Subgroups — #34181256

Order 20224.

456 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763438	C₂	1	1	yes	maximal, characteristic, complemented
2	#23763425	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761943	C₂	1	2	yes	maximal
4	#23761305	C₂	1	2	yes	maximal, complemented
5	C₇₉×#2929	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579097	—	2	1	no	—
7	C₂×#14579093	—	2	1	no	—
8	#19098527	C₂²	1	2	yes	—
9	#19098423	C₂²	1	2	yes	—
10	#19098328	—	2	1	no	—
11	#19098326	—	2	1	no	—
12	#19098300	—	2	2	no	—
13	#19098297	—	2	2	no	—
14	C₂×#131×C₇₉	C₄	1	1	yes	characteristic, complemented
15	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	#525×C₇₉	C₂²	1	1	yes	characteristic
17	#435×C₇₉	—	2	1	no	—
18	#435×C₇₉	—	2	1	no	—
19	C₈×D₄×C₇₉	—	2	1	no	—
20	C₈×D₄×C₇₉	—	2	1	no	—
21	#424×C₇₉	C₄	1	1	yes	characteristic, complemented
22	#408×C₇₉	C₄	1	1	yes	characteristic, complemented
23	#407×C₇₉	C₂²	1	1	yes	characteristic
24	#405×C₇₉	C₄	1	1	yes	characteristic, complemented
25	C₂×#11929608	—	2	2	no	—
26	C₂×C₄⋊Dic₇₉	—	2	2	no	—
27	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
28	#14579097	—	4	2	no	—
29	#14579093	—	4	2	no	—
30	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
31	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
32	C₇₉⋊#95	—	2	2	no	—
33	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
35	#131×C₇₉	—	4	2	no	—
36	C₂×M₁₆×C₇₉	—	2	1	no	—
37	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₉	—	2	1	no	—
40	C₂²×C₈×C₇₉	—	4	2	no	—
41	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
47	#123×C₇₉	—	2	1	no	—
48	C₂²⋊Q₈×C₇₉	D₄	1	1	yes	characteristic
49	C₂²⋊Q₈×C₇₉	Q₈	1	1	yes	characteristic
50	#121×C₇₉	D₄	1	1	yes	characteristic
51	#121×C₇₉	Q₈	1	1	yes	characteristic
52	#120×C₇₉	—	2	1	no	—
53	C₄×D₄×C₇₉	—	2	1	no	—
54	C₄×D₄×C₇₉	—	2	1	no	—
55	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₈×C₇₉	—	2	1	no	—
57	C₄⋊C₈×C₇₉	—	2	1	no	—
58	C₂²⋊C₈×C₇₉	—	2	1	no	—
59	C₂²⋊C₈×C₇₉	—	2	1	no	—
60	C₂²⋊C₈×C₇₉	—	2	1	no	—
61	C₂²⋊C₈×C₇₉	—	2	1	no	—
62	#97×C₇₉	—	2	1	no	—
63	C₄×C₈×C₇₉	—	2	1	no	—
64	C₂²×Dic₇₉	—	4	2	no	—
65	#11929608	—	4	2	no	—
66	#11929608	—	4	2	no	—
67	C₂×C₇₉⋊C₈	—	4	2	no	—
68	C₄⋊Dic₇₉	—	4	2	no	—
69	C₄⋊Dic₇₉	—	4	2	no	—
70	C₂²⋊Dic₇₉	—	8	2	no	—
71	C₂⁴×C₇₉	#41	1	1	yes	characteristic
72	#41×C₇₉	—	4	1	no	—
73	#41×C₇₉	—	4	1	no	—
74	C₂×Q₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
75	C₂×D₄×C₇₉	—	2	1	no	—
76	C₂×D₄×C₇₉	—	2	1	no	—
77	C₂×D₄×C₇₉	—	2	1	no	—
78	C₂×D₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	—	2	1	no	—
80	C₂²×C₄×C₇₉	—	4	1	no	—
81	C₂²×C₄×C₇₉	—	4	1	no	—
82	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
88	M₁₆×C₇₉	—	4	1	no	—
89	M₁₆×C₇₉	—	4	1	no	—
90	M₁₆×C₇₉	—	4	2	no	—
91	C₂×C₈×C₇₉	—	2	2	no	—
92	C₂×C₈×C₇₉	—	2	2	no	—
93	C₂×C₈×C₇₉	—	2	2	no	—
94	C₂×C₈×C₇₉	—	2	1	no	—
95	C₂×C₈×C₇₉	—	2	1	no	—
96	C₂×C₈×C₇₉	—	4	1	no	—
97	C₂×C₈×C₇₉	—	4	1	no	—
98	C₂×C₈×C₇₉	—	4	1	no	—
99	C₂×C₈×C₇₉	—	4	2	no	—
100	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
104	C₄⋊C₄×C₇₉	—	2	1	no	—
105	C₄⋊C₄×C₇₉	—	2	1	no	—
106	C₄⋊C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
107	C₄⋊C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
108	C₂²⋊C₄×C₇₉	—	2	1	no	—
109	C₂²⋊C₄×C₇₉	—	2	1	no	—
110	C₂²⋊C₄×C₇₉	—	2	1	no	—
111	C₂²⋊C₄×C₇₉	—	2	1	no	—
112	C₂²⋊C₄×C₇₉	—	2	1	no	—
113	C₄²×C₇₉	—	2	1	no	—
114	C₂³×C₇₉	—	2	2	no	—
115	C₂³×C₇₉	—	2	1	no	—
116	C₂³×C₇₉	—	4	1	no	—
117	C₂³×C₇₉	—	4	1	no	—
