Subgroups — #34193338

Order 20224.

494 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23764207	C₂	1	1	yes	maximal, characteristic
2	#23764207	C₂	1	1	yes	maximal, characteristic
3	#23763556	C₂	1	1	yes	maximal, characteristic, complemented
4	#23763528	C₂	1	1	yes	maximal, characteristic, complemented
5	#23763527	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1957	C₂	1	1	yes	maximal, characteristic
8	C₂×#14579092	—	2	2	no	—
9	C₂×#14579082	—	2	2	no	—
10	C₂×#14579082	—	2	2	no	—
11	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	#19098433	C₂²	1	1	yes	characteristic
14	#19098433	C₂²	1	1	yes	characteristic
15	#19098326	—	2	2	no	—
16	#19098325	—	2	2	no	—
17	C₂×#127×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×#102×C₇₉	—	2	2	no	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	#384×C₇₉	C₂²	1	1	yes	characteristic
21	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
22	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
23	C₂×C₄⋊Dic₇₉	—	2	1	no	—
24	C₂×C₄⋊Dic₇₉	—	2	1	no	—
25	C₂×C₄⋊Dic₇₉	—	2	1	no	—
26	C₂×C₄⋊Dic₇₉	—	2	1	no	—
27	C₂×C₄⋊Dic₇₉	—	2	1	no	—
28	C₂×C₄⋊Dic₇₉	—	2	1	no	—
29	#14579092	—	4	4	no	—
30	#14579082	—	4	4	no	—
31	#14579082	—	4	4	no	—
32	C₄×C₇₉⋊C₈	—	2	1	no	—
33	C₄×C₇₉⋊C₈	—	2	1	no	—
34	C₄×C₇₉⋊C₈	—	2	1	no	—
35	C₄×C₇₉⋊C₈	—	2	1	no	—
36	C₂²×D₄×C₇₉	—	2	1	no	—
37	C₂²×D₄×C₇₉	—	2	1	no	—
38	C₂²×D₄×C₇₉	—	2	1	no	—
39	C₂²×C₈×C₇₉	D₄	1	2	yes	—
40	#127×C₇₉	—	2	1	no	—
41	#127×C₇₉	—	2	1	no	—
42	#127×C₇₉	—	2	1	no	—
43	#127×C₇₉	Q₈	1	2	yes	—
44	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
47	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
48	#102×C₇₉	—	4	4	no	—
49	C₄×C₈×C₇₉	—	2	1	no	—
50	C₄×C₈×C₇₉	—	2	1	no	—
51	C₂²×Dic₇₉	—	4	1	no	—
52	C₂²×Dic₇₉	—	4	1	no	—
53	C₂×C₇₉⋊C₈	—	2	2	no	—
54	C₂×C₇₉⋊C₈	—	2	2	no	—
55	C₂×C₇₉⋊C₈	—	2	2	no	—
56	C₂×C₇₉⋊C₈	—	2	2	no	—
57	C₂×C₇₉⋊C₈	—	2	2	no	—
58	C₂×C₇₉⋊C₈	SD₁₆	1	4	yes	—
59	C₄⋊Dic₇₉	—	4	2	no	—
60	C₄⋊Dic₇₉	—	4	2	no	—
61	C₄⋊Dic₇₉	—	4	2	no	—
62	C₄⋊Dic₇₉	—	4	2	no	—
63	C₄⋊Dic₇₉	—	4	2	no	—
64	C₄⋊Dic₇₉	—	4	2	no	—
65	C₂⁴×C₇₉	—	4	1	no	—
66	C₂×D₄×C₇₉	—	2	2	no	—
67	C₂×D₄×C₇₉	—	2	2	no	—
68	C₂×D₄×C₇₉	—	2	2	no	—
69	C₂×D₄×C₇₉	—	4	1	no	—
70	C₂×D₄×C₇₉	—	4	1	no	—
71	C₂×D₄×C₇₉	—	4	1	no	—
72	C₂×D₄×C₇₉	—	4	2	no	—
73	C₂×D₄×C₇₉	—	4	2	no	—
74	C₂×D₄×C₇₉	—	4	2	no	—
75	C₂×D₄×C₇₉	—	4	2	no	—
76	C₂×D₄×C₇₉	—	4	2	no	—
77	C₂×D₄×C₇₉	—	4	2	no	—
