Subgroups — #34179117

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×#19098642	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14579083	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098292	C₂	1	1	yes	maximal, characteristic
5	#23761240	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761235	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#322×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11929607	—	2	1	no	—
9	C₂×C₄×D₃₁₆	C₂²	1	1	yes	characteristic
10	#19098642	—	2	2	no	—
11	#19098642	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#14579083	C₄	1	2	yes	—
15	C₂×(C₂×D₇₉)⋊C₈	—	2	1	no	—
16	C₂×C₈×Dic₇₉	—	2	1	no	—
17	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
18	#19098326	C₄	1	2	yes	—
19	#19098317	—	2	1	no	—
20	#19098292	C₄	1	4	yes	complemented
21	C₂×C₄×C₈×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	#322×C₇₉	—	2	2	no	—
25	C₂²×D₃₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₇₉	—	2	1	no	—
27	C₂×#11929607	—	2	4	no	—
28	C₂×#11929607	—	4	2	no	—
29	C₂×#11929607	—	4	2	no	—
30	C₂²×C₇₉⋊C₈	—	2	1	no	—
31	C₂²×C₇₉⋊C₈	—	2	1	no	—
32	C₄×D₃₁₆	—	2	1	no	—
33	C₄×D₃₁₆	—	2	1	no	—
34	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
35	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₇₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
38	C₂×(C₂×D₇₉)⋊C₄	—	2	1	no	—
39	C₂×C₄×Dic₇₉	—	2	1	no	—
40	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
41	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
42	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
43	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
44	#14579083	—	2	4	no	—
45	(C₂×D₇₉)⋊C₈	—	2	4	no	—
46	C₈×Dic₇₉	—	4	2	no	—
47	C₇₉⋊#95	—	2	1	no	—
48	C₂²×C₈×C₇₉	—	2	1	no	—
49	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₇₉	—	2	1	no	—
54	C₄×C₈×C₇₉	—	2	1	no	—
55	C₄×C₈×C₇₉	—	2	1	no	—
56	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₇₉	D₄	1	2	yes	—
59	C₂³×D₇₉	—	2	1	no	—
60	C₂×D₃₁₆	—	2	1	no	—
61	C₂×D₃₁₆	—	2	2	no	—
62	C₂×D₃₁₆	—	2	2	no	—
63	C₂×D₃₁₆	C₄²	1	2	yes	—
64	C₂²×Dic₇₉	—	2	2	no	—
65	C₂²×Dic₇₉	—	2	1	no	—
66	C₂×C₄×D₇₉	—	2	2	no	—
67	C₂×C₄×D₇₉	—	2	2	no	—
68	C₂×C₄×D₇₉	—	4	1	no	—
69	C₂×C₄×D₇₉	—	4	1	no	—
70	C₂×C₄×D₇₉	—	4	1	no	—
71	C₂×C₄×D₇₉	—	4	1	no	—
72	#11929607	—	4	8	no	—
73	C₂×C₇₉⋊C₈	—	2	2	no	—
74	C₂×C₇₉⋊C₈	—	2	2	no	—
75	C₂×C₇₉⋊C₈	—	4	1	no	—
76	C₂×C₇₉⋊C₈	—	4	1	no	—
77	C₂×C₇₉⋊C₈	—	4	1	no	—
78	C₂×C₇₉⋊C₈	—	4	1	no	—
79	C₄⋊Dic₇₉	—	2	1	no	—
80	C₄⋊Dic₇₉	—	2	2	no	—
81	C₄⋊Dic₇₉	C₄²	1	2	yes	—
82	C₄⋊Dic₇₉	M₁₆	1	4	yes	complemented
83	(C₂×D₇₉)⋊C₄	—	2	2	no	—
84	(C₂×D₇₉)⋊C₄	—	2	2	no	—
85	C₄×Dic₇₉	—	4	1	no	—
86	C₄×Dic₇₉	—	4	1	no	—
87	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
90	C₂×C₈×C₇₉	—	2	2	no	—
91	C₂×C₈×C₇₉	—	2	2	no	—
92	C₂×C₈×C₇₉	—	2	2	no	—
93	C₂×C₈×C₇₉	—	2	2	no	—
94	C₂×C₈×C₇₉	—	2	1	no	—
95	C₂×C₈×C₇₉	—	2	1	no	—
96	C₂×C₈×C₇₉	—	4	2	no	—
97	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
98	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
99	C₂×C₈×C₇₉	D₈	1	2	yes	—
100	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
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	C₄×D₄	1	1	yes	characteristic
106	C₂²×D₇₉	—	2	2	no	—
107	C₂²×D₇₉	—	4	2	no	—
108	C₂²×D₇₉	—	4	1	no	—
109	C₂²×D₇₉	—	4	1	no	—
110	C₂²×D₇₉	—	4	2	no	—
111	C₂×C₄×C₇₉	—	2	2	no	—
112	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₇₉	#131	1	2	yes	—
120	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
