Subgroups — #34176678

Order 20224.

468 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763468	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#23761223	C₂	1	2	yes	maximal, complemented
4	#23761223	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2851	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
9	#19098318	—	2	1	no	—
10	#19098318	C₂²	1	2	yes	—
11	#19098318	C₂²	1	2	yes	—
12	C₂×#125×C₇₉	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
14	#389×C₇₉	C₂²	1	1	yes	characteristic
15	#389×C₇₉	C₂²	1	1	yes	characteristic
16	#387×C₇₉	—	2	1	no	—
17	#387×C₇₉	—	2	1	no	—
18	#385×C₇₉	—	2	1	no	—
19	#385×C₇₉	—	2	1	no	—
20	#382×C₇₉	C₄	1	1	yes	characteristic
21	#380×C₇₉	C₄	1	1	yes	characteristic
22	#380×C₇₉	C₄	1	1	yes	characteristic
23	C₂²×C₇₉⋊C₈	—	2	2	no	—
24	C₂²×C₇₉⋊C₈	—	2	2	no	—
25	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
26	C₂²⋊(C₇₉⋊C₈)	—	4	2	no	—
27	C₄×C₇₉⋊C₈	—	4	2	no	—
28	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
31	#125×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	1	yes	characteristic
37	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
41	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
42	C₂×C₄²×C₇₉	—	4	2	no	—
43	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
47	#95×C₇₉	—	2	1	no	—
48	#95×C₇₉	—	2	1	no	—
49	#95×C₇₉	—	2	1	no	—
50	#95×C₇₉	—	2	1	no	—
51	#95×C₇₉	—	2	1	no	—
52	#95×C₇₉	—	2	1	no	—
53	C₂×C₇₉⋊C₈	—	2	4	no	—
54	C₂×C₇₉⋊C₈	—	4	2	no	—
55	C₂×C₇₉⋊C₈	—	4	2	no	—
56	C₂×C₇₉⋊C₈	—	4	2	no	—
57	C₂×C₇₉⋊C₈	—	4	2	no	—
58	C₂×C₇₉⋊C₈	—	4	2	no	—
59	C₂⁴×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	—	2	2	no	—
61	C₂²×C₄×C₇₉	—	2	2	no	—
62	C₂²×C₄×C₇₉	—	2	2	no	—
63	C₂²×C₄×C₇₉	—	2	2	no	—
64	C₂²×C₄×C₇₉	—	2	2	no	—
65	C₂²×C₄×C₇₉	—	2	2	no	—
66	C₂²×C₄×C₇₉	—	2	1	no	—
67	C₂²×C₄×C₇₉	—	2	1	no	—
68	C₂²×C₄×C₇₉	—	2	1	no	—
69	C₂²×C₄×C₇₉	—	2	1	no	—
70	C₂²×C₄×C₇₉	—	4	1	no	—
71	C₂²×C₄×C₇₉	—	4	1	no	—
72	C₂²×C₄×C₇₉	—	4	1	no	—
73	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
79	C₄⋊C₄×C₇₉	—	2	2	no	—
80	C₄⋊C₄×C₇₉	—	2	2	no	—
81	C₄⋊C₄×C₇₉	—	4	2	no	—
82	C₄⋊C₄×C₇₉	—	4	1	no	—
83	C₄⋊C₄×C₇₉	—	4	1	no	—
84	C₄⋊C₄×C₇₉	—	4	2	no	—
85	C₄⋊C₄×C₇₉	—	4	2	no	—
86	C₄⋊C₄×C₇₉	—	4	2	no	—
87	C₂²⋊C₄×C₇₉	—	4	2	no	—
88	C₂²⋊C₄×C₇₉	—	4	2	no	—
89	C₄²×C₇₉	—	2	2	no	—
90	C₄²×C₇₉	—	2	2	no	—
91	C₄²×C₇₉	—	4	1	no	—
92	C₄²×C₇₉	—	4	1	no	—
93	C₂³×C₇₉	—	2	1	no	—
94	C₂³×C₇₉	—	2	1	no	—
95	C₂³×C₇₉	—	2	1	no	—
96	C₂³×C₇₉	—	2	1	no	—
97	C₂³×C₇₉	—	2	1	no	—
98	C₂³×C₇₉	—	2	1	no	—
99	C₂³×C₇₉	#104	1	2	yes	—
100	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₉	—	2	1	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	—	2	1	no	—
