Subgroups — #34183770

Order 20224.

498 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772481	C₂	1	1	yes	maximal, characteristic
2	#23761795	C₂	1	2	yes	maximal
3	#23761724	C₂	1	1	yes	maximal, characteristic
4	#23761229	C₂	1	2	yes	maximal
5	C₇₉×#2327	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₇₉⋊C₈	—	2	4	no	—
7	#19098616	—	2	2	no	—
8	#19098616	C₂²	1	2	yes	—
9	#19098616	C₂²	1	2	yes	—
10	Q₈⋊(C₇₉⋊C₈)	—	2	4	no	—
11	#19098289	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₇₉	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
15	#391×C₇₉	C₄	1	1	yes	characteristic
16	#391×C₇₉	C₄	1	1	yes	characteristic
17	#387×C₇₉	—	2	2	no	—
18	#387×C₇₉	—	2	2	no	—
19	#382×C₇₉	C₄	1	1	yes	characteristic
20	#376×C₇₉	C₂²	1	1	yes	characteristic
21	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
26	C₄×C₇₉⋊C₈	—	2	4	no	—
27	C₄×C₇₉⋊C₈	—	2	2	no	—
28	C₄×C₇₉⋊C₈	D₄	1	4	yes	—
29	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₇₉	—	2	2	no	—
32	C₄⋊Q₈×C₇₉	C₈	1	4	yes	complemented
33	C₄×Q₈×C₇₉	—	2	2	no	—
34	C₄×Q₈×C₇₉	—	2	2	no	—
35	C₄×Q₈×C₇₉	—	2	2	no	—
36	C₄×Q₈×C₇₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
42	C₂×C₄²×C₇₉	—	2	1	no	—
43	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₉	D₄	1	2	yes	—
46	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
47	#95×C₇₉	—	2	2	no	—
48	#95×C₇₉	—	2	2	no	—
49	#95×C₇₉	—	2	2	no	—
50	#95×C₇₉	—	2	1	no	—
51	#95×C₇₉	—	2	1	no	—
52	C₂×C₇₉⋊C₈	—	2	4	no	—
53	C₂×C₇₉⋊C₈	—	4	2	no	—
54	C₂×Q₈×C₇₉	—	2	2	no	—
55	C₂×Q₈×C₇₉	—	2	1	no	—
56	C₂×Q₈×C₇₉	—	2	2	no	—
57	C₂×Q₈×C₇₉	—	2	2	no	—
58	C₂×Q₈×C₇₉	—	2	2	no	—
59	C₂×Q₈×C₇₉	C₂×C₈	1	2	yes	—
60	C₂×Q₈×C₇₉	C₂×C₈	1	2	yes	—
61	C₂×Q₈×C₇₉	M₁₆	1	2	yes	—
62	C₂²×C₄×C₇₉	—	2	1	no	—
63	C₂²×C₄×C₇₉	—	2	1	no	—
64	C₂²×C₄×C₇₉	—	2	1	no	—
65	C₂²×C₄×C₇₉	—	2	1	no	—
66	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₇₉	—	2	8	no	—
72	C₄⋊C₄×C₇₉	—	2	2	no	—
73	C₄⋊C₄×C₇₉	—	2	4	no	—
74	C₄⋊C₄×C₇₉	—	2	2	no	—
75	C₄⋊C₄×C₇₉	—	4	1	no	—
76	C₄⋊C₄×C₇₉	—	4	1	no	—
77	C₄²×C₇₉	—	2	2	no	—
78	C₄²×C₇₉	—	2	2	no	—
79	C₄²×C₇₉	—	2	1	no	—
80	C₄²×C₇₉	—	2	2	no	—
81	C₄²×C₇₉	—	4	2	no	—
82	C₄²×C₇₉	C₂×C₈	1	2	yes	—
83	C₄²×C₇₉	C₂×D₄	1	2	yes	—
84	C₄²×C₇₉	#41	1	2	yes	—
85	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₇₉	—	2	4	no	—
87	Q₈×C₇₉	—	2	4	no	—
88	Q₈×C₇₉	—	2	4	no	—
89	Q₈×C₇₉	—	4	2	no	—
90	C₂×C₄×C₇₉	—	2	2	no	—
91	C₂×C₄×C₇₉	—	2	2	no	—
92	C₂×C₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	—	2	2	no	—
94	C₂×C₄×C₇₉	—	2	2	no	—
95	C₂×C₄×C₇₉	—	4	1	no	—
96	C₂×C₄×C₇₉	—	4	1	no	—
97	C₂×C₄×C₇₉	—	4	1	no	—
98	C₂×C₄×C₇₉	—	4	1	no	—
99	C₂×C₄×C₇₉	—	4	1	no	—
100	C₂×C₄×C₇₉	—	4	1	no	—
101	C₂×C₄×C₇₉	—	4	2	no	—
102	C₂×C₄×C₇₉	—	4	1	no	—
103	C₂×C₄×C₇₉	—	4	1	no	—
104	C₂×C₄×C₇₉	—	4	2	no	—
105	C₂×C₄×C₇₉	—	4	1	no	—
106	C₂×C₄×C₇₉	—	4	1	no	—
107	C₂×C₄×C₇₉	C₂²⋊C₈	1	4	yes	—
108	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
110	C₂×C₄×C₇₉	#121	1	2	yes	—
111	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₇₉	#131	1	2	yes	—
115	C₇₉⋊C₈	—	4	4	no	—
116	C₇₉⋊C₈	—	8	2	no	—
117	C₂²×C₇₉	—	2	1	no	—
118	C₂²×C₇₉	—	2	1	no	—
119	C₂²×C₇₉	#386	1	1	yes	characteristic
120	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₂²×C₇₉	#408	1	1	yes	characteristic
122	C₄×C₇₉	—	2	4	no	—
123	C₄×C₇₉	—	2	2	no	—
124	C₄×C₇₉	—	2	1	no	—
125	C₄×C₇₉	—	2	1	no	—
126	C₄×C₇₉	—	4	2	no	—
127	C₄×C₇₉	—	4	2	no	—
128	C₄×C₇₉	—	4	2	no	—
129	C₄×C₇₉	—	8	1	no	—
130	C₄×C₇₉	—	8	1	no	—
131	C₄×C₇₉	C₈×D₄	1	2	yes	—
132	C₄×C₇₉	#435	1	2	yes	—
133	#9100	—	79	1	no	maximal, Sylow
134	C₂×C₇₉	—	2	1	no	—
135	C₂×C₇₉	—	2	1	no	—
136	C₂×C₇₉	#1869	1	1	yes	characteristic
137	C₂×C₇₉	#1909	1	1	yes	characteristic
138	C₂×C₇₉	#1912	1	1	yes	characteristic
139	#2982	—	79	1	no	—
140	#2327	D₇₉	1	1	yes	characteristic
