Subgroups — C₇×#20787 #13198003

Order 1792.

616 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic
2	C₇×#1841	C₂	1	2	yes	maximal, complemented
3	C₇×#1833	C₂	1	2	yes	maximal, complemented
4	C₇×#1827	C₂	1	2	yes	maximal
5	C₇×#1813	C₂	1	1	yes	maximal, characteristic
6	C₇×#1812	C₂	1	1	yes	maximal, characteristic
7	C₇×#1772	C₂	1	2	yes	maximal, complemented
8	C₇×#1757	C₂	1	2	yes	maximal
9	C₇×#1747	C₂	1	2	yes	maximal, complemented
10	C₂×C₇×#131	C₂²	1	1	yes	characteristic
11	C₂×C₇×#131	C₂²	1	1	yes	characteristic
12	C₄×C₇×#41	C₂²	1	1	yes	characteristic
13	C₂×C₇×#108	C₂²	1	2	yes	—
14	C₇×#427	C₂²	1	2	yes	—
15	C₇×#413	C₂²	1	2	yes	—
16	C₇×#410	C₂²	1	2	yes	—
17	C₇×#405	C₄	1	2	yes	complemented
18	C₇×#405	C₄	1	2	yes	complemented
19	C₇×#405	C₂²	1	4	yes	—
20	C₇×#405	C₂²	1	1	yes	characteristic
21	C₇×#405	C₂²	1	1	yes	characteristic
22	C₄×C₇×M₁₆	C₄	1	2	yes	complemented
23	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₇×C₈	C₄	1	2	yes	complemented
25	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
26	C₇×#344	C₂²	1	4	yes	—
27	C₇×#339	C₂²	1	4	yes	—
28	C₇×#338	C₂²	1	2	yes	—
29	C₇×H(3,ℤ/4)	C₂²	1	2	yes	—
30	C₇×H(3,ℤ/4)	C₂²	1	4	yes	—
31	#20787	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₇×#41	C₂³	1	1	yes	characteristic
33	C₇×#131	—	2	2	no	—
34	C₇×#131	—	2	2	no	—
35	C₇×#131	—	2	2	no	—
36	C₇×#131	—	2	2	no	—
37	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₇×M₁₆	C₂³	1	4	yes	—
40	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
42	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
43	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
44	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
46	C₄×C₇×Q₈	—	2	1	no	—
47	C₄×C₇×D₄	—	2	2	no	—
48	C₄×C₇×D₄	—	2	1	no	—
49	C₇×#117	C₂×C₄	1	2	yes	—
50	C₇×#117	C₂³	1	4	yes	—
51	C₇×#117	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
54	C₇×#108	—	2	2	no	—
55	C₇×#108	—	2	2	no	—
56	C₇×#107	—	2	2	no	—
57	C₇×#106	—	2	2	no	—
58	C₇×#101	—	2	2	no	—
59	C₇×#100	—	2	2	no	—
60	C₇×#99	—	2	4	no	—
61	C₇×#97	—	2	4	no	—
62	C₇×#97	C₂×C₄	1	2	yes	—
63	C₇×#97	C₂×C₄	1	2	yes	—
64	C₇×#97	C₂×C₄	1	2	yes	—
65	C₄×C₇×C₈	—	2	4	no	—
66	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
68	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
69	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
70	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
72	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
73	C₄×#131	C₂×C₇	1	1	yes	characteristic
74	#1841	C₂×C₇	1	2	yes	complemented
75	#1833	C₂×C₇	1	2	yes	complemented
76	#1827	C₂×C₇	1	2	yes	—
77	#1813	C₂×C₇	1	1	yes	characteristic
78	#1812	C₂×C₇	1	1	yes	characteristic
79	#1772	C₂×C₇	1	2	yes	complemented
80	#1757	C₂×C₇	1	2	yes	—
81	#1747	C₂×C₇	1	2	yes	complemented
82	C₇×#41	—	2	2	no	—
83	C₇×#41	—	2	2	no	—
84	C₂×C₇×Q₈	#41	1	1	yes	characteristic
85	C₂×C₇×D₄	#41	1	2	yes	—
86	C₂×C₇×D₄	#41	1	1	yes	characteristic
87	C₂²×C₄×C₇	—	2	2	no	—
88	C₂²×C₄×C₇	—	2	1	no	—
89	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
91	C₇×M₁₆	—	2	2	no	—
92	C₇×M₁₆	—	2	2	no	—
93	C₇×M₁₆	—	2	4	no	—
94	C₇×M₁₆	—	2	4	no	—
95	C₇×M₁₆	—	2	4	no	—
96	C₇×M₁₆	—	2	4	no	—
97	C₂×C₇×C₈	—	2	2	no	—
98	C₂×C₇×C₈	—	2	2	no	—
99	C₂×C₇×C₈	—	2	2	no	—
100	C₂×C₇×C₈	—	2	2	no	—
101	C₂×C₇×C₈	—	2	4	no	—
102	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
103	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₇×C₈	#41	1	2	yes	—
109	C₂×C₇×C₈	#41	1	2	yes	—
