Subgroups — C₇₉×#3018 #23760640

Order 10112.

724 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#585×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#575×C₇₉	C₂	1	2	yes	maximal, complemented
3	#532×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#456×C₇₉	C₂	1	2	yes	maximal, complemented
5	#455×C₇₉	C₂	1	2	yes	maximal, complemented
6	#454×C₇₉	C₂	1	2	yes	maximal, complemented
7	#453×C₇₉	C₂	1	2	yes	maximal, complemented
8	C₂×#104×C₇₉	C₂	1	2	yes	maximal, complemented
9	#413×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	2₋¹⁺⁴×C₇₉	—	2	2	no	—
11	2₋¹⁺⁴×C₇₉	C₂²	1	2	yes	—
12	H(5,𝔽₂)×C₇₉	—	2	2	no	—
13	H(5,𝔽₂)×C₇₉	—	2	2	no	—
14	H(5,𝔽₂)×C₇₉	C₂²	1	2	yes	—
15	C₂×#41×C₇₉	—	2	4	no	—
16	C₂×#41×C₇₉	—	2	2	no	—
17	C₂×#41×C₇₉	C₂²	1	2	yes	—
18	C₂×#41×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×#41×C₇₉	C₂²	1	1	yes	characteristic, complemented
20	#137×C₇₉	—	2	2	no	—
21	#137×C₇₉	C₂²	1	4	yes	complemented
22	AGL(1,ℤ/8)×C₇₉	—	2	2	no	—
23	AGL(1,ℤ/8)×C₇₉	C₂²	1	4	yes	complemented
24	#135×C₇₉	—	2	4	no	—
25	C₂×M₁₆×C₇₉	C₂²	1	2	yes	complemented
26	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic, complemented
27	#127×C₇₉	C₂²	1	1	yes	characteristic, complemented
28	#124×C₇₉	C₂²	1	1	yes	characteristic, complemented
29	#124×C₇₉	C₂²	1	1	yes	characteristic, complemented
30	#121×C₇₉	—	2	2	no	—
31	C₄×D₄×C₇₉	—	2	2	no	—
32	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic, complemented
33	#104×C₇₉	C₂²	1	4	yes	complemented
34	#104×C₇₉	C₂²	1	4	yes	complemented
35	#101×C₇₉	C₂²	1	2	yes	complemented
36	#100×C₇₉	C₂²	1	2	yes	complemented
37	#41×C₇₉	—	2	4	no	—
38	#41×C₇₉	—	2	8	no	—
39	#41×C₇₉	—	2	4	no	—
40	#41×C₇₉	—	2	4	no	—
41	#41×C₇₉	—	2	4	no	—
42	#41×C₇₉	—	2	2	no	—
43	#41×C₇₉	—	2	2	no	—
44	#41×C₇₉	—	2	4	no	—
45	#41×C₇₉	—	2	2	no	—
46	#41×C₇₉	—	2	2	no	—
47	#41×C₇₉	—	2	2	no	—
48	#41×C₇₉	—	2	2	no	—
49	#41×C₇₉	D₄	1	4	yes	complemented
50	#41×C₇₉	C₂³	1	4	yes	—
51	C₂×Q₈×C₇₉	—	2	2	no	—
52	C₂×Q₈×C₇₉	—	2	4	no	—
53	C₂×Q₈×C₇₉	D₄	1	2	yes	complemented
54	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
55	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₇₉	—	2	8	no	—
57	C₂×D₄×C₇₉	—	2	2	no	—
58	C₂×D₄×C₇₉	—	2	4	no	—
59	C₂×D₄×C₇₉	—	2	2	no	—
60	C₂×D₄×C₇₉	—	2	4	no	—
61	C₂×D₄×C₇₉	—	2	2	no	—
62	C₂×D₄×C₇₉	—	2	1	no	—
63	C₂×D₄×C₇₉	D₄	1	2	yes	—
64	C₂×D₄×C₇₉	D₄	1	2	yes	complemented
65	C₂×D₄×C₇₉	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₇₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	—	2	4	no	—
68	C₂²×C₄×C₇₉	—	2	2	no	—
69	C₂²×C₄×C₇₉	—	2	1	no	—
70	C₂²×C₄×C₇₉	—	2	1	no	—
71	C₂²×C₄×C₇₉	D₄	1	2	yes	—
