Subgroups — #34297036

Order 20224.

750 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098960	C₂	1	1	yes	maximal, characteristic
2	#23766144	C₂	1	2	yes	maximal
3	#23764371	C₂	1	2	yes	maximal
4	#23764142	C₂	1	2	yes	maximal
5	#23764129	C₂	1	1	yes	maximal, characteristic
6	#23764126	C₂	1	1	yes	maximal, characteristic
7	#23763561	C₂	1	1	yes	maximal, characteristic
8	#23763298	C₂	1	2	yes	maximal
9	C₄×#14579093	C₂	1	1	yes	maximal, characteristic
10	#23763086	C₂	1	1	yes	maximal, characteristic
11	C₄×#106×C₇₉	C₂	1	1	yes	maximal, characteristic
12	C₂×#14579160	C₂²	1	2	yes	—
13	#19098960	C₄	1	8	yes	complemented
14	C₂×#14579094	C₂²	1	2	yes	—
15	C₂×#14579093	C₂²	1	1	yes	characteristic
16	C₂×#14579093	C₂²	1	1	yes	characteristic
17	C₂×#14579092	C₂²	1	2	yes	—
18	C₂×#14579072	C₂²	1	1	yes	characteristic
19	C₂×#14579071	C₂²	1	1	yes	characteristic
20	C₂×#14579071	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
23	#19098469	—	2	2	no	—
24	#19098467	—	2	2	no	—
25	#19098462	C₂²	1	2	yes	—
26	#19098459	C₂²	1	2	yes	—
27	#19098450	—	2	2	no	—
28	#19098447	—	2	2	no	—
29	#19098428	C₂²	1	2	yes	—
30	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
31	#19098409	C₂²	1	1	yes	characteristic
32	#19098326	C₂²	1	2	yes	—
33	#19098325	C₂²	1	2	yes	—
34	#19098325	C₂²	1	2	yes	—
35	C₂×#106×C₇₉	C₂²	1	1	yes	characteristic
36	C₂×#106×C₇₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
39	#340×C₇₉	C₂²	1	2	yes	—
40	C₂²×Dic₁₅₈	—	2	2	no	—
41	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
43	#14579160	—	2	4	no	—
44	#14579160	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊Dic₇₉	—	2	2	no	—
46	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
48	C₂×Dic₇₉⋊C₄	—	2	2	no	—
49	C₂×Dic₇₉⋊C₄	—	2	2	no	—
50	C₂×C₄×Dic₇₉	—	2	2	no	—
51	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
52	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
53	#14579094	—	2	4	no	—
54	#14579093	—	2	2	no	—
55	#14579093	C₂×C₄	1	4	yes	—
56	#14579092	C₂×C₄	1	8	yes	—
57	#14579072	Q₈	1	4	yes	complemented
58	#14579071	—	2	2	no	—
59	#14579071	—	2	2	no	—
60	C₈×Dic₇₉	—	2	2	no	—
61	C₈×Dic₇₉	C₂×C₄	1	4	yes	—
62	C₇₉⋊#95	—	2	2	no	—
63	C₇₉⋊#95	—	2	2	no	—
64	C₇₉⋊#95	—	2	2	no	—
65	C₇₉⋊#95	—	2	2	no	—
66	C₇₉⋊#95	—	2	2	no	—
67	C₇₉⋊#95	D₄	1	2	yes	—
68	C₇₉⋊#95	D₄	1	2	yes	—
69	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
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	#106×C₇₉	—	2	2	no	—
76	#106×C₇₉	C₂×C₄	1	4	yes	—
77	C₄×C₈×C₇₉	Q₈	1	4	yes	complemented
78	#95×C₇₉	—	2	2	no	—
79	C₂×Dic₁₅₈	—	2	8	no	—
80	C₂×Dic₁₅₈	—	8	2	no	—
81	C₂×Dic₁₅₈	—	8	2	no	—
82	C₂²×Dic₇₉	—	2	2	no	—
83	C₂²×Dic₇₉	—	2	2	no	—
84	C₂²×Dic₇₉	C₂×D₄	1	2	yes	—
