Subgroups — #34178985

Order 20224.

498 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#23763146	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761782	C₂	1	1	yes	maximal, characteristic
4	M₁₆×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761238	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761233	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#104×C₇₉	C₂	1	1	yes	maximal, characteristic
8	Q₈×C₇₉⋊C₈	C₄	1	2	yes	complemented
9	D₄×C₇₉⋊C₈	—	2	2	no	—
10	D₄×C₇₉⋊C₈	C₄	1	2	yes	complemented
11	#19098616	—	2	1	no	—
12	#19098616	C₂²	1	1	yes	characteristic
13	#19098616	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	#19098318	—	2	2	no	—
18	Q₈⋊(C₇₉⋊C₈)	C₄	1	2	yes	complemented
19	D₄⋊(C₇₉⋊C₈)	C₄	1	2	yes	complemented
20	#19098284	—	2	1	no	—
21	C₈×C₇₉⋊C₈	—	2	1	no	—
22	C₄×#41×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×#104×C₇₉	C₄	1	2	yes	—
24	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
25	C₄³×C₇₉	C₂²	1	1	yes	characteristic
26	#339×C₇₉	C₄	1	2	yes	—
27	C₂²×C₇₉⋊C₈	—	2	1	no	—
28	C₂²×C₇₉⋊C₈	—	2	2	no	—
29	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
30	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
31	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
32	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
33	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
34	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
35	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
36	C₄×C₇₉⋊C₈	—	2	1	no	—
37	C₄×C₇₉⋊C₈	—	2	1	no	—
38	C₄×C₇₉⋊C₈	—	2	1	no	—
39	C₄×C₇₉⋊C₈	—	2	2	no	—
40	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
41	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
42	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
43	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
44	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
46	C₄×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
47	C₄×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
48	C₄×D₄×C₇₉	—	2	1	no	—
49	C₄×D₄×C₇₉	—	2	1	no	—
50	C₄×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₄×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
52	#117×C₇₉	—	2	1	no	—
53	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₉	—	2	1	no	—
55	C₂×C₄²×C₇₉	—	2	1	no	—
56	C₂×C₄²×C₇₉	—	2	1	no	—
57	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
59	#104×C₇₉	C₈	1	8	yes	complemented
60	#97×C₇₉	—	2	1	no	—
61	C₄×C₈×C₇₉	—	2	1	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	—	4	2	no	—
66	C₂×C₇₉⋊C₈	—	4	2	no	—
67	C₂×C₇₉⋊C₈	—	4	4	no	—
68	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
70	#41×C₇₉	—	2	2	no	—
71	#41×C₇₉	C₂×C₈	1	4	yes	—
72	C₂×Q₈×C₇₉	C₄²	1	1	yes	characteristic
73	C₂×D₄×C₇₉	—	2	1	no	—
74	C₂×D₄×C₇₉	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	—	2	1	no	—
76	C₂²×C₄×C₇₉	—	2	2	no	—
77	C₂²×C₄×C₇₉	—	2	1	no	—
78	C₂²×C₄×C₇₉	—	2	1	no	—
79	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
82	M₁₆×C₇₉	—	2	2	no	—
83	M₁₆×C₇₉	C₂×C₈	1	4	yes	complemented
84	C₂×C₈×C₇₉	—	2	2	no	—
85	C₄⋊C₄×C₇₉	—	2	1	no	—
86	C₄⋊C₄×C₇₉	—	2	1	no	—
87	C₄⋊C₄×C₇₉	C₄²	1	1	yes	characteristic
88	C₄⋊C₄×C₇₉	C₄²	1	1	yes	characteristic
89	C₂²⋊C₄×C₇₉	—	2	1	no	—
90	C₂²⋊C₄×C₇₉	—	2	1	no	—
91	C₂²⋊C₄×C₇₉	—	2	1	no	—