72	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
73	Q₁₆×C₇₉	—	2	4	no	—
74	SD₁₆×C₇₉	—	2	4	no	—
75	SD₁₆×C₇₉	—	2	4	no	—
76	D₈×C₇₉	—	2	4	no	—
77	M₁₆×C₇₉	—	2	2	no	—
78	M₁₆×C₇₉	C₂³	1	4	yes	—
79	C₂×C₈×C₇₉	—	2	2	no	—
80	C₄⋊C₄×C₇₉	—	2	2	no	—
81	C₂²⋊C₄×C₇₉	—	2	2	no	—
82	C₂²⋊C₄×C₇₉	—	2	2	no	—
83	C₄²×C₇₉	—	2	1	no	—
84	C₄²×C₇₉	C₂³	1	1	yes	characteristic
85	C₄²×C₇₉	C₂³	1	1	yes	characteristic
86	C₂³×C₇₉	—	2	4	no	—
87	C₂³×C₇₉	—	2	2	no	—
88	C₂³×C₇₉	—	2	1	no	—
89	C₂³×C₇₉	C₂×D₄	1	2	yes	—
90	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
91	Q₈×C₇₉	—	2	4	no	—
92	Q₈×C₇₉	—	2	2	no	—
93	Q₈×C₇₉	—	2	2	no	—
94	Q₈×C₇₉	—	2	2	no	—
95	Q₈×C₇₉	C₂×D₄	1	4	yes	—
96	D₄×C₇₉	—	2	8	no	—
97	D₄×C₇₉	—	2	4	no	—
98	D₄×C₇₉	—	2	4	no	—
99	D₄×C₇₉	—	2	4	no	—
100	D₄×C₇₉	—	2	4	no	—
101	D₄×C₇₉	—	2	2	no	—
102	D₄×C₇₉	—	2	2	no	—
103	D₄×C₇₉	—	2	2	no	—
104	D₄×C₇₉	—	4	4	no	—
105	D₄×C₇₉	—	4	2	no	—
106	D₄×C₇₉	C₂×D₄	1	4	yes	—
107	C₂×C₄×C₇₉	—	2	4	no	—
108	C₂×C₄×C₇₉	—	2	8	no	—
109	C₂×C₄×C₇₉	—	2	2	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	2	no	—
112	C₂×C₄×C₇₉	—	2	4	no	—
113	C₂×C₄×C₇₉	—	2	2	no	—
114	C₂×C₄×C₇₉	—	2	4	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	2	1	no	—
117	C₂×C₄×C₇₉	—	4	2	no	—
118	C₂×C₄×C₇₉	—	4	1	no	—
119	C₂×C₄×C₇₉	—	4	1	no	—
120	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
123	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
124	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
125	C₈×C₇₉	—	2	4	no	—
126	C₂²×C₇₉	—	2	2	no	—
127	C₂²×C₇₉	—	2	4	no	—
128	C₂²×C₇₉	—	4	1	no	—
129	C₂²×C₇₉	—	4	2	no	—
130	C₂²×C₇₉	—	4	2	no	—
131	C₂²×C₇₉	—	4	1	no	—
132	C₂²×C₇₉	—	8	2	no	—
133	C₂²×C₇₉	—	8	4	no	—
134	C₂²×C₇₉	#120	1	2	yes	—
135	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₇₉	—	2	2	no	—
137	C₄×C₇₉	—	2	4	no	—
138	C₄×C₇₉	—	2	2	no	—
139	C₄×C₇₉	—	4	1	no	—
140	C₄×C₇₉	—	4	1	no	—
141	C₄×C₇₉	#120	1	1	yes	characteristic
142	C₄×C₇₉	#120	1	1	yes	characteristic
143	C₄×C₇₉	C₂²×D₄	1	2	yes	—
144	C₂×C₇₉	—	2	1	no	—
145	C₂×C₇₉	—	2	2	no	—
146	C₂×C₇₉	—	4	2	no	—
147	C₂×C₇₉	—	4	4	no	—
148	C₂×C₇₉	—	8	1	no	—
149	C₂×C₇₉	C₂×#120	1	1	yes	characteristic
150	#3018	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₇₉	#3018	1	1	yes	characteristic, direct product factor, Sylow
152	#585	C₂×C₇₉	1	1	yes	characteristic, complemented
153	#575	C₂×C₇₉	1	2	yes	complemented
154	#532	C₂×C₇₉	1	1	yes	characteristic, complemented
155	#456	C₂×C₇₉	1	2	yes	complemented
156	#455	C₂×C₇₉	1	2	yes	complemented
157	#454	C₂×C₇₉	1	2	yes	complemented
158	#453	C₂×C₇₉	1	2	yes	complemented
159	C₂×#104	C₂×C₇₉	1	2	yes	complemented
