Subgroups — #34253982

Order 20224.

806 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	#23772031	C₂	1	2	yes	maximal
2	C₄×M₁₆×D₇₉	C₂	1	1	yes	maximal, characteristic
3	#23761703	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761703	C₂	1	2	yes	maximal, complemented
5	#23761698	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761696	C₂	1	2	yes	maximal, complemented
7	#23761694	C₂	1	1	yes	maximal, characteristic, complemented
8	#23761691	C₂	1	2	yes	maximal, complemented
9	#23761685	C₂	1	1	yes	maximal, characteristic, complemented
10	#23761682	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#1455	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₇₉	C₄	1	2	yes	—
13	C₂×C₄²×D₇₉	C₂²	1	1	yes	characteristic
14	#19098618	C₄	1	4	yes	—
15	#19098618	C₄	1	4	yes	—
16	#19098614	C₂²	1	2	yes	—
17	#19098614	C₂²	1	2	yes	—
18	#19098614	C₂²	1	2	yes	—
19	#19098614	C₂²	1	2	yes	—
20	#19098586	C₄	1	4	yes	—
21	#19098526	C₄	1	4	yes	—
22	C₄×#11929608	C₂²	1	2	yes	—
23	C₄×#11929608	C₂²	1	1	yes	characteristic
24	#19098515	C₄	1	2	yes	—
25	M₁₆×Dic₇₉	C₄	1	2	yes	—
26	C₄×#11929607	C₂²	1	1	yes	characteristic
27	C₄×#11929607	C₂²	1	1	yes	characteristic
28	#97×D₇₉	C₄	1	2	yes	—
29	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
31	#19098285	C₂²	1	2	yes	complemented
32	#19098285	C₂²	1	2	yes	complemented
33	#19098285	C₂²	1	2	yes	complemented
34	#19098284	C₂²	1	1	yes	characteristic, complemented
35	#19098284	C₂²	1	2	yes	complemented
36	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
37	C₈×C₇₉⋊C₈	C₂²	1	2	yes	complemented
38	C₄×M₁₆×C₇₉	C₂²	1	2	yes	—
39	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	#322×C₇₉	C₂²	1	1	yes	characteristic, complemented
41	#322×C₇₉	C₂²	1	2	yes	complemented
42	C₈²×C₇₉	C₂²	1	1	yes	characteristic, complemented
43	C₂²×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×D₇₉	C₂×C₄	1	2	yes	—
45	C₂×#11929608	C₂×C₄	1	2	yes	—
46	C₂×#11929608	C₂×C₄	1	4	yes	—
47	C₂×#11929607	C₂×C₄	1	2	yes	—
48	C₂×#11929607	C₂×C₄	1	2	yes	—
49	M₁₆×D₇₉	—	2	8	no	—
50	C₂×C₈×D₇₉	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₇₉	C₂×C₄	1	2	yes	—
52	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×Dic₇₉	C₂×C₄	1	2	yes	—
54	C₄²×D₇₉	C₂×C₄	1	1	yes	characteristic
55	C₄²×D₇₉	C₂×C₄	1	1	yes	characteristic
56	C₄²×D₇₉	C₂×C₄	1	2	yes	—
57	C₄²×D₇₉	C₂×C₄	1	2	yes	—
58	C₄²×D₇₉	C₂³	1	1	yes	characteristic
59	C₄²×D₇₉	C₂³	1	1	yes	characteristic
60	Dic₇₉⋊C₈	C₂×C₄	1	4	yes	—
61	Dic₇₉⋊C₈	C₂×C₄	1	4	yes	—
62	Dic₇₉⋊C₈	C₂×C₄	1	4	yes	—
63	Dic₇₉⋊C₈	C₂×C₄	1	4	yes	—
64	(C₂×D₇₉)⋊C₈	C₂×C₄	1	4	yes	—
65	(C₂×D₇₉)⋊C₈	C₂×C₄	1	4	yes	—
66	(C₂×D₇₉)⋊C₈	C₂×C₄	1	4	yes	—
67	(C₂×D₇₉)⋊C₈	C₂×C₄	1	4	yes	—
68	#14579072	C₂×C₄	1	2	yes	—
69	#14579072	C₂×C₄	1	4	yes	—
70	#14579071	C₂×C₄	1	2	yes	—
71	#14579071	C₂×C₄	1	2	yes	—
72	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
75	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
76	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
77	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
78	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
79	C₂×M₁₆×C₇₉	C₂×C₄	1	4	yes	—