110	C₂×C₇×C₈	#41	1	2	yes	—
111	C₂×C₇×C₈	#41	1	2	yes	—
112	C₇×C₄⋊C₄	—	2	2	no	—
113	C₇×C₄⋊C₄	—	2	1	no	—
114	C₇×C₄⋊C₄	—	2	4	no	—
115	C₇×C₂²⋊C₄	—	2	2	no	—
116	C₇×C₂²⋊C₄	—	2	1	no	—
117	C₇×C₂²⋊C₄	—	2	4	no	—
118	C₄²×C₇	—	2	2	no	—
119	C₄²×C₇	—	2	1	no	—
120	C₄²×C₇	—	2	4	no	—
121	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₇	C₂²×C₄	1	2	yes	—
123	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×#131	C₂²×C₇	1	1	yes	characteristic
125	C₂×#131	C₂²×C₇	1	1	yes	characteristic
126	C₄×#41	C₂²×C₇	1	1	yes	characteristic
127	C₂×#108	C₂²×C₇	1	2	yes	—
128	#427	C₂²×C₇	1	2	yes	—
129	#413	C₂²×C₇	1	2	yes	—
130	#410	C₂²×C₇	1	2	yes	—
131	#405	C₄×C₇	1	2	yes	complemented
132	#405	C₄×C₇	1	2	yes	complemented
133	#405	C₂²×C₇	1	4	yes	—
134	#405	C₂²×C₇	1	1	yes	characteristic
135	#405	C₂²×C₇	1	1	yes	characteristic
136	C₄×M₁₆	C₄×C₇	1	2	yes	complemented
137	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
138	C₂×C₄×C₈	C₄×C₇	1	2	yes	complemented
139	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
140	#344	C₂²×C₇	1	4	yes	—
141	#339	C₂²×C₇	1	4	yes	—
142	#338	C₂²×C₇	1	2	yes	—
143	H(3,ℤ/4)	C₂²×C₇	1	2	yes	—
144	H(3,ℤ/4)	C₂²×C₇	1	4	yes	—
145	C₂³×C₇	#117	1	2	yes	—
146	C₂³×C₇	C₂×#41	1	1	yes	characteristic
147	C₇×Q₈	—	2	2	no	—
148	C₇×D₄	—	2	4	no	—
149	C₇×D₄	—	2	2	no	—
150	C₂×C₄×C₇	—	2	1	no	—
151	C₂×C₄×C₇	—	2	2	no	—
152	C₂×C₄×C₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	2	1	no	—
155	C₂×C₄×C₇	—	4	2	no	—
156	C₂×C₄×C₇	—	4	4	no	—
157	C₂×C₄×C₇	—	4	4	no	—
158	C₂×C₄×C₇	#117	1	2	yes	—
159	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂×#41	1	2	yes	—
161	C₂×C₄×C₇	C₂×#41	1	2	yes	—
162	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
163	C₇×C₈	—	2	2	no	—
164	C₇×C₈	—	2	2	no	—
165	C₇×C₈	—	4	4	no	—
166	C₇×C₈	#117	1	1	yes	characteristic
167	C₇×C₈	#117	1	1	yes	characteristic
168	C₇×C₈	#117	1	1	yes	characteristic
169	C₇×C₈	#117	1	2	yes	—
170	C₇×C₈	#117	1	2	yes	—
171	C₇×C₈	#117	1	1	yes	characteristic
172	C₂×#41	C₂³×C₇	1	1	yes	characteristic
173	#131	—	2	2	no	—
174	#131	—	2	2	no	—
175	#131	—	2	2	no	—
176	#131	—	2	2	no	—
177	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
178	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
179	C₂×M₁₆	C₂³×C₇	1	4	yes	—
180	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
181	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
182	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
183	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
184	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
186	C₄×Q₈	—	2	1	no	—
187	C₄×D₄	—	2	2	no	—
188	C₄×D₄	—	2	1	no	—
189	#117	C₂×C₄×C₇	1	2	yes	—
190	#117	C₂³×C₇	1	4	yes	—
191	#117	C₂³×C₇	1	1	yes	characteristic
192	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
193	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
194	#108	—	2	2	no	—
195	#108	—	2	2	no	—
196	#107	—	2	2	no	—
197	#106	—	2	2	no	—
198	#101	—	2	2	no	—
199	#100	—	2	2	no	—
200	#99	—	2	4	no	—
201	#97	—	2	4	no	—
202	#97	C₂×C₄×C₇	1	2	yes	—
203	#97	C₂×C₄×C₇	1	2	yes	—
204	#97	C₂×C₄×C₇	1	2	yes	—
205	C₄×C₈	—	2	4	no	—
206	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
207	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
208	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
209	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
210	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
211	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