92	C₄²×C₇₉	—	2	1	no	—
93	C₄²×C₇₉	—	2	1	no	—
94	C₄²×C₇₉	—	2	1	no	—
95	C₄²×C₇₉	—	2	2	no	—
96	C₄²×C₇₉	—	4	2	no	—
97	C₄²×C₇₉	—	4	2	no	—
98	C₄²×C₇₉	C₂×C₈	1	4	yes	complemented
99	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇₉	#41	1	1	yes	characteristic
103	C₂³×C₇₉	—	2	1	no	—
104	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
105	Q₈×C₇₉	—	2	1	no	—
106	Q₈×C₇₉	C₄×C₈	1	2	yes	—
107	D₄×C₇₉	—	2	4	no	—
108	D₄×C₇₉	—	2	1	no	—
109	D₄×C₇₉	C₄×C₈	1	2	yes	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	2	no	—
112	C₂×C₄×C₇₉	—	2	4	no	—
113	C₂×C₄×C₇₉	—	2	2	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	2	4	no	—
117	C₂×C₄×C₇₉	—	4	2	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	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₄×D₄	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	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₇₉	#131	1	2	yes	—
129	C₈×C₇₉	—	2	4	no	—
130	C₇₉⋊C₈	—	2	2	no	—
131	C₇₉⋊C₈	—	2	2	no	—
132	C₇₉⋊C₈	—	4	2	no	—
133	C₇₉⋊C₈	—	4	2	no	—
134	C₇₉⋊C₈	—	4	2	no	—
135	C₂²×C₇₉	—	2	1	no	—
136	C₂²×C₇₉	—	2	1	no	—
137	C₂²×C₇₉	—	2	2	no	—
138	C₂²×C₇₉	—	4	1	no	—
139	C₂²×C₇₉	—	4	1	no	—
140	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₇₉	C₈×D₄	1	2	yes	—
142	C₄×C₇₉	—	2	2	no	—
143	C₄×C₇₉	—	2	1	no	—
144	C₄×C₇₉	—	2	1	no	—
145	C₄×C₇₉	—	4	4	no	—
146	C₄×C₇₉	—	4	1	no	—
147	C₄×C₇₉	—	4	1	no	—
148	C₄×C₇₉	C₂×C₄×C₈	1	1	yes	characteristic
149	C₄×C₇₉	#405	1	1	yes	characteristic
150	C₄×C₇₉	#421	1	2	yes	—
151	C₄×C₇₉	C₈×D₄	1	2	yes	—
152	C₄×C₇₉	#444	1	2	yes	—
153	#8191	—	79	1	no	maximal, Sylow
154	C₂×C₇₉	—	2	2	no	—
155	C₂×C₇₉	—	4	2	no	—
156	C₂×C₇₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₇₉	#1759	1	1	yes	characteristic
158	C₂×C₇₉	#1765	1	1	yes	characteristic
159	C₈×#41	—	79	1	no	—
160	C₄×#104	D₇₉	1	1	yes	characteristic
161	#1744	—	79	1	no	—
162	#1476	—	79	1	no	—
163	C₈×M₁₆	—	79	1	no	—
164	#1278	—	79	1	no	—
165	#1276	—	79	1	no	—
166	C₇₉	#8191	1	1	yes	characteristic, complemented, Sylow
167	C₄×#41	C₂×D₇₉	1	1	yes	characteristic
168	C₈×Q₈	—	79	2	no	—
169	C₈×D₄	—	79	2	no	—
170	C₈×D₄	—	158	2	no	—
171	#431	—	79	1	no	—
172	#431	—	79	1	no	—
173	#431	—	158	1	no	—
174	C₂×#104	Dic₇₉	1	2	yes	—
175	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
176	C₂×C₄×C₈	—	79	1	no	—
177	C₂×C₄×C₈	—	79	1	no	—
178	C₂×C₄×C₈	—	158	1	no	—
179	C₄³	C₂×D₇₉	1	1	yes	characteristic
180	#339	Dic₇₉	1	2	yes	—
181	#336	—	158	2	no	—
182	Q₈⋊C₈	—	79	2	no	—
183	D₄⋊C₈	—	79	2	no	—
184	#322	—	158	1	no	—
185	C₈²	—	158	1	no	—
186	C₂×#41	C₂×Dic₇₉	1	1	yes	characteristic
187	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
188	C₂²×C₈	—	79	2	no	—
189	C₂²×C₈	—	158	2	no	—
190	C₂²×C₈	—	158	1	no	—
191	C₄×Q₈	C₄×D₇₉	1	1	yes	characteristic
192	C₄×Q₈	C₄×D₇₉	1	1	yes	characteristic
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	C₄×D₇₉	1	1	yes	characteristic
196	C₄×D₄	C₄×D₇₉	1	1	yes	characteristic
197	#117	—	2	1	no	—
198	#117	C₂×Dic₇₉	1	1	yes	characteristic
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
203	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
204	C₄⋊C₈	—	79	2	no	—
205	C₄⋊C₈	—	79	2	no	—
206	C₄⋊C₈	—	158	2	no	—
207	#104	C₇₉⋊C₈	1	8	yes	complemented
208	C₂²⋊C₈	—	158	2	no	—
209	C₂²⋊C₈	—	158	1	no	—
210	C₂²⋊C₈	—	158	1	no	—