85	C₂²×Dic₇₉	C₂×D₄	1	2	yes	—
86	C₂×C₇₉⋊C₈	—	2	1	no	—
87	C₂×C₇₉⋊C₈	—	2	1	no	—
88	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
90	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
91	C₂×C₇₉⋊C₈	#41	1	2	yes	—
92	C₄⋊Dic₇₉	—	2	2	no	—
93	C₄⋊Dic₇₉	—	2	2	no	—
94	C₄⋊Dic₇₉	—	2	2	no	—
95	C₄⋊Dic₇₉	—	8	2	no	—
96	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
97	Dic₇₉⋊C₄	—	2	8	no	—
98	Dic₇₉⋊C₄	—	8	2	no	—
99	C₄×Dic₇₉	—	2	1	no	—
100	C₄×Dic₇₉	—	2	1	no	—
101	C₄×Dic₇₉	—	8	2	no	—
102	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
103	C₄×Dic₇₉	#41	1	2	yes	—
104	C₂²×C₄×C₇₉	—	2	2	no	—
105	C₂²×C₄×C₇₉	—	2	2	no	—
106	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₇₉	—	2	1	no	—
110	C₂×C₈×C₇₉	—	2	1	no	—
111	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
113	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
114	C₂×C₈×C₇₉	#41	1	2	yes	—
115	C₄⋊C₄×C₇₉	—	2	2	no	—
116	C₄⋊C₄×C₇₉	—	2	2	no	—
117	C₄⋊C₄×C₇₉	—	2	2	no	—
118	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
119	C₄²×C₇₉	—	8	2	no	—
120	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
121	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
122	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	—	2	2	no	—
124	C₂×C₄×C₇₉	—	2	2	no	—
125	C₂×C₄×C₇₉	—	2	4	no	—
126	C₂×C₄×C₇₉	—	4	2	no	—
127	C₂×C₄×C₇₉	—	4	2	no	—
128	C₂×C₄×C₇₉	—	4	2	no	—
129	C₂×C₄×C₇₉	—	4	2	no	—
130	C₂×C₄×C₇₉	—	4	2	no	—
131	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
132	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
133	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₇₉	—	2	2	no	—
140	C₂×Dic₇₉	—	2	2	no	—
141	C₂×Dic₇₉	—	2	4	no	—
142	C₂×Dic₇₉	—	2	2	no	—
143	C₂×Dic₇₉	—	2	2	no	—
144	C₂×Dic₇₉	—	2	2	no	—
145	C₂×Dic₇₉	—	4	2	no	—
146	C₂×Dic₇₉	—	4	2	no	—
147	C₂×Dic₇₉	—	4	2	no	—
148	C₂×Dic₇₉	—	4	2	no	—
149	C₂×Dic₇₉	—	4	2	no	—
150	C₂×Dic₇₉	C₄×D₄	1	4	yes	—
151	Dic₁₅₈	—	8	8	no	—
152	C₈×C₇₉	—	2	2	no	—
153	C₈×C₇₉	C₄×Q₈	1	4	yes	—
154	C₇₉⋊C₈	—	2	2	no	—
155	C₇₉⋊C₈	C₄×Q₈	1	4	yes	—
156	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	C₂²×C₇₉	#532	1	1	yes	characteristic
159	C₂²×C₇₉	#577	1	2	yes	—
160	C₂²×C₇₉	#577	1	2	yes	—
161	C₄×C₇₉	—	2	2	no	—
162	C₄×C₇₉	—	2	2	no	—
163	C₄×C₇₉	—	4	4	no	—
164	C₄×C₇₉	—	8	2	no	—
165	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
166	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄×C₇₉	#533	1	1	yes	characteristic
168	C₄×C₇₉	#533	1	1	yes	characteristic
169	Dic₇₉	—	2	4	no	—
170	Dic₇₉	—	4	2	no	—
171	Dic₇₉	—	4	4	no	—
172	Dic₇₉	—	8	2	no	—
173	#21724	—	79	1	no	maximal, Sylow
174	C₂×C₇₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
175	C₂×C₇₉	#2949	1	2	yes	—
176	C₂×C₇₉	#3163	1	1	yes	characteristic
