Subgroups — #34443790

Order 20224.

514 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	#23773911	C₂	1	1	yes	maximal, characteristic
2	#500×D₇₉	C₂	1	1	yes	maximal, characteristic
3	#23765685	C₂	1	4	yes	maximal, complemented
4	#23765680	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765675	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765673	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765187	C₂	1	4	yes	maximal
8	C₄×#14579100	C₂	1	1	yes	maximal, characteristic
9	C₇₉×#2254	C₂	1	1	yes	maximal, characteristic, complemented
10	#19099118	—	2	1	no	—
11	C₄⋊Q₈×D₇₉	—	2	1	no	—
12	C₂×#14579100	C₂²	1	2	yes	—
13	#19098968	C₂²	1	1	yes	characteristic
14	#19098968	C₂²	1	1	yes	characteristic
15	#19098967	—	2	1	no	—
16	#19098967	C₂²	1	2	yes	—
17	#19098966	—	2	1	no	—
18	#19098964	—	2	1	no	—
19	#19098964	C₂²	1	2	yes	—
20	#19098959	—	2	1	no	—
21	#19098600	—	2	1	no	—
22	#19098600	C₂²	1	2	yes	—
23	#107×D₇₉	—	2	1	no	—
24	#107×D₇₉	C₂²	1	2	yes	—
25	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
26	#19098390	C₂²	1	4	yes	—
27	#19098389	C₂²	1	4	yes	—
28	#19098388	C₂²	1	4	yes	—
29	#19098338	C₂²	1	2	yes	—
30	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
31	#500×C₇₉	C₂²	1	1	yes	characteristic
32	#500×C₇₉	C₂²	1	1	yes	characteristic
33	#367×C₇₉	C₂²	1	4	yes	—
34	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
35	C₂×Q₈×D₇₉	—	4	1	no	—
36	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
37	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
38	#14579161	—	2	1	no	—
39	#14579161	—	2	1	no	—
40	#14579160	—	2	2	no	—
41	#14579160	—	4	1	no	—
42	#14579159	—	4	1	no	—
43	#14579152	—	2	2	no	—
44	#14579141	—	4	1	no	—
45	#14579137	—	4	1	no	—
46	Q₈×Dic₇₉	—	4	1	no	—
47	#14579126	—	4	1	no	—
48	#14579119	—	2	2	no	—
49	C₄⋊C₄×D₇₉	—	2	2	no	—
50	C₄⋊C₄×D₇₉	—	4	1	no	—
51	C₄²×D₇₉	D₄	1	1	yes	characteristic
52	#14579100	Q₈	1	8	yes	complemented
53	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
54	#14579095	—	2	1	no	—
55	#14579095	—	2	1	no	—
56	#14579095	C₂³	1	2	yes	—
57	#14579095	C₂³	1	2	yes	—
58	#14579094	—	2	2	no	—
59	#14579094	—	2	2	no	—
60	#14579092	—	2	2	no	—
61	#14579092	—	2	2	no	—
62	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
63	C₈×Dic₇₉	D₄	1	1	yes	characteristic
64	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₇₉	—	2	1	no	—
66	C₄⋊Q₈×C₇₉	—	2	1	no	—
67	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
68	#107×C₇₉	—	2	1	no	—
69	#107×C₇₉	—	2	1	no	—
70	#107×C₇₉	C₂³	1	2	yes	—
71	#107×C₇₉	C₂³	1	2	yes	—
72	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
73	C₂×Dic₁₅₈	—	4	2	no	—
74	C₂×Dic₁₅₈	—	4	1	no	—
75	C₂×Dic₁₅₈	—	4	1	no	—
76	Q₈×D₇₉	—	8	4	no	—
77	C₂×C₄×D₇₉	—	4	2	no	—
78	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
79	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
80	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
83	C₈×D₇₉	Q₁₆	1	4	yes	—
84	C₈×D₇₉	C₂×Q₈	1	4	yes	—
85	C₄⋊Dic₇₉	—	2	2	no	—
86	C₄⋊Dic₇₉	—	2	2	no	—
87	C₄⋊Dic₇₉	—	4	2	no	—
88	C₄⋊Dic₇₉	—	4	1	no	—
89	C₄⋊Dic₇₉	—	4	1	no	—
90	Dic₇₉⋊C₄	—	4	2	no	—
91	Dic₇₉⋊C₄	—	4	2	no	—
92	Dic₇₉⋊C₄	—	4	2	no	—
93	(C₂×D₇₉)⋊C₄	—	4	2	no	—
94	C₄×Dic₇₉	—	4	2	no	—
95	C₄×Dic₇₉	D₈	1	1	yes	characteristic