109	C₂×C₄×C₇₉	—	2	1	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	1	no	—
112	C₂×C₄×C₇₉	—	2	1	no	—
113	C₂×C₄×C₇₉	—	2	1	no	—
114	C₂×C₄×C₇₉	—	4	4	no	—
115	C₂×C₄×C₇₉	—	4	4	no	—
116	C₂×C₄×C₇₉	—	4	4	no	—
117	C₂×C₄×C₇₉	—	4	1	no	—
118	C₂×C₄×C₇₉	—	4	1	no	—
119	C₂×C₄×C₇₉	—	4	1	no	—
120	C₂×C₄×C₇₉	—	4	1	no	—
121	C₂×C₄×C₇₉	—	4	1	no	—
122	C₂×C₄×C₇₉	—	4	1	no	—
123	C₂×C₄×C₇₉	—	4	1	no	—
124	C₂×C₄×C₇₉	—	4	1	no	—
125	C₂×C₄×C₇₉	—	4	1	no	—
126	C₂×C₄×C₇₉	—	4	1	no	—
127	C₂×C₄×C₇₉	—	4	1	no	—
128	C₂×C₄×C₇₉	—	4	1	no	—
129	C₂×C₄×C₇₉	#99	1	2	yes	—
130	C₂×C₄×C₇₉	#101	1	2	yes	—
131	C₂×C₄×C₇₉	#131	1	2	yes	—
132	C₂×C₄×C₇₉	#135	1	2	yes	—
133	C₂×C₄×C₇₉	#135	1	2	yes	—
134	C₇₉⋊C₈	—	4	4	no	—
135	C₇₉⋊C₈	—	8	2	no	—
136	C₂²×C₇₉	—	2	2	no	—
137	C₂²×C₇₉	—	2	2	no	—
138	C₂²×C₇₉	—	2	2	no	—
139	C₂²×C₇₉	—	4	1	no	—
140	C₂²×C₇₉	—	4	1	no	—
141	C₂²×C₇₉	—	4	1	no	—
142	C₂²×C₇₉	—	4	1	no	—
143	C₂²×C₇₉	#408	1	1	yes	characteristic
144	C₂²×C₇₉	C₂×#99	1	1	yes	characteristic
145	C₂²×C₇₉	C₂×#101	1	1	yes	characteristic
146	C₂²×C₇₉	#416	1	1	yes	characteristic
147	C₂²×C₇₉	#416	1	1	yes	characteristic
148	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
149	C₂²×C₇₉	#421	1	1	yes	characteristic
150	C₄×C₇₉	—	2	2	no	—
151	C₄×C₇₉	—	2	2	no	—
152	C₄×C₇₉	—	2	2	no	—
153	C₄×C₇₉	—	2	2	no	—
154	C₄×C₇₉	—	2	2	no	—
155	C₄×C₇₉	—	4	1	no	—
156	C₄×C₇₉	—	8	1	no	—
157	C₄×C₇₉	—	8	1	no	—
158	C₄×C₇₉	—	8	1	no	—
159	C₄×C₇₉	—	8	1	no	—
160	#7671	—	79	1	no	maximal, Sylow
161	C₂×C₇₉	—	4	2	no	—
162	C₂×C₇₉	#1472	1	1	yes	characteristic
163	C₂×C₇₉	#1482	1	1	yes	characteristic
164	C₂×C₇₉	#1494	1	1	yes	characteristic
165	C₂×C₇₉	#1501	1	1	yes	characteristic
166	C₂×C₇₉	#1516	1	1	yes	characteristic
167	C₂×C₇₉	#1528	1	1	yes	characteristic
168	C₂×C₇₉	#1544	1	1	yes	characteristic
169	#2851	D₇₉	1	1	yes	characteristic
170	#1817	—	79	1	no	—
171	C₄×C₂²⋊C₈	—	79	1	no	—
172	#1271	—	79	2	no	—
173	#1271	—	79	2	no	—
174	C₇₉	#7671	1	1	yes	characteristic, complemented, Sylow
175	C₂×#125	Dic₇₉	1	1	yes	characteristic
176	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
177	C₂×C₂²⋊C₈	—	158	1	no	—
178	C₂×C₂²⋊C₈	—	158	1	no	—
179	C₂×C₄×C₈	—	158	1	no	—
180	#389	C₂×D₇₉	1	1	yes	characteristic
181	#389	C₂×D₇₉	1	1	yes	characteristic
182	#387	—	2	1	no	—
183	#387	—	2	1	no	—
184	#385	—	2	1	no	—
185	#385	—	2	1	no	—
186	#382	Dic₇₉	1	1	yes	characteristic
187	#380	Dic₇₉	1	1	yes	characteristic
188	#380	Dic₇₉	1	1	yes	characteristic
189	#336	—	79	2	no	—
190	#336	—	79	2	no	—
191	#336	—	158	1	no	—
192	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