160	#413	C₂×C₇₉	1	1	yes	characteristic, complemented
161	2₋¹⁺⁴	—	2	2	no	—
162	2₋¹⁺⁴	C₂²×C₇₉	1	2	yes	—
163	H(5,𝔽₂)	—	2	2	no	—
164	H(5,𝔽₂)	—	2	2	no	—
165	H(5,𝔽₂)	C₂²×C₇₉	1	2	yes	—
166	C₂×#41	—	2	4	no	—
167	C₂×#41	—	2	2	no	—
168	C₂×#41	C₂²×C₇₉	1	2	yes	—
169	C₂×#41	C₂²×C₇₉	1	1	yes	characteristic
170	C₂×#41	C₂²×C₇₉	1	1	yes	characteristic, complemented
171	#137	—	2	2	no	—
172	#137	C₂²×C₇₉	1	4	yes	complemented
173	AGL(1,ℤ/8)	—	2	2	no	—
174	AGL(1,ℤ/8)	C₂²×C₇₉	1	4	yes	complemented
175	#135	—	2	4	no	—
176	C₂×M₁₆	C₂²×C₇₉	1	2	yes	complemented
177	C₄⋊Q₈	C₂²×C₇₉	1	1	yes	characteristic, complemented
178	#127	C₂²×C₇₉	1	1	yes	characteristic, complemented
179	#124	C₂²×C₇₉	1	1	yes	characteristic, complemented
180	#124	C₂²×C₇₉	1	1	yes	characteristic, complemented
181	#121	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₂×C₄²	C₂²×C₇₉	1	1	yes	characteristic, complemented
184	#104	C₂²×C₇₉	1	4	yes	complemented
185	#104	C₂²×C₇₉	1	4	yes	complemented
186	#101	C₂²×C₇₉	1	2	yes	complemented
187	#100	C₂²×C₇₉	1	2	yes	complemented
188	#41	—	2	4	no	—
189	#41	—	2	8	no	—
190	#41	—	2	4	no	—
191	#41	—	2	4	no	—
192	#41	—	2	4	no	—
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	#41	—	2	4	no	—
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	#41	D₄×C₇₉	1	4	yes	complemented
201	#41	C₂³×C₇₉	1	4	yes	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	D₄×C₇₉	1	2	yes	complemented
205	C₂×Q₈	C₂³×C₇₉	1	1	yes	characteristic
206	C₂×Q₈	C₂³×C₇₉	1	1	yes	characteristic
207	C₂×D₄	—	2	8	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	D₄×C₇₉	1	2	yes	—
215	C₂×D₄	D₄×C₇₉	1	2	yes	complemented
216	C₂×D₄	C₂³×C₇₉	1	1	yes	characteristic
217	C₂×D₄	C₂³×C₇₉	1	1	yes	characteristic
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	D₄×C₇₉	1	2	yes	—
223	C₂²×C₄	C₂³×C₇₉	1	1	yes	characteristic
224	Q₁₆	—	2	4	no	—
225	SD₁₆	—	2	4	no	—
226	SD₁₆	—	2	4	no	—
227	D₈	—	2	4	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	C₂³×C₇₉	1	4	yes	—
230	C₂×C₈	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₄²	—	2	1	no	—
235	C₄²	C₂³×C₇₉	1	1	yes	characteristic
236	C₄²	C₂³×C₇₉	1	1	yes	characteristic
237	C₂³	—	2	4	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	1	no	—
240	C₂³	C₂×D₄×C₇₉	1	2	yes	—
241	C₂³	C₂×D₄×C₇₉	1	1	yes	characteristic
242	Q₈	—	2	4	no	—
243	Q₈	—	2	2	no	—
244	Q₈	—	2	2	no	—
245	Q₈	—	2	2	no	—
246	Q₈	C₂×D₄×C₇₉	1	4	yes	—
247	D₄	—	2	8	no	—
248	D₄	—	2	4	no	—
249	D₄	—	2	4	no	—
250	D₄	—	2	4	no	—
251	D₄	—	2	4	no	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	2	no	—
255	D₄	—	4	4	no	—
256	D₄	—	4	2	no	—
