Subgroups — C₇×#5768 #13061484

Order 1344.

524 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×C₇×#41	C₂	1	2	yes	maximal, complemented
2	C₃×C₇×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#950	C₂	1	2	yes	maximal, complemented
4	C₇×#918	C₂	1	2	yes	maximal, complemented
5	C₇×#915	C₂	1	2	yes	maximal, complemented
6	C₇×#914	C₂	1	2	yes	maximal, complemented
7	C₇×#911	C₂	1	2	yes	maximal, complemented
8	C₇×#904	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#575	—	3	1	no	maximal, Hall
11	C₃×C₇×#41	C₂²	1	2	yes	complemented
12	C₇×#239	C₂²	1	2	yes	—
13	D₃×C₇×Q₈	C₂²	1	2	yes	complemented
14	C₇×#237	—	2	2	no	—
15	C₇×#237	C₂²	1	2	yes	—
16	D₃×C₇×D₄	—	2	2	no	—
17	D₃×C₇×D₄	C₂²	1	2	yes	complemented
18	C₇×#235	—	2	2	no	—
19	C₇×#235	C₂²	1	2	yes	complemented
20	C₂×C₄×D₃×C₇	—	2	2	no	—
21	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
22	C₃×C₇×Q₁₆	C₂²	1	1	yes	characteristic, complemented
23	C₃×C₇×SD₁₆	C₂²	1	2	yes	complemented
24	C₃×C₇×D₈	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
26	C₇×#216	C₂²	1	2	yes	complemented
27	C₇×#215	C₂²	1	2	yes	complemented
28	C₇×#214	C₂²	1	2	yes	complemented
29	C₇×#213	C₂²	1	2	yes	complemented
30	C₂×C₇×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
31	C₇×Dic₁₂	C₂²	1	1	yes	characteristic, complemented
32	C₇×D₂₄	C₂²	1	1	yes	characteristic, complemented
33	C₇×#204	C₂²	1	2	yes	complemented
34	C₇×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
35	C₇×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
36	C₂×C₇×#41	—	3	2	no	—
37	C₇×#137	—	3	2	no	—
38	C₇×#137	—	3	2	no	—
39	C₇×AGL(1,ℤ/8)	—	3	2	no	—
40	C₇×AGL(1,ℤ/8)	—	3	2	no	—
41	C₇×#135	—	3	2	no	—
42	C₇×#135	—	3	1	no	—
43	C₇×#135	D₃	1	1	yes	characteristic, complemented
44	C₂×C₇×M₁₆	—	3	1	no	—
45	#5768	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
46	C₂²×D₃×C₇	—	2	2	no	—
47	C₂²×D₃×C₇	D₄	1	1	yes	characteristic
48	C₃×C₇×Q₈	C₂³	1	2	yes	—
49	C₃×C₇×D₄	—	2	2	no	—
50	C₃×C₇×D₄	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₇	—	2	2	no	—
52	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₇×#67	—	2	2	no	—
54	C₇×#67	—	2	2	no	—
55	C₇×#67	—	2	2	no	—
56	C₂×C₇×Dic₃	—	2	2	no	—
57	C₂×C₇×Dic₃	D₄	1	1	yes	characteristic
58	C₇×D₁₂	—	2	2	no	—
59	C₇×D₁₂	C₂³	1	2	yes	—
60	C₄×D₃×C₇	—	2	2	no	—
61	C₄×D₃×C₇	—	2	2	no	—
62	C₄×D₃×C₇	—	2	2	no	—
63	C₄×D₃×C₇	D₄	1	2	yes	—
64	C₄×D₃×C₇	C₂³	1	2	yes	—
65	C₇×Dic₆	—	2	2	no	—
66	C₇×Dic₆	C₂³	1	2	yes	—
