Subgroups — #34182149

Order 20224.

466 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772508	C₂	1	1	yes	maximal, characteristic
2	#23763551	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14579087	C₂	1	1	yes	maximal, characteristic
4	#23761785	C₂	1	1	yes	maximal, characteristic
5	#23761251	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761232	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1565	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11929624	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₅₈	—	2	1	no	—
10	C₈×D₃₁₆	—	2	1	no	—
11	(C₂²⋊D₇₉)⋊C₈	—	2	2	no	—
12	#19098616	C₂²	1	1	yes	characteristic
13	#19098615	—	2	1	no	—
14	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×Dic₇₉⋊C₈	—	2	2	no	—
16	C₄⋊C₈×D₇₉	—	2	1	no	—
17	C₂×#14579087	C₄	1	1	yes	characteristic
18	C₂×#14579087	C₄	1	1	yes	characteristic
19	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
20	#19098324	C₄	1	1	yes	characteristic
21	#19098324	C₄	1	1	yes	characteristic
22	#19098293	—	2	1	no	—
23	#19098290	—	2	1	no	—
24	#431×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
27	#335×C₇₉	—	2	1	no	—
28	#334×C₇₉	—	2	1	no	—
29	C₂×#11929624	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
31	C₂×C₈×D₇₉	—	2	2	no	—
32	C₄×Dic₁₅₈	D₄	1	1	yes	characteristic
33	C₄×Dic₁₅₈	Q₈	1	1	yes	characteristic
34	C₄×D₃₁₆	D₄	1	1	yes	characteristic
35	C₄×D₃₁₆	Q₈	1	1	yes	characteristic
36	C₄×C₂²⋊D₇₉	—	2	1	no	—
37	C₄×C₂²⋊D₇₉	—	2	1	no	—
38	#14579120	C₂×C₄	1	1	yes	characteristic
39	#14579118	—	2	1	no	—
40	C₂×C₄×Dic₇₉	—	2	1	no	—
41	C₂×C₄×Dic₇₉	—	2	1	no	—
42	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
44	C₄²×D₇₉	—	2	1	no	—
45	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
46	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
47	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
48	Dic₇₉⋊C₈	—	2	2	no	—
49	Dic₇₉⋊C₈	—	4	4	no	—
50	#14579087	—	2	2	no	—
51	#14579087	C₈	1	4	yes	complemented
52	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
53	(C₂×D₇₉)⋊C₈	—	2	2	no	—
54	C₄×C₇₉⋊C₈	—	2	1	no	—
55	C₈×Dic₇₉	—	2	2	no	—
56	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
57	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₇₉	—	2	1	no	—
59	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₇₉	—	2	1	no	—
61	C₄⋊C₈×C₇₉	—	2	1	no	—
62	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
63	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
64	C₂²⋊C₈×C₇₉	—	2	1	no	—
65	#97×C₇₉	—	2	1	no	—
66	C₄×C₈×C₇₉	—	2	1	no	—
67	C₄×C₈×C₇₉	—	2	1	no	—
68	#11929624	—	2	1	no	—
69	#11929624	—	2	1	no	—
70	#11929624	C₂×C₈	1	2	yes	—
71	#11929624	M₁₆	1	2	yes	—
72	C₂×Dic₁₅₈	C₄⋊C₄	1	1	yes	characteristic
73	C₂×D₃₁₆	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₇₉	—	2	1	no	—
75	C₂²×Dic₇₉	—	2	1	no	—
76	C₂²×Dic₇₉	—	2	1	no	—
77	C₂×C₄×D₇₉	—	2	1	no	—
78	C₂×C₄×D₇₉	—	2	1	no	—
79	C₂×C₄×D₇₉	—	2	1	no	—
80	C₂×C₇₉⋊C₈	—	2	2	no	—
