Subgroups — #34296438

Order 20224.

722 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772341	C₂	1	1	yes	maximal, characteristic, complemented
2	#23772118	C₂	1	1	yes	maximal, characteristic
3	#23764382	C₂	1	1	yes	maximal, characteristic, complemented
4	#23764233	C₂	1	1	yes	maximal, characteristic
5	#23764113	C₂	1	2	yes	maximal, complemented
6	#23764106	C₂	1	2	yes	maximal, complemented
7	#23763538	C₂	1	1	yes	maximal, characteristic, complemented
8	#23763247	C₂	1	2	yes	maximal, complemented
9	#23763240	C₂	1	2	yes	maximal, complemented
10	#23763086	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#3131	C₂	1	1	yes	maximal, characteristic
12	C₂×#14579168	C₂²	1	2	yes	—
13	#19098694	—	2	4	no	—
14	#19098635	—	2	4	no	—
15	#19098590	C₂²	1	2	yes	—
16	#19098587	C₂²	1	1	yes	characteristic
17	#19098587	C₂²	1	1	yes	characteristic
18	C₂×#14579088	C₂²	1	2	yes	—
19	C₂×#14579088	C₂²	1	2	yes	—
20	#19098530	C₂²	1	2	yes	—
21	#19098519	C₂²	1	2	yes	—
22	C₂×#14579072	C₂²	1	1	yes	characteristic
23	C₂×#14579071	C₂²	1	1	yes	characteristic
24	C₂×#14579071	C₂²	1	1	yes	characteristic
25	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
27	#19098409	C₂²	1	1	yes	characteristic
28	#19098328	C₂²	1	2	yes	—
29	#19098328	C₂²	1	2	yes	—
30	#19098322	C₂²	1	2	yes	—
31	#19098322	C₂²	1	2	yes	—
32	#533×C₇₉	C₂²	1	1	yes	characteristic
33	#532×C₇₉	C₂²	1	1	yes	characteristic
34	#488×C₇₉	C₄	1	4	yes	complemented
35	#487×C₇₉	C₄	1	2	yes	complemented
36	#486×C₇₉	C₄	1	2	yes	complemented
37	#416×C₇₉	C₂²	1	2	yes	—
38	#416×C₇₉	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
40	#14579168	—	2	4	no	—
41	#14579168	—	2	4	no	—
42	C₂×#11929608	—	2	2	no	—
43	C₂×#11929608	C₂³	1	2	yes	—
44	C₂×#11929608	C₂³	1	2	yes	—
45	C₂²×C₇₉⋊C₈	—	2	2	no	—
46	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
50	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
51	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
52	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
53	#14579088	—	2	4	no	—
54	#14579088	—	2	4	no	—
55	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
56	#14579072	—	2	2	no	—
57	#14579072	D₄	1	4	yes	—
58	#14579071	—	2	2	no	—
59	#14579071	—	2	2	no	—
60	C₄×C₇₉⋊C₈	—	2	2	no	—
61	C₈×Dic₇₉	—	2	2	no	—
62	C₈×Dic₇₉	—	2	2	no	—
63	C₇₉⋊#95	—	2	2	no	—
64	C₂×#41×C₇₉	C₂³	1	2	yes	—
65	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
68	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
69	#127×C₇₉	C₂×C₄	1	1	yes	characteristic
70	#125×C₇₉	C₂×C₄	1	2	yes	—
71	#124×C₇₉	C₂×C₄	1	2	yes	—
72	C₂²⋊Q₈×C₇₉	—	2	2	no	—
73	#121×C₇₉	—	2	2	no	—
74	C₄×Q₈×C₇₉	—	2	2	no	—
75	C₄×D₄×C₇₉	—	2	2	no	—
76	#117×C₇₉	C₂³	1	2	yes	—
77	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
78	#103×C₇₉	C₂×C₄	1	4	yes	—
79	#103×C₇₉	C₂×C₄	1	4	yes	—
80	#102×C₇₉	C₂×C₄	1	4	yes	—
81	#102×C₇₉	C₂×C₄	1	4	yes	—
82	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
84	C₂²×Dic₇₉	—	2	1	no	—
85	C₂²×Dic₇₉	—	2	1	no	—
