C₂²×C₇	1	2	yes	—
131	D₄⋊C₈	C₂²×C₇	1	4	yes	—
132	D₄⋊C₈	C₂²×C₇	1	4	yes	—
133	#322	C₂²×C₇	1	1	yes	characteristic
134	#322	C₂²×C₇	1	1	yes	characteristic
135	C₈²	C₂²×C₇	1	1	yes	characteristic
136	C₈²	C₂²×C₇	1	1	yes	characteristic
137	C₂³×C₇	—	2	4	no	—
138	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
139	C₇×Q₈	—	8	2	no	—
140	C₇×D₄	—	2	4	no	—
141	C₇×D₄	—	2	4	no	—
142	C₇×D₄	—	8	2	no	—
143	C₇×D₄	—	8	2	no	—
144	C₇×D₄	—	8	4	no	—
145	C₂×C₄×C₇	—	2	1	no	—
146	C₂×C₄×C₇	—	2	1	no	—
147	C₂×C₄×C₇	—	2	2	no	—
148	C₂×C₄×C₇	—	2	4	no	—
149	C₂×C₄×C₇	—	4	1	no	—
150	C₂×C₄×C₇	—	4	4	no	—
151	C₂×C₄×C₇	—	4	4	no	—
152	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
158	C₇×C₈	—	2	1	no	—
159	C₇×C₈	—	2	1	no	—
160	C₇×C₈	—	2	2	no	—
161	C₇×C₈	—	2	2	no	—
162	C₇×C₈	—	4	1	no	—
163	C₇×C₈	—	4	1	no	—
164	C₇×C₈	—	4	4	no	—
165	C₇×C₈	#131	1	2	yes	—
166	C₇×C₈	#131	1	2	yes	—
167	C₂×#41	—	2	2	no	—
168	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
169	C₂×D₈	C₂×C₄×C₇	1	2	yes	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂²×C₈	—	2	4	no	—
172	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
173	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
174	C₄⋊Q₈	C₂×C₄×C₇	1	2	yes	—
175	#127	C₂×C₄×C₇	1	2	yes	—
176	#124	—	2	2	no	—
177	#121	—	2	2	no	—
178	#121	C₂×C₄×C₇	1	4	yes	—
179	C₄×D₄	—	2	2	no	—
180	C₄×D₄	C₂³×C₇	1	4	yes	—
181	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
182	#107	C₂×C₄×C₇	1	2	yes	—
183	#107	C₂×C₄×C₇	1	2	yes	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	C₇×D₄	1	2	yes	—
186	C₄⋊C₈	C₂³×C₇	1	4	yes	—
187	#102	C₂×C₄×C₇	1	4	yes	—
188	#102	C₂×C₄×C₇	1	4	yes	—
189	C₂²⋊C₈	—	2	2	no	—
190	C₂²⋊C₈	—	2	1	no	—
191	C₂²⋊C₈	—	2	4	no	—
192	C₂²⋊C₈	—	2	4	no	—
193	C₂²⋊C₈	C₇×D₄	1	2	yes	—
194	#97	—	2	2	no	—
195	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
196	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
197	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
198	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
199	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
200	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
202	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
203	C₂²×C₇	—	2	1	no	—
204	C₂²×C₇	—	2	1	no	—
205	C₂²×C₇	—	2	1	no	—
206	C₂²×C₇	—	4	4	no	—
207	C₂²×C₇	—	4	4	no	—
208	C₂²×C₇	—	4	4	no	—
209	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
210	C₄×C₇	—	2	1	no	—
211	C₄×C₇	—	2	1	no	—
212	C₄×C₇	—	2	1	no	—
213	C₄×C₇	—	2	1	no	—
214	C₄×C₇	—	4	4	no	—
215	C₄×C₇	C₂×#131	1	1	yes	characteristic
216	C₄×C₇	C₂×#131	1	1	yes	characteristic
217	C₄×C₇	#576	1	2	yes	—
218	#41	—	8	4	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
227	C₂²×C₄	C₇×#41	1	1	yes	characteristic
228	D₈	—	2	4	no	—
229	D₈	—	2	4	no	—
230	M₁₆	—	8	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	8	4	no	—
237	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
239	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
240	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
241	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
242	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
248	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
249	C₄²	C₇×#41	1	1	yes	characteristic
250	C₄²	C₂⁴×C₇	1	1	yes	characteristic
251	C₂×C₇	—	4	1	no	—
252	C₂×C₇	—	8	4	no	—
253	C₂×C₇	#2931	1	1	yes	characteristic
254	C₂×C₇	#2948	1	1	yes	characteristic
255	C₂×C₇	#2957	1	1	yes	characteristic
256	C₂³	—	2	4	no	—
257	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
258	Q₈	—	8	2	no	—
259	D₄	—	2	4	no	—
260	D₄	—	2	4	no	—
261	D₄	—	8	2	no	—
262	D₄	—	8	2	no	—
263	D₄	—	8	4	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
272	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
273	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
274	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
276	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	4	1	no	—
282	C₈	—	4	1	no	—
283	C₈	—	4	4	no	—
284	C₈	C₇×#131	1	2	yes	—
285	C₈	C₇×#131	1	2	yes	—
286	C₇	#17970	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	4	no	—
299	C₄	C₂×C₇×#131	1	1	yes	characteristic
300	C₄	C₂×C₇×#131	1	1	yes	characteristic
301	C₄	C₇×#576	1	2	yes	—
302	C₂	—	4	1	no	—
303	C₂	—	8	4	no	—
304	C₂	C₇×#2931	1	1	yes	characteristic
305	C₂	C₇×#2948	1	1	yes	characteristic
306	C₂	C₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₇×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₇×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2981	C₂	semidirect
3. C₇×#1594	C₂	semidirect
4. C₇×#1589	C₂	semidirect
5. C₇×#1580	C₂	semidirect
6. C₇×C₈×D₈	C₂	semidirect
7. C₇×#1568	C₂	semidirect
8. C₇×#1496	C₂	semidirect
9. C₇×#1450	C₂	semidirect
34. #17970	C₇	direct (N × H)
72. #2981	C₂×C₇	semidirect
73. #1594	C₂×C₇	semidirect
74. #1589	C₂×C₇	semidirect
75. #1580	C₂×C₇	semidirect
76. C₈×D₈	C₂×C₇	semidirect
77. #1568	C₂×C₇	semidirect
78. #1496	C₂×C₇	semidirect
79. #1450	C₂×C₇	semidirect
286. C₇	#17970	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3169	2
C₇×#2981	2
C₇×#1594	2
C₇×#1589	2
C₇×#1580	2
C₇×C₈×D₈	2
C₇×#1568	2
C₇×#1496	2
C₇×#1450	2
#17970	7

Subgroups — C7×#17970 #13195186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17970 #13195186