Subgroups — #34181696

Order 20224.

514 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	#23763380	C₂	1	1	yes	maximal, characteristic
2	#23763380	C₂	1	1	yes	maximal, characteristic
3	#23761283	C₂	1	2	yes	maximal, complemented
4	#23761283	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2285	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
9	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
10	#19098318	—	2	1	no	—
11	#19098318	—	2	1	no	—
12	C₂×#117×C₇₉	—	2	1	no	—
13	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
14	#384×C₇₉	C₂²	1	1	yes	characteristic
15	#384×C₇₉	C₂²	1	1	yes	characteristic
16	#382×C₇₉	—	2	1	no	—
17	#380×C₇₉	—	2	1	no	—
18	#380×C₇₉	C₄	1	2	yes	—
19	C₄×C₄⋊C₄×C₇₉	—	2	1	no	—
20	C₄×C₄⋊C₄×C₇₉	C₄	1	2	yes	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	C₂²×C₇₉⋊C₈	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
25	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
26	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
27	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
30	#117×C₇₉	—	4	2	no	—
31	#117×C₇₉	—	4	2	no	—
32	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
38	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
39	C₂×C₄²×C₇₉	—	2	4	no	—
40	C₂×C₄²×C₇₉	—	2	2	no	—
41	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
45	#95×C₇₉	—	2	2	no	—
46	#95×C₇₉	—	2	2	no	—
47	C₂×C₇₉⋊C₈	—	4	2	no	—
48	C₂×C₇₉⋊C₈	—	4	2	no	—
49	C₂×C₇₉⋊C₈	—	4	2	no	—
50	C₂×C₇₉⋊C₈	—	4	2	no	—
51	C₂×C₇₉⋊C₈	—	4	2	no	—
52	C₂×C₇₉⋊C₈	—	4	2	no	—
53	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	—	2	2	no	—
55	C₂²×C₄×C₇₉	—	2	2	no	—
56	C₂²×C₄×C₇₉	—	2	2	no	—
57	C₂²×C₄×C₇₉	—	2	2	no	—
58	C₂²×C₄×C₇₉	—	2	2	no	—
59	C₂²×C₄×C₇₉	—	2	1	no	—
60	C₂²×C₄×C₇₉	—	2	1	no	—
61	C₂²×C₄×C₇₉	—	2	2	no	—
62	C₂²×C₄×C₇₉	—	2	2	no	—
63	C₂²×C₄×C₇₉	—	4	1	no	—
64	C₂²×C₄×C₇₉	—	4	1	no	—
65	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	D₈	1	2	yes	—
69	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₇₉	—	4	2	no	—
74	C₄⋊C₄×C₇₉	—	4	2	no	—
75	C₄⋊C₄×C₇₉	—	4	1	no	—
76	C₄⋊C₄×C₇₉	—	4	1	no	—
77	C₄⋊C₄×C₇₉	—	4	1	no	—
78	C₄⋊C₄×C₇₉	—	4	1	no	—
79	C₄⋊C₄×C₇₉	—	4	2	no	—
80	C₄⋊C₄×C₇₉	—	4	1	no	—
81	C₄⋊C₄×C₇₉	—	4	1	no	—
82	C₂²⋊C₄×C₇₉	—	4	4	no	—
83	C₄²×C₇₉	—	2	2	no	—
84	C₄²×C₇₉	—	2	2	no	—
85	C₄²×C₇₉	—	2	2	no	—
86	C₄²×C₇₉	—	4	4	no	—
87	C₄²×C₇₉	#41	1	4	yes	—
88	C₂³×C₇₉	—	2	1	no	—
89	C₂³×C₇₉	—	2	1	no	—
90	C₂³×C₇₉	—	2	1	no	—
91	C₂³×C₇₉	—	2	1	no	—
92	C₂³×C₇₉	—	2	1	no	—
93	C₂³×C₇₉	—	2	1	no	—
94	C₂³×C₇₉	#102	1	2	yes	—
95	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇₉	—	2	1	no	—
97	C₂×C₄×C₇₉	—	2	1	no	—
98	C₂×C₄×C₇₉	—	2	1	no	—
99	C₂×C₄×C₇₉	—	2	1	no	—
