Subgroups — #34179120

Order 20224.

550 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098641	C₂	1	1	yes	maximal, characteristic
2	#23763265	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14579082	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098285	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761240	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761234	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11929608	—	2	1	no	—
9	#19098641	—	2	2	no	—
10	#19098641	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#14579082	C₄	1	2	yes	—
13	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
14	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₇₉	—	2	1	no	—
17	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
18	#19098325	C₄	1	2	yes	—
19	#19098317	—	2	1	no	—
20	#19098285	—	2	2	no	—
21	C₂×C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	D₄⋊C₈×C₇₉	C₄	1	4	yes	complemented
25	C₂×#11929608	—	2	4	no	—
26	C₂×#11929608	—	4	2	no	—
27	C₂×#11929608	—	4	2	no	—
28	C₂²×C₇₉⋊C₈	—	2	1	no	—
29	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
31	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₇₉	—	2	1	no	—
34	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
35	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
36	#14579082	—	2	4	no	—
37	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
38	C₄×C₇₉⋊C₈	—	2	1	no	—
39	C₄×C₇₉⋊C₈	—	2	1	no	—
40	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
41	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
42	C₈×Dic₇₉	—	4	2	no	—
43	C₇₉⋊#95	—	2	1	no	—
44	C₂²×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₇₉	—	2	1	no	—
46	C₂²×C₈×C₇₉	—	2	1	no	—
47	C₂²×C₈×C₇₉	—	2	1	no	—
48	C₄×D₄×C₇₉	—	2	1	no	—
49	C₄×D₄×C₇₉	—	2	1	no	—
50	C₄×D₄×C₇₉	C₂×C₄	1	2	yes	—
51	C₄×D₄×C₇₉	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
54	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₇₉	—	2	1	no	—
56	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₇₉	—	2	1	no	—
58	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
59	C₂²×Dic₇₉	—	2	2	no	—
60	#11929608	—	4	8	no	—
61	C₂×C₇₉⋊C₈	—	2	2	no	—
62	C₂×C₇₉⋊C₈	—	2	2	no	—
63	C₂×C₇₉⋊C₈	—	2	2	no	—
64	C₂×C₇₉⋊C₈	—	2	2	no	—
65	C₂×C₇₉⋊C₈	—	2	1	no	—
66	C₂×C₇₉⋊C₈	—	2	1	no	—
67	C₂×C₇₉⋊C₈	—	4	1	no	—
68	C₂×C₇₉⋊C₈	—	4	1	no	—
69	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
71	C₂×C₇₉⋊C₈	D₈	1	2	yes	—
72	C₂×C₇₉⋊C₈	SD₁₆	1	2	yes	—
73	C₄⋊Dic₇₉	—	2	2	no	—
74	C₄⋊Dic₇₉	M₁₆	1	4	yes	complemented
75	C₄×Dic₇₉	—	4	1	no	—
76	C₄×Dic₇₉	—	4	1	no	—
77	C₂⁴×C₇₉	—	2	1	no	—
78	C₂×D₄×C₇₉	—	2	1	no	—
79	C₂×D₄×C₇₉	—	2	2	no	—
80	C₂×D₄×C₇₉	—	2	2	no	—
81	C₂×D₄×C₇₉	C₄²	1	2	yes	—
82	C₂²×C₄×C₇₉	—	2	1	no	—
83	C₂²×C₄×C₇₉	—	2	2	no	—
84	C₂²×C₄×C₇₉	—	2	2	no	—
85	C₂²×C₄×C₇₉	—	4	1	no	—
86	C₂²×C₄×C₇₉	—	4	1	no	—
87	C₂²×C₄×C₇₉	—	4	1	no	—
88	C₂²×C₄×C₇₉	—	4	1	no	—
89	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
92	C₂×C₈×C₇₉	—	2	2	no	—
93	C₂×C₈×C₇₉	—	2	2	no	—
94	C₂×C₈×C₇₉	—	4	2	no	—
95	C₂×C₈×C₇₉	—	4	1	no	—
96	C₂×C₈×C₇₉	—	4	1	no	—
97	C₄⋊C₄×C₇₉	—	2	1	no	—
98	C₄⋊C₄×C₇₉	C₄²	1	2	yes	—
99	C₂²⋊C₄×C₇₉	—	2	2	no	—
100	C₂²⋊C₄×C₇₉	—	2	2	no	—
101	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₇₉	#41	1	1	yes	characteristic
105	C₂³×C₇₉	—	2	2	no	—
106	C₂³×C₇₉	—	4	2	no	—
107	C₂³×C₇₉	—	4	1	no	—
108	C₂³×C₇₉	—	4	1	no	—
109	C₂³×C₇₉	—	4	2	no	—
110	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
111	D₄×C₇₉	—	2	2	no	—
112	D₄×C₇₉	—	2	2	no	—