80	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
81	#97×C₇₉	C₂×C₄	1	2	yes	—
82	#97×C₇₉	C₂×C₄	1	4	yes	—
83	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₇₉	C₂³	1	2	yes	—
86	C₄×C₈×C₇₉	C₂³	1	2	yes	—
87	C₂³×D₇₉	C₄²	1	1	yes	characteristic
88	C₂²×Dic₇₉	C₄²	1	1	yes	characteristic
89	C₂²×Dic₇₉	C₄²	1	2	yes	—
90	C₂×C₄×D₇₉	—	2	4	no	—
91	C₂×C₄×D₇₉	—	4	4	no	—
92	C₂×C₄×D₇₉	C₄²	1	1	yes	characteristic
93	C₂×C₄×D₇₉	C₄²	1	1	yes	characteristic
94	C₂×C₄×D₇₉	C₄²	1	2	yes	—
95	C₂×C₄×D₇₉	C₄²	1	2	yes	—
96	C₂×C₄×D₇₉	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₇₉	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₇₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×D₇₉	C₂²×C₄	1	1	yes	characteristic
100	#11929608	—	2	4	no	—
101	#11929608	—	4	4	no	—
102	#11929607	—	2	4	no	—
103	#11929607	—	2	4	no	—
104	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
106	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
107	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
108	C₈×D₇₉	—	2	4	no	—
109	C₈×D₇₉	M₁₆	1	8	yes	complemented
110	C₄×Dic₇₉	C₄²	1	2	yes	—
111	C₄×Dic₇₉	C₄²	1	2	yes	—
112	C₄×Dic₇₉	C₄²	1	2	yes	—
113	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
114	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
115	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
116	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
119	M₁₆×C₇₉	—	2	4	no	—
120	M₁₆×C₇₉	—	4	4	no	—
121	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
123	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
124	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
125	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
127	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
128	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
129	C₂²×D₇₉	—	2	2	no	—
130	C₂²×D₇₉	—	4	2	no	—
131	C₂²×D₇₉	C₂×C₄²	1	1	yes	characteristic
132	C₂²×D₇₉	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₇₉	—	2	2	no	—
134	C₂×C₄×C₇₉	—	2	4	no	—
135	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
137	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
139	C₂×Dic₇₉	—	2	2	no	—
140	C₂×Dic₇₉	—	4	2	no	—
141	C₂×Dic₇₉	C₂×C₄²	1	1	yes	characteristic
142	C₂×Dic₇₉	C₂×C₄²	1	2	yes	—
143	C₂×Dic₇₉	C₂×C₄²	1	2	yes	—
144	C₂×Dic₇₉	C₂×C₄²	1	1	yes	characteristic
145	C₄×D₇₉	—	2	2	no	—
146	C₄×D₇₉	—	2	2	no	—
147	C₄×D₇₉	—	2	4	no	—
148	C₄×D₇₉	—	2	4	no	—
149	C₄×D₇₉	—	2	2	no	—
150	C₄×D₇₉	—	4	2	no	—
151	C₄×D₇₉	—	4	2	no	—
152	C₄×D₇₉	C₂×M₁₆	1	4	yes	—
153	C₈×C₇₉	—	2	2	no	—
154	C₈×C₇₉	—	2	4	no	—
155	C₈×C₇₉	—	2	4	no	—
156	C₈×C₇₉	C₂×M₁₆	1	4	yes	—
157	C₇₉⋊C₈	—	2	2	no	—
158	C₇₉⋊C₈	—	2	4	no	—
159	C₇₉⋊C₈	—	2	4	no	—
160	C₇₉⋊C₈	C₂×M₁₆	1	4	yes	—
161	C₂²×C₇₉	—	2	1	no	—
162	C₂²×C₇₉	—	2	2	no	—
163	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
164	C₂×D₇₉	—	2	2	no	—
165	C₂×D₇₉	—	2	1	no	—
166	C₂×D₇₉	—	2	2	no	—
167	C₂×D₇₉	—	4	4	no	—
168	C₂×D₇₉	C₄×M₁₆	1	2	yes	—
169	C₄×C₇₉	—	2	1	no	—
170	C₄×C₇₉	—	2	2	no	—
171	C₄×C₇₉	C₂²×M₁₆	1	2	yes	—
172	C₄×C₇₉	#568	1	4	yes	—
173	Dic₇₉	—	2	1	no	—
174	Dic₇₉	—	2	2	no	—
175	Dic₇₉	—	2	2	no	—
176	Dic₇₉	C₄×M₁₆	1	2	yes	—