100	C₂×C₄×C₇₉	—	2	4	no	—
101	C₂×C₄×C₇₉	—	2	4	no	—
102	C₂×C₄×C₇₉	—	2	1	no	—
103	C₂×C₄×C₇₉	—	2	1	no	—
104	C₂×C₄×C₇₉	—	2	1	no	—
105	C₂×C₄×C₇₉	—	2	1	no	—
106	C₂×C₄×C₇₉	—	2	1	no	—
107	C₂×C₄×C₇₉	—	2	1	no	—
108	C₂×C₄×C₇₉	—	4	4	no	—
109	C₂×C₄×C₇₉	—	4	2	no	—
110	C₂×C₄×C₇₉	—	4	2	no	—
111	C₂×C₄×C₇₉	—	4	2	no	—
112	C₂×C₄×C₇₉	—	4	2	no	—
113	C₂×C₄×C₇₉	—	4	2	no	—
114	C₂×C₄×C₇₉	—	4	2	no	—
115	C₂×C₄×C₇₉	—	4	2	no	—
116	C₂×C₄×C₇₉	—	4	2	no	—
117	C₂×C₄×C₇₉	#102	1	2	yes	—
118	C₂×C₄×C₇₉	#117	1	2	yes	—
119	C₂×C₄×C₇₉	#123	1	2	yes	—
120	C₂×C₄×C₇₉	#123	1	2	yes	—
121	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₇₉	#135	1	2	yes	—
124	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
126	C₂×C₄×C₇₉	#137	1	2	yes	—
127	C₇₉⋊C₈	—	8	2	no	—
128	C₇₉⋊C₈	—	8	2	no	—
129	C₂²×C₇₉	—	2	2	no	—
130	C₂²×C₇₉	—	2	2	no	—
131	C₂²×C₇₉	—	2	2	no	—
132	C₂²×C₇₉	—	4	1	no	—
133	C₂²×C₇₉	—	4	1	no	—
134	C₂²×C₇₉	—	4	1	no	—
135	C₂²×C₇₉	—	4	1	no	—
136	C₂²×C₇₉	#381	1	1	yes	characteristic
137	C₂²×C₇₉	#413	1	1	yes	characteristic
138	C₂²×C₇₉	#413	1	1	yes	characteristic
139	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
140	C₂²×C₇₉	#416	1	1	yes	characteristic
141	C₂²×C₇₉	#417	1	1	yes	characteristic
142	C₂²×C₇₉	#419	1	1	yes	characteristic
143	C₄×C₇₉	—	2	2	no	—
144	C₄×C₇₉	—	2	2	no	—
145	C₄×C₇₉	—	4	1	no	—
146	C₄×C₇₉	—	4	1	no	—
147	C₄×C₇₉	—	8	2	no	—
148	C₄×C₇₉	—	8	2	no	—
149	C₄×C₇₉	#480	1	2	yes	—
150	C₄×C₇₉	#482	1	2	yes	—
151	C₄×C₇₉	#483	1	2	yes	—
152	C₄×C₇₉	#485	1	2	yes	—
153	#8726	—	79	1	no	maximal, Sylow
154	C₂×C₇₉	—	4	2	no	—
155	C₂×C₇₉	#1540	1	1	yes	characteristic
156	C₂×C₇₉	#1547	1	1	yes	characteristic
157	C₂×C₇₉	#1554	1	1	yes	characteristic
158	C₂×C₇₉	#1559	1	1	yes	characteristic
159	C₂×C₇₉	#1822	1	1	yes	characteristic
160	C₂×C₇₉	#1823	1	1	yes	characteristic
161	C₂×C₇₉	#1825	1	1	yes	characteristic
162	#2285	D₇₉	1	1	yes	characteristic
163	#1797	—	79	1	no	—
164	#1797	—	79	1	no	—
165	#1296	—	79	2	no	—
166	#1296	—	79	2	no	—
167	C₇₉	#8726	1	1	yes	characteristic, complemented, Sylow
168	C₂×#117	—	2	1	no	—
169	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
170	C₂×C₄⋊C₈	—	79	2	no	—
171	C₂×C₄⋊C₈	—	79	2	no	—
172	C₂×C₂²⋊C₈	—	158	1	no	—
173	C₂×C₂²⋊C₈	—	158	1	no	—
174	#384	C₂×D₇₉	1	1	yes	characteristic
175	#384	C₂×D₇₉	1	1	yes	characteristic
176	#382	—	2	1	no	—
177	#380	—	2	1	no	—
178	#380	Dic₇₉	1	2	yes	—
179	C₄×C₄⋊C₄	—	2	1	no	—
180	C₄×C₄⋊C₄	Dic₇₉	1	2	yes	—
181	#336	—	158	1	no	—
182	#336	—	158	1	no	—
183	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
184	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
185	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
186	C₂²×C₈	—	158	2	no	—
