Subgroups — #34297034

Order 20224.

750 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098959	C₂	1	1	yes	maximal, characteristic
2	#23766148	C₂	1	1	yes	maximal, characteristic
3	#23764374	C₂	1	1	yes	maximal, characteristic
4	#23764372	C₂	1	1	yes	maximal, characteristic
5	#23764140	C₂	1	2	yes	maximal
6	#23764125	C₂	1	2	yes	maximal
7	#23763562	C₂	1	1	yes	maximal, characteristic
8	#23763298	C₂	1	2	yes	maximal
9	C₄×#14579092	C₂	1	2	yes	maximal
10	#23763087	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#2331	C₂	1	1	yes	maximal, characteristic
12	C₂×#14579161	C₂²	1	1	yes	characteristic
13	#19098959	C₄	1	8	yes	complemented
14	C₂×#14579094	C₂²	1	2	yes	—
15	C₂×#14579092	C₂²	1	2	yes	—
16	C₂×#14579092	C₂²	1	2	yes	—
17	C₂×#14579092	C₂²	1	2	yes	—
18	C₂×#14579072	C₂²	1	1	yes	characteristic
19	C₂×#14579072	C₂²	1	1	yes	characteristic
20	C₂×#14579072	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
24	#19098467	—	2	2	no	—
25	#19098461	C₂²	1	2	yes	—
26	#19098446	—	2	2	no	—
27	#19098429	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
29	#19098325	C₂²	1	2	yes	—
30	#19098325	C₂²	1	2	yes	—
31	#19098325	C₂²	1	2	yes	—
32	#19098325	C₂²	1	2	yes	—
33	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
34	#391×C₇₉	—	2	2	no	—
35	#389×C₇₉	C₂²	1	2	yes	—
36	#387×C₇₉	—	2	2	no	—
37	#382×C₇₉	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
39	#376×C₇₉	C₂²	1	1	yes	characteristic
40	C₂²×Dic₁₅₈	—	2	1	no	—
41	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
45	#14579161	—	2	2	no	—
46	#14579161	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊Dic₇₉	—	2	1	no	—
48	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
50	C₂×Dic₇₉⋊C₄	—	2	2	no	—
51	C₂×C₄×Dic₇₉	—	2	2	no	—
52	#14579094	—	2	4	no	—
53	#14579092	—	2	4	no	—
54	#14579092	C₂×C₄	1	8	yes	—
55	#14579092	C₂×C₄	1	8	yes	—
56	#14579072	—	2	2	no	—
57	#14579072	—	2	2	no	—
58	#14579072	Q₈	1	4	yes	complemented
59	C₄×C₇₉⋊C₈	—	2	2	no	—
60	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
61	C₄×C₇₉⋊C₈	Q₈	1	4	yes	complemented
62	C₇₉⋊#95	—	2	2	no	—
63	C₇₉⋊#95	—	2	1	no	—
64	C₇₉⋊#95	—	2	1	no	—
65	C₇₉⋊#95	—	2	2	no	—
66	C₂²×Q₈×C₇₉	—	2	1	no	—
67	C₄⋊Q₈×C₇₉	—	2	2	no	—
68	C₄⋊Q₈×C₇₉	C₂×C₄	1	4	yes	—
69	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
70	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
71	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
73	C₂×C₄²×C₇₉	—	2	2	no	—
74	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
77	#95×C₇₉	—	2	2	no	—
78	#95×C₇₉	—	2	1	no	—
79	#95×C₇₉	—	2	1	no	—
80	#95×C₇₉	—	2	2	no	—
81	#95×C₇₉	D₄	1	2	yes	—
82	#95×C₇₉	D₄	1	2	yes	—
83	C₂×Dic₁₅₈	—	2	4	no	—
84	C₂×Dic₁₅₈	—	8	2	no	—
85	C₂²×Dic₇₉	—	2	2	no	—
86	C₂²×Dic₇₉	—	2	2	no	—
87	C₂×C₇₉⋊C₈	—	2	1	no	—
88	C₂×C₇₉⋊C₈	—	2	1	no	—
