Subgroups — #34292694

Order 20224.

730 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772044	C₂	1	2	yes	maximal, complemented
2	#23764294	C₂	1	1	yes	maximal, characteristic
3	#23764289	C₂	1	1	yes	maximal, characteristic
4	#23764154	C₂	1	1	yes	maximal, characteristic, complemented
5	#23764149	C₂	1	1	yes	maximal, characteristic, complemented
6	#23764113	C₂	1	2	yes	maximal, complemented
7	#23764106	C₂	1	2	yes	maximal, complemented
8	#23763306	C₂	1	1	yes	maximal, characteristic, complemented
9	#23763210	C₂	1	2	yes	maximal, complemented
10	#23763088	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2943	C₂	1	1	yes	maximal, characteristic
12	#19098624	—	2	4	no	—
13	#19098616	—	2	4	no	—
14	C₂×#14579097	C₂²	1	1	yes	characteristic
15	C₂×#14579097	C₂²	1	1	yes	characteristic
16	#19098590	C₂²	1	2	yes	—
17	#19098590	C₂²	1	2	yes	—
18	C₂×#14579088	C₂²	1	2	yes	—
19	C₂×#14579088	C₂²	1	2	yes	—
20	#19098530	C₂²	1	2	yes	—
21	#19098530	C₂²	1	2	yes	—
22	C₄×#11929608	C₂²	1	2	yes	—
23	C₂×#14579071	C₂²	1	1	yes	characteristic
24	C₂×#14579071	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
28	#19098409	C₂²	1	1	yes	characteristic
29	#19098328	C₂²	1	2	yes	—
30	#19098322	C₂²	1	2	yes	—
31	#19098322	C₂²	1	2	yes	—
32	C₂×#135×C₇₉	C₂²	1	1	yes	characteristic
33	C₄×#41×C₇₉	C₂²	1	2	yes	—
34	#442×C₇₉	C₄	1	2	yes	complemented
35	#441×C₇₉	C₄	1	2	yes	complemented
36	#440×C₇₉	C₄	1	4	yes	complemented
37	#427×C₇₉	C₂²	1	1	yes	characteristic
38	#416×C₇₉	C₂²	1	2	yes	—
39	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
40	C₂×#11929608	C₂³	1	2	yes	—
41	C₂×#11929608	C₂³	1	2	yes	—
42	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
44	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
45	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
46	#14579097	—	2	2	no	—
47	#14579097	—	2	2	no	—
48	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
49	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
50	#14579088	—	2	4	no	—
51	#14579088	—	2	4	no	—
52	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
53	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
54	#14579072	—	2	2	no	—
55	#14579071	—	2	2	no	—
56	#14579071	—	2	2	no	—
57	C₄×C₇₉⋊C₈	—	2	2	no	—
58	C₄×C₇₉⋊C₈	D₄	1	4	yes	—
59	C₈×Dic₇₉	—	2	2	no	—
60	C₈×Dic₇₉	—	2	2	no	—
61	C₇₉⋊#95	—	2	2	no	—
62	C₂×#41×C₇₉	C₂³	1	2	yes	—
63	#135×C₇₉	—	2	2	no	—
64	#135×C₇₉	—	2	2	no	—
65	C₂×Q₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
66	C₂×SD₁₆×C₇₉	C₂×C₄	1	2	yes	—
67	C₂×D₈×C₇₉	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
70	C₄×Q₈×C₇₉	C₂×C₄	1	4	yes	—
71	C₄×D₄×C₇₉	—	2	4	no	—
72	C₄×D₄×C₇₉	C₂×C₄	1	4	yes	—
73	#117×C₇₉	—	2	2	no	—
74	#117×C₇₉	C₂³	1	2	yes	—
75	C₂×C₄²×C₇₉	—	2	2	no	—
76	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
77	#107×C₇₉	C₂×C₄	1	2	yes	—
78	#106×C₇₉	C₂×C₄	1	2	yes	—
79	#103×C₇₉	C₂×C₄	1	4	yes	—
80	#102×C₇₉	C₂×C₄	1	4	yes	—
81	#97×C₇₉	C₂×C₄	1	2	yes	—