187	C₂²×C₈	—	158	2	no	—
188	#117	—	4	2	no	—
189	#117	—	4	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
195	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₄²	—	2	4	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
203	C₄⋊C₈	—	158	4	no	—
204	C₄⋊C₈	—	158	4	no	—
205	C₂²⋊C₈	—	316	2	no	—
206	C₂²⋊C₈	—	316	2	no	—
207	#95	—	2	2	no	—
208	#95	—	2	2	no	—
209	C₂⁴	C₂²⋊Dic₇₉	1	1	yes	characteristic
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
224	C₂²×C₄	#11929609	1	2	yes	—
225	C₂²×C₄	#11929612	1	2	yes	—
226	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
229	C₂×C₈	—	316	2	no	—
230	C₂×C₈	—	316	2	no	—
231	C₂×C₈	—	316	2	no	—
232	C₂×C₈	—	316	2	no	—
233	C₂×C₈	—	316	2	no	—
234	C₂×C₈	—	316	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	4	1	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	4	4	no	—
249	C₄²	#11929624	1	4	yes	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	#14579082	1	2	yes	—
257	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	#14579082	1	2	yes	—
280	C₂×C₄	#14579120	1	2	yes	—
281	C₂×C₄	#14579144	1	2	yes	—
282	C₂×C₄	#14579144	1	2	yes	—
283	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
284	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
285	C₂×C₄	#14579181	1	2	yes	—
286	C₂×C₄	#14579186	1	1	yes	characteristic
287	C₂×C₄	#14579191	1	1	yes	characteristic
288	C₂×C₄	#14579192	1	2	yes	—
289	C₈	—	632	2	no	—
290	C₈	—	632	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	#19098426	1	1	yes	characteristic
299	C₂²	#19098547	1	1	yes	characteristic
300	C₂²	#19098547	1	1	yes	characteristic
301	C₂²	C₂×#14579082	1	1	yes	characteristic
302	C₂²	#19098559	1	1	yes	characteristic
303	C₂²	#19098564	1	1	yes	characteristic
304	C₂²	#19098572	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	#19098876	1	2	yes	—
312	C₄	#19098887	1	2	yes	—
313	C₄	#19098893	1	2	yes	—
314	C₄	#19098904	1	2	yes	—
315	C₂	—	4	2	no	—
316	C₂	#23762049	1	1	yes	characteristic
317	C₂	#23762077	1	1	yes	characteristic
318	C₂	#23762110	1	1	yes	characteristic
319	C₂	#23762130	1	1	yes	characteristic
320	C₂	#23763483	1	1	yes	characteristic
321	C₂	#23763486	1	1	yes	characteristic
322	C₂	#23763495	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₉
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2285	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2285	D₇₉
hypocenter	C₇₉	#8726
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #23761283	C₂	semidirect
4. #23761283	C₂	semidirect
167. C₇₉	#8726	semidirect

Maximal subgroups

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

subgroup	index
#23763380	2
#23761283	2
C₇₉×#2285	2
#8726	79