86	#11929608	—	2	2	no	—
87	#11929608	—	2	2	no	—
88	#11929608	—	2	2	no	—
89	#11929608	—	2	2	no	—
90	#11929608	—	4	4	no	—
91	C₂×C₇₉⋊C₈	—	2	2	no	—
92	C₂×C₇₉⋊C₈	—	2	2	no	—
93	C₂×C₇₉⋊C₈	—	2	1	no	—
94	C₂×C₇₉⋊C₈	—	2	1	no	—
95	C₂×C₇₉⋊C₈	—	4	4	no	—
96	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
97	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
98	C₂×C₇₉⋊C₈	#41	1	2	yes	—
99	C₂×C₇₉⋊C₈	#41	1	2	yes	—
100	C₄×Dic₇₉	—	2	1	no	—
101	C₄×Dic₇₉	—	2	1	no	—
102	C₄×Dic₇₉	#41	1	2	yes	—
103	C₄×Dic₇₉	#41	1	2	yes	—
104	#41×C₇₉	—	2	2	no	—
105	#41×C₇₉	—	2	2	no	—
106	#41×C₇₉	—	2	2	no	—
107	#41×C₇₉	—	2	2	no	—
108	C₂×Q₈×C₇₉	—	2	2	no	—
109	C₂×Q₈×C₇₉	C₂²×C₄	1	2	yes	—
110	C₂×D₄×C₇₉	—	2	2	no	—
111	C₂×D₄×C₇₉	—	2	2	no	—
112	C₂×D₄×C₇₉	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₇₉	—	2	2	no	—
114	C₂²×C₄×C₇₉	#41	1	2	yes	—
115	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
116	C₂×C₈×C₇₉	—	2	1	no	—
117	C₂×C₈×C₇₉	—	2	1	no	—
118	C₂×C₈×C₇₉	—	2	1	no	—
119	C₂×C₈×C₇₉	—	2	1	no	—
120	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
122	C₄⋊C₄×C₇₉	—	2	2	no	—
123	C₄⋊C₄×C₇₉	—	2	4	no	—
124	C₄⋊C₄×C₇₉	—	2	2	no	—
125	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
126	C₂²⋊C₄×C₇₉	—	2	2	no	—
127	C₂²⋊C₄×C₇₉	—	2	4	no	—
128	C₄²×C₇₉	—	2	2	no	—
129	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₇₉	C₂×D₄	1	2	yes	—
132	C₂³×C₇₉	—	2	2	no	—
133	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
134	Q₈×C₇₉	—	2	2	no	—
135	Q₈×C₇₉	—	2	2	no	—
136	Q₈×C₇₉	—	8	2	no	—
137	D₄×C₇₉	—	2	2	no	—
138	D₄×C₇₉	—	2	2	no	—
139	D₄×C₇₉	—	4	2	no	—
140	D₄×C₇₉	—	4	2	no	—
141	D₄×C₇₉	—	8	2	no	—
142	C₂×C₄×C₇₉	—	2	4	no	—
143	C₂×C₄×C₇₉	—	2	2	no	—
144	C₂×C₄×C₇₉	—	4	2	no	—
145	C₂×C₄×C₇₉	—	4	2	no	—
146	C₂×C₄×C₇₉	—	4	2	no	—
147	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
148	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
155	C₂×Dic₇₉	—	2	2	no	—
156	C₂×Dic₇₉	—	2	2	no	—
157	C₂×Dic₇₉	—	4	2	no	—
158	C₂×Dic₇₉	—	4	2	no	—
159	C₈×C₇₉	—	2	2	no	—
160	C₈×C₇₉	—	2	2	no	—
161	C₇₉⋊C₈	—	2	2	no	—
162	C₇₉⋊C₈	—	4	2	no	—
163	C₇₉⋊C₈	—	4	2	no	—
164	C₇₉⋊C₈	#124	1	4	yes	—
165	C₂²×C₇₉	—	2	2	no	—
166	C₂²×C₇₉	—	4	4	no	—
167	C₂²×C₇₉	—	4	2	no	—
168	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₇₉	#529	1	1	yes	characteristic
170	C₂²×C₇₉	#529	1	1	yes	characteristic
171	C₄×C₇₉	—	2	2	no	—
172	C₄×C₇₉	—	2	2	no	—
173	C₄×C₇₉	—	4	4	no	—
174	C₄×C₇₉	—	4	2	no	—
175	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
176	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
177	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
178	C₄×C₇₉	#532	1	1	yes	characteristic
179	Dic₇₉	—	4	2	no	—
180	Dic₇₉	—	4	2	no	—
181	#21659	—	79	1	no	maximal, Sylow
182	C₂×C₇₉	—	2	1	no	—
183	C₂×C₇₉	—	2	1	no	—
184	C₂×C₇₉	—	8	2	no	—
185	C₂×C₇₉	C₄×#124	1	1	yes	characteristic
186	C₂×C₇₉	#2957	1	2	yes	—
187	#3131	D₇₉	1	1	yes	characteristic
188	#2961	—	79	1	no	—
189	#2932	—	79	1	no	—
190	#1995	—	79	1	no	—
191	#1963	—	79	1	no	—