67	C₃×C₇×C₈	C₂³	1	1	yes	characteristic
68	C₃×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₇×C₃⋊C₈	C₂³	1	2	yes	—
70	C₇×#41	—	3	2	no	—
71	C₇×#41	—	3	2	no	—
72	C₇×#41	—	3	2	no	—
73	C₇×#41	—	6	2	no	—
74	C₇×#41	—	6	2	no	—
75	C₇×#41	C₂×D₃	1	2	yes	complemented
76	C₂×C₇×Q₈	—	3	2	no	—
77	C₂×C₇×D₄	—	3	2	no	—
78	C₂×C₇×D₄	—	6	2	no	—
79	C₂²×C₄×C₇	—	3	1	no	—
80	C₂²×C₄×C₇	—	6	2	no	—
81	C₇×Q₁₆	—	3	2	no	—
82	C₇×Q₁₆	—	3	1	no	—
83	C₇×Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
84	C₇×SD₁₆	—	3	2	no	—
85	C₇×SD₁₆	—	3	2	no	—
86	C₇×SD₁₆	—	3	2	no	—
87	C₇×SD₁₆	C₂×D₃	1	2	yes	complemented
88	C₇×D₈	—	3	2	no	—
89	C₇×D₈	—	3	1	no	—
90	C₇×D₈	C₂×D₃	1	1	yes	characteristic, complemented
91	C₇×M₁₆	—	3	2	no	—
92	C₇×M₁₆	—	3	2	no	—
93	C₂×C₇×C₈	—	3	1	no	—
94	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic, complemented
95	D₃×#41	C₂×C₇	1	2	yes	complemented
96	C₃×#135	C₂×C₇	1	1	yes	characteristic, complemented
97	#950	C₂×C₇	1	2	yes	complemented
98	#918	C₂×C₇	1	2	yes	complemented
99	#915	C₂×C₇	1	2	yes	complemented
100	#914	C₂×C₇	1	2	yes	complemented
101	#911	C₂×C₇	1	2	yes	complemented
102	#904	C₂×C₇	1	1	yes	characteristic, complemented
103	C₂×ΓL(1,𝔽₂₅)	C₂×C₇	1	1	yes	characteristic, complemented
104	C₂²×C₃×C₇	—	2	2	no	—
105	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
106	C₂×D₃×C₇	—	2	2	no	—
107	C₂×D₃×C₇	—	4	1	no	—
108	C₂×D₃×C₇	—	4	4	no	—
109	C₂×D₃×C₇	C₂×D₄	1	2	yes	—
110	C₃×C₄×C₇	—	2	2	no	—
111	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
112	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
113	C₇×Dic₃	—	2	2	no	—
114	C₇×Dic₃	C₂×D₄	1	2	yes	—
115	#575	—	3	1	no	Sylow
116	C₂³×C₇	—	3	1	no	—
117	C₂³×C₇	—	6	2	no	—
118	C₇×Q₈	—	3	2	no	—
119	C₇×Q₈	—	6	2	no	—
120	C₇×Q₈	C₂²×D₃	1	2	yes	—
121	C₇×D₄	—	2	2	no	—
122	C₇×D₄	—	3	2	no	—
123	C₇×D₄	—	6	2	no	—
124	C₇×D₄	—	6	2	no	—
125	C₇×D₄	—	6	2	no	—
126	C₇×D₄	—	6	2	no	—
127	C₇×D₄	C₂²×D₃	1	2	yes	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	—	3	1	no	—
130	C₂×C₄×C₇	—	3	2	no	—
131	C₂×C₄×C₇	—	3	2	no	—
132	C₂×C₄×C₇	—	6	2	no	—
133	C₂×C₄×C₇	—	6	2	no	—
134	C₂×C₄×C₇	—	6	2	no	—
135	C₂×C₄×C₇	—	6	2	no	—
136	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
137	C₇×C₈	—	3	2	no	—
138	C₇×C₈	C₂²×D₃	1	1	yes	characteristic
139	C₇×C₈	C₂²×D₃	1	1	yes	characteristic
140	C₃×#41	C₂²×C₇	1	2	yes	complemented
141	#239	C₂²×C₇	1	2	yes	—
142	D₃×Q₈	C₂²×C₇	1	2	yes	complemented
143	#237	—	2	2	no	—
144	#237	C₂²×C₇	1	2	yes	—