81	C₂×C₇₉⋊C₈	—	4	2	no	—
82	C₂×C₇₉⋊C₈	—	4	1	no	—
83	C₈×D₇₉	—	8	2	no	—
84	C₄⋊Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
85	C₄⋊Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
86	Dic₇₉⋊C₄	—	2	1	no	—
87	Dic₇₉⋊C₄	—	2	1	no	—
88	C₂²⋊Dic₇₉	—	2	1	no	—
89	(C₂×D₇₉)⋊C₄	—	2	1	no	—
90	(C₂×D₇₉)⋊C₄	—	2	1	no	—
91	C₄×Dic₇₉	—	2	2	no	—
92	C₄×Dic₇₉	—	2	2	no	—
93	C₄×Dic₇₉	—	2	1	no	—
94	C₄×Dic₇₉	—	2	1	no	—
95	C₄×Dic₇₉	—	2	1	no	—
96	C₄×Dic₇₉	—	2	1	no	—
97	C₄×Dic₇₉	—	2	1	no	—
98	C₄×Dic₇₉	—	2	1	no	—
99	C₄×Dic₇₉	—	4	2	no	—
100	C₄×Dic₇₉	C₂×C₈	1	2	yes	complemented
101	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
104	M₁₆×C₇₉	—	2	1	no	—
105	M₁₆×C₇₉	—	2	1	no	—
106	M₁₆×C₇₉	—	2	1	no	—
107	M₁₆×C₇₉	C₂×C₈	1	2	yes	complemented
108	C₂×C₈×C₇₉	—	2	1	no	—
109	C₂×C₈×C₇₉	—	2	1	no	—
110	C₂×C₈×C₇₉	—	2	1	no	—
111	C₂×C₈×C₇₉	—	2	1	no	—
112	C₂×C₈×C₇₉	—	4	2	no	—
113	C₂×C₈×C₇₉	—	4	1	no	—
114	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
117	C₄²×C₇₉	#41	1	1	yes	characteristic
118	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
119	C₂²×D₇₉	—	2	1	no	—
120	C₂×C₄×C₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	—	2	1	no	—
122	C₂×C₄×C₇₉	—	4	1	no	—
123	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
126	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
132	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
133	C₂²⋊D₇₉	—	2	2	no	—
134	C₂²⋊D₇₉	—	2	2	no	—
135	C₂×Dic₇₉	—	2	2	no	—
136	C₂×Dic₇₉	—	2	2	no	—
137	C₂×Dic₇₉	—	2	1	no	—
138	C₂×Dic₇₉	—	2	2	no	—
139	C₂×Dic₇₉	—	2	2	no	—
140	C₂×Dic₇₉	—	2	1	no	—
141	C₂×Dic₇₉	—	2	1	no	—
142	C₂×Dic₇₉	—	4	1	no	—
143	C₂×Dic₇₉	—	4	1	no	—
144	D₃₁₆	—	2	1	no	—
145	D₃₁₆	C₄⋊C₈	1	2	yes	complemented
146	C₄×D₇₉	—	2	2	no	—
147	C₄×D₇₉	—	2	2	no	—
148	C₄×D₇₉	—	4	1	no	—
149	C₄×D₇₉	—	4	1	no	—
150	C₄×D₇₉	—	8	1	no	—
151	Dic₁₅₈	—	2	1	no	—
152	Dic₁₅₈	C₄⋊C₈	1	2	yes	complemented
153	C₈×C₇₉	—	2	2	no	—
154	C₈×C₇₉	—	4	1	no	—
155	C₈×C₇₉	—	4	1	no	—
156	C₈×C₇₉	—	4	1	no	—
157	C₈×C₇₉	—	8	1	no	—
158	C₇₉⋊C₈	—	4	2	no	—
159	C₇₉⋊C₈	—	8	1	no	—
160	C₂²×C₇₉	—	2	1	no	—
161	C₂²×C₇₉	—	2	1	no	—
162	C₂²×C₇₉	#385	1	1	yes	characteristic
163	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
164	C₂²×C₇₉	#435	1	1	yes	characteristic
165	C₂×D₇₉	—	2	2	no	—
166	C₂×D₇₉	—	4	1	no	—
167	C₂×D₇₉	—	4	1	no	—
168	C₄×C₇₉	—	2	1	no	—
169	C₄×C₇₉	—	2	1	no	—
170	C₄×C₇₉	—	2	1	no	—
171	C₄×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
172	C₄×C₇₉	#424	1	1	yes	characteristic
173	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
174	C₄×C₇₉	#435	1	1	yes	characteristic
175	C₄×C₇₉	#453	1	1	yes	characteristic
176	C₄×C₇₉	#456	1	1	yes	characteristic
177	Dic₇₉	—	2	4	no	—
178	Dic₇₉	—	2	2	no	—
179	Dic₇₉	—	4	2	no	—
180	Dic₇₉	—	4	1	no	—
181	Dic₇₉	—	4	1	no	—
182	#8839	—	79	1	no	maximal, Sylow
183	C₂×C₇₉	—	2	1	no	—
184	C₂×C₇₉	—	2	1	no	—
185	C₂×C₇₉	#1850	1	1	yes	characteristic
186	C₂×C₇₉	#1859	1	1	yes	characteristic
187	C₂×C₇₉	#1861	1	1	yes	characteristic