192	#1938	—	79	2	no	—
193	#1937	—	79	2	no	—
194	#1837	—	79	1	no	—
195	#1765	—	79	2	no	—
196	#1764	—	79	2	no	—
197	#1726	—	79	1	no	—
198	C₇₉	#21659	1	1	yes	characteristic, complemented, Sylow
199	C₂×#131	—	79	2	no	—
200	#533	C₂×D₇₉	1	1	yes	characteristic
201	#532	C₂×D₇₉	1	1	yes	characteristic
202	#488	Dic₇₉	1	4	yes	complemented
203	#487	Dic₇₉	1	2	yes	complemented
204	#486	Dic₇₉	1	2	yes	complemented
205	#446	—	158	4	no	—
206	#435	—	158	4	no	—
207	#424	—	79	2	no	—
208	#423	—	79	1	no	—
209	#423	—	79	1	no	—
210	C₂×#104	—	79	2	no	—
211	C₂×#104	—	79	2	no	—
212	#416	C₂×D₇₉	1	2	yes	—
213	#416	C₂×D₇₉	1	2	yes	—
214	#408	—	79	2	no	—
215	#405	—	79	2	no	—
216	C₂×#97	—	79	1	no	—
217	C₂×#97	—	79	1	no	—
218	C₂×#97	—	79	1	no	—
219	C₂×C₄×C₈	—	79	1	no	—
220	C₂×C₄×C₈	—	79	1	no	—
221	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
222	#376	—	79	1	no	—
223	#341	—	79	2	no	—
224	#341	—	79	2	no	—
225	#339	—	79	2	no	—
226	#339	—	79	2	no	—
227	C₂×#41	C₂²×D₇₉	1	2	yes	—
228	#131	—	158	4	no	—
229	#131	—	158	4	no	—
230	C₂×M₁₆	—	79	2	no	—
231	C₂×M₁₆	—	79	2	no	—
232	C₂×M₁₆	—	158	2	no	—
233	C₂²×C₈	—	79	1	no	—
234	C₂²×C₈	—	79	1	no	—
235	C₂²×C₈	—	158	2	no	—
236	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
237	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
238	C₄⋊Q₈	C₂×Dic₇₉	1	1	yes	characteristic
239	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
240	#127	C₂×Dic₇₉	1	1	yes	characteristic
241	#125	C₂×Dic₇₉	1	2	yes	—
242	#124	C₂×Dic₇₉	1	2	yes	—
243	C₂²⋊Q₈	—	2	2	no	—
244	#121	—	2	2	no	—
245	C₄×Q₈	—	2	2	no	—
246	C₄×D₄	—	2	2	no	—
247	#117	C₂²×D₇₉	1	2	yes	—
248	C₂×C₄²	—	79	1	no	—
249	C₂×C₄²	—	79	1	no	—
250	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
251	C₄⋊C₈	—	158	4	no	—
252	C₄⋊C₈	—	158	2	no	—
253	C₄⋊C₈	—	158	2	no	—
254	#104	—	158	4	no	—
255	#104	—	158	4	no	—
256	#103	C₂×Dic₇₉	1	4	yes	—
257	#103	C₂×Dic₇₉	1	4	yes	—
258	#102	C₂×Dic₇₉	1	4	yes	—
259	#102	C₂×Dic₇₉	1	4	yes	—
260	C₂²⋊C₈	—	158	4	no	—
261	#97	—	79	4	no	—
262	#97	—	158	2	no	—
263	#97	—	158	2	no	—
264	#97	—	158	2	no	—
265	C₄×C₈	—	158	2	no	—
266	C₄×C₈	—	158	2	no	—
267	C₄×C₈	—	158	2	no	—
268	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
269	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
270	#95	—	158	2	no	—
271	#41	—	2	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	#41	—	2	2	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	C₂²×Dic₇₉	1	2	yes	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	C₂²×Dic₇₉	1	2	yes	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	158	1	no	—
282	C₂²×C₄	—	158	1	no	—
283	C₂²×C₄	#11929623	1	2	yes	—
284	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
285	M₁₆	—	316	4	no	—
286	M₁₆	—	158	2	no	—
287	M₁₆	—	158	2	no	—
288	M₁₆	—	158	2	no	—
289	M₁₆	—	158	2	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	316	4	no	—
295	C₂×C₈	—	79	2	no	—
296	C₂×C₈	—	79	2	no	—
297	C₂×C₈	—	79	2	no	—
298	C₂×C₈	—	79	2	no	—
299	C₂×C₈	—	158	2	no	—