82	#97×C₇₉	C₂×C₄	1	2	yes	—
83	C₂²×Dic₇₉	—	2	1	no	—
84	C₂²×Dic₇₉	—	2	1	no	—
85	#11929608	—	2	2	no	—
86	#11929608	—	2	2	no	—
87	#11929608	—	2	2	no	—
88	#11929608	—	2	2	no	—
89	C₂×C₇₉⋊C₈	—	2	2	no	—
90	C₂×C₇₉⋊C₈	—	2	2	no	—
91	C₂×C₇₉⋊C₈	—	2	1	no	—
92	C₂×C₇₉⋊C₈	—	2	1	no	—
93	C₂×C₇₉⋊C₈	—	2	1	no	—
94	C₂×C₇₉⋊C₈	—	2	1	no	—
95	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
96	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
97	C₄×Dic₇₉	—	2	1	no	—
98	C₄×Dic₇₉	—	2	1	no	—
99	C₄×Dic₇₉	#41	1	2	yes	—
100	C₄×Dic₇₉	#41	1	2	yes	—
101	#41×C₇₉	—	2	2	no	—
102	#41×C₇₉	—	2	2	no	—
103	#41×C₇₉	—	2	2	no	—
104	#41×C₇₉	—	2	2	no	—
105	C₂×Q₈×C₇₉	C₂²×C₄	1	2	yes	—
106	C₂×D₄×C₇₉	—	2	2	no	—
107	C₂×D₄×C₇₉	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₇₉	—	2	4	no	—
109	C₂²×C₄×C₇₉	—	2	2	no	—
110	C₂²×C₄×C₇₉	#41	1	2	yes	—
111	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
112	Q₁₆×C₇₉	—	2	2	no	—
113	SD₁₆×C₇₉	—	2	4	no	—
114	D₈×C₇₉	—	2	2	no	—
115	C₂×C₈×C₇₉	—	2	1	no	—
116	C₂×C₈×C₇₉	—	2	1	no	—
117	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₇₉	#41	1	2	yes	—
121	C₂×C₈×C₇₉	#41	1	2	yes	—
122	C₄⋊C₄×C₇₉	—	2	4	no	—
123	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
124	C₂²⋊C₄×C₇₉	—	2	2	no	—
125	C₂²⋊C₄×C₇₉	—	2	4	no	—
126	C₄²×C₇₉	—	2	4	no	—
127	C₄²×C₇₉	—	2	2	no	—
128	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
129	C₄²×C₇₉	C₂×D₄	1	2	yes	—
130	C₂³×C₇₉	—	2	2	no	—
131	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
132	Q₈×C₇₉	—	2	2	no	—
133	Q₈×C₇₉	—	2	2	no	—
134	D₄×C₇₉	—	2	2	no	—
135	D₄×C₇₉	—	2	2	no	—
136	D₄×C₇₉	—	4	2	no	—
137	D₄×C₇₉	—	4	2	no	—
138	C₂×C₄×C₇₉	—	2	4	no	—
139	C₂×C₄×C₇₉	—	2	2	no	—
140	C₂×C₄×C₇₉	—	4	2	no	—
141	C₂×C₄×C₇₉	—	4	8	no	—
142	C₂×C₄×C₇₉	—	4	2	no	—
143	C₂×C₄×C₇₉	—	4	2	no	—
144	C₂×C₄×C₇₉	C₄×D₄	1	4	yes	—
145	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
151	C₂×Dic₇₉	—	2	2	no	—
152	C₂×Dic₇₉	—	2	2	no	—
153	C₂×Dic₇₉	—	4	2	no	—
154	C₂×Dic₇₉	—	4	2	no	—
155	C₈×C₇₉	—	2	2	no	—
156	C₈×C₇₉	#117	1	2	yes	—
157	C₈×C₇₉	#117	1	2	yes	—
158	C₇₉⋊C₈	—	2	2	no	—
159	C₇₉⋊C₈	—	2	2	no	—
160	C₇₉⋊C₈	—	4	2	no	—
161	C₇₉⋊C₈	—	4	2	no	—
162	C₂²×C₇₉	—	2	2	no	—
163	C₂²×C₇₉	—	4	4	no	—
164	C₂²×C₇₉	—	4	2	no	—
165	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₇₉	#529	1	1	yes	characteristic
167	C₂²×C₇₉	#529	1	1	yes	characteristic
168	C₄×C₇₉	—	2	4	no	—
169	C₄×C₇₉	—	4	4	no	—
170	C₄×C₇₉	—	4	2	no	—
171	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
172	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
173	C₄×C₇₉	#532	1	1	yes	characteristic
174	C₄×C₇₉	#532	1	1	yes	characteristic
175	Dic₇₉	—	4	2	no	—
176	Dic₇₉	—	4	2	no	—
177	#21296	—	79	1	no	maximal, Sylow
178	C₂×C₇₉	—	2	1	no	—
179	C₂×C₇₉	—	2	1	no	—
180	C₂×C₇₉	—	8	2	no	—
181	C₂×C₇₉	#2293	1	1	yes	characteristic
182	C₂×C₇₉	#2955	1	2	yes	—
183	#2943	D₇₉	1	1	yes	characteristic
184	#2922	—	79	2	no	—
185	#1975	—	79	1	no	—
186	#1974	—	79	1	no	—