89	C₂×C₇₉⋊C₈	—	2	1	no	—
90	C₂×C₇₉⋊C₈	—	2	1	no	—
91	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
94	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
95	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
96	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
97	C₂×C₇₉⋊C₈	#41	1	2	yes	—
98	C₂×C₇₉⋊C₈	#41	1	2	yes	—
99	C₄⋊Dic₇₉	—	2	4	no	—
100	C₄⋊Dic₇₉	—	2	2	no	—
101	C₄⋊Dic₇₉	—	2	2	no	—
102	C₄⋊Dic₇₉	—	2	2	no	—
103	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
104	Dic₇₉⋊C₄	—	8	2	no	—
105	C₄×Dic₇₉	—	8	2	no	—
106	C₂×Q₈×C₇₉	—	2	4	no	—
107	C₂×Q₈×C₇₉	—	8	2	no	—
108	C₂²×C₄×C₇₉	—	2	2	no	—
109	C₂²×C₄×C₇₉	—	2	2	no	—
110	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₇₉	—	2	4	no	—
116	C₄⋊C₄×C₇₉	—	2	2	no	—
117	C₄⋊C₄×C₇₉	—	2	2	no	—
118	C₄⋊C₄×C₇₉	—	2	2	no	—
119	C₄⋊C₄×C₇₉	—	8	2	no	—
120	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
121	C₄²×C₇₉	—	2	1	no	—
122	C₄²×C₇₉	—	2	1	no	—
123	C₄²×C₇₉	—	8	2	no	—
124	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
125	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
126	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
127	C₄²×C₇₉	#41	1	2	yes	—
128	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
129	Q₈×C₇₉	—	8	4	no	—
130	C₂×C₄×C₇₉	—	2	2	no	—
131	C₂×C₄×C₇₉	—	2	2	no	—
132	C₂×C₄×C₇₉	—	2	2	no	—
133	C₂×C₄×C₇₉	—	2	2	no	—
134	C₂×C₄×C₇₉	—	2	4	no	—
135	C₂×C₄×C₇₉	—	2	2	no	—
136	C₂×C₄×C₇₉	—	2	2	no	—
137	C₂×C₄×C₇₉	—	2	2	no	—
138	C₂×C₄×C₇₉	—	4	2	no	—
139	C₂×C₄×C₇₉	—	4	2	no	—
140	C₂×C₄×C₇₉	—	4	2	no	—
141	C₂×C₄×C₇₉	—	4	2	no	—
142	C₂×C₄×C₇₉	—	4	2	no	—
143	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
145	C₂×C₄×C₇₉	C₄⋊Q₈	1	4	yes	—
146	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₇₉	C₂²×Q₈	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×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₇₉	—	2	4	no	—
153	C₂×Dic₇₉	—	4	2	no	—
154	C₂×Dic₇₉	—	4	2	no	—
155	C₂×Dic₇₉	—	4	2	no	—
156	C₂×Dic₇₉	—	4	2	no	—
157	C₂×Dic₇₉	—	4	2	no	—
158	Dic₁₅₈	—	8	4	no	—
159	C₇₉⋊C₈	—	2	2	no	—
160	C₇₉⋊C₈	—	2	2	no	—
161	C₇₉⋊C₈	C₄×Q₈	1	4	yes	—
162	C₇₉⋊C₈	C₄×Q₈	1	4	yes	—
163	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
165	C₂²×C₇₉	#532	1	1	yes	characteristic
166	C₂²×C₇₉	#577	1	2	yes	—
167	C₂²×C₇₉	#577	1	2	yes	—
168	C₄×C₇₉	—	2	4	no	—
169	C₄×C₇₉	—	2	2	no	—
170	C₄×C₇₉	—	2	2	no	—
171	C₄×C₇₉	—	4	2	no	—
172	C₄×C₇₉	—	4	4	no	—
173	C₄×C₇₉	—	8	2	no	—
174	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
175	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
176	C₄×C₇₉	#533	1	1	yes	characteristic
177	C₄×C₇₉	#533	1	1	yes	characteristic
178	Dic₇₉	—	4	4	no	—
179	Dic₇₉	—	8	2	no	—
180	#21724	—	79	1	no	maximal, Sylow
181	C₂×C₇₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
182	C₂×C₇₉	#2949	1	1	yes	characteristic