177	C₂×C₇₉	#3163	1	1	yes	characteristic
178	C₂×C₇₉	#3173	1	2	yes	—
179	C₂×#498	—	79	1	no	—
180	#2331	—	79	2	no	—
181	#1993	—	79	2	no	—
182	#1944	—	79	2	no	—
183	#1941	—	79	1	no	—
184	#1941	—	79	1	no	—
185	#1846	—	79	1	no	—
186	#1776	—	79	2	no	—
187	C₄×#106	—	79	1	no	—
188	C₄×#106	D₇₉	1	1	yes	characteristic
189	#1726	—	79	1	no	—
190	C₇₉	#21724	1	1	yes	characteristic, complemented, Sylow
191	C₂×C₄⋊Q₈	—	79	2	no	—
192	#498	—	79	8	no	—
193	C₂×#107	—	79	2	no	—
194	C₂×#106	—	79	1	no	—
195	C₂×#106	—	79	2	no	—
196	C₂×#106	—	79	1	no	—
197	C₂×#106	C₂×D₇₉	1	1	yes	characteristic
198	C₂×#106	C₂×D₇₉	1	1	yes	characteristic
199	C₂×#97	—	79	1	no	—
200	C₂×#97	—	79	1	no	—
201	C₂×#97	—	79	1	no	—
202	C₂×C₄×C₈	—	79	1	no	—
203	C₂×C₄×C₈	—	79	1	no	—
204	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
205	#391	—	158	2	no	—
206	#391	—	158	2	no	—
207	#389	—	79	2	no	—
208	#389	—	79	2	no	—
209	#387	—	158	2	no	—
210	#387	—	158	2	no	—
211	#382	—	79	2	no	—
212	C₄×C₄⋊C₄	—	79	2	no	—
213	C₄×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
214	#376	—	79	1	no	—
215	#340	—	79	2	no	—
216	#340	—	79	2	no	—
217	#340	—	79	2	no	—
218	#340	C₂×D₇₉	1	2	yes	—
219	C₂²×Q₈	—	158	2	no	—
220	C₂²×C₈	—	79	1	no	—
221	C₂²×C₈	—	79	1	no	—
222	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
223	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
224	C₄⋊Q₈	—	79	8	no	—
225	C₄⋊Q₈	—	158	4	no	—
226	C₂×C₄⋊C₄	—	79	2	no	—
227	C₂×C₄⋊C₄	—	79	2	no	—
228	C₂×C₄⋊C₄	—	158	2	no	—
229	C₂×C₄⋊C₄	—	158	2	no	—
230	C₂×C₄⋊C₄	—	158	2	no	—
231	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
232	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	—	79	1	no	—
235	C₂×C₄²	—	79	1	no	—
236	C₂×C₄²	—	158	2	no	—
237	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
238	#107	—	158	4	no	—
239	#106	—	2	2	no	—
240	#106	—	79	4	no	—
241	#106	—	79	8	no	—
242	#106	—	158	2	no	—
243	#106	C₄×D₇₉	1	4	yes	—
244	#97	—	79	4	no	—
245	#97	—	158	2	no	—
246	#97	—	158	2	no	—
247	C₄×C₈	—	79	4	no	—
248	C₄×C₈	—	158	2	no	—
249	C₄×C₈	Dic₁₅₈	1	4	yes	complemented
250	#95	—	2	2	no	—
251	#95	—	79	2	no	—
252	#95	—	79	2	no	—
253	#95	—	158	2	no	—
254	#95	—	158	2	no	—
255	#95	—	158	2	no	—
256	#95	—	158	2	no	—
257	#95	—	158	2	no	—
258	C₂×Q₈	—	632	2	no	—
259	C₂×Q₈	—	632	2	no	—
260	C₂×Q₈	—	158	8	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	79	2	no	—
264	C₂²×C₄	—	79	2	no	—
265	C₂²×C₄	—	158	2	no	—
266	C₂²×C₄	—	158	2	no	—
267	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
268	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
269	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	79	2	no	—
273	C₂×C₈	—	79	2	no	—
274	C₂×C₈	—	79	2	no	—
275	C₂×C₈	—	79	2	no	—
276	C₂×C₈	—	158	1	no	—