211	#97	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	79	2	no	—
214	C₄×C₈	—	79	2	no	—
215	C₄×C₈	—	79	1	no	—
216	C₄×C₈	—	79	1	no	—
217	C₄×C₈	—	158	2	no	—
218	C₄×C₈	—	158	1	no	—
219	C₄×C₈	—	158	1	no	—
220	C₄×C₈	—	158	1	no	—
221	#41	—	2	2	no	—
222	#41	C₂×C₇₉⋊C₈	1	4	yes	—
223	C₂×Q₈	C₄×Dic₇₉	1	1	yes	characteristic
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	C₄×Dic₇₉	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
232	C₂²×C₄	#11929623	1	1	yes	characteristic
233	M₁₆	—	2	2	no	—
234	M₁₆	C₂×C₇₉⋊C₈	1	4	yes	complemented
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	316	2	no	—
237	C₂×C₈	—	316	4	no	—
238	C₂×C₈	—	316	2	no	—
239	C₂×C₈	—	79	2	no	—
240	C₂×C₈	—	79	2	no	—
241	C₂×C₈	—	158	2	no	—
242	C₂×C₈	—	158	2	no	—
243	C₂×C₈	—	158	2	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	C₄×Dic₇₉	1	1	yes	characteristic
247	C₄⋊C₄	C₄×Dic₇₉	1	1	yes	characteristic
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	4	2	no	—
256	C₄²	—	4	2	no	—
257	C₄²	C₂×C₇₉⋊C₈	1	4	yes	complemented
258	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
260	C₄²	D₄×D₇₉	1	1	yes	characteristic
261	C₄²	#11929623	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	C₂³	D₄×Dic₇₉	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	C₄×C₇₉⋊C₈	1	2	yes	—
266	D₄	—	2	4	no	—
267	D₄	—	2	1	no	—
268	D₄	C₄×C₇₉⋊C₈	1	2	yes	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
282	C₂×C₄	#14579117	1	1	yes	characteristic
283	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
285	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₇₉⋊C₈	1	2	yes	—
287	C₂×C₄	#14579168	1	2	yes	—
288	C₈	—	2	4	no	—
289	C₈	—	316	2	no	—
290	C₈	—	316	2	no	—
291	C₈	—	316	2	no	—
292	C₈	—	158	2	no	—
293	C₈	—	158	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	C₂²⋊C₄×Dic₇₉	1	1	yes	characteristic
300	C₂²	D₄×C₇₉⋊C₈	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	C₂×C₄×C₇₉⋊C₈	1	1	yes	characteristic
308	C₄	#19098519	1	1	yes	characteristic
309	C₄	#19098578	1	2	yes	—
310	C₄	D₄×C₇₉⋊C₈	1	2	yes	—
311	C₄	#19098682	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	4	2	no	—
314	C₂	C₂²⋊C₄×C₇₉⋊C₈	1	1	yes	characteristic
315	C₂	#23763212	1	1	yes	characteristic
316	C₂	#23763245	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₇₉
commutator	C₄×C₇₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×#104×C₇₉	C₂
socle	C₂²×C₇₉	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₇₉	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×#104	D₇₉
hypocenter	C₇₉	#8191
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#104

Complements

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

N	H ≅ G/N	type
2. #23763146	C₂	semidirect
4. M₁₆×C₇₉⋊C₈	C₂	semidirect
5. #23761238	C₂	semidirect
6. #23761233	C₂	semidirect
8. Q₈×C₇₉⋊C₈	C₄	semidirect
10. D₄×C₇₉⋊C₈	C₄	semidirect
18. Q₈⋊(C₇₉⋊C₈)	C₄	semidirect
19. D₄⋊(C₇₉⋊C₈)	C₄	semidirect
59. #104×C₇₉	C₈	semidirect
83. M₁₆×C₇₉	C₂×C₈	semidirect
98. C₄²×C₇₉	C₂×C₈	semidirect
166. C₇₉	#8191	semidirect
207. #104	C₇₉⋊C₈	semidirect
234. M₁₆	C₂×C₇₉⋊C₈	semidirect
257. C₄²	C₂×C₇₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#41×C₇₉⋊C₈	2
#23763146	2
#23761782	2
M₁₆×C₇₉⋊C₈	2
#23761238	2
#23761233	2
C₄×#104×C₇₉	2
#8191	79