118	C₂³×C₇₉	#117	1	1	yes	characteristic
119	C₂³×C₇₉	#121	1	1	yes	characteristic
120	C₂³×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
121	Q₈×C₇₉	—	4	1	no	—
122	D₄×C₇₉	—	4	1	no	—
123	D₄×C₇₉	—	4	2	no	—
124	D₄×C₇₉	—	4	2	no	—
125	D₄×C₇₉	—	4	2	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	—	2	1	no	—
130	C₂×C₄×C₇₉	—	2	1	no	—
131	C₂×C₄×C₇₉	—	2	1	no	—
132	C₂×C₄×C₇₉	—	2	1	no	—
133	C₂×C₄×C₇₉	—	2	1	no	—
134	C₂×C₄×C₇₉	—	2	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	—	4	1	no	—
139	C₂×C₄×C₇₉	—	4	1	no	—
140	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×Dic₇₉	—	8	2	no	—
145	C₂×Dic₇₉	—	8	2	no	—
146	C₂×Dic₇₉	—	8	2	no	—
147	C₈×C₇₉	—	2	1	no	—
148	C₈×C₇₉	—	2	1	no	—
149	C₈×C₇₉	—	2	1	no	—
150	C₈×C₇₉	—	4	1	no	—
151	C₈×C₇₉	—	4	1	no	—
152	C₈×C₇₉	#99	1	2	yes	—
153	C₇₉⋊C₈	—	8	2	no	—
154	C₂²×C₇₉	—	2	1	no	—
155	C₂²×C₇₉	—	2	1	no	—
156	C₂²×C₇₉	—	2	1	no	—
157	C₂²×C₇₉	—	2	1	no	—
158	C₂²×C₇₉	—	2	1	no	—
159	C₂²×C₇₉	—	2	1	no	—
160	C₂²×C₇₉	—	2	1	no	—
161	C₂²×C₇₉	—	2	1	no	—
162	C₂²×C₇₉	—	4	4	no	—
163	C₂²×C₇₉	—	4	1	no	—
164	C₂²×C₇₉	—	4	1	no	—
165	C₂²×C₇₉	—	4	1	no	—
166	C₂²×C₇₉	#380	1	1	yes	characteristic
167	C₂²×C₇₉	#471	1	2	yes	—
168	C₄×C₇₉	—	2	1	no	—
169	C₄×C₇₉	—	2	1	no	—
170	C₄×C₇₉	—	2	1	no	—
171	C₄×C₇₉	—	4	1	no	—
172	C₄×C₇₉	—	4	1	no	—
173	C₄×C₇₉	—	4	1	no	—
174	C₄×C₇₉	C₂×#99	1	1	yes	characteristic
175	C₄×C₇₉	#410	1	1	yes	characteristic
176	Dic₇₉	—	16	2	no	—
177	#8629	—	79	1	no	maximal, Sylow
178	C₂×C₇₉	—	2	2	no	—
179	C₂×C₇₉	—	4	1	no	—
180	C₂×C₇₉	—	4	1	no	—
181	C₂×C₇₉	—	8	1	no	—
182	C₂×C₇₉	#1799	1	1	yes	characteristic
183	C₂×C₇₉	#1814	1	1	yes	characteristic
184	C₂×C₇₉	#1815	1	1	yes	characteristic
185	#2929	D₇₉	1	1	yes	characteristic
186	#1810	—	79	1	no	—
187	#1807	—	79	1	no	—
188	#1513	—	79	2	no	—
189	#1305	—	79	2	no	—
190	C₇₉	#8629	1	1	yes	characteristic, complemented, Sylow
191	C₂×#131	Dic₇₉	1	1	yes	characteristic, complemented
192	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
193	#525	C₂×D₇₉	1	1	yes	characteristic
194	#435	—	2	1	no	—
195	#435	—	2	1	no	—
196	C₈×D₄	—	2	1	no	—
197	C₈×D₄	—	2	1	no	—
198	C₂×#108	—	158	1	no	—
199	C₂×#106	—	158	1	no	—
200	#424	Dic₇₉	1	1	yes	characteristic, complemented
201	#408	Dic₇₉	1	1	yes	characteristic, complemented
202	#407	—	79	2	no	—
203	#407	C₂×D₇₉	1	1	yes	characteristic
204	#405	Dic₇₉	1	1	yes	characteristic, complemented
205	#380	—	79	2	no	—
206	#341	—	158	1	no	—
207	#340	—	158	1	no	—
208	#328	—	158	2	no	—
209	#327	—	158	2	no	—
210	C₂×#41	C₂×Dic₇₉	1	1	yes	characteristic
211	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
212	#131	—	4	2	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	158	2	no	—
215	C₂×M₁₆	C₂×Dic₇₉	1	1	yes	characteristic
216	C₂×M₁₆	C₂×Dic₇₉	1	1	yes	characteristic
217	C₂²×C₈	—	2	1	no	—
218	C₂²×C₈	—	4	2	no	—
219	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
220	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
221	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
222	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
223	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
224	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
225	#123	—	2	1	no	—
226	C₂²⋊Q₈	Dic₁₅₈	1	1	yes	characteristic
227	C₂²⋊Q₈	D₃₁₆	1	1	yes	characteristic
228	#121	Dic₁₅₈	1	1	yes	characteristic
229	#121	D₃₁₆	1	1	yes	characteristic
230	#120	—	2	1	no	—
231	C₄×D₄	—	2	1	no	—
232	C₄×D₄	—	2	1	no	—
233	#117	C₂×Dic₇₉	1	1	yes	characteristic
