Subgroups — #34187288

Order 20224.

482 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763253	C₂	1	1	yes	maximal, characteristic, complemented
2	#23763149	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098305	C₂	1	2	yes	maximal, complemented
4	C₂×#19098303	C₂	1	2	yes	maximal
5	C₇₉×#2861	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
7	C₂×#14579072	C₂²	1	2	yes	—
8	C₂×#14579072	C₂²	1	2	yes	—
9	#19098318	—	2	1	no	—
10	#19098318	—	2	1	no	—
11	#19098318	—	2	1	no	—
12	#19098305	—	2	4	no	—
13	#19098303	—	2	4	no	—
14	C₂×#125×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#124×C₇₉	C₂²	1	1	yes	characteristic
16	#391×C₇₉	C₄	1	1	yes	characteristic
17	#391×C₇₉	C₄	1	1	yes	characteristic
18	#390×C₇₉	C₄	1	1	yes	characteristic
19	#390×C₇₉	C₄	1	1	yes	characteristic
20	#388×C₇₉	—	2	1	no	—
21	#388×C₇₉	—	2	1	no	—
22	#388×C₇₉	—	2	1	no	—
23	#388×C₇₉	—	2	1	no	—
24	C₄³×C₇₉	C₂²	1	1	yes	characteristic
25	C₂²×C₇₉⋊C₈	—	2	2	no	—
26	C₂²×C₇₉⋊C₈	—	2	2	no	—
27	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
28	#14579072	—	2	2	no	—
29	#14579072	—	2	2	no	—
30	#14579072	—	2	2	no	—
31	#14579072	D₄	1	4	yes	—
32	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
34	#125×C₇₉	—	2	2	no	—
35	#125×C₇₉	—	2	2	no	—
36	#124×C₇₉	—	2	2	no	—
37	#124×C₇₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
47	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
48	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
55	#95×C₇₉	—	2	1	no	—
56	#95×C₇₉	—	2	1	no	—
57	#95×C₇₉	—	2	1	no	—
58	#95×C₇₉	—	2	1	no	—
59	C₂×C₇₉⋊C₈	—	2	4	no	—
60	C₂×C₇₉⋊C₈	—	2	4	no	—
61	C₂×C₇₉⋊C₈	—	4	2	no	—
62	C₂×C₇₉⋊C₈	—	4	2	no	—
63	C₂×C₇₉⋊C₈	—	4	2	no	—
64	C₂×C₇₉⋊C₈	—	4	2	no	—
65	C₂⁴×C₇₉	—	2	1	no	—
66	C₂×Q₈×C₇₉	—	2	2	no	—
67	C₂×Q₈×C₇₉	—	4	1	no	—
68	C₂×Q₈×C₇₉	—	4	1	no	—
69	C₂×D₄×C₇₉	—	2	2	no	—
70	C₂×D₄×C₇₉	—	4	1	no	—
71	C₂×D₄×C₇₉	—	4	1	no	—
72	C₂²×C₄×C₇₉	—	2	1	no	—
73	C₂²×C₄×C₇₉	—	2	1	no	—
74	C₂²×C₄×C₇₉	—	2	1	no	—
75	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	C₂²×C₄	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	C₄⋊C₄×C₇₉	—	2	2	no	—
83	C₄⋊C₄×C₇₉	—	2	2	no	—
84	C₄⋊C₄×C₇₉	—	4	2	no	—
85	C₄⋊C₄×C₇₉	—	4	1	no	—
86	C₄⋊C₄×C₇₉	—	4	1	no	—
87	C₂²⋊C₄×C₇₉	—	4	2	no	—
88	C₂²⋊C₄×C₇₉	—	4	2	no	—
89	C₂²⋊C₄×C₇₉	—	4	2	no	—
90	C₂²⋊C₄×C₇₉	—	4	2	no	—
91	C₂²⋊C₄×C₇₉	—	4	2	no	—
92	C₂²⋊C₄×C₇₉	—	4	1	no	—
93	C₂²⋊C₄×C₇₉	—	4	1	no	—
94	C₄²×C₇₉	—	2	2	no	—
95	C₄²×C₇₉	—	4	1	no	—
96	C₄²×C₇₉	—	4	1	no	—
97	C₄²×C₇₉	—	4	1	no	—
98	C₄²×C₇₉	—	4	1	no	—
99	C₄²×C₇₉	—	4	1	no	—
100	C₄²×C₇₉	C₂×D₄	1	2	yes	—
101	C₄²×C₇₉	#41	1	2	yes	—
102	C₂³×C₇₉	—	4	1	no	—
103	C₂³×C₇₉	—	4	1	no	—
104	C₂³×C₇₉	—	4	1	no	—
105	C₂³×C₇₉	—	4	1	no	—
106	C₂³×C₇₉	—	4	1	no	—
107	C₂³×C₇₉	—	4	1	no	—
108	C₂³×C₇₉	—	4	1	no	—
109	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	Q₈×C₇₉	—	8	2	no	—
111	D₄×C₇₉	—	8	2	no	—
112	C₂×C₄×C₇₉	—	2	1	no	—