187	#1946	—	79	1	no	—
188	#1945	—	79	1	no	—
189	#1938	—	79	2	no	—
190	#1937	—	79	2	no	—
191	#1778	—	79	1	no	—
192	#1759	—	79	2	no	—
193	#1726	—	79	1	no	—
194	C₇₉	#21296	1	1	yes	characteristic, complemented, Sylow
195	C₂×#135	C₂×D₇₉	1	1	yes	characteristic
196	C₄×#41	C₂×D₇₉	1	2	yes	—
197	#442	Dic₇₉	1	2	yes	complemented
198	#441	Dic₇₉	1	2	yes	complemented
199	#440	Dic₇₉	1	4	yes	complemented
200	#433	—	158	4	no	—
201	#431	—	158	4	no	—
202	C₂×#108	—	79	1	no	—
203	C₂×#108	—	79	1	no	—
204	#427	C₂×D₇₉	1	1	yes	characteristic
205	#424	—	79	2	no	—
206	#424	—	79	2	no	—
207	C₂×#104	—	79	2	no	—
208	C₂×#104	—	79	2	no	—
209	#416	C₂×D₇₉	1	2	yes	—
210	#408	—	79	2	no	—
211	#408	—	79	2	no	—
212	C₄×M₁₆	—	79	2	no	—
213	C₂×#97	—	79	1	no	—
214	C₂×#97	—	79	1	no	—
215	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
216	C₂×C₄×C₈	—	79	1	no	—
217	C₂×C₄×C₈	—	79	1	no	—
218	C₂×C₄×C₈	—	79	1	no	—
219	#376	—	79	1	no	—
220	#341	—	79	2	no	—
221	#339	—	79	2	no	—
222	#339	—	79	2	no	—
223	C₂×#41	C₂²×D₇₉	1	2	yes	—
224	#135	—	2	2	no	—
225	#135	—	2	2	no	—
226	C₂×Q₁₆	C₂×Dic₇₉	1	1	yes	characteristic
227	C₂×SD₁₆	C₂×Dic₇₉	1	2	yes	—
228	C₂×D₈	C₂×Dic₇₉	1	1	yes	characteristic
229	C₂×M₁₆	—	79	2	no	—
230	C₂×M₁₆	—	79	2	no	—
231	C₂²×C₈	—	79	1	no	—
232	C₂²×C₈	—	79	1	no	—
233	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
235	C₄×Q₈	C₂×Dic₇₉	1	4	yes	—
236	C₄×D₄	—	2	4	no	—
237	C₄×D₄	C₂×Dic₇₉	1	4	yes	—
238	#117	—	2	2	no	—
239	#117	C₂²×D₇₉	1	2	yes	—
240	C₂×C₄²	—	2	2	no	—
241	C₂×C₄²	—	79	1	no	—
242	C₂×C₄²	—	79	1	no	—
243	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
244	#108	—	158	2	no	—
245	#108	—	158	2	no	—
246	#107	C₂×Dic₇₉	1	2	yes	—
247	#106	C₂×Dic₇₉	1	2	yes	—
248	C₄⋊C₈	—	158	4	no	—
249	C₄⋊C₈	—	158	4	no	—
250	#104	—	158	4	no	—
251	#104	—	158	4	no	—
252	#103	C₂×Dic₇₉	1	4	yes	—
253	#102	C₂×Dic₇₉	1	4	yes	—
254	C₂²⋊C₈	—	158	4	no	—
255	C₂²⋊C₈	—	158	4	no	—
256	#97	—	158	2	no	—
257	#97	—	158	2	no	—
258	#97	—	158	2	no	—
259	#97	C₂×Dic₇₉	1	2	yes	—
260	#97	C₂×Dic₇₉	1	2	yes	—
261	C₄×C₈	—	79	4	no	—
262	C₄×C₈	—	158	2	no	—
263	C₄×C₈	—	158	2	no	—
264	C₄×C₈	—	158	2	no	—
265	#95	—	158	2	no	—
266	#41	—	2	2	no	—
267	#41	—	2	2	no	—
268	#41	—	2	2	no	—
269	#41	—	2	2	no	—
270	C₂×Q₈	C₂²×Dic₇₉	1	2	yes	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	C₂²×Dic₇₉	1	2	yes	—
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	158	1	no	—
276	C₂²×C₄	—	158	1	no	—
277	C₂²×C₄	#11929623	1	2	yes	—
278	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
279	Q₁₆	—	2	2	no	—
280	SD₁₆	—	2	4	no	—
281	D₈	—	2	2	no	—
282	M₁₆	—	158	2	no	—
283	M₁₆	—	158	2	no	—
284	M₁₆	—	158	2	no	—
285	M₁₆	—	158	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	79	2	no	—
289	C₂×C₈	—	79	2	no	—
290	C₂×C₈	—	158	2	no	—
291	C₂×C₈	—	158	2	no	—
292	C₂×C₈	—	158	1	no	—
293	C₂×C₈	—	158	1	no	—
294	C₂×C₈	—	158	1	no	—
295	C₂×C₈	—	158	1	no	—
296	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