183	C₂×C₇₉	#2949	1	1	yes	characteristic
184	C₂×C₇₉	#3163	1	2	yes	—
185	C₂×C₇₉	#3173	1	2	yes	—
186	C₂×#498	—	79	1	no	—
187	#2331	—	79	1	no	—
188	#2331	D₇₉	1	1	yes	characteristic
189	#1993	—	79	1	no	—
190	#1993	—	79	1	no	—
191	#1944	—	79	2	no	—
192	#1941	—	79	2	no	—
193	#1846	—	79	1	no	—
194	#1776	—	79	2	no	—
195	C₄×#106	—	79	2	no	—
196	#1726	—	79	1	no	—
197	C₇₉	#21724	1	1	yes	characteristic, complemented, Sylow
198	C₂×C₄⋊Q₈	—	79	1	no	—
199	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
200	#498	—	79	8	no	—
201	C₂×#107	—	79	2	no	—
202	C₂×#106	—	79	2	no	—
203	C₂×#106	—	79	2	no	—
204	C₂×#106	—	79	2	no	—
205	C₂×#97	—	79	1	no	—
206	C₂×#97	—	79	1	no	—
207	C₂×#97	—	79	1	no	—
208	C₂×C₄×C₈	—	79	1	no	—
209	C₂×C₄×C₈	—	79	1	no	—
210	C₂×C₄×C₈	—	79	1	no	—
211	#391	—	2	2	no	—
212	#391	—	158	2	no	—
213	#389	—	79	2	no	—
214	#389	C₂×D₇₉	1	2	yes	—
215	#387	—	2	2	no	—
216	#387	—	158	2	no	—
217	#382	—	79	1	no	—
218	#382	C₂×D₇₉	1	1	yes	characteristic
219	C₄×C₄⋊C₄	—	79	2	no	—
220	C₄×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
221	#376	C₂×D₇₉	1	1	yes	characteristic
222	#340	—	79	2	no	—
223	#340	—	79	2	no	—
224	#340	—	79	2	no	—
225	#340	—	79	2	no	—
226	C₂²×Q₈	—	2	1	no	—
227	C₂²×Q₈	—	158	1	no	—
228	C₂²×C₈	—	79	1	no	—
229	C₂²×C₈	—	79	1	no	—
230	C₂²×C₈	—	79	1	no	—
231	C₂²×C₈	—	79	1	no	—
232	C₄⋊Q₈	—	2	2	no	—
233	C₄⋊Q₈	—	79	4	no	—
234	C₄⋊Q₈	—	158	2	no	—
235	C₄⋊Q₈	C₄×D₇₉	1	4	yes	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	2	no	—
238	C₂×C₄⋊C₄	—	79	2	no	—
239	C₂×C₄⋊C₄	—	79	2	no	—
240	C₂×C₄⋊C₄	—	158	2	no	—
241	C₂×C₄⋊C₄	—	158	1	no	—
242	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
243	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
244	C₂×C₄²	—	2	2	no	—
245	C₂×C₄²	—	158	2	no	—
246	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
247	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
248	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
249	#107	—	158	4	no	—
250	#106	—	79	8	no	—
251	#106	—	79	8	no	—
252	#106	—	158	4	no	—
253	#97	—	79	4	no	—
254	#97	—	158	2	no	—
255	#97	—	158	2	no	—
256	C₄×C₈	—	79	4	no	—
257	C₄×C₈	—	79	4	no	—
258	C₄×C₈	—	158	2	no	—
259	#95	—	2	2	no	—
260	#95	—	2	1	no	—
261	#95	—	2	1	no	—
262	#95	—	2	2	no	—
263	#95	—	158	2	no	—
264	#95	—	158	1	no	—
265	#95	—	158	1	no	—
266	#95	—	158	2	no	—
267	#95	C₂²⋊D₇₉	1	2	yes	—
268	#95	C₂²⋊D₇₉	1	2	yes	—
269	C₂×Q₈	—	2	4	no	—
270	C₂×Q₈	—	8	2	no	—
271	C₂×Q₈	—	632	2	no	—
272	C₂×Q₈	—	158	4	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	158	2	no	—
276	C₂²×C₄	—	158	2	no	—
277	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
278	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
280	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
281	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