78	C₂²×C₄×C₇₉	—	4	1	no	—
79	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₇₉	—	2	2	no	—
83	C₂×C₈×C₇₉	—	2	2	no	—
84	C₂×C₈×C₇₉	—	2	2	no	—
85	C₄⋊C₄×C₇₉	—	4	2	no	—
86	C₄⋊C₄×C₇₉	—	4	2	no	—
87	C₄⋊C₄×C₇₉	—	4	2	no	—
88	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
89	C₄²×C₇₉	#41	1	1	yes	characteristic
90	C₄²×C₇₉	#41	1	1	yes	characteristic
91	C₄²×C₇₉	#41	1	1	yes	characteristic
92	C₂³×C₇₉	—	4	2	no	—
93	C₂³×C₇₉	—	8	1	no	—
94	C₂³×C₇₉	—	8	1	no	—
95	C₂³×C₇₉	—	8	1	no	—
96	C₂³×C₇₉	—	8	1	no	—
97	C₂³×C₇₉	—	8	1	no	—
98	C₂³×C₇₉	—	8	1	no	—
99	C₂³×C₇₉	#127	1	1	yes	characteristic
100	D₄×C₇₉	—	4	4	no	—
101	D₄×C₇₉	—	4	4	no	—
102	D₄×C₇₉	—	4	4	no	—
103	D₄×C₇₉	—	8	2	no	—
104	D₄×C₇₉	—	8	2	no	—
105	D₄×C₇₉	—	8	2	no	—
106	C₂×C₄×C₇₉	—	8	1	no	—
107	C₂×C₄×C₇₉	—	8	1	no	—
108	C₂×C₄×C₇₉	—	8	1	no	—
109	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
113	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
114	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
116	C₂×C₄×C₇₉	#135	1	2	yes	—
117	C₂×C₄×C₇₉	#135	1	2	yes	—
118	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×Dic₇₉	—	8	1	no	—
122	C₂×Dic₇₉	—	8	1	no	—
123	C₂×Dic₇₉	—	8	1	no	—
124	C₂×Dic₇₉	—	8	1	no	—
125	C₂×Dic₇₉	—	8	1	no	—
126	C₂×Dic₇₉	—	8	1	no	—
127	C₈×C₇₉	—	4	2	no	—
128	C₇₉⋊C₈	—	2	4	no	—
129	C₇₉⋊C₈	—	4	2	no	—
130	C₂²×C₇₉	—	8	2	no	—
131	C₂²×C₇₉	—	8	2	no	—
132	C₂²×C₇₉	—	8	2	no	—
133	C₂²×C₇₉	—	16	1	no	—
134	C₂²×C₇₉	—	16	1	no	—
135	C₂²×C₇₉	—	16	1	no	—
136	C₂²×C₇₉	—	16	1	no	—
137	C₂²×C₇₉	#399	1	1	yes	characteristic
138	C₂²×C₇₉	#492	1	1	yes	characteristic
139	C₂²×C₇₉	#495	1	1	yes	characteristic
140	C₂²×C₇₉	#495	1	1	yes	characteristic
141	C₂²×C₇₉	#496	1	1	yes	characteristic
142	C₂²×C₇₉	#496	1	1	yes	characteristic
143	C₂²×C₇₉	#496	1	1	yes	characteristic
144	C₄×C₇₉	—	16	1	no	—
145	C₄×C₇₉	#476	1	2	yes	—
146	C₄×C₇₉	#478	1	2	yes	—
147	C₄×C₇₉	#478	1	2	yes	—
148	C₄×C₇₉	#481	1	2	yes	—
149	C₄×C₇₉	#482	1	2	yes	—
150	C₄×C₇₉	#482	1	2	yes	—
151	Dic₇₉	—	16	1	no	—
152	Dic₇₉	—	16	1	no	—
153	#10677	—	79	1	no	maximal, Sylow
154	C₂×C₇₉	—	16	2	no	—
155	C₂×C₇₉	#1708	1	1	yes	characteristic
156	C₂×C₇₉	#1715	1	1	yes	characteristic
157	C₂×C₇₉	#1717	1	1	yes	characteristic
158	C₂×C₇₉	#1717	1	1	yes	characteristic
159	C₂×C₇₉	#2092	1	1	yes	characteristic
160	C₂×C₇₉	#2092	1	1	yes	characteristic
161	C₂×C₇₉	#2094	1	1	yes	characteristic