141	#1479	—	79	2	no	—
142	#1461	—	79	1	no	—
143	#1274	—	79	2	no	—
144	C₇₉	#9100	1	1	yes	characteristic, complemented, Sylow
145	C₂×C₄⋊Q₈	Dic₇₉	1	1	yes	characteristic
146	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
147	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
148	C₈×Q₈	—	158	4	no	—
149	#431	—	79	2	no	—
150	#431	—	79	2	no	—
151	#431	—	158	2	no	—
152	#391	Dic₇₉	1	1	yes	characteristic
153	#391	Dic₇₉	1	1	yes	characteristic
154	#387	—	2	2	no	—
155	#387	—	2	2	no	—
156	#382	Dic₇₉	1	1	yes	characteristic
157	#376	C₂×D₇₉	1	1	yes	characteristic
158	Q₈⋊C₈	—	158	4	no	—
159	#324	—	158	2	no	—
160	C₂²×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
161	C₂²×Q₈	C₂×Dic₇₉	1	1	yes	characteristic
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	C₇₉⋊C₈	1	4	yes	complemented
164	C₄×Q₈	—	2	2	no	—
165	C₄×Q₈	—	2	2	no	—
166	C₄×Q₈	—	2	2	no	—
167	C₄×Q₈	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂×Dic₇₉	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
173	C₂×C₄²	—	2	1	no	—
174	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
175	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
177	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
178	C₄⋊C₈	—	158	4	no	—
179	C₄⋊C₈	—	158	2	no	—
180	C₄⋊C₈	—	158	2	no	—
181	C₂²⋊C₈	—	158	2	no	—
182	C₂²⋊C₈	—	158	2	no	—
183	C₄×C₈	—	79	4	no	—
184	C₄×C₈	—	158	4	no	—
185	C₄×C₈	—	158	2	no	—
186	#95	—	2	2	no	—
187	#95	—	2	2	no	—
188	#95	—	2	2	no	—
189	#95	—	2	1	no	—
190	#95	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	2	1	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	C₂×C₇₉⋊C₈	1	2	yes	—
197	C₂×Q₈	C₂×C₇₉⋊C₈	1	2	yes	—
198	C₂×Q₈	#11929608	1	2	yes	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
204	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
205	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
208	C₂×C₈	—	316	2	no	—
209	C₂×C₈	—	158	4	no	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	4	2	no	—
221	C₄²	C₂×C₇₉⋊C₈	1	2	yes	—
222	C₄²	D₄×D₇₉	1	2	yes	—
223	C₄²	#11929623	1	2	yes	—
224	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
225	Q₈	—	2	4	no	—
226	Q₈	—	2	4	no	—
227	Q₈	—	2	4	no	—
228	Q₈	—	4	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	1	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	4	yes	—
247	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
248	C₂×C₄	#14579129	1	1	yes	characteristic
249	C₂×C₄	#14579133	1	2	yes	—
250	C₂×C₄	#14579142	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
252	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
253	C₂×C₄	#14579168	1	2	yes	—
254	C₈	—	316	4	no	—
255	C₈	—	632	2	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	#19098443	1	1	yes	characteristic
259	C₂²	C₂×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
260	C₂²	#19098530	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	4	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	1	no	—
269	C₄	—	8	1	no	—
270	C₄	D₄×C₇₉⋊C₈	1	2	yes	—
271	C₄	#19098635	1	2	yes	—
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	#23763693	1	1	yes	characteristic
275	C₂	#23763923	1	1	yes	characteristic
276	C₂	#23763938	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098443
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2327	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2327	D₇₉
hypocenter	C₇₉	#9100
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2327

Complements

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

N	H ≅ G/N	type
32. C₄⋊Q₈×C₇₉	C₈	semidirect
144. C₇₉	#9100	semidirect
163. C₄⋊Q₈	C₇₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#23772481	2
#23761795	2
#23761724	2
#23761229	2
C₇₉×#2327	2
#9100	79