113	D₄×C₇₉	—	4	2	no	—
114	C₂×C₄×C₇₉	—	2	2	no	—
115	C₂×C₄×C₇₉	—	2	2	no	—
116	C₂×C₄×C₇₉	—	4	1	no	—
117	C₂×C₄×C₇₉	—	4	1	no	—
118	C₂×C₄×C₇₉	—	4	2	no	—
119	C₂×C₄×C₇₉	—	4	2	no	—
120	C₂×C₄×C₇₉	—	4	2	no	—
121	C₂×C₄×C₇₉	—	4	2	no	—
122	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₇₉	#131	1	2	yes	—
130	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
133	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
134	C₂×Dic₇₉	—	2	4	no	—
135	C₂×Dic₇₉	—	4	2	no	—
136	C₂×Dic₇₉	—	4	2	no	—
137	C₈×C₇₉	—	4	2	no	—
138	C₈×C₇₉	—	4	2	no	—
139	C₇₉⋊C₈	—	2	2	no	—
140	C₇₉⋊C₈	—	4	2	no	—
141	C₇₉⋊C₈	—	4	2	no	—
142	C₇₉⋊C₈	#102	1	4	yes	complemented
143	C₂²×C₇₉	—	4	2	no	—
144	C₂²×C₇₉	—	4	2	no	—
145	C₂²×C₇₉	—	4	2	no	—
146	C₂²×C₇₉	—	8	2	no	—
147	C₂²×C₇₉	—	8	2	no	—
148	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₇₉	#435	1	2	yes	—
150	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
151	C₂²×C₇₉	C₄×SD₁₆	1	1	yes	characteristic
152	C₂²×C₇₉	#444	1	2	yes	—
153	C₄×C₇₉	—	2	1	no	—
154	C₄×C₇₉	—	2	1	no	—
155	C₄×C₇₉	—	4	2	no	—
156	C₄×C₇₉	C₄×M₁₆	1	1	yes	characteristic
157	C₄×C₇₉	#405	1	1	yes	characteristic
158	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
159	C₄×C₇₉	#416	1	1	yes	characteristic
160	C₄×C₇₉	#436	1	2	yes	—
161	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
162	C₄×C₇₉	C₄×SD₁₆	1	1	yes	characteristic
163	Dic₇₉	—	4	4	no	—
164	#8210	—	79	1	no	maximal, Sylow
165	C₂×C₇₉	—	8	2	no	—
166	C₂×C₇₉	#1581	1	2	yes	—
167	C₂×C₇₉	#1583	1	2	yes	—
168	C₂×C₇₉	#1753	1	1	yes	characteristic
169	C₂×C₇₉	C₄×#102	1	1	yes	characteristic
170	C₂×C₇₉	#1765	1	1	yes	characteristic
171	C₂×#436	—	79	1	no	—
172	C₈×C₄⋊C₄	—	79	1	no	—
173	C₄×#102	—	79	1	no	—
174	C₂×D₄⋊C₈	D₇₉	1	1	yes	characteristic
175	C₂×#322	—	79	1	no	—
176	#1279	—	79	1	no	—
177	#1277	—	79	1	no	—
178	C₇₉	#8210	1	1	yes	characteristic, complemented, Sylow
179	C₂²×M₁₆	—	158	1	no	—
180	C₂×C₄×D₄	C₂×D₇₉	1	1	yes	characteristic
181	#436	—	79	4	no	—
182	#436	—	158	2	no	—
183	C₂×C₄⋊C₈	—	79	1	no	—
184	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
185	C₂×#102	—	79	2	no	—
186	C₂×C₂²⋊C₈	—	158	1	no	—
187	C₂×C₄×C₈	—	79	1	no	—
188	C₂×C₄×C₈	—	79	1	no	—
189	C₂×C₄×C₈	—	158	1	no	—
190	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
191	C₄×C₄⋊C₄	—	79	1	no	—
192	#340	—	79	2	no	—
193	#336	—	158	1	no	—
194	D₄⋊C₈	Dic₇₉	1	4	yes	complemented
195	#322	—	158	2	no	—
196	C₂²×D₄	C₄×D₇₉	1	1	yes	characteristic
197	C₂³×C₄	—	2	1	no	—
198	C₂×M₁₆	—	316	2	no	—
199	C₂×M₁₆	—	316	2	no	—
200	C₂×M₁₆	—	158	4	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	—	79	1	no	—
204	C₂²×C₈	—	79	1	no	—
205	C₂²×C₈	—	79	2	no	—
206	C₂²×C₈	—	158	1	no	—
207	C₄×D₄	—	2	1	no	—
208	C₄×D₄	—	2	1	no	—
209	C₄×D₄	C₄×D₇₉	1	2	yes	—
210	C₄×D₄	C₂×Dic₇₉	1	2	yes	—
211	C₂×C₄⋊C₄	—	79	2	no	—
212	C₂×C₄⋊C₄	C₄×D₇₉	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	—	2	1	no	—
214	C₂×C₄²	—	158	1	no	—
215	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	79	2	no	—
218	C₄⋊C₈	—	158	1	no	—
219	C₄⋊C₈	C₂×Dic₇₉	1	2	yes	—
220	#102	—	158	4	no	—
221	C₂²⋊C₈	—	158	4	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	316	2	no	—
224	C₄×C₈	—	79	2	no	—
225	C₄×C₈	—	79	2	no	—
226	C₄×C₈	—	158	1	no	—
227	C₄×C₈	—	158	1	no	—
228	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
229	#95	—	158	1	no	—
230	C₂⁴	—	2	1	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	C₄×Dic₇₉	1	2	yes	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	158	2	no	—