234	C₂×C₄⋊C₄	—	158	2	no	—
235	C₂×C₂²⋊C₄	—	158	2	no	—
236	#108	—	316	2	no	—
237	#106	—	316	2	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₂²⋊C₈	—	2	1	no	—
241	C₂²⋊C₈	—	2	1	no	—
242	C₂²⋊C₈	—	2	1	no	—
243	C₂²⋊C₈	—	2	1	no	—
244	C₂²⋊C₈	—	158	2	no	—
245	C₂²⋊C₈	—	158	2	no	—
246	#97	—	2	1	no	—
247	C₄×C₈	—	2	1	no	—
248	#95	—	158	2	no	—
249	C₂⁴	#11929624	1	1	yes	characteristic
250	#41	—	4	1	no	—
251	#41	—	4	1	no	—
252	C₂×Q₈	C₄⋊Dic₇₉	1	1	yes	characteristic
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	C₄⋊Dic₇₉	1	1	yes	characteristic
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	—	316	2	no	—
261	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
265	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
266	C₂²×C₄	#11929624	1	1	yes	characteristic
267	M₁₆	—	4	2	no	—
268	M₁₆	—	4	1	no	—
269	M₁₆	—	4	1	no	—
270	M₁₆	—	316	2	no	—
271	M₁₆	—	316	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	—	316	2	no	—
282	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
283	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
284	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
285	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	316	2	no	—
289	C₄⋊C₄	—	316	2	no	—
290	C₄⋊C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
291	C₄⋊C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
292	C₂²⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₂²⋊C₄	—	2	1	no	—
295	C₂²⋊C₄	—	2	1	no	—
296	C₂²⋊C₄	—	2	1	no	—
297	C₂²⋊C₄	—	632	2	no	—
298	C₄²	—	2	1	no	—
299	C₂³	—	2	2	no	—
300	C₂³	—	2	1	no	—
301	C₂³	—	4	1	no	—
302	C₂³	—	4	1	no	—
303	C₂³	#14579120	1	1	yes	characteristic
304	C₂³	#14579135	1	1	yes	characteristic
305	C₂³	#14579140	1	1	yes	characteristic
306	Q₈	—	4	1	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	632	2	no	—
326	C₂×C₄	—	632	2	no	—
327	C₂×C₄	—	632	2	no	—
328	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
329	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
330	C₂×C₄	#14579135	1	1	yes	characteristic
331	C₂×C₄	#14579140	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	1	no	—
337	C₈	—	632	2	no	—
338	C₈	#14579076	1	2	yes	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	#19098423	1	1	yes	characteristic
352	C₂²	#19098826	1	2	yes	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	1264	2	no	—
360	C₄	C₂×#14579076	1	1	yes	characteristic
361	C₄	#19098537	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	—	4	1	no	—
365	C₂	—	8	1	no	—
366	C₂	#23763388	1	1	yes	characteristic
367	C₂	#23763454	1	1	yes	characteristic
368	C₂	#23763459	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098423
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₇₉×#2929	C₂
socle	C₂²×C₇₉	#380
prime radical	#525×C₇₉	C₂²
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	#2929	D₇₉
hypocenter	C₇₉	#8629
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929

Complements

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

N	H ≅ G/N	type
1. #23763438	C₂	semidirect
2. #23763425	C₂	semidirect
4. #23761305	C₂	semidirect
14. C₂×#131×C₇₉	C₄	semidirect
21. #424×C₇₉	C₄	semidirect
22. #408×C₇₉	C₄	semidirect
24. #405×C₇₉	C₄	semidirect
190. C₇₉	#8629	semidirect
191. C₂×#131	Dic₇₉	semidirect
200. #424	Dic₇₉	semidirect
201. #408	Dic₇₉	semidirect
204. #405	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23763438	2
#23763425	2
#23761943	2
#23761305	2
C₇₉×#2929	2
#8629	79