193	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
194	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
195	C₂²×C₈	—	158	2	no	—
196	C₂²×C₈	—	158	2	no	—
197	#125	—	4	2	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
203	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	—	2	1	no	—
207	C₂×C₂²⋊C₄	—	2	1	no	—
208	C₂×C₄²	—	4	2	no	—
209	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
212	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
213	C₂²⋊C₈	—	316	2	no	—
214	C₂²⋊C₈	—	158	4	no	—
215	C₄×C₈	—	316	2	no	—
216	#95	—	2	1	no	—
217	#95	—	2	1	no	—
218	#95	—	2	1	no	—
219	#95	—	2	1	no	—
220	#95	—	2	1	no	—
221	#95	—	2	1	no	—
222	C₂⁴	C₂²⋊Dic₇₉	1	1	yes	characteristic
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
242	C₂×C₈	—	316	2	no	—
243	C₂×C₈	—	316	2	no	—
244	C₂×C₈	—	316	2	no	—
245	C₂×C₈	—	316	2	no	—
246	C₂×C₈	—	316	2	no	—
247	C₂×C₈	—	158	4	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	4	1	no	—
261	C₄²	—	4	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	#14579088	1	2	yes	—
269	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	#14579076	1	2	yes	—
299	C₂×C₄	#14579080	1	2	yes	—
300	C₂×C₄	#14579168	1	2	yes	—
301	C₂×C₄	#14579181	1	2	yes	—
302	C₂×C₄	#14579181	1	2	yes	—
303	C₈	—	316	4	no	—
304	C₈	—	632	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	#19098530	1	1	yes	characteristic
313	C₂²	C₂×#14579076	1	1	yes	characteristic
314	C₂²	C₂×#14579080	1	1	yes	characteristic
315	C₂²	#19098559	1	1	yes	characteristic
316	C₂²	#19098559	1	1	yes	characteristic
317	C₂²	C₂×#14579088	1	1	yes	characteristic
318	C₂²	#19098580	1	1	yes	characteristic
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	8	1	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	—	8	1	no	—
329	C₂	—	4	2	no	—
330	C₂	#23761764	1	1	yes	characteristic
331	C₂	#23761808	1	1	yes	characteristic
332	C₂	#23761860	1	1	yes	characteristic
333	C₂	#23761891	1	1	yes	characteristic
334	C₂	#23761956	1	1	yes	characteristic
335	C₂	#23762002	1	1	yes	characteristic
336	C₂	#23762065	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×#2851	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2851	D₇₉
hypocenter	C₇₉	#7671
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2851

Complements

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

N	H ≅ G/N	type
3. #23761223	C₂	semidirect
4. #23761223	C₂	semidirect
174. C₇₉	#7671	semidirect

Maximal subgroups

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

subgroup	index
#23763468	2
C₄×C₂²⋊(C₇₉⋊C₈)	2
#23761223	2
C₇₉×#2851	2
#7671	79