96	C₄×Dic₇₉	D₈	1	1	yes	characteristic
97	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
98	C₇₉⋊C₁₆	C₂×Q₈	1	4	yes	complemented
99	C₂×Q₈×C₇₉	—	4	1	no	—
100	C₂×Q₈×C₇₉	—	4	1	no	—
101	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₇₉	—	2	2	no	—
104	C₄⋊C₄×C₇₉	—	2	2	no	—
105	C₄⋊C₄×C₇₉	—	4	1	no	—
106	C₄⋊C₄×C₇₉	—	4	1	no	—
107	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
108	C₁₆×C₇₉	C₂×Q₈	1	4	yes	complemented
109	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
110	Q₈×C₇₉	—	8	2	no	—
111	Q₈×C₇₉	—	8	2	no	—
112	C₂×C₄×C₇₉	—	4	2	no	—
113	C₂×C₄×C₇₉	—	4	2	no	—
114	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₇₉	—	4	2	no	—
118	C₂×Dic₇₉	—	4	2	no	—
119	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
120	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
121	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
122	C₄×D₇₉	—	2	2	no	—
123	C₄×D₇₉	—	2	2	no	—
124	C₄×D₇₉	—	8	4	no	—
125	C₄×D₇₉	C₄⋊Q₈	1	2	yes	—
126	C₄×D₇₉	C₂×Q₁₆	1	2	yes	—
127	Dic₁₅₈	—	8	4	no	—
128	Dic₁₅₈	—	8	2	no	—
129	Dic₁₅₈	—	8	2	no	—
130	C₈×C₇₉	C₂×Q₁₆	1	2	yes	—
131	C₈×C₇₉	C₂²×Q₈	1	2	yes	—
132	C₇₉⋊C₈	C₄⋊Q₈	1	2	yes	—
133	C₇₉⋊C₈	C₂×Q₁₆	1	2	yes	—
134	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
135	C₂×D₇₉	—	2	1	no	—
136	C₂×D₇₉	—	2	1	no	—
137	C₂×D₇₉	#500	1	2	yes	—
138	C₄×C₇₉	—	8	2	no	—
139	C₄×C₇₉	—	8	2	no	—
140	C₄×C₇₉	#509	1	2	yes	—
141	C₄×C₇₉	#510	1	2	yes	—
142	C₄×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
143	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
144	Dic₇₉	—	2	1	no	—
145	Dic₇₉	—	2	1	no	—
146	Dic₇₉	—	8	2	no	—
147	Dic₇₉	—	8	2	no	—
148	Dic₇₉	#500	1	2	yes	—
149	#33731	—	79	1	no	maximal, Sylow
150	C₂×C₇₉	C₂×#500	1	1	yes	characteristic
151	C₂×C₇₉	C₂×#509	1	1	yes	characteristic
152	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
153	D₇₉	—	2	2	no	—
154	#3160	—	79	1	no	—
155	C₂×#500	—	79	1	no	—
156	#2255	—	79	4	no	—
157	#2254	—	79	1	no	—
158	#2254	D₇₉	1	1	yes	characteristic, complemented
159	#2252	—	79	1	no	—
160	#2252	—	79	1	no	—
161	#2159	—	79	4	no	—
162	C₄×M₃₂	—	79	1	no	—
163	C₇₉	#33731	1	1	yes	characteristic, complemented, Sylow
164	#533	—	158	1	no	—
165	C₂×C₄⋊Q₈	—	158	1	no	—
166	C₂×M₃₂	—	79	2	no	—
167	#500	—	79	2	no	—
168	#500	—	79	1	no	—
169	#500	—	79	1	no	—
170	#500	—	158	1	no	—
171	#500	—	158	1	no	—
172	#500	C₂×D₇₉	1	1	yes	characteristic
173	#500	C₂×D₇₉	1	1	yes	characteristic
174	#499	—	79	2	no	—
175	#499	—	158	1	no	—
176	#498	—	158	1	no	—
177	#427	—	79	2	no	—
178	#427	—	158	1	no	—
179	C₂×#107	—	79	2	no	—
180	C₂×#107	—	158	1	no	—
181	C₂×C₄×C₈	—	79	1	no	—
182	#367	—	79	4	no	—
183	#367	C₂×D₇₉	1	4	yes	—
184	#366	—	79	4	no	—
185	#366	—	79	4	no	—
186	#346	—	79	2	no	—
187	C₄×C₁₆	—	79	1	no	—
188	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
189	C₂²×Q₈	—	316	1	no	—
190	C₂²×C₈	—	79	1	no	—
191	C₂²×C₈	—	79	1	no	—
192	C₄⋊Q₈	—	2	1	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	316	1	no	—
195	C₄⋊Q₈	—	316	1	no	—
196	C₄⋊Q₈	—	158	2	no	—
197	C₄⋊Q₈	—	158	1	no	—
198	C₄⋊Q₈	—	158	1	no	—
199	#125	—	158	2	no	—
200	C₂²⋊Q₈	—	316	1	no	—
201	C₂²⋊Q₈	—	316	1	no	—
202	C₄×Q₈	—	316	1	no	—
203	C₄×Q₈	—	316	1	no	—
204	#117	—	158	2	no	—
205	C₂×C₄⋊C₄	—	316	1	no	—