188	D₇₉	—	4	2	no	—
189	#2988	—	79	1	no	—
190	#1842	—	79	1	no	—
191	C₄×#104	—	79	1	no	—
192	#1565	D₇₉	1	1	yes	characteristic, complemented
193	#1477	—	79	1	no	—
194	#1283	—	79	1	no	—
195	#1276	—	79	1	no	—
196	C₇₉	#8839	1	1	yes	characteristic, complemented, Sylow
197	C₄×#41	—	79	1	no	—
198	C₈×Q₈	—	158	1	no	—
199	C₈×D₄	—	158	1	no	—
200	C₈×D₄	—	158	2	no	—
201	#431	—	79	1	no	—
202	#431	—	158	1	no	—
203	#431	C₂×D₇₉	1	1	yes	characteristic
204	C₂×C₄⋊C₈	—	79	1	no	—
205	C₂×C₄⋊C₈	—	158	2	no	—
206	C₂×C₄⋊C₈	—	158	1	no	—
207	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
208	C₂×#104	—	79	1	no	—
209	C₂×#104	—	79	1	no	—
210	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
211	C₄³	—	79	1	no	—
212	#339	—	79	1	no	—
213	#339	—	79	1	no	—
214	#335	—	2	1	no	—
215	#334	—	2	1	no	—
216	Q₈⋊C₈	—	158	1	no	—
217	D₄⋊C₈	—	158	1	no	—
218	C₂×#41	—	79	1	no	—
219	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
220	C₂×M₁₆	C₄×D₇₉	1	1	yes	characteristic
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	—	79	2	no	—
223	C₂²×C₈	—	158	2	no	—
224	C₄×Q₈	—	79	1	no	—
225	C₄×Q₈	—	79	1	no	—
226	C₄×D₄	—	79	1	no	—
227	C₄×D₄	—	79	1	no	—
228	C₄×D₄	—	158	1	no	—
229	C₄×D₄	—	158	1	no	—
230	#117	—	79	1	no	—
231	#117	—	158	1	no	—
232	C₂×C₄²	—	79	1	no	—
233	C₂×C₄²	—	79	1	no	—
234	C₂×C₄²	—	158	1	no	—
235	C₂×C₄²	—	158	1	no	—
236	C₂×C₄²	—	158	1	no	—
237	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	316	4	no	—
241	C₄⋊C₈	—	158	2	no	—
242	C₄⋊C₈	—	158	1	no	—
243	C₄⋊C₈	—	158	1	no	—
244	C₄⋊C₈	—	158	1	no	—
245	C₄⋊C₈	C₂²⋊D₇₉	1	1	yes	characteristic
246	C₄⋊C₈	C₂²⋊D₇₉	1	1	yes	characteristic
247	#104	—	79	4	no	—
248	#104	—	158	2	no	—
249	C₂²⋊C₈	—	2	1	no	—
250	C₂²⋊C₈	—	158	2	no	—
251	C₂²⋊C₈	—	158	1	no	—
252	#97	—	2	1	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	158	2	no	—
256	C₄×C₈	—	158	1	no	—
257	#41	—	79	2	no	—
258	#41	—	79	2	no	—
259	#41	—	158	1	no	—
260	#41	—	158	1	no	—
261	C₂×Q₈	—	79	1	no	—
262	C₂×D₄	—	79	1	no	—
263	C₂×D₄	—	158	1	no	—
264	C₂²×C₄	—	158	1	no	—
265	C₂²×C₄	—	158	1	no	—
266	C₂²×C₄	—	158	1	no	—
267	C₂²×C₄	—	158	1	no	—
268	C₂²×C₄	—	158	1	no	—
269	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
270	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
271	C₂²×C₄	#11929625	1	1	yes	characteristic
272	M₁₆	—	2	1	no	—
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	C₈×D₇₉	1	2	yes	complemented
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	316	2	no	—
283	C₂×C₈	—	316	1	no	—
284	C₂×C₈	—	632	2	no	—
285	C₂×C₈	—	158	2	no	—
286	C₄⋊C₄	—	79	1	no	—
287	C₄⋊C₄	—	79	1	no	—
288	C₄⋊C₄	—	158	1	no	—
289	C₄⋊C₄	—	158	1	no	—
290	C₂²⋊C₄	—	158	1	no	—
291	C₂²⋊C₄	—	158	1	no	—
292	C₂²⋊C₄	—	158	1	no	—
293	C₄²	—	316	2	no	—
294	C₄²	—	79	2	no	—
295	C₄²	—	158	2	no	—
296	C₄²	—	158	2	no	—
297	C₄²	—	158	1	no	—
298	C₄²	—	158	1	no	—
299	C₄²	—	158	1	no	—
300	C₄²	—	158	1	no	—
301	C₄²	—	158	1	no	—
302	C₄²	—	158	1	no	—