257	D₄	C₂×D₄×C₇₉	1	4	yes	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	C₂×D₄×C₇₉	1	2	yes	—
272	C₂×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×C₇₉	1	2	yes	—
274	C₂×C₄	C₂×D₄×C₇₉	1	2	yes	—
275	C₂×C₄	C₂⁴×C₇₉	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	4	no	—
285	C₂²	#120×C₇₉	1	2	yes	—
286	C₂²	C₂²×D₄×C₇₉	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	#120×C₇₉	1	1	yes	characteristic
293	C₄	#120×C₇₉	1	1	yes	characteristic
294	C₄	C₂²×D₄×C₇₉	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	4	2	no	—
298	C₂	—	4	4	no	—
299	C₂	—	8	1	no	—
300	C₂	C₂×#120×C₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇₉	#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇₉
socle	C₂×C₇₉	C₂×#120
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #585×C₇₉	C₂	semidirect
2. #575×C₇₉	C₂	semidirect
3. #532×C₇₉	C₂	semidirect
4. #456×C₇₉	C₂	semidirect
5. #455×C₇₉	C₂	semidirect
6. #454×C₇₉	C₂	semidirect
7. #453×C₇₉	C₂	semidirect
8. C₂×#104×C₇₉	C₂	semidirect
9. #413×C₇₉	C₂	semidirect
19. C₂×#41×C₇₉	C₂²	semidirect
21. #137×C₇₉	C₂²	semidirect
23. AGL(1,ℤ/8)×C₇₉	C₂²	semidirect
25. C₂×M₁₆×C₇₉	C₂²	semidirect
26. C₄⋊Q₈×C₇₉	C₂²	semidirect
27. #127×C₇₉	C₂²	semidirect
28. #124×C₇₉	C₂²	semidirect
29. #124×C₇₉	C₂²	semidirect
32. C₂×C₄²×C₇₉	C₂²	semidirect
33. #104×C₇₉	C₂²	semidirect
34. #104×C₇₉	C₂²	semidirect
35. #101×C₇₉	C₂²	semidirect
36. #100×C₇₉	C₂²	semidirect
49. #41×C₇₉	D₄	semidirect
53. C₂×Q₈×C₇₉	D₄	semidirect
64. C₂×D₄×C₇₉	D₄	semidirect
150. #3018	C₇₉	direct (N × H)
151. C₇₉	#3018	direct (N × H)
152. #585	C₂×C₇₉	semidirect
153. #575	C₂×C₇₉	semidirect
154. #532	C₂×C₇₉	semidirect
155. #456	C₂×C₇₉	semidirect
156. #455	C₂×C₇₉	semidirect
157. #454	C₂×C₇₉	semidirect
158. #453	C₂×C₇₉	semidirect
159. C₂×#104	C₂×C₇₉	semidirect
160. #413	C₂×C₇₉	semidirect
170. C₂×#41	C₂²×C₇₉	semidirect
172. #137	C₂²×C₇₉	semidirect
174. AGL(1,ℤ/8)	C₂²×C₇₉	semidirect
176. C₂×M₁₆	C₂²×C₇₉	semidirect
177. C₄⋊Q₈	C₂²×C₇₉	semidirect
178. #127	C₂²×C₇₉	semidirect
179. #124	C₂²×C₇₉	semidirect
180. #124	C₂²×C₇₉	semidirect
183. C₂×C₄²	C₂²×C₇₉	semidirect
184. #104	C₂²×C₇₉	semidirect
185. #104	C₂²×C₇₉	semidirect
186. #101	C₂²×C₇₉	semidirect
187. #100	C₂²×C₇₉	semidirect
200. #41	D₄×C₇₉	semidirect
204. C₂×Q₈	D₄×C₇₉	semidirect
215. C₂×D₄	D₄×C₇₉	semidirect

Maximal subgroups

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

subgroup	index
#585×C₇₉	2
#575×C₇₉	2
#532×C₇₉	2
#456×C₇₉	2
#455×C₇₉	2
#454×C₇₉	2
#453×C₇₉	2
C₂×#104×C₇₉	2
#413×C₇₉	2
#3018	79

Subgroups — C79×#3018 #23760640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#3018 #23760640