206	C₂×C₄⋊C₄	—	158	2	no	—
207	C₂×C₄²	—	79	1	no	—
208	M₃₂	—	79	8	no	—
209	C₂×C₁₆	—	79	2	no	—
210	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	—	79	2	no	—
214	#107	—	79	2	no	—
215	#107	—	158	2	no	—
216	#107	—	158	2	no	—
217	#107	—	158	1	no	—
218	#107	—	158	1	no	—
219	#107	C₂²×D₇₉	1	2	yes	—
220	#107	C₂²×D₇₉	1	2	yes	—
221	#106	—	158	2	no	—
222	#106	—	158	2	no	—
223	C₄×C₈	—	79	1	no	—
224	C₄×C₈	—	79	1	no	—
225	C₄×C₈	—	79	1	no	—
226	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	316	2	no	—
230	C₂×Q₈	—	316	1	no	—
231	C₂×Q₈	—	316	1	no	—
232	C₂×Q₈	—	632	4	no	—
233	C₂²×C₄	—	316	2	no	—
234	C₂²×C₄	—	79	1	no	—
235	C₂²×C₄	—	79	1	no	—
236	C₂²×C₄	—	79	1	no	—
237	C₂×C₈	—	79	1	no	—
238	C₂×C₈	—	79	1	no	—
239	C₂×C₈	—	79	4	no	—
240	C₂×C₈	—	79	4	no	—
241	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
242	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	316	2	no	—
248	C₄⋊C₄	—	316	2	no	—
249	C₄⋊C₄	—	316	2	no	—
250	C₄⋊C₄	—	316	2	no	—
251	C₄⋊C₄	—	316	1	no	—
252	C₄⋊C₄	—	316	1	no	—
253	C₄⋊C₄	—	158	2	no	—
254	C₄⋊C₄	—	158	2	no	—
255	C₂²⋊C₄	—	316	2	no	—
256	C₄²	—	316	2	no	—
257	C₄²	—	79	1	no	—
258	C₄²	—	79	1	no	—
259	C₄²	—	79	1	no	—
260	C₄²	D₄×D₇₉	1	1	yes	characteristic
261	C₁₆	—	79	4	no	—
262	C₁₆	Q₈×D₇₉	1	4	yes	complemented
263	C₂³	—	79	1	no	—
264	Q₈	—	8	2	no	—
265	Q₈	—	8	2	no	—
266	Q₈	—	632	4	no	—
267	Q₈	—	632	2	no	—
268	Q₈	—	632	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	316	2	no	—
272	C₂×C₄	—	316	2	no	—
273	C₂×C₄	—	79	1	no	—
274	C₂×C₄	—	79	2	no	—
275	C₂×C₄	—	79	2	no	—
276	C₂×C₄	—	79	1	no	—
277	C₂×C₄	—	79	1	no	—
278	C₂×C₄	—	632	4	no	—
279	C₂×C₄	—	158	2	no	—
280	C₂×C₄	—	158	2	no	—
281	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
282	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
283	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
284	C₈	—	79	2	no	—
285	C₈	—	79	2	no	—
286	C₈	Q₁₆×D₇₉	1	2	yes	—
287	C₈	C₂×Q₈×D₇₉	1	2	yes	—
288	C₂²	—	79	2	no	—
289	C₂²	—	158	1	no	—
290	C₂²	—	158	1	no	—
291	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	—	79	2	no	—
295	C₄	—	632	2	no	—
296	C₄	—	632	2	no	—
297	C₄	—	158	1	no	—
298	C₄	—	158	1	no	—
299	C₄	#19099000	1	2	yes	—
300	C₄	#19099002	1	2	yes	—
301	C₄	C₄⋊Q₈×D₇₉	1	1	yes	characteristic
302	C₄	C₂×Q₁₆×D₇₉	1	1	yes	characteristic
303	C₂	—	158	2	no	—
304	C₂	#500×D₇₉	1	1	yes	characteristic
305	C₂	C₂×#19099000	1	1	yes	characteristic
306	C₂	C₂×#19099002	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2254	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂²×D₇₉	C₂×D₈
hypercenter	#2254	D₇₉
hypocenter	C₇₉	#33731
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

Complements

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

N	H ≅ G/N	type
3. #23765685	C₂	semidirect
4. #23765680	C₂	semidirect
5. #23765675	C₂	semidirect
6. #23765673	C₂	semidirect
9. C₇₉×#2254	C₂	semidirect
52. #14579100	Q₈	semidirect
98. C₇₉⋊C₁₆	C₂×Q₈	semidirect
108. C₁₆×C₇₉	C₂×Q₈	semidirect
158. #2254	D₇₉	semidirect
163. C₇₉	#33731	semidirect
262. C₁₆	Q₈×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23773911	2
#500×D₇₉	2
#23765685	2
#23765680	2
#23765675	2
#23765673	2
#23765187	2
C₄×#14579100	2
C₇₉×#2254	2
#33731	79