Subgroups — #34296989

Order 20224.

774 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098694	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#23764341	C₂	1	1	yes	maximal, characteristic
4	#23764341	C₂	1	1	yes	maximal, characteristic
5	#23764119	C₂	1	4	yes	maximal
6	#23763554	C₂	1	1	yes	maximal, characteristic
7	#23763275	C₂	1	4	yes	maximal
8	#23763087	C₂	1	1	yes	maximal, characteristic
9	C₇₉×#2399	C₂	1	1	yes	maximal, characteristic
10	#19098694	—	2	4	no	—
11	Q₈×C₇₉⋊C₈	—	2	4	no	—
12	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
13	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
17	C₂×#14579072	C₂²	1	2	yes	—
18	C₂×#14579072	C₂²	1	1	yes	characteristic
19	C₂×#14579072	C₂²	1	2	yes	—
20	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
23	#19098318	C₂²	1	4	yes	—
24	#19098318	C₂²	1	4	yes	—
25	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
27	#398×C₇₉	C₄	1	2	yes	—
28	#398×C₇₉	C₄	1	2	yes	—
29	#395×C₇₉	C₄	1	2	yes	—
30	#395×C₇₉	C₄	1	2	yes	—
31	C₄×C₄⋊C₄×C₇₉	C₂²	1	4	yes	—
32	#376×C₇₉	C₂²	1	1	yes	characteristic
33	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
34	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
35	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
38	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
39	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
40	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
41	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
42	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
43	#14579072	—	2	2	no	—
44	#14579072	—	2	4	no	—
45	#14579072	Q₈	1	8	yes	complemented
46	C₄×C₇₉⋊C₈	—	2	4	no	—
47	C₄×C₇₉⋊C₈	—	2	4	no	—
48	C₄×C₇₉⋊C₈	D₄	1	4	yes	—
49	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₄×Q₈×C₇₉	—	2	2	no	—
52	C₄×Q₈×C₇₉	—	2	2	no	—
53	C₄×Q₈×C₇₉	—	2	2	no	—
54	C₄×Q₈×C₇₉	—	2	2	no	—
55	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
60	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
62	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
63	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
65	#95×C₇₉	C₂×C₄	1	2	yes	—
66	#95×C₇₉	C₂×C₄	1	2	yes	—
67	#95×C₇₉	C₂×C₄	1	4	yes	—
68	#95×C₇₉	C₂×C₄	1	4	yes	—
69	C₂×C₇₉⋊C₈	—	2	2	no	—
70	C₂×C₇₉⋊C₈	—	2	2	no	—
71	C₂×C₇₉⋊C₈	—	2	2	no	—
72	C₂×C₇₉⋊C₈	—	2	2	no	—
73	C₂×C₇₉⋊C₈	—	2	2	no	—
74	C₂×C₇₉⋊C₈	—	2	4	no	—
75	C₂×C₇₉⋊C₈	—	2	4	no	—
76	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
77	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
78	C₂×C₇₉⋊C₈	C₂×Q₈	1	4	yes	—
79	C₂×C₇₉⋊C₈	C₂×Q₈	1	4	yes	—
80	C₂×Q₈×C₇₉	—	2	2	no	—
81	C₂×Q₈×C₇₉	—	2	2	no	—
82	C₂×Q₈×C₇₉	—	2	4	no	—
83	C₂×Q₈×C₇₉	—	4	2	no	—
84	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
90	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₇₉	—	2	4	no	—
93	C₄⋊C₄×C₇₉	—	2	2	no	—
94	C₄⋊C₄×C₇₉	—	2	2	no	—
95	C₄⋊C₄×C₇₉	—	2	4	no	—
96	C₄⋊C₄×C₇₉	—	2	8	no	—
97	C₄⋊C₄×C₇₉	—	4	4	no	—
98	C₄²×C₇₉	—	2	2	no	—
99	C₄²×C₇₉	—	2	2	no	—
100	C₄²×C₇₉	—	2	2	no	—
101	C₄²×C₇₉	—	2	2	no	—
102	C₄²×C₇₉	C₂×D₄	1	2	yes	—
103	C₄²×C₇₉	C₂×Q₈	1	4	yes	—
104	C₄²×C₇₉	#41	1	2	yes	—