145	D₃×D₄	—	2	2	no	—
146	D₃×D₄	C₂²×C₇	1	2	yes	complemented
147	#235	—	2	2	no	—
148	#235	C₂²×C₇	1	2	yes	complemented
149	C₂×C₄×D₃	—	2	2	no	—
150	C₂×C₄×D₃	C₂²×C₇	1	1	yes	characteristic
151	C₃×Q₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
152	C₃×SD₁₆	C₂²×C₇	1	2	yes	complemented
153	C₃×D₈	C₂²×C₇	1	1	yes	characteristic, complemented
154	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
155	#216	C₂²×C₇	1	2	yes	complemented
156	#215	C₂²×C₇	1	2	yes	complemented
157	#214	C₂²×C₇	1	2	yes	complemented
158	#213	C₂²×C₇	1	2	yes	complemented
159	C₂×C₃⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
160	Dic₁₂	C₂²×C₇	1	1	yes	characteristic, complemented
161	D₂₄	C₂²×C₇	1	1	yes	characteristic, complemented
162	#204	C₂²×C₇	1	2	yes	complemented
163	ΓL(1,𝔽₂₅)	C₂²×C₇	1	2	yes	complemented
164	ΓL(1,𝔽₂₅)	C₂²×C₇	1	2	yes	complemented
165	C₂×C₃×C₇	—	2	1	no	—
166	C₂×C₃×C₇	—	4	2	no	—
167	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
168	D₃×C₇	—	2	2	no	—
169	D₃×C₇	—	4	2	no	—
170	C₂×#41	—	3	2	no	—
171	#137	—	3	2	no	—
172	#137	—	3	2	no	—
173	AGL(1,ℤ/8)	—	3	2	no	—
174	AGL(1,ℤ/8)	—	3	2	no	—
175	#135	—	3	2	no	—
176	#135	—	3	1	no	—
177	#135	D₃×C₇	1	1	yes	characteristic, complemented
178	C₂×M₁₆	—	3	1	no	—
179	C₂²×C₇	—	2	2	no	—
180	C₂²×C₇	—	3	2	no	—
181	C₂²×C₇	—	6	2	no	—
182	C₂²×C₇	—	12	1	no	—
183	C₂²×C₇	—	12	4	no	—
184	C₂²×C₇	D₃×D₄	1	1	yes	characteristic
185	C₄×C₇	—	2	2	no	—
186	C₄×C₇	—	3	2	no	—
187	C₄×C₇	—	6	2	no	—
188	C₄×C₇	D₃×D₄	1	1	yes	characteristic
189	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
190	C₂²×D₃	—	2	2	no	—
191	C₂²×D₃	C₇×D₄	1	1	yes	characteristic
192	C₃×Q₈	C₂³×C₇	1	2	yes	—
193	C₃×D₄	—	2	2	no	—
194	C₃×D₄	C₂³×C₇	1	2	yes	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
197	#67	—	2	2	no	—
198	#67	—	2	2	no	—
199	#67	—	2	2	no	—
200	C₂×Dic₃	—	2	2	no	—
201	C₂×Dic₃	C₇×D₄	1	1	yes	characteristic
202	D₁₂	—	2	2	no	—
203	D₁₂	C₂³×C₇	1	2	yes	—
204	C₄×D₃	—	2	2	no	—
205	C₄×D₃	—	2	2	no	—
206	C₄×D₃	—	2	2	no	—
207	C₄×D₃	C₇×D₄	1	2	yes	—
208	C₄×D₃	C₂³×C₇	1	2	yes	—
209	Dic₆	—	2	2	no	—
210	Dic₆	C₂³×C₇	1	2	yes	—
211	C₃×C₈	C₂³×C₇	1	1	yes	characteristic
212	C₃×C₈	C₂³×C₇	1	1	yes	characteristic
213	C₃⋊C₈	C₂³×C₇	1	2	yes	—
214	C₃×C₇	#575	1	1	yes	characteristic, complemented, Hall
215	#41	—	3	2	no	—
216	#41	—	3	2	no	—
217	#41	—	3	2	no	—
218	#41	—	6	2	no	—
219	#41	—	6	2	no	—
220	#41	C₂×D₃×C₇	1	2	yes	complemented
221	C₂×Q₈	—	3	2	no	—