123	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
124	C₂×Dic₇₉	—	2	2	no	—
125	C₂×Dic₇₉	—	2	4	no	—
126	C₂×Dic₇₉	—	4	2	no	—
127	C₂×Dic₇₉	—	4	1	no	—
128	C₂×Dic₇₉	—	4	1	no	—
129	C₂×Dic₇₉	—	4	2	no	—
130	D₃₁₆	—	2	2	no	—
131	D₃₁₆	—	2	2	no	—
132	D₃₁₆	—	4	2	no	—
133	C₄×D₇₉	—	4	2	no	—
134	C₄×D₇₉	—	4	2	no	—
135	C₄×D₇₉	—	4	2	no	—
136	C₄×D₇₉	—	4	2	no	—
137	C₈×C₇₉	—	2	2	no	—
138	C₈×C₇₉	—	4	2	no	—
139	C₈×C₇₉	—	4	2	no	—
140	C₈×C₇₉	#102	1	4	yes	complemented
141	C₇₉⋊C₈	—	4	2	no	—
142	C₇₉⋊C₈	—	4	2	no	—
143	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₇₉	#435	1	2	yes	—
145	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
146	C₂²×C₇₉	C₄×SD₁₆	1	1	yes	characteristic
147	C₂²×C₇₉	#444	1	2	yes	—
148	C₂×D₇₉	—	4	2	no	—
149	C₂×D₇₉	—	4	2	no	—
150	C₂×D₇₉	—	4	2	no	—
151	C₂×D₇₉	—	8	2	no	—
152	C₂×D₇₉	—	8	2	no	—
153	C₄×C₇₉	—	2	1	no	—
154	C₄×C₇₉	—	2	1	no	—
155	C₄×C₇₉	C₄×M₁₆	1	1	yes	characteristic
156	C₄×C₇₉	#405	1	1	yes	characteristic
157	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
158	C₄×C₇₉	#416	1	1	yes	characteristic
159	C₄×C₇₉	#436	1	2	yes	—
160	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
161	C₄×C₇₉	C₄×SD₁₆	1	1	yes	characteristic
162	Dic₇₉	—	4	4	no	—
163	Dic₇₉	—	4	2	no	—
164	#8210	—	79	1	no	maximal, Sylow
165	C₂×C₇₉	#1581	1	2	yes	—
166	C₂×C₇₉	#1583	1	2	yes	—
167	C₂×C₇₉	#1753	1	1	yes	characteristic
168	C₂×C₇₉	C₄×#102	1	1	yes	characteristic
169	C₂×C₇₉	#1765	1	1	yes	characteristic
170	D₇₉	—	8	2	no	—
171	C₂×#436	—	79	1	no	—
172	C₈×C₄⋊C₄	—	79	1	no	—
173	C₄×#102	—	79	1	no	—
174	C₂×D₄⋊C₈	—	79	1	no	—
175	C₂×#322	D₇₉	1	1	yes	characteristic, complemented
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	—	79	1	no	—
181	#436	—	79	4	no	—
182	#436	—	158	2	no	—
183	C₂×C₄⋊C₈	—	79	1	no	—
184	C₂×C₄⋊C₈	—	79	1	no	—
185	C₂×#102	—	79	2	no	—
186	C₂×C₂²⋊C₈	—	158	1	no	—
187	C₂×C₄×C₈	—	158	1	no	—
188	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
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	—	79	4	no	—
195	#322	—	2	2	no	—
196	C₂²×D₄	—	79	1	no	—
197	C₂³×C₄	—	158	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	—	158	1	no	—
203	C₂²×C₈	—	158	1	no	—
204	C₂²×C₈	C₄×D₇₉	1	2	yes	—
205	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
206	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
207	C₄×D₄	—	79	2	no	—
208	C₄×D₄	—	79	2	no	—
209	C₄×D₄	—	158	1	no	—
210	C₄×D₄	—	158	1	no	—
211	C₂×C₄⋊C₄	—	79	1	no	—
212	C₂×C₄⋊C₄	—	79	2	no	—
213	C₂×C₂²⋊C₄	—	158	1	no	—
214	C₂×C₄²	—	158	1	no	—
215	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
216	C₄⋊C₈	—	79	2	no	—
217	C₄⋊C₈	—	79	2	no	—
218	C₄⋊C₈	—	158	1	no	—
219	C₄⋊C₈	—	158	1	no	—
220	#102	—	158	4	no	—
221	C₂²⋊C₈	—	158	4	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	316	2	no	—
226	C₄×C₈	C₄×D₇₉	1	2	yes	—
227	C₄×C₈	C₄×D₇₉	1	2	yes	—
228	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
229	#95	—	158	1	no	—
230	C₂⁴	—	158	1	no	—
231	C₂×D₄	—	79	2	no	—
232	C₂×D₄	—	158	1	no	—
233	C₂×D₄	—	158	2	no	—
234	C₂×D₄	—	158	2	no	—
235	C₂²×C₄	—	316	1	no	—
236	C₂²×C₄	—	316	1	no	—
237	C₂²×C₄	—	316	1	no	—
238	C₂²×C₄	—	316	1	no	—
239	C₂²×C₄	—	158	2	no	—
240	C₂²×C₄	—	158	1	no	—
241	C₂²×C₄	—	158	2	no	—
242	C₂²×C₄	—	158	2	no	—
243	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
245	C₂²×C₄	#11929624	1	1	yes	characteristic
246	M₁₆	—	316	8	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	316	1	no	—