177	#17334	—	79	1	no	maximal, Sylow
178	C₂×C₇₉	—	4	1	no	—
179	C₂×C₇₉	C₂×C₄×M₁₆	1	1	yes	characteristic
180	C₂×C₇₉	#2873	1	2	yes	—
181	D₇₉	—	2	2	no	—
182	D₇₉	—	4	1	no	—
183	#2920	—	79	2	no	—
184	C₂×C₄×M₁₆	—	79	1	no	—
185	#1455	—	79	1	no	—
186	#1455	—	79	2	no	—
187	#1455	D₇₉	1	1	yes	characteristic, complemented
188	#1453	—	79	2	no	—
189	#1453	—	79	1	no	—
190	#1453	—	79	1	no	—
191	#1452	—	79	2	no	—
192	#1450	—	79	1	no	—
193	#1450	—	79	1	no	—
194	C₇₉	#17334	1	1	yes	characteristic, complemented, Sylow
195	C₂²×M₁₆	—	79	2	no	—
196	C₂²×C₄²	—	79	1	no	—
197	#432	—	79	4	no	—
198	#432	—	79	4	no	—
199	#431	—	79	2	no	—
200	#431	—	79	2	no	—
201	#431	—	79	2	no	—
202	#431	—	79	2	no	—
203	#423	—	79	4	no	—
204	#407	—	79	4	no	—
205	C₄×M₁₆	—	79	2	no	—
206	C₄×M₁₆	—	79	1	no	—
207	C₄×M₁₆	—	79	2	no	—
208	C₄×M₁₆	—	79	2	no	—
209	C₄×M₁₆	—	79	1	no	—
210	C₄×M₁₆	—	79	1	no	—
211	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
212	C₄×M₁₆	C₂×D₇₉	1	2	yes	—
213	C₂×#97	—	79	2	no	—
214	C₂×C₄×C₈	—	79	1	no	—
215	C₂×C₄×C₈	—	79	1	no	—
216	#322	—	79	1	no	—
217	#322	—	79	2	no	—
218	#322	—	79	2	no	—
219	#322	—	79	2	no	—
220	#322	—	79	2	no	—
221	#322	C₂×D₇₉	1	1	yes	characteristic, complemented
222	#322	C₂×D₇₉	1	2	yes	complemented
223	C₈²	—	79	2	no	—
224	C₈²	—	79	1	no	—
225	C₈²	C₂×D₇₉	1	1	yes	characteristic, complemented
226	C₂³×C₄	—	79	2	no	—
227	C₂³×C₄	—	79	1	no	—
228	C₂×M₁₆	—	79	4	no	—
229	C₂×M₁₆	—	79	2	no	—
230	C₂×M₁₆	—	79	2	no	—
231	C₂×M₁₆	—	79	2	no	—
232	C₂×M₁₆	—	158	8	no	—
233	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
234	C₂×M₁₆	C₄×D₇₉	1	4	yes	—
235	C₂²×C₈	—	79	2	no	—
236	C₂²×C₈	—	79	2	no	—
237	C₂×C₄²	—	79	1	no	—
238	C₂×C₄²	—	79	2	no	—
239	C₂×C₄²	—	79	1	no	—
240	C₂×C₄²	—	79	1	no	—
241	C₂×C₄²	—	79	2	no	—
242	C₂×C₄²	—	79	2	no	—
243	C₂×C₄²	—	79	1	no	—
244	C₂×C₄²	—	79	1	no	—
245	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
246	C₄⋊C₈	—	79	4	no	—
247	C₄⋊C₈	—	79	4	no	—
248	C₄⋊C₈	—	79	4	no	—
249	C₄⋊C₈	—	79	4	no	—
250	C₂²⋊C₈	—	79	4	no	—
251	C₂²⋊C₈	—	79	4	no	—
252	C₂²⋊C₈	—	79	4	no	—
253	C₂²⋊C₈	—	79	4	no	—
254	#97	—	79	2	no	—
255	#97	—	79	2	no	—
256	#97	—	79	2	no	—
257	#97	—	79	4	no	—
258	#97	C₄×D₇₉	1	2	yes	—
259	#97	C₄×D₇₉	1	4	yes	—
260	C₄×C₈	—	79	2	no	—
261	C₄×C₈	—	79	2	no	—
262	C₄×C₈	—	79	1	no	—
263	C₄×C₈	—	79	2	no	—
264	C₄×C₈	—	79	2	no	—
265	C₄×C₈	—	79	1	no	—
266	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₇₉	1	2	yes	—
268	C₄×C₈	C₂²×D₇₉	1	2	yes	—
269	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
270	C₂⁴	—	79	1	no	—
271	C₂²×C₄	—	316	4	no	—
272	C₂²×C₄	—	79	2	no	—
273	C₂²×C₄	—	79	2	no	—
274	C₂²×C₄	—	79	2	no	—
275	C₂²×C₄	—	79	1	no	—
276	C₂²×C₄	—	79	1	no	—
277	C₂²×C₄	—	79	1	no	—
278	C₂²×C₄	—	79	1	no	—
279	C₂²×C₄	—	79	2	no	—
280	C₂²×C₄	—	79	2	no	—
281	C₂²×C₄	—	79	1	no	—
282	C₂²×C₄	—	158	4	no	—
283	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
284	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
285	M₁₆	—	2	4	no	—
286	M₁₆	—	4	4	no	—
287	M₁₆	—	316	4	no	—
288	M₁₆	—	158	4	no	—
289	M₁₆	—	158	4	no	—
290	M₁₆	—	158	4	no	—
291	C₂×C₈	—	79	2	no	—
292	C₂×C₈	—	79	8	no	—