113	C₂×C₄×C₇₉	—	2	1	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	2	1	no	—
117	C₂×C₄×C₇₉	—	2	1	no	—
118	C₂×C₄×C₇₉	—	2	1	no	—
119	C₂×C₄×C₇₉	—	2	1	no	—
120	C₂×C₄×C₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	—	2	1	no	—
122	C₂×C₄×C₇₉	—	2	1	no	—
123	C₂×C₄×C₇₉	—	2	1	no	—
124	C₂×C₄×C₇₉	—	2	1	no	—
125	C₂×C₄×C₇₉	—	2	1	no	—
126	C₂×C₄×C₇₉	—	2	1	no	—
127	C₂×C₄×C₇₉	—	2	1	no	—
128	C₂×C₄×C₇₉	—	2	1	no	—
129	C₂×C₄×C₇₉	—	4	1	no	—
130	C₂×C₄×C₇₉	—	4	1	no	—
131	C₂×C₄×C₇₉	—	4	1	no	—
132	C₂×C₄×C₇₉	—	4	1	no	—
133	C₂×C₄×C₇₉	—	4	1	no	—
134	C₂×C₄×C₇₉	—	4	1	no	—
135	C₂×C₄×C₇₉	—	4	1	no	—
136	C₂×C₄×C₇₉	—	4	1	no	—
137	C₂×C₄×C₇₉	—	4	1	no	—
138	C₂×C₄×C₇₉	#104	1	2	yes	—
139	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₇₉	#121	1	2	yes	—
141	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
142	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
143	C₇₉⋊C₈	—	4	4	no	—
144	C₇₉⋊C₈	—	4	4	no	—
145	C₂²×C₇₉	—	8	1	no	—
146	C₂²×C₇₉	—	8	1	no	—
147	C₂²×C₇₉	—	8	1	no	—
148	C₂²×C₇₉	—	8	1	no	—
149	C₂²×C₇₉	—	8	1	no	—
150	C₂²×C₇₉	—	8	1	no	—
151	C₂²×C₇₉	—	8	1	no	—
152	C₂²×C₇₉	#390	1	1	yes	characteristic
153	C₂²×C₇₉	#413	1	1	yes	characteristic
154	C₂²×C₇₉	#413	1	1	yes	characteristic
155	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
156	C₂²×C₇₉	#421	1	1	yes	characteristic
157	C₂²×C₇₉	#421	1	1	yes	characteristic
158	C₂²×C₇₉	#421	1	1	yes	characteristic
159	C₄×C₇₉	—	2	2	no	—
160	C₄×C₇₉	—	2	2	no	—
161	C₄×C₇₉	—	2	1	no	—
162	C₄×C₇₉	—	2	1	no	—
163	C₄×C₇₉	—	2	2	no	—
164	C₄×C₇₉	—	4	1	no	—
165	C₄×C₇₉	—	4	1	no	—
166	C₄×C₇₉	—	4	1	no	—
167	C₄×C₇₉	—	8	1	no	—
168	C₄×C₇₉	—	8	1	no	—
169	C₄×C₇₉	—	8	1	no	—
170	#9688	—	79	1	no	maximal, Sylow
171	C₂×C₇₉	—	16	1	no	—
172	C₂×C₇₉	#1529	1	1	yes	characteristic
173	C₂×C₇₉	#1532	1	1	yes	characteristic
174	C₂×C₇₉	#1535	1	1	yes	characteristic
175	C₂×C₇₉	#1535	1	1	yes	characteristic
176	C₂×C₇₉	#1963	1	1	yes	characteristic
177	C₂×C₇₉	#1967	1	1	yes	characteristic
178	C₂×C₇₉	#1980	1	1	yes	characteristic
179	#2861	D₇₉	1	1	yes	characteristic
180	#1767	—	79	1	no	—
181	#1745	—	79	1	no	—
182	C₂×#330	—	79	2	no	—
183	C₂×#329	—	79	2	no	—
184	C₇₉	#9688	1	1	yes	characteristic, complemented, Sylow
185	C₂×#125	C₂×D₇₉	1	1	yes	characteristic
186	C₂×#124	C₂×D₇₉	1	1	yes	characteristic
187	C₂×C₄⋊C₈	—	158	1	no	—
188	C₂×#97	—	79	2	no	—
189	C₂×#97	—	79	2	no	—
190	#391	Dic₇₉	1	1	yes	characteristic
191	#391	Dic₇₉	1	1	yes	characteristic
192	#390	Dic₇₉	1	1	yes	characteristic
193	#390	Dic₇₉	1	1	yes	characteristic
194	#388	—	2	1	no	—
195	#388	—	2	1	no	—
196	#388	—	2	1	no	—
197	#388	—	2	1	no	—
198	C₄³	C₂×D₇₉	1	1	yes	characteristic
199	#336	—	158	1	no	—
200	#336	—	158	1	no	—
201	#336	—	158	1	no	—
202	#330	—	158	4	no	—
203	#329	—	158	4	no	—
204	C₂²×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
205	C₂²×D₄	C₂×Dic₇₉	1	1	yes	characteristic
206	C₂²×C₈	—	158	2	no	—
207	C₂²×C₈	—	158	2	no	—