222	C₂×D₄	—	3	2	no	—
223	C₂×D₄	—	6	2	no	—
224	C₂²×C₄	—	3	1	no	—
225	C₂²×C₄	—	6	2	no	—
226	Q₁₆	—	3	2	no	—
227	Q₁₆	—	3	1	no	—
228	Q₁₆	C₂×D₃×C₇	1	1	yes	characteristic, complemented
229	SD₁₆	—	3	2	no	—
230	SD₁₆	—	3	2	no	—
231	SD₁₆	—	3	2	no	—
232	SD₁₆	C₂×D₃×C₇	1	2	yes	complemented
233	D₈	—	3	2	no	—
234	D₈	—	3	1	no	—
235	D₈	C₂×D₃×C₇	1	1	yes	characteristic, complemented
236	M₁₆	—	3	2	no	—
237	M₁₆	—	3	2	no	—
238	C₂×C₈	—	3	1	no	—
239	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic, complemented
240	C₂×C₇	—	2	1	no	—
241	C₂×C₇	—	4	2	no	—
242	C₂×C₇	—	6	2	no	—
243	C₂×C₇	—	12	2	no	—
244	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
247	C₂×D₃	—	2	2	no	—
248	C₂×D₃	—	4	1	no	—
249	C₂×D₃	—	4	4	no	—
250	C₂×D₃	C₂×C₇×D₄	1	2	yes	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
253	C₃×C₄	C₂⁴×C₇	1	1	yes	characteristic
254	Dic₃	—	2	2	no	—
255	Dic₃	C₂×C₇×D₄	1	2	yes	—
256	C₂³	—	3	1	no	—
257	C₂³	—	6	2	no	—
258	Q₈	—	3	2	no	—
259	Q₈	—	6	2	no	—
260	Q₈	C₂²×D₃×C₇	1	2	yes	—
261	D₄	—	2	2	no	—
262	D₄	—	3	2	no	—
263	D₄	—	6	2	no	—
264	D₄	—	6	2	no	—
265	D₄	—	6	2	no	—
266	D₄	—	6	2	no	—
267	D₄	C₂²×D₃×C₇	1	2	yes	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	3	1	no	—
270	C₂×C₄	—	3	2	no	—
271	C₂×C₄	—	3	2	no	—
272	C₂×C₄	—	6	2	no	—
273	C₂×C₄	—	6	2	no	—
274	C₂×C₄	—	6	2	no	—
275	C₂×C₄	—	6	2	no	—
276	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
277	C₈	—	3	2	no	—
278	C₈	C₂²×D₃×C₇	1	1	yes	characteristic
279	C₈	C₂²×D₃×C₇	1	1	yes	characteristic
280	C₇	#5768	1	1	yes	characteristic, direct product factor, Sylow
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	4	2	no	—
283	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
284	D₃	—	2	2	no	—
285	D₃	—	4	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	3	2	no	—
288	C₂²	—	6	2	no	—
289	C₂²	—	12	1	no	—
290	C₂²	—	12	4	no	—
291	C₂²	D₃×C₇×D₄	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	3	2	no	—
294	C₄	—	6	2	no	—
295	C₄	D₃×C₇×D₄	1	1	yes	characteristic
296	C₄	C₂³×D₃×C₇	1	1	yes	characteristic
297	C₃	C₇×#575	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	2	1	no	—
299	C₂	—	4	2	no	—
300	C₂	—	6	2	no	—
301	C₂	—	12	2	no	—
302	C₂	C₂×D₃×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁴×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₃×C₇×#135	C₂
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₇×#135	D₃
hypocenter	C₃	C₇×#575
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#135