212	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
213	C₂²×C₇	—	2	2	no	—
214	C₂²×C₇	—	4	1	no	—
215	C₂²×C₇	—	4	2	no	—
216	C₂²×C₇	C₂×#117	1	1	yes	characteristic
217	C₂²×C₇	#529	1	2	yes	—
218	C₄×C₇	—	2	2	no	—
219	C₄×C₇	—	2	2	no	—
220	C₄×C₇	—	4	2	no	—
221	C₄×C₇	—	4	1	no	—
222	C₄×C₇	—	8	4	no	—
223	C₄×C₇	C₂×#117	1	1	yes	characteristic
224	C₄×C₇	C₂×#117	1	1	yes	characteristic
225	C₄×C₇	#529	1	2	yes	—
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	C₇×#41	1	1	yes	characteristic
229	C₂×D₄	C₇×#41	1	2	yes	—
230	C₂×D₄	C₇×#41	1	1	yes	characteristic
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	2	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	2	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
247	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
248	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
249	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
250	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
251	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
252	C₂×C₈	C₇×#41	1	2	yes	—
253	C₂×C₈	C₇×#41	1	2	yes	—
254	C₂×C₈	C₇×#41	1	2	yes	—
255	C₂×C₈	C₇×#41	1	2	yes	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	4	no	—
265	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
266	C₄²	C₂²×C₄×C₇	1	2	yes	—
267	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
268	C₂×C₇	—	2	1	no	—
269	C₂×C₇	—	2	2	no	—
270	C₂×C₇	—	4	2	no	—
271	C₂×C₇	#2283	1	1	yes	characteristic
272	C₂³	C₇×#117	1	2	yes	—
273	C₂³	C₂×C₇×#41	1	1	yes	characteristic
274	Q₈	—	2	2	no	—
275	D₄	—	2	4	no	—
276	D₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	C₇×#117	1	2	yes	—
286	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
287	C₂×C₄	C₂×C₇×#41	1	2	yes	—
288	C₂×C₄	C₂×C₇×#41	1	2	yes	—
289	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	2	2	no	—
292	C₈	—	4	4	no	—
293	C₈	C₇×#117	1	1	yes	characteristic
294	C₈	C₇×#117	1	1	yes	characteristic
295	C₈	C₇×#117	1	1	yes	characteristic
296	C₈	C₇×#117	1	2	yes	—
297	C₈	C₇×#117	1	2	yes	—
298	C₈	C₇×#117	1	1	yes	characteristic
299	C₇	#20787	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₂×C₇×#117	1	1	yes	characteristic
304	C₂²	C₇×#529	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	8	4	no	—
310	C₄	C₂×C₇×#117	1	1	yes	characteristic
311	C₄	C₂×C₇×#117	1	1	yes	characteristic
312	C₄	C₇×#529	1	2	yes	—
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	4	2	no	—
316	C₂	C₇×#2283	1	1	yes	characteristic

	subgroup	G/H
center	C₇×C₈	#117
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂×C₇	#2283
prime radical	C₂×C₇×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₇×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#1841	C₂	semidirect
3. C₇×#1833	C₂	semidirect
7. C₇×#1772	C₂	semidirect
9. C₇×#1747	C₂	semidirect
17. C₇×#405	C₄	semidirect
18. C₇×#405	C₄	semidirect
22. C₄×C₇×M₁₆	C₄	semidirect
24. C₂×C₄×C₇×C₈	C₄	semidirect
31. #20787	C₇	direct (N × H)
74. #1841	C₂×C₇	semidirect
75. #1833	C₂×C₇	semidirect
79. #1772	C₂×C₇	semidirect
81. #1747	C₂×C₇	semidirect
131. #405	C₄×C₇	semidirect
132. #405	C₄×C₇	semidirect
136. C₄×M₁₆	C₄×C₇	semidirect
138. C₂×C₄×C₈	C₄×C₇	semidirect
299. C₇	#20787	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×#131	2
C₇×#1841	2
C₇×#1833	2
C₇×#1827	2
C₇×#1813	2
C₇×#1812	2
C₇×#1772	2
C₇×#1757	2
C₇×#1747	2
#20787	7

Subgroups — C7×#20787 #13198003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20787 #13198003