255	C₂×C₈	—	316	1	no	—
256	C₂×C₈	—	316	1	no	—
257	C₂×C₈	—	316	1	no	—
258	C₂×C₈	—	158	2	no	—
259	C₂×C₈	—	158	2	no	—
260	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
261	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
262	C₂×C₈	D₆₃₂	1	2	yes	—
263	C₂×C₈	#11929613	1	2	yes	—
264	C₄⋊C₄	—	79	2	no	—
265	C₄⋊C₄	—	79	4	no	—
266	C₄⋊C₄	—	158	1	no	—
267	C₄⋊C₄	—	158	2	no	—
268	C₂²⋊C₄	—	158	2	no	—
269	C₂²⋊C₄	—	158	2	no	—
270	C₄²	—	316	1	no	—
271	C₄²	—	316	1	no	—
272	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
274	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
275	C₄²	#11929624	1	1	yes	characteristic
276	C₂³	—	316	2	no	—
277	C₂³	—	316	1	no	—
278	C₂³	—	316	1	no	—
279	C₂³	—	316	2	no	—
280	C₂³	—	158	2	no	—
281	C₂³	C₄×D₃₁₆	1	1	yes	characteristic
282	D₄	—	316	2	no	—
283	D₄	—	158	2	no	—
284	D₄	—	158	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	316	2	no	—
287	C₂×C₄	—	316	1	no	—
288	C₂×C₄	—	316	1	no	—
289	C₂×C₄	—	316	2	no	—
290	C₂×C₄	—	316	2	no	—
291	C₂×C₄	—	316	2	no	—
292	C₂×C₄	—	316	2	no	—
293	C₂×C₄	—	316	2	no	—
294	C₂×C₄	—	158	4	no	—
295	C₂×C₄	—	158	2	no	—
296	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
297	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
298	C₂×C₄	#14579118	1	1	yes	characteristic
299	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
300	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
301	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
302	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
303	C₂×C₄	#14579169	1	2	yes	—
304	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
305	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
306	C₂×C₄	#14579183	1	1	yes	characteristic
307	C₂×C₄	#14579183	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	316	2	no	—
312	C₈	—	316	2	no	—
313	C₈	#14579083	1	4	yes	complemented
314	C₂²	—	316	2	no	—
315	C₂²	—	316	2	no	—
316	C₂²	—	316	2	no	—
317	C₂²	—	632	2	no	—
318	C₂²	—	632	2	no	—
319	C₂²	C₄×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
320	C₂²	#19098636	1	2	yes	—
321	C₂²	C₄×D₆₃₂	1	1	yes	characteristic
322	C₂²	C₄×#11929613	1	1	yes	characteristic
323	C₂²	#19098686	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	316	2	no	—
327	C₄	—	316	4	no	—
328	C₄	#19098515	1	1	yes	characteristic
329	C₄	#19098520	1	1	yes	characteristic
330	C₄	C₂×#14579083	1	1	yes	characteristic
331	C₄	#19098560	1	1	yes	characteristic
332	C₄	#19098639	1	2	yes	—
333	C₄	C₄×D₆₃₂	1	1	yes	characteristic
334	C₄	C₄×#11929613	1	1	yes	characteristic
335	C₂	—	632	2	no	—
336	C₂	#23762231	1	2	yes	—
337	C₂	#23762245	1	2	yes	—
338	C₂	#23763185	1	1	yes	characteristic
339	C₂	C₄×#14579083	1	1	yes	characteristic
340	C₂	#23763244	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#322×C₇₉	C₂
socle	C₂³×C₇₉	C₄×D₄
prime radical	C₂²×D₃₁₆	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#322	D₇₉
hypocenter	C₇₉	#8210
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. C₈×C₄⋊Dic₇₉	C₂	semidirect
5. #23761240	C₂	semidirect
6. #23761235	C₂	semidirect
7. C₂×#322×C₇₉	C₂	semidirect
11. #19098642	C₄	semidirect
20. #19098292	C₄	semidirect
82. C₄⋊Dic₇₉	M₁₆	semidirect
140. C₈×C₇₉	#102	semidirect
175. C₂×#322	D₇₉	semidirect
178. C₇₉	#8210	semidirect
313. C₈	#14579083	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098642	2
C₈×C₄⋊Dic₇₉	2
C₄×#14579083	2
C₂×#19098292	2
#23761240	2
#23761235	2
C₂×#322×C₇₉	2
#8210	79