300	C₂×C₈	—	158	2	no	—
301	C₂×C₈	—	158	1	no	—
302	C₂×C₈	—	158	1	no	—
303	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
304	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	4	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	C₂²×Dic₇₉	1	4	yes	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	2	4	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	79	2	no	—
313	C₄²	—	79	2	no	—
314	C₄²	—	158	1	no	—
315	C₄²	—	158	1	no	—
316	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
317	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
318	C₄²	D₄×D₇₉	1	2	yes	—
319	C₂³	—	2	2	no	—
320	C₂³	C₂×#11929623	1	1	yes	characteristic
321	Q₈	—	2	2	no	—
322	Q₈	—	2	2	no	—
323	Q₈	—	8	2	no	—
324	D₄	—	2	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	8	2	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	316	2	no	—
335	C₂×C₄	—	316	2	no	—
336	C₂×C₄	—	158	2	no	—
337	C₂×C₄	—	158	2	no	—
338	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
339	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
340	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
342	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
343	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
344	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
345	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	2	2	no	—
348	C₈	—	316	2	no	—
349	C₈	—	316	2	no	—
350	C₈	—	79	4	no	—
351	C₈	—	158	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	4	4	no	—
354	C₂²	—	4	2	no	—
355	C₂²	C₂×D₄×Dic₇₉	1	1	yes	characteristic
356	C₂²	#19099092	1	1	yes	characteristic
357	C₂²	#19099092	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	4	4	no	—
361	C₄	—	4	2	no	—
362	C₄	—	316	2	no	—
363	C₄	—	316	2	no	—
364	C₄	#41×Dic₇₉	1	1	yes	characteristic
365	C₄	#41×Dic₇₉	1	1	yes	characteristic
366	C₄	#124×D₇₉	1	1	yes	characteristic
367	C₄	#19099112	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	8	2	no	—
371	C₂	#124×Dic₇₉	1	1	yes	characteristic
372	C₂	#23772317	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×#3131	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	#532×C₇₉	C₂²
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#3131	D₇₉
hypocenter	C₇₉	#21659
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

Complements

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

N	H ≅ G/N	type
1. #23772341	C₂	semidirect
3. #23764382	C₂	semidirect
5. #23764113	C₂	semidirect
6. #23764106	C₂	semidirect
7. #23763538	C₂	semidirect
8. #23763247	C₂	semidirect
9. #23763240	C₂	semidirect
10. #23763086	C₂	semidirect
34. #488×C₇₉	C₄	semidirect
35. #487×C₇₉	C₄	semidirect
36. #486×C₇₉	C₄	semidirect
198. C₇₉	#21659	semidirect
202. #488	Dic₇₉	semidirect
203. #487	Dic₇₉	semidirect
204. #486	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23772341	2
#23772118	2
#23764382	2
#23764233	2
#23764113	2
#23764106	2
#23763538	2
#23763247	2
#23763240	2
#23763086	2
C₇₉×#3131	2
#21659	79