162	#1957	—	79	1	no	—
163	#1957	—	79	1	no	—
164	#1957	D₇₉	1	1	yes	characteristic
165	#1844	—	79	1	no	—
166	#1834	—	79	1	no	—
167	#1834	—	79	1	no	—
168	#1273	—	79	1	no	—
169	C₇₉	#10677	1	1	yes	characteristic, complemented, Sylow
170	C₂×#127	C₂×D₇₉	1	1	yes	characteristic
171	C₂×#106	—	158	2	no	—
172	C₂×#102	—	2	2	no	—
173	C₂×#102	—	158	2	no	—
174	C₂×#102	—	158	2	no	—
175	C₂×C₄×C₈	—	79	1	no	—
176	C₂×C₄×C₈	—	79	1	no	—
177	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
178	#384	—	79	1	no	—
179	#384	—	79	1	no	—
180	#384	C₂×D₇₉	1	1	yes	characteristic
181	#340	—	158	2	no	—
182	#340	—	158	2	no	—
183	C₂²×D₄	—	2	1	no	—
184	C₂²×D₄	—	2	1	no	—
185	C₂²×D₄	—	2	1	no	—
186	C₂²×C₈	—	79	2	no	—
187	C₂²×C₈	—	79	2	no	—
188	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
189	#127	—	2	1	no	—
190	#127	—	2	1	no	—
191	#127	—	2	1	no	—
192	#127	Dic₁₅₈	1	2	yes	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	158	1	no	—
197	C₂×C₄⋊C₄	—	158	1	no	—
198	C₂×C₄⋊C₄	—	158	1	no	—
199	C₂×C₄⋊C₄	—	158	1	no	—
200	C₂×C₄⋊C₄	—	158	1	no	—
201	C₂×C₄⋊C₄	—	158	1	no	—
202	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
203	#106	—	316	4	no	—
204	#102	—	4	4	no	—
205	#102	—	316	4	no	—
206	#102	—	316	4	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	158	1	no	—
210	C₄×C₈	—	158	1	no	—
211	C₄×C₈	—	158	1	no	—
212	C₄×C₈	—	158	1	no	—
213	C₂⁴	—	4	1	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	4	1	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	316	1	no	—
228	C₂²×C₄	—	316	1	no	—
229	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
230	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	79	4	no	—
236	C₂×C₈	—	158	2	no	—
237	C₂×C₈	—	158	2	no	—
238	C₂×C₈	—	158	2	no	—
239	C₂×C₈	—	158	2	no	—
240	C₂×C₈	—	158	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	316	2	no	—
245	C₄⋊C₄	—	316	2	no	—
246	C₄⋊C₄	—	316	2	no	—
247	C₄⋊C₄	—	316	2	no	—
248	C₄⋊C₄	—	316	2	no	—
249	C₄⋊C₄	—	316	2	no	—
250	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
251	C₄²	#11929623	1	1	yes	characteristic
252	C₄²	#11929623	1	1	yes	characteristic
253	C₄²	#11929624	1	1	yes	characteristic
254	C₂³	—	4	2	no	—
255	C₂³	—	8	1	no	—
256	C₂³	—	8	1	no	—
257	C₂³	—	8	1	no	—
258	C₂³	—	8	1	no	—
259	C₂³	—	8	1	no	—
260	C₂³	—	8	1	no	—
261	C₂³	#14579157	1	1	yes	characteristic
262	D₄	—	4	4	no	—
263	D₄	—	4	4	no	—
264	D₄	—	4	4	no	—