Complements

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

N	H ≅ G/N	type
1. D₃×C₇×#41	C₂	semidirect
2. C₃×C₇×#135	C₂	semidirect
3. C₇×#950	C₂	semidirect
4. C₇×#918	C₂	semidirect
5. C₇×#915	C₂	semidirect
6. C₇×#914	C₂	semidirect
7. C₇×#911	C₂	semidirect
8. C₇×#904	C₂	semidirect
9. C₂×C₇×ΓL(1,𝔽₂₅)	C₂	semidirect
11. C₃×C₇×#41	C₂²	semidirect
13. D₃×C₇×Q₈	C₂²	semidirect
17. D₃×C₇×D₄	C₂²	semidirect
19. C₇×#235	C₂²	semidirect
22. C₃×C₇×Q₁₆	C₂²	semidirect
23. C₃×C₇×SD₁₆	C₂²	semidirect
24. C₃×C₇×D₈	C₂²	semidirect
25. C₂×C₃×C₇×C₈	C₂²	semidirect
26. C₇×#216	C₂²	semidirect
27. C₇×#215	C₂²	semidirect
28. C₇×#214	C₂²	semidirect
29. C₇×#213	C₂²	semidirect
30. C₂×C₇×C₃⋊C₈	C₂²	semidirect
31. C₇×Dic₁₂	C₂²	semidirect
32. C₇×D₂₄	C₂²	semidirect
33. C₇×#204	C₂²	semidirect
34. C₇×ΓL(1,𝔽₂₅)	C₂²	semidirect
35. C₇×ΓL(1,𝔽₂₅)	C₂²	semidirect
43. C₇×#135	D₃	semidirect
45. #5768	C₇	direct (N × H)
75. C₇×#41	C₂×D₃	semidirect
83. C₇×Q₁₆	C₂×D₃	semidirect
87. C₇×SD₁₆	C₂×D₃	semidirect
90. C₇×D₈	C₂×D₃	semidirect
94. C₂×C₇×C₈	C₂×D₃	semidirect
95. D₃×#41	C₂×C₇	semidirect
96. C₃×#135	C₂×C₇	semidirect
97. #950	C₂×C₇	semidirect
98. #918	C₂×C₇	semidirect
99. #915	C₂×C₇	semidirect
100. #914	C₂×C₇	semidirect
101. #911	C₂×C₇	semidirect
102. #904	C₂×C₇	semidirect
103. C₂×ΓL(1,𝔽₂₅)	C₂×C₇	semidirect
140. C₃×#41	C₂²×C₇	semidirect
142. D₃×Q₈	C₂²×C₇	semidirect
146. D₃×D₄	C₂²×C₇	semidirect
148. #235	C₂²×C₇	semidirect
151. C₃×Q₁₆	C₂²×C₇	semidirect
152. C₃×SD₁₆	C₂²×C₇	semidirect
153. C₃×D₈	C₂²×C₇	semidirect
154. C₂×C₃×C₈	C₂²×C₇	semidirect
155. #216	C₂²×C₇	semidirect
156. #215	C₂²×C₇	semidirect
157. #214	C₂²×C₇	semidirect
158. #213	C₂²×C₇	semidirect
159. C₂×C₃⋊C₈	C₂²×C₇	semidirect
160. Dic₁₂	C₂²×C₇	semidirect
161. D₂₄	C₂²×C₇	semidirect
162. #204	C₂²×C₇	semidirect
163. ΓL(1,𝔽₂₅)	C₂²×C₇	semidirect
164. ΓL(1,𝔽₂₅)	C₂²×C₇	semidirect
177. #135	D₃×C₇	semidirect
214. C₃×C₇	#575	semidirect
220. #41	C₂×D₃×C₇	semidirect
228. Q₁₆	C₂×D₃×C₇	semidirect
232. SD₁₆	C₂×D₃×C₇	semidirect
235. D₈	C₂×D₃×C₇	semidirect
239. C₂×C₈	C₂×D₃×C₇	semidirect
280. C₇	#5768	direct (N × H)
297. C₃	C₇×#575	semidirect

Maximal subgroups

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

subgroup	index
D₃×C₇×#41	2
C₃×C₇×#135	2
C₇×#950	2
C₇×#918	2
C₇×#915	2
C₇×#914	2
C₇×#911	2
C₇×#904	2
C₂×C₇×ΓL(1,𝔽₂₅)	2
C₇×#575	3
#5768	7

Subgroups — C7×#5768 #13061484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5768 #13061484