282	C₂×C₈	—	79	2	no	—
283	C₂×C₈	—	79	2	no	—
284	C₂×C₈	—	79	2	no	—
285	C₂×C₈	—	79	2	no	—
286	C₂×C₈	—	79	2	no	—
287	C₂×C₈	—	79	2	no	—
288	C₂×C₈	—	79	2	no	—
289	C₂×C₈	—	79	2	no	—
290	C₂×C₈	—	158	1	no	—
291	C₂×C₈	—	158	1	no	—
292	C₂×C₈	—	158	1	no	—
293	C₂×C₈	—	158	1	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	4	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	8	2	no	—
299	C₄⋊C₄	—	79	4	no	—
300	C₄⋊C₄	—	632	2	no	—
301	C₄⋊C₄	—	158	2	no	—
302	C₄⋊C₄	—	158	2	no	—
303	C₄⋊C₄	—	158	2	no	—
304	C₄⋊C₄	—	158	4	no	—
305	C₄⋊C₄	C₂×C₄×D₇₉	1	4	yes	—
306	C₄²	—	2	1	no	—
307	C₄²	—	2	1	no	—
308	C₄²	—	8	2	no	—
309	C₄²	—	632	2	no	—
310	C₄²	C₂×C₄×D₇₉	1	2	yes	—
311	C₄²	Q₈×D₇₉	1	2	yes	—
312	C₄²	Q₈×D₇₉	1	2	yes	—
313	C₄²	#11929624	1	2	yes	—
314	C₂³	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
315	Q₈	—	8	4	no	—
316	Q₈	—	632	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	316	2	no	—
331	C₂×C₄	—	316	2	no	—
332	C₂×C₄	—	316	2	no	—
333	C₂×C₄	—	316	2	no	—
334	C₂×C₄	—	316	2	no	—
335	C₂×C₄	—	158	4	no	—
336	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
337	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
338	C₂×C₄	#14579159	1	4	yes	—
339	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
340	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
341	C₂×C₄	C₂×Q₈×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	C₂×#11929624	1	1	yes	characteristic
345	C₈	—	79	4	no	—
346	C₈	—	79	4	no	—
347	C₈	—	158	2	no	—
348	C₈	—	158	2	no	—
349	C₂²	C₂×C₄×C₂²⋊D₇₉	1	1	yes	characteristic
350	C₂²	C₂×#14579159	1	1	yes	characteristic
351	C₂²	#19099109	1	1	yes	characteristic
352	C₂²	#19099371	1	2	yes	—
353	C₂²	#19099374	1	2	yes	—
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	4	no	—
357	C₄	—	4	2	no	—
358	C₄	—	4	4	no	—
359	C₄	—	8	2	no	—
360	C₄	—	316	4	no	—
361	C₄	—	632	2	no	—
362	C₄	C₄×Q₈×D₇₉	1	1	yes	characteristic
363	C₄	C₄×Q₈×D₇₉	1	1	yes	characteristic
364	C₄	#19099117	1	1	yes	characteristic
365	C₄	#19099117	1	1	yes	characteristic
366	C₂	C₄×#14579159	1	1	yes	characteristic
367	C₂	#23772257	1	1	yes	characteristic
368	C₂	#23772259	1	1	yes	characteristic
369	C₂	#23773926	1	2	yes	—
370	C₂	#23774011	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#2331	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#2331	D₇₉
hypocenter	C₇₉	#21724
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #2331

Complements

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

N	H ≅ G/N	type
13. #19098959	C₄	semidirect
58. #14579072	Q₈	semidirect
61. C₄×C₇₉⋊C₈	Q₈	semidirect
197. C₇₉	#21724	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098959	2
#23766148	2
#23764374	2
#23764372	2
#23764140	2
#23764125	2
#23763562	2
#23763298	2
C₄×#14579092	2
#23763087	2
C₇₉×#2331	2
#21724	79