Subgroups — #34443445

Order 20224.

556 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23774059	C₂	1	1	yes	maximal, characteristic, complemented
2	#23765608	C₂	1	2	yes	maximal, complemented
3	#23765557	C₂	1	2	yes	maximal, complemented
4	#23765445	C₂	1	2	yes	maximal, complemented
5	#23765329	C₂	1	2	yes	maximal
6	#23765264	C₂	1	2	yes	maximal, complemented
7	C₄×#14579105	C₂	1	2	yes	maximal, complemented
8	#23765207	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₉×#2948	C₂	1	1	yes	maximal, characteristic
10	#19099175	—	2	1	no	—
11	C₂×#14579105	C₂²	1	2	yes	—
12	C₂×#14579105	C₂²	1	2	yes	—
13	#19098965	C₂²	1	1	yes	characteristic, complemented
14	#19098943	C₂²	1	1	yes	characteristic, complemented
15	#19098932	—	2	1	no	—
16	#19098821	—	2	1	no	—
17	#19098804	C₂²	1	2	yes	—
18	#19098671	—	2	1	no	—
19	C₄×Dic₃₁₆	C₂²	1	2	yes	complemented
20	#19098390	C₂²	1	2	yes	complemented
21	#19098388	C₂²	1	2	yes	complemented
22	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	complemented
23	#19098366	C₂²	1	2	yes	complemented
24	#19098366	C₂²	1	2	yes	complemented
25	#19098364	C₂²	1	2	yes	—
26	#19098364	C₂²	1	2	yes	—
27	C₂²⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
28	#19098339	C₂²	1	1	yes	characteristic, complemented
29	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
30	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
31	#518×C₇₉	—	2	1	no	—
32	#442×C₇₉	—	2	1	no	—
33	#442×C₇₉	—	2	1	no	—
34	C₄×D₈×C₇₉	C₂²	1	2	yes	—
35	C₄×D₈×C₇₉	C₂²	1	2	yes	—
36	#427×C₇₉	C₂²	1	1	yes	characteristic
37	#418×C₇₉	—	2	1	no	—
38	#405×C₇₉	C₂²	1	1	yes	characteristic
39	C₂×Dic₃₁₆	—	2	1	no	—
40	C₂×Dic₃₁₆	C₂³	1	2	yes	—
41	#14579161	—	2	1	no	—
42	#14579154	—	4	2	no	—
43	#14579153	—	2	1	no	—
44	#14579151	—	4	2	no	—
45	#14579140	—	2	2	no	—
46	C₄×Dic₁₅₈	—	2	2	no	—
47	#14579105	—	2	4	no	—
48	#14579105	—	2	4	no	—
49	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
50	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
51	#14579095	C₂³	1	2	yes	—
52	#14579093	—	2	1	no	—
53	#14579086	—	2	2	no	—
54	#14579086	—	2	2	no	—
55	C₂²×D₄×C₇₉	—	2	1	no	—
56	C₂×D₈×C₇₉	—	2	4	no	—
57	C₂×D₈×C₇₉	—	2	1	no	—
58	C₂×D₈×C₇₉	C₂³	1	2	yes	—
59	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
61	C₄×D₄×C₇₉	—	2	2	no	—
62	C₄×D₄×C₇₉	—	2	2	no	—
63	C₄×D₄×C₇₉	—	4	2	no	—
64	#117×C₇₉	—	2	1	no	—
65	#117×C₇₉	D₄	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₇₉	—	4	2	no	—
67	#107×C₇₉	C₂³	1	2	yes	—
68	#106×C₇₉	—	2	1	no	—
69	#102×C₇₉	—	2	2	no	—
70	#102×C₇₉	—	2	2	no	—
71	#97×C₇₉	D₄	1	1	yes	characteristic
72	#97×C₇₉	D₄	1	1	yes	characteristic
73	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₅₈	—	2	2	no	—
76	Dic₃₁₆	—	2	2	no	—
77	Dic₃₁₆	—	2	2	no	—
78	Dic₃₁₆	—	4	2	no	—
79	C₄⋊Dic₇₉	—	2	2	no	—
80	C₄⋊Dic₇₉	—	4	2	no	—
81	Dic₇₉⋊C₄	—	4	2	no	—
82	Dic₇₉⋊C₄	—	4	2	no	—
83	Dic₇₉⋊C₄	—	4	2	no	—
84	C₂²⋊Dic₇₉	—	4	2	no	—
85	C₄×Dic₇₉	—	4	2	no	—
86	C₇₉⋊C₁₆	—	2	2	no	—
87	C₇₉⋊C₁₆	—	2	2	no	—
88	C₂⁴×C₇₉	—	4	1	no	—
89	C₂×D₄×C₇₉	—	2	4	no	—
90	C₂×D₄×C₇₉	—	2	2	no	—
91	C₂×D₄×C₇₉	—	4	2	no	—
92	C₂×D₄×C₇₉	—	4	1	no	—
93	C₂²×C₄×C₇₉	—	4	2	no	—
94	C₂²×C₄×C₇₉	—	4	2	no	—