208	#125	—	2	2	no	—
209	#125	—	2	2	no	—
210	#124	—	2	2	no	—
211	#124	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	2	1	no	—
217	C₂×C₂²⋊C₄	—	2	1	no	—
218	C₂×C₂²⋊C₄	—	2	1	no	—
219	C₂×C₂²⋊C₄	—	2	1	no	—
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	2	1	no	—
222	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
223	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
225	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
226	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
227	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
229	C₄⋊C₈	—	158	4	no	—
230	#97	—	79	4	no	—
231	#97	—	158	2	no	—
232	#97	—	158	2	no	—
233	#97	—	158	2	no	—
234	#95	—	2	1	no	—
235	#95	—	2	1	no	—
236	#95	—	2	1	no	—
237	#95	—	2	1	no	—
238	C₂⁴	—	2	1	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
255	C₂×C₈	—	316	2	no	—
256	C₂×C₈	—	316	2	no	—
257	C₂×C₈	—	316	2	no	—
258	C₂×C₈	—	316	2	no	—
259	C₂×C₈	—	158	4	no	—
260	C₂×C₈	—	158	4	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	4	1	no	—
279	C₄²	D₄×D₇₉	1	2	yes	—
280	C₄²	#11929623	1	2	yes	—
281	C₂³	—	4	1	no	—
282	C₂³	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	—	4	1	no	—
288	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
289	Q₈	—	8	2	no	—
290	D₄	—	8	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	#14579088	1	2	yes	—
318	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
319	C₂×C₄	#14579132	1	2	yes	—
320	C₂×C₄	#14579148	1	1	yes	characteristic
321	C₂×C₄	#14579157	1	1	yes	characteristic
322	C₈	—	316	4	no	—
323	C₈	—	316	4	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	#19098466	1	1	yes	characteristic
332	C₂²	#19098547	1	1	yes	characteristic
333	C₂²	#19098547	1	1	yes	characteristic
334	C₂²	C₂×#14579088	1	1	yes	characteristic
335	C₂²	#19098580	1	1	yes	characteristic
336	C₂²	#19098580	1	1	yes	characteristic
337	C₂²	#19098580	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	1	no	—
349	C₂	—	16	1	no	—
350	C₂	#23762007	1	1	yes	characteristic
351	C₂	#23762020	1	1	yes	characteristic
352	C₂	#23762031	1	1	yes	characteristic
353	C₂	#23762031	1	1	yes	characteristic
354	C₂	#23764233	1	1	yes	characteristic
355	C₂	#23764250	1	1	yes	characteristic
356	C₂	#23764320	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×#2861	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₇₉	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2861	D₇₉
hypocenter	C₇₉	#9688
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2861

Complements

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

N	H ≅ G/N	type
1. #23763253	C₂	semidirect
2. #23763149	C₂	semidirect
3. C₂×#19098305	C₂	semidirect
184. C₇₉	#9688	semidirect

Maximal subgroups

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

subgroup	index
#23763253	2
#23763149	2
C₂×#19098305	2
C₂×#19098303	2
C₇₉×#2861	2
#9688	79