277	C₂×C₈	—	158	1	no	—
278	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
279	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
280	C₂×C₈	C₂×Dic₁₅₈	1	2	yes	—
281	C₂×C₈	#11929624	1	2	yes	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	79	4	no	—
286	C₄⋊C₄	—	632	2	no	—
287	C₄⋊C₄	—	632	2	no	—
288	C₄⋊C₄	—	158	8	no	—
289	C₄⋊C₄	—	158	2	no	—
290	C₄⋊C₄	—	158	2	no	—
291	C₄⋊C₄	—	158	2	no	—
292	C₄⋊C₄	C₂×C₄×D₇₉	1	4	yes	—
293	C₄²	—	8	2	no	—
294	C₄²	—	79	2	no	—
295	C₄²	—	79	2	no	—
296	C₄²	—	632	2	no	—
297	C₄²	—	158	1	no	—
298	C₄²	—	158	1	no	—
299	C₄²	Q₈×D₇₉	1	2	yes	—
300	C₄²	C₂×Dic₁₅₈	1	2	yes	—
301	C₂³	C₂×D₄×D₇₉	1	1	yes	characteristic
302	Q₈	—	632	8	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	316	2	no	—
312	C₂×C₄	—	316	2	no	—
313	C₂×C₄	—	316	2	no	—
314	C₂×C₄	—	316	2	no	—
315	C₂×C₄	—	316	2	no	—
316	C₂×C₄	—	79	4	no	—
317	C₂×C₄	—	158	4	no	—
318	C₂×C₄	—	158	2	no	—
319	C₂×C₄	—	158	2	no	—
320	C₂×C₄	—	158	2	no	—
321	C₂×C₄	—	158	2	no	—
322	C₂×C₄	—	158	2	no	—
323	C₂×C₄	#14579160	1	2	yes	—
324	C₂×C₄	#14579160	1	2	yes	—
325	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
326	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
327	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
328	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
329	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
330	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	79	4	no	—
333	C₈	—	158	2	no	—
334	C₈	C₄×Dic₁₅₈	1	4	yes	—
335	C₂²	C₄×D₄×D₇₉	1	1	yes	characteristic
336	C₂²	C₂×#14579160	1	1	yes	characteristic
337	C₂²	#19099110	1	1	yes	characteristic
338	C₂²	#19099369	1	2	yes	—
339	C₂²	#19099370	1	2	yes	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	4	4	no	—
343	C₄	—	8	2	no	—
344	C₄	—	316	2	no	—
345	C₄	—	316	4	no	—
346	C₄	—	632	2	no	—
347	C₄	—	158	4	no	—
348	C₄	C₄×Q₈×D₇₉	1	1	yes	characteristic
349	C₄	C₂×C₄×Dic₁₅₈	1	1	yes	characteristic
350	C₄	#19099116	1	1	yes	characteristic
351	C₄	#19099117	1	1	yes	characteristic
352	C₂	C₄×#14579160	1	1	yes	characteristic
353	C₂	#23772254	1	2	yes	—
354	C₂	#23773928	1	1	yes	characteristic
355	C₂	#23773932	1	1	yes	characteristic
356	C₂	#23774013	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₄×#106×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₄×#106	D₇₉
hypocenter	C₇₉	#21724
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
13. #19098960	C₄	semidirect
57. #14579072	Q₈	semidirect
77. C₄×C₈×C₇₉	Q₈	semidirect
190. C₇₉	#21724	semidirect
249. C₄×C₈	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098960	2
#23766144	2
#23764371	2
#23764142	2
#23764129	2
#23764126	2
#23763561	2
#23763298	2
C₄×#14579093	2
#23763086	2
C₄×#106×C₇₉	2
#21724	79