297	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
298	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
299	C₂×C₈	#11929624	1	2	yes	—
300	C₂×C₈	#11929624	1	2	yes	—
301	C₄⋊C₄	—	2	4	no	—
302	C₄⋊C₄	C₂²×Dic₇₉	1	4	yes	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	4	no	—
305	C₄²	—	2	4	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	79	2	no	—
308	C₄²	—	79	2	no	—
309	C₄²	—	158	1	no	—
310	C₄²	—	158	1	no	—
311	C₄²	C₂²×Dic₇₉	1	2	yes	—
312	C₄²	D₄×D₇₉	1	2	yes	—
313	C₂³	—	2	2	no	—
314	C₂³	C₂×#11929623	1	1	yes	characteristic
315	Q₈	—	2	2	no	—
316	Q₈	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	2	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	2	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	8	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	316	2	no	—
328	C₂×C₄	—	316	2	no	—
329	C₂×C₄	—	158	2	no	—
330	C₂×C₄	—	158	2	no	—
331	C₂×C₄	D₄×Dic₇₉	1	4	yes	—
332	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
334	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
335	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
336	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
337	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
338	C₈	—	2	2	no	—
339	C₈	—	316	2	no	—
340	C₈	—	316	2	no	—
341	C₈	—	158	2	no	—
342	C₈	—	158	2	no	—
343	C₈	#14579120	1	2	yes	—
344	C₈	#14579120	1	2	yes	—
345	C₂²	—	2	2	no	—
346	C₂²	—	4	2	no	—
347	C₂²	—	4	4	no	—
348	C₂²	C₂×D₄×Dic₇₉	1	1	yes	characteristic
349	C₂²	#19099095	1	1	yes	characteristic
350	C₂²	#19099095	1	1	yes	characteristic
351	C₄	—	2	4	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	4	no	—
354	C₄	—	316	2	no	—
355	C₄	—	316	2	no	—
356	C₄	C₂×#14579120	1	1	yes	characteristic
357	C₄	#41×Dic₇₉	1	1	yes	characteristic
358	C₄	#19099110	1	1	yes	characteristic
359	C₄	#19099110	1	1	yes	characteristic
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	8	2	no	—
363	C₂	#23765865	1	1	yes	characteristic
364	C₂	#23772303	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#2943	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂×#135×C₇₉	C₂²
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#2943	D₇₉
hypocenter	C₇₉	#21296
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2943

Complements

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

N	H ≅ G/N	type
1. #23772044	C₂	semidirect
4. #23764154	C₂	semidirect
5. #23764149	C₂	semidirect
6. #23764113	C₂	semidirect
7. #23764106	C₂	semidirect
8. #23763306	C₂	semidirect
9. #23763210	C₂	semidirect
10. #23763088	C₂	semidirect
34. #442×C₇₉	C₄	semidirect
35. #441×C₇₉	C₄	semidirect
36. #440×C₇₉	C₄	semidirect
194. C₇₉	#21296	semidirect
197. #442	Dic₇₉	semidirect
198. #441	Dic₇₉	semidirect
199. #440	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23772044	2
#23764294	2
#23764289	2
#23764154	2
#23764149	2
#23764113	2
#23764106	2
#23763306	2
#23763210	2
#23763088	2
C₇₉×#2943	2
#21296	79