303	C₄²	D₄×D₇₉	1	1	yes	characteristic
304	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
305	C₄²	Q₈×D₇₉	1	1	yes	characteristic
306	C₄²	#11929624	1	1	yes	characteristic
307	C₂³	—	158	1	no	—
308	C₂³	#14579123	1	1	yes	characteristic
309	Q₈	—	79	2	no	—
310	Q₈	—	158	1	no	—
311	D₄	—	79	2	no	—
312	D₄	—	158	2	no	—
313	D₄	—	158	2	no	—
314	D₄	—	158	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	316	1	no	—
319	C₂×C₄	—	316	1	no	—
320	C₂×C₄	—	316	1	no	—
321	C₂×C₄	—	316	1	no	—
322	C₂×C₄	—	632	1	no	—
323	C₂×C₄	—	158	2	no	—
324	C₂×C₄	—	158	1	no	—
325	C₂×C₄	—	158	1	no	—
326	C₂×C₄	—	158	2	no	—
327	C₂×C₄	—	158	2	no	—
328	C₂×C₄	—	158	2	no	—
329	C₂×C₄	—	158	1	no	—
330	C₂×C₄	—	158	2	no	—
331	C₂×C₄	—	158	2	no	—
332	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
333	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
334	C₂×C₄	#14579129	1	1	yes	characteristic
335	C₂×C₄	#14579137	1	1	yes	characteristic
336	C₂×C₄	#14579139	1	1	yes	characteristic
337	C₂×C₄	#14579146	1	1	yes	characteristic
338	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
339	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
340	C₂×C₄	#14579167	1	1	yes	characteristic
341	C₂×C₄	#14579169	1	1	yes	characteristic
342	C₈	—	2	2	no	—
343	C₈	—	4	1	no	—
344	C₈	—	4	1	no	—
345	C₈	—	4	1	no	—
346	C₈	—	8	1	no	—
347	C₈	—	316	2	no	—
348	C₈	—	632	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	316	1	no	—
352	C₂²	—	316	1	no	—
353	C₂²	—	158	2	no	—
354	C₂²	#19098434	1	1	yes	characteristic
355	C₂²	#19098628	1	1	yes	characteristic
356	C₂²	#19098633	1	1	yes	characteristic
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	316	2	no	—
361	C₄	—	316	1	no	—
362	C₄	—	316	1	no	—
363	C₄	—	158	4	no	—
364	C₄	—	158	2	no	—
365	C₄	C₄⋊C₈×D₇₉	1	1	yes	characteristic
366	C₄	#19098588	1	1	yes	characteristic
367	C₄	C₈×C₂²⋊D₇₉	1	1	yes	characteristic
368	C₄	#19098632	1	1	yes	characteristic
369	C₄	#19098729	1	1	yes	characteristic
370	C₄	#19098744	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	316	2	no	—
374	C₂	#23763576	1	1	yes	characteristic
375	C₂	#23763623	1	1	yes	characteristic
376	C₂	#23763637	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14579129
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1565	C₂
socle	C₂²×C₇₉	#385
prime radical	C₂×#11929624	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1565	D₇₉
hypocenter	C₇₉	#8839
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
2. #23763551	C₂	semidirect
5. #23761251	C₂	semidirect
6. #23761232	C₂	semidirect
7. C₇₉×#1565	C₂	semidirect
51. #14579087	C₈	semidirect
100. C₄×Dic₇₉	C₂×C₈	semidirect
107. M₁₆×C₇₉	C₂×C₈	semidirect
145. D₃₁₆	C₄⋊C₈	semidirect
152. Dic₁₅₈	C₄⋊C₈	semidirect
192. #1565	D₇₉	semidirect
196. C₇₉	#8839	semidirect
275. M₁₆	C₈×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23772508	2
#23763551	2
C₄×#14579087	2
#23761785	2
#23761251	2
#23761232	2
C₇₉×#1565	2
#8839	79