293	C₂×C₈	—	79	2	no	—
294	C₂×C₈	—	79	4	no	—
295	C₂×C₈	—	79	4	no	—
296	C₂×C₈	—	158	4	no	—
297	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
298	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
299	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
300	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
301	C₄²	—	79	2	no	—
302	C₄²	—	79	2	no	—
303	C₄²	—	79	2	no	—
304	C₄²	—	79	1	no	—
305	C₄²	—	79	1	no	—
306	C₄²	—	79	2	no	—
307	C₄²	—	79	2	no	—
308	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
309	C₄²	C₂×C₄×D₇₉	1	2	yes	—
310	C₄²	C₂³×D₇₉	1	1	yes	characteristic
311	C₂³	—	316	2	no	—
312	C₂³	—	79	1	no	—
313	C₂³	—	79	1	no	—
314	C₂³	—	158	2	no	—
315	C₂³	C₄²×D₇₉	1	1	yes	characteristic
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	316	2	no	—
319	C₂×C₄	—	316	2	no	—
320	C₂×C₄	—	316	2	no	—
321	C₂×C₄	—	79	1	no	—
322	C₂×C₄	—	79	4	no	—
323	C₂×C₄	—	79	1	no	—
324	C₂×C₄	—	79	2	no	—
325	C₂×C₄	—	79	2	no	—
326	C₂×C₄	—	158	2	no	—
327	C₂×C₄	—	158	2	no	—
328	C₂×C₄	—	158	2	no	—
329	C₂×C₄	—	158	2	no	—
330	C₂×C₄	—	158	4	no	—
331	C₂×C₄	—	158	4	no	—
332	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
333	C₂×C₄	C₄²×D₇₉	1	2	yes	—
334	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
335	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
336	C₈	—	2	2	no	—
337	C₈	—	2	4	no	—
338	C₈	—	2	4	no	—
339	C₈	—	79	4	no	—
340	C₈	—	158	2	no	—
341	C₈	—	158	4	no	—
342	C₈	—	158	4	no	—
343	C₈	M₁₆×D₇₉	1	4	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	316	4	no	—
347	C₂²	—	79	2	no	—
348	C₂²	—	158	1	no	—
349	C₂²	—	158	2	no	—
350	C₂²	—	158	2	no	—
351	C₂²	C₂×C₄²×D₇₉	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	2	no	—
354	C₄	—	79	2	no	—
355	C₄	—	158	1	no	—
356	C₄	—	158	2	no	—
357	C₄	—	158	2	no	—
358	C₄	C₂×M₁₆×D₇₉	1	2	yes	—
359	C₄	#19099329	1	4	yes	—
360	C₂	—	4	1	no	—
361	C₂	—	316	1	no	—
362	C₂	—	158	2	no	—
363	C₂	C₄×M₁₆×D₇₉	1	1	yes	characteristic
364	C₂	#23771817	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₇₉
commutator, socle	C₂²×C₇₉	C₂²×C₄²
Fitting	C₇₉×#1455	C₂
prime radical	C₂³×D₇₉	C₄²
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1455	D₇₉
hypocenter	C₇₉	#17334
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
3. #23761703	C₂	semidirect
4. #23761703	C₂	semidirect
5. #23761698	C₂	semidirect
6. #23761696	C₂	semidirect
7. #23761694	C₂	semidirect
8. #23761691	C₂	semidirect
9. #23761685	C₂	semidirect
10. #23761682	C₂	semidirect
11. C₇₉×#1455	C₂	semidirect
31. #19098285	C₂²	semidirect
32. #19098285	C₂²	semidirect
33. #19098285	C₂²	semidirect
34. #19098284	C₂²	semidirect
35. #19098284	C₂²	semidirect
36. C₈×C₇₉⋊C₈	C₂²	semidirect
37. C₈×C₇₉⋊C₈	C₂²	semidirect
40. #322×C₇₉	C₂²	semidirect
41. #322×C₇₉	C₂²	semidirect
42. C₈²×C₇₉	C₂²	semidirect
109. C₈×D₇₉	M₁₆	semidirect
187. #1455	D₇₉	semidirect
194. C₇₉	#17334	semidirect
221. #322	C₂×D₇₉	semidirect
222. #322	C₂×D₇₉	semidirect
225. C₈²	C₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23772031	2
C₄×M₁₆×D₇₉	2
#23761703	2
#23761698	2
#23761696	2
#23761694	2
#23761691	2
#23761685	2
#23761682	2
C₇₉×#1455	2
#17334	79