105	C₄²×C₇₉	#41	1	4	yes	—
106	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
107	Q₈×C₇₉	—	2	8	no	—
108	Q₈×C₇₉	—	4	4	no	—
109	C₂×C₄×C₇₉	—	2	2	no	—
110	C₂×C₄×C₇₉	—	2	4	no	—
111	C₂×C₄×C₇₉	—	2	2	no	—
112	C₂×C₄×C₇₉	—	2	2	no	—
113	C₂×C₄×C₇₉	—	2	4	no	—
114	C₂×C₄×C₇₉	—	2	2	no	—
115	C₂×C₄×C₇₉	—	2	2	no	—
116	C₂×C₄×C₇₉	—	2	2	no	—
117	C₂×C₄×C₇₉	—	2	2	no	—
118	C₂×C₄×C₇₉	—	2	4	no	—
119	C₂×C₄×C₇₉	—	2	2	no	—
120	C₂×C₄×C₇₉	—	2	4	no	—
121	C₂×C₄×C₇₉	—	2	2	no	—
122	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₇₉	C₄×Q₈	1	8	yes	—
125	C₂×C₄×C₇₉	#131	1	4	yes	—
126	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂²×Q₈	1	2	yes	—
128	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
130	C₇₉⋊C₈	—	2	2	no	—
131	C₇₉⋊C₈	—	2	4	no	—
132	C₇₉⋊C₈	—	4	2	no	—
133	C₇₉⋊C₈	—	4	2	no	—
134	C₇₉⋊C₈	C₄⋊Q₈	1	4	yes	—
135	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₇₉	C₂×C₄×Q₈	1	2	yes	—
137	C₂²×C₇₉	C₂×#131	1	2	yes	—
138	C₂²×C₇₉	#568	1	2	yes	—
139	C₄×C₇₉	—	2	2	no	—
140	C₄×C₇₉	—	2	4	no	—
141	C₄×C₇₉	—	2	2	no	—
142	C₄×C₇₉	—	2	8	no	—
143	C₄×C₇₉	—	4	2	no	—
144	C₄×C₇₉	—	4	2	no	—
145	C₄×C₇₉	—	4	2	no	—
146	C₄×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	C₄×C₇₉	#532	1	1	yes	characteristic
148	C₄×C₇₉	#533	1	2	yes	—
149	#21718	—	79	1	no	maximal, Sylow
150	C₂×C₇₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
151	C₂×C₇₉	#2929	1	1	yes	characteristic
152	C₂×C₇₉	#2930	1	1	yes	characteristic
153	C₂×C₇₉	#2962	1	4	yes	—
154	C₂×#446	—	79	1	no	—
155	C₂×C₈×Q₈	—	79	1	no	—
156	#2399	D₇₉	1	1	yes	characteristic
157	#1985	—	79	1	no	—
158	#1985	—	79	1	no	—
159	#1939	—	79	4	no	—
160	#1843	—	79	1	no	—
161	#1771	—	79	4	no	—
162	#1726	—	79	1	no	—
163	C₇₉	#21718	1	1	yes	characteristic, complemented, Sylow
164	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
165	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
166	#446	—	158	4	no	—
167	C₈×Q₈	—	158	4	no	—
168	C₂×C₄⋊C₈	—	79	1	no	—
169	C₂×C₄⋊C₈	—	79	1	no	—
170	C₂×C₄⋊C₈	—	79	4	no	—
171	C₂×C₄⋊C₈	—	79	2	no	—
172	C₂×C₄⋊C₈	—	79	2	no	—
173	C₂×#97	—	79	2	no	—
174	C₂×#97	—	79	2	no	—
175	C₂×#97	—	79	1	no	—
176	C₂×C₄×C₈	—	79	1	no	—
177	C₂×C₄×C₈	—	79	2	no	—
178	C₂×C₄×C₈	—	79	2	no	—
179	#398	Dic₇₉	1	2	yes	—
180	#398	Dic₇₉	1	2	yes	—
181	#395	Dic₇₉	1	2	yes	—
182	#395	Dic₇₉	1	2	yes	—
183	C₄×C₄⋊C₄	C₂×D₇₉	1	4	yes	—
184	#376	C₂×D₇₉	1	1	yes	characteristic
185	#336	—	79	4	no	—
186	#336	—	79	4	no	—
187	C₂²×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
188	C₂²×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
189	C₂²×C₈	—	79	1	no	—
190	C₂²×C₈	—	79	2	no	—
191	C₂²×C₈	—	79	1	no	—
192	C₂²×C₈	—	79	2	no	—
193	C₂²×C₈	—	79	2	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₄×Q₈	—	2	2	no	—
196	C₄×Q₈	—	2	2	no	—
197	C₄×Q₈	—	2	2	no	—
198	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
199	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
200	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
202	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
203	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