95	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
96	D₈×C₇₉	—	2	4	no	—
97	D₈×C₇₉	—	2	2	no	—
98	D₈×C₇₉	#41	1	4	yes	—
99	M₁₆×C₇₉	—	4	1	no	—
100	C₂×C₈×C₇₉	—	2	2	no	—
101	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₇₉	—	2	2	no	—
106	C₄⋊C₄×C₇₉	D₈	1	2	yes	—
107	C₂²⋊C₄×C₇₉	—	4	1	no	—
108	C₂²⋊C₄×C₇₉	—	4	2	no	—
109	C₂²⋊C₄×C₇₉	—	4	2	no	—
110	C₂²⋊C₄×C₇₉	D₈	1	2	yes	—
111	C₄²×C₇₉	—	4	1	no	—
112	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂³×C₇₉	—	4	4	no	—
115	C₂³×C₇₉	—	4	2	no	—
116	C₂³×C₇₉	—	16	2	no	—
117	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
118	D₄×C₇₉	—	2	4	no	—
119	D₄×C₇₉	—	4	4	no	—
120	D₄×C₇₉	—	4	2	no	—
121	C₂×C₄×C₇₉	—	2	1	no	—
122	C₂×C₄×C₇₉	—	2	1	no	—
123	C₂×C₄×C₇₉	—	4	2	no	—
124	C₂×C₄×C₇₉	—	16	2	no	—
125	C₂×C₄×C₇₉	—	16	2	no	—
126	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₇₉	—	4	2	no	—
131	C₂×Dic₇₉	—	4	2	no	—
132	Dic₁₅₈	—	4	2	no	—
133	Dic₁₅₈	—	4	2	no	—
134	C₈×C₇₉	—	2	1	no	—
135	C₈×C₇₉	—	2	1	no	—
136	C₈×C₇₉	—	2	1	no	—
137	C₈×C₇₉	C₂×#41	1	2	yes	—
138	C₂²×C₇₉	—	2	1	no	—
139	C₂²×C₇₉	—	2	2	no	—
140	C₂²×C₇₉	—	4	4	no	—
141	C₂²×C₇₉	—	8	2	no	—
142	C₂²×C₇₉	—	8	2	no	—
143	C₂²×C₇₉	—	16	4	no	—
144	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
145	C₄×C₇₉	—	2	1	no	—
146	C₄×C₇₉	—	2	2	no	—
147	C₄×C₇₉	—	2	2	no	—
148	C₄×C₇₉	—	8	2	no	—
149	C₄×C₇₉	#525	1	1	yes	characteristic
150	C₄×C₇₉	#575	1	1	yes	characteristic
151	Dic₇₉	—	8	2	no	—
152	Dic₇₉	—	8	2	no	—
153	#33683	—	79	1	no	maximal, Sylow
154	C₂×C₇₉	—	4	1	no	—
155	C₂×C₇₉	—	8	4	no	—
156	C₂×C₇₉	#3100	1	1	yes	characteristic
157	C₂×C₇₉	#3416	1	2	yes	—
158	#3177	—	79	1	no	—
159	#2948	D₇₉	1	1	yes	characteristic
160	#2237	—	79	2	no	—
161	#2228	—	79	2	no	—
162	#2208	—	79	2	no	—
163	#2187	—	79	2	no	—
164	#2175	—	79	2	no	—
165	C₄×SD₃₂	—	79	2	no	—
166	#2164	—	79	1	no	—
167	C₇₉	#33683	1	1	yes	characteristic, complemented, Sylow
168	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
169	#541	—	158	1	no	—
170	#518	—	2	1	no	—
171	C₂×SD₃₂	—	79	2	no	—
172	C₂×SD₃₂	—	79	2	no	—
173	#499	—	79	1	no	—
174	#494	—	79	1	no	—
175	#491	—	158	1	no	—
176	#470	—	158	1	no	—
177	#467	—	79	2	no	—
178	#442	—	2	1	no	—
179	#442	—	2	1	no	—
180	#441	—	158	1	no	—
181	C₄×Q₁₆	—	79	2	no	—
182	C₄×D₈	C₂×D₇₉	1	2	yes	—
183	C₄×D₈	C₂×D₇₉	1	2	yes	—
184	#427	C₂×D₇₉	1	1	yes	characteristic
185	#418	—	2	1	no	—
186	#405	C₂×D₇₉	1	1	yes	characteristic
187	#367	—	79	2	no	—
188	#366	—	79	2	no	—
189	C₄⋊C₁₆	—	79	2	no	—
190	#358	—	79	2	no	—
191	#358	—	79	2	no	—
192	#357	—	79	2	no	—
193	#357	—	79	2	no	—
194	C₂²⋊C₁₆	—	79	2	no	—
195	#346	—	79	1	no	—
196	C₄×C₁₆	—	79	1	no	—
197	C₂²×D₄	—	2	1	no	—
198	C₂×Q₁₆	—	79	2	no	—
199	C₂×Q₁₆	—	158	1	no	—
200	C₂×D₈	—	2	4	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	C₂²×D₇₉	1	2	yes	—
203	C₂×M₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
205	C₄⋊Q₈	—	158	1	no	—
206	#126	—	316	2	no	—
207	#125	—	316	2	no	—
208	#125	—	158	1	no	—
209	C₂²⋊Q₈	—	158	2	no	—
210	C₄×Q₈	—	158	2	no	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	4	2	no	—
214	#117	—	2	1	no	—
215	#117	C₂²⋊D₇₉	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	4	2	no	—