265	D₄	—	8	2	no	—
266	D₄	—	8	2	no	—
267	D₄	—	8	2	no	—
268	C₂×C₄	—	8	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	632	1	no	—
272	C₂×C₄	—	632	1	no	—
273	C₂×C₄	—	632	1	no	—
274	C₂×C₄	—	632	1	no	—
275	C₂×C₄	—	632	1	no	—
276	C₂×C₄	—	632	1	no	—
277	C₂×C₄	#14579138	1	1	yes	characteristic
278	C₂×C₄	#14579138	1	1	yes	characteristic
279	C₂×C₄	#14579140	1	1	yes	characteristic
280	C₂×C₄	#14579142	1	1	yes	characteristic
281	C₂×C₄	#14579143	1	1	yes	characteristic
282	C₂×C₄	#14579143	1	1	yes	characteristic
283	C₂×C₄	SD₁₆×D₇₉	1	2	yes	—
284	C₂×C₄	#14579180	1	2	yes	—
285	C₂×C₄	#14579181	1	2	yes	—
286	C₂×C₄	#14579185	1	2	yes	—
287	C₂×C₄	#14579187	1	2	yes	—
288	C₂×C₄	#14579188	1	2	yes	—
289	C₈	—	4	2	no	—
290	C₈	—	316	2	no	—
291	C₈	—	158	4	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	16	1	no	—
296	C₂²	—	16	1	no	—
297	C₂²	—	16	1	no	—
298	C₂²	—	16	1	no	—
299	C₂²	#19098499	1	1	yes	characteristic
300	C₂²	#19098934	1	1	yes	characteristic
301	C₂²	#19098945	1	1	yes	characteristic
302	C₂²	#19098946	1	1	yes	characteristic
303	C₂²	#19098949	1	1	yes	characteristic
304	C₂²	#19098950	1	1	yes	characteristic
305	C₂²	#19098951	1	1	yes	characteristic
306	C₄	—	16	1	no	—
307	C₄	—	1264	1	no	—
308	C₄	—	1264	1	no	—
309	C₄	#19098851	1	2	yes	—
310	C₄	#19098861	1	2	yes	—
311	C₄	#19098863	1	2	yes	—
312	C₄	#19098880	1	2	yes	—
313	C₄	#19098884	1	2	yes	—
314	C₄	#19098885	1	2	yes	—
315	C₂	—	16	2	no	—
316	C₂	#23763000	1	1	yes	characteristic
317	C₂	#23763027	1	1	yes	characteristic
318	C₂	#23763038	1	1	yes	characteristic
319	C₂	#23763040	1	1	yes	characteristic
320	C₂	#23764919	1	1	yes	characteristic
321	C₂	#23764921	1	1	yes	characteristic
322	C₂	#23764931	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579157
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1957	C₂
socle	C₂³×C₇₉	#127
prime radical	C₂×#127×C₇₉	C₂²
hypercenter	#1957	D₇₉
hypocenter	C₇₉	#10677
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1957

Complements

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

N	H ≅ G/N	type
3. #23763556	C₂	semidirect
4. #23763528	C₂	semidirect
5. #23763527	C₂	semidirect
6. C₇₉⋊#1273	C₂	semidirect
169. C₇₉	#10677	semidirect

Maximal subgroups

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

subgroup	index
#23764207	2
#23763556	2
#23763528	2
#23763527	2
C₇₉⋊#1273	2
C₇₉×#1957	2
#10677	79