204	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
205	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
206	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
207	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
208	C₄⋊C₈	—	158	4	no	—
209	C₄⋊C₈	—	158	4	no	—
210	C₄⋊C₈	—	158	2	no	—
211	C₄⋊C₈	—	158	2	no	—
212	C₄⋊C₈	—	158	8	no	—
213	#97	—	79	8	no	—
214	#97	—	158	4	no	—
215	#97	—	158	2	no	—
216	C₄×C₈	—	79	4	no	—
217	C₄×C₈	—	158	4	no	—
218	C₄×C₈	—	158	4	no	—
219	#95	C₂×Dic₇₉	1	2	yes	—
220	#95	C₂×Dic₇₉	1	2	yes	—
221	#95	C₂×Dic₇₉	1	4	yes	—
222	#95	C₂×Dic₇₉	1	4	yes	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	4	2	no	—
227	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
228	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
229	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
230	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
231	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
233	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
234	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
235	C₂×C₈	—	79	4	no	—
236	C₂×C₈	—	79	2	no	—
237	C₂×C₈	—	79	4	no	—
238	C₂×C₈	—	79	2	no	—
239	C₂×C₈	—	158	4	no	—
240	C₂×C₈	—	158	4	no	—
241	C₂×C₈	—	158	2	no	—
242	C₂×C₈	—	158	2	no	—
243	C₂×C₈	—	158	2	no	—
244	C₂×C₈	—	158	2	no	—
245	C₂×C₈	—	158	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	D₄×D₇₉	1	2	yes	—
257	C₄²	Q₈×D₇₉	1	4	yes	—
258	C₄²	#11929623	1	2	yes	—
259	C₄²	#11929625	1	4	yes	—
260	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
261	Q₈	—	2	8	no	—
262	Q₈	—	4	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
277	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
278	C₂×C₄	Q₈×Dic₇₉	1	8	yes	—
279	C₂×C₄	#14579168	1	4	yes	—
280	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
281	C₂×C₄	C₂×Q₈×D₇₉	1	2	yes	—
282	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
283	C₂×C₄	C₂×#11929625	1	2	yes	—
284	C₈	—	316	2	no	—
285	C₈	—	316	2	no	—
286	C₈	—	79	4	no	—
287	C₈	—	158	2	no	—
288	C₈	—	158	4	no	—
289	C₂²	C₂×D₄×Dic₇₉	1	1	yes	characteristic
290	C₂²	C₂×Q₈×Dic₇₉	1	2	yes	—
291	C₂²	C₂×#14579168	1	2	yes	—
292	C₂²	#19099330	1	2	yes	—
293	C₄	—	2	8	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	4	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	C₄⋊Q₈×D₇₉	1	1	yes	characteristic
301	C₄	#19099114	1	1	yes	characteristic
302	C₄	#19099118	1	2	yes	—
303	C₂	C₄⋊Q₈×Dic₇₉	1	1	yes	characteristic
304	C₂	#23772095	1	1	yes	characteristic
305	C₂	#23772104	1	1	yes	characteristic
306	C₂	#23772349	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₉
commutator, socle, prime radical	C₂³×C₇₉	C₂³×C₄
Fitting	C₇₉×#2399	C₂
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#2399	D₇₉
hypocenter	C₇₉	#21718
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ #2399

Complements

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

N	H ≅ G/N	type
45. #14579072	Q₈	semidirect
163. C₇₉	#21718	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098694	2
C₂×Q₈×C₇₉⋊C₈	2
#23764341	2
#23764119	2
#23763554	2
#23763275	2
#23763087	2
C₇₉×#2399	2
#21718	79