217	SD₃₂	—	158	4	no	—
218	SD₃₂	—	158	4	no	—
219	C₂×C₁₆	—	79	2	no	—
220	C₂×C₁₆	—	79	2	no	—
221	#107	—	79	2	no	—
222	#107	C₂²×D₇₉	1	2	yes	—
223	#106	—	2	1	no	—
224	#106	—	158	1	no	—
225	#103	—	158	2	no	—
226	#103	—	158	2	no	—
227	#102	—	2	2	no	—
228	#102	—	2	2	no	—
229	#97	C₂²⋊D₇₉	1	1	yes	characteristic
230	#97	C₂²⋊D₇₉	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
233	C₂⁴	—	4	1	no	—
234	C₂×Q₈	—	158	2	no	—
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
242	Q₁₆	—	316	2	no	—
243	Q₁₆	—	158	2	no	—
244	Q₁₆	—	158	2	no	—
245	D₈	—	2	4	no	—
246	D₈	—	2	2	no	—
247	D₈	#11929624	1	4	yes	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
252	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	316	2	no	—
256	C₄⋊C₄	—	316	2	no	—
257	C₄⋊C₄	—	316	2	no	—
258	C₄⋊C₄	—	316	2	no	—
259	C₄⋊C₄	—	158	2	no	—
260	C₄⋊C₄	#11929609	1	2	yes	—
261	C₂²⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	316	2	no	—
265	C₂²⋊C₄	#11929609	1	2	yes	—
266	C₄²	—	4	1	no	—
267	C₄²	—	316	2	no	—
268	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
270	C₁₆	—	158	2	no	—
271	C₁₆	—	158	2	no	—
272	C₂³	—	4	4	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	16	2	no	—
275	C₂³	C₂×#11929609	1	1	yes	characteristic
276	Q₈	—	316	2	no	—
277	Q₈	—	316	2	no	—
278	D₄	—	2	4	no	—
279	D₄	—	4	4	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	16	2	no	—
285	C₂×C₄	—	16	2	no	—
286	C₂×C₄	—	316	2	no	—
287	C₂×C₄	—	316	2	no	—
288	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
289	C₂×C₄	C₂×#11929609	1	2	yes	—
290	C₂×C₄	C₂×#11929609	1	2	yes	—
291	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	C₂×#11929624	1	2	yes	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	16	4	no	—
302	C₂²	C₂²×#11929609	1	1	yes	characteristic
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	8	2	no	—
307	C₄	—	632	2	no	—
308	C₄	—	632	2	no	—
309	C₄	#19099074	1	1	yes	characteristic
310	C₄	#19099361	1	1	yes	characteristic
311	C₂	—	4	1	no	—
312	C₂	—	8	4	no	—
313	C₂	#23773400	1	1	yes	characteristic
314	C₂	#23776547	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2948	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂²×D₈×C₇₉	C₂²
hypercenter	#2948	D₇₉
hypocenter	C₇₉	#33683
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2948

Complements

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

N	H ≅ G/N	type
1. #23774059	C₂	semidirect
2. #23765608	C₂	semidirect
3. #23765557	C₂	semidirect
4. #23765445	C₂	semidirect
6. #23765264	C₂	semidirect
7. C₄×#14579105	C₂	semidirect
8. #23765207	C₂	semidirect
13. #19098965	C₂²	semidirect
14. #19098943	C₂²	semidirect
19. C₄×Dic₃₁₆	C₂²	semidirect
20. #19098390	C₂²	semidirect
21. #19098388	C₂²	semidirect
22. C₄⋊(C₇₉⋊C₁₆)	C₂²	semidirect
23. #19098366	C₂²	semidirect
24. #19098366	C₂²	semidirect
28. #19098339	C₂²	semidirect
29. C₄×C₇₉⋊C₁₆	C₂²	semidirect
167. C₇₉	#33683	semidirect

Maximal subgroups

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

subgroup	index
#23774059	2
#23765608	2
#23765557	2
#23765445	2
#23765329	2
#23765264	2
C₄×#14579105	2
#23765207	2
C₇₉×#2948	2
#33683	79