243	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
244	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
245	C₂²×C₄	#11929623	1	1	yes	characteristic
246	M₁₆	—	316	8	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	316	1	no	—
253	C₂×C₈	—	316	1	no	—
254	C₂×C₈	—	79	2	no	—
255	C₂×C₈	—	79	2	no	—
256	C₂×C₈	—	79	2	no	—
257	C₂×C₈	—	79	2	no	—
258	C₂×C₈	—	158	2	no	—
259	C₂×C₈	—	158	2	no	—
260	C₂×C₈	—	158	2	no	—
261	C₂×C₈	—	158	2	no	—
262	C₂×C₈	—	158	1	no	—
263	C₂×C₈	—	158	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	79	4	no	—
266	C₄⋊C₄	—	158	2	no	—
267	C₄⋊C₄	C₄×Dic₇₉	1	2	yes	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₄²	—	316	1	no	—
271	C₄²	—	316	1	no	—
272	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
273	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
274	C₄²	D₄×D₇₉	1	1	yes	characteristic
275	C₄²	#11929623	1	1	yes	characteristic
276	C₂³	—	2	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	—	4	2	no	—
281	C₂³	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
282	D₄	—	2	2	no	—
283	D₄	—	2	2	no	—
284	D₄	—	4	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	316	2	no	—
294	C₂×C₄	—	316	2	no	—
295	C₂×C₄	—	158	4	no	—
296	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
297	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
298	C₂×C₄	#14579117	1	1	yes	characteristic
299	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
300	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
301	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
302	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
303	C₂×C₄	#14579167	1	2	yes	—
304	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
305	C₂×C₄	SD₁₆×D₇₉	1	1	yes	characteristic
306	C₂×C₄	#14579180	1	1	yes	characteristic
307	C₂×C₄	#14579182	1	1	yes	characteristic
308	C₈	—	4	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	316	2	no	—
311	C₈	—	316	2	no	—
312	C₈	—	79	4	no	—
313	C₈	—	158	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	C₂²⋊C₄×Dic₇₉	1	1	yes	characteristic
320	C₂²	#19098637	1	2	yes	—
321	C₂²	#19098643	1	1	yes	characteristic
322	C₂²	#19098654	1	1	yes	characteristic
323	C₂²	#19098682	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	316	4	no	—
328	C₄	M₁₆×Dic₇₉	1	1	yes	characteristic
329	C₄	#19098518	1	1	yes	characteristic
330	C₄	#102×D₇₉	1	1	yes	characteristic
331	C₄	#19098556	1	1	yes	characteristic
332	C₄	#19098638	1	2	yes	—
333	C₄	D₈×Dic₇₉	1	1	yes	characteristic
334	C₄	SD₁₆×Dic₇₉	1	1	yes	characteristic
335	C₂	—	8	2	no	—
336	C₂	#23762232	1	2	yes	—
337	C₂	#23762246	1	2	yes	—
338	C₂	#23763186	1	1	yes	characteristic
339	C₂	#102×Dic₇₉	1	1	yes	characteristic
340	C₂	#23763245	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×D₄⋊C₈×C₇₉	C₂
socle	C₂³×C₇₉	C₄×D₄
prime radical	C₂²×D₄×C₇₉	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×D₄⋊C₈	D₇₉
hypocenter	C₇₉	#8210
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×D₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #23763265	C₂	semidirect
4. C₂×#19098285	C₂	semidirect
5. #23761240	C₂	semidirect
6. #23761234	C₂	semidirect
10. #19098641	C₄	semidirect
24. D₄⋊C₈×C₇₉	C₄	semidirect
74. C₄⋊Dic₇₉	M₁₆	semidirect
142. C₇₉⋊C₈	#102	semidirect
178. C₇₉	#8210	semidirect
194. D₄⋊C₈	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098641	2
#23763265	2
C₄×#14579082	2
C₂×#19098285	2
#23761240	2
#23761234	2
C₂×D₄⋊C₈×C₇₉	2
#8210	79