Subgroups — #34192986

Order 20224.

366 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23764336	C₂	1	1	yes	maximal, characteristic
2	#23763560	C₂	1	1	yes	maximal, characteristic
3	#23763406	C₂	1	2	yes	maximal
4	#23761284	C₂	1	2	yes	maximal
5	C₇₉×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
7	C₂×#14579095	—	2	1	no	—
8	C₂×#14579095	—	2	1	no	—
9	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#14579086	—	2	2	no	—
11	#19098433	C₂²	1	2	yes	—
12	#19098326	—	2	2	no	—
13	#19098317	—	2	1	no	—
14	#19098317	—	2	1	no	—
15	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₅₈	D₄	1	1	yes	characteristic
18	C₂²×C₇₉⋊C₈	—	2	1	no	—
19	C₄×Dic₁₅₈	—	2	2	no	—
20	C₄×Dic₁₅₈	—	2	2	no	—
21	C₂×C₄⋊Dic₇₉	—	2	2	no	—
22	C₂×C₄⋊Dic₇₉	—	2	2	no	—
23	C₂×C₄⋊Dic₇₉	—	2	2	no	—
24	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
25	C₂×Dic₇₉⋊C₄	—	2	1	no	—
26	C₂×C₄×Dic₇₉	—	2	1	no	—
27	#14579095	—	4	2	no	—
28	#14579095	—	4	2	no	—
29	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
30	#14579086	—	4	4	no	—
31	C₂²×C₈×C₇₉	—	2	2	no	—
32	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₉	—	2	2	no	—
36	C₄⋊C₈×C₇₉	—	2	2	no	—
37	C₄×C₈×C₇₉	—	2	2	no	—
38	C₂×Dic₁₅₈	—	2	2	no	—
39	C₂×Dic₁₅₈	—	2	1	no	—
40	C₂×Dic₁₅₈	—	2	2	no	—
41	C₂×Dic₁₅₈	Q₁₆	1	2	yes	—
42	C₂²×Dic₇₉	—	2	1	no	—
43	C₂²×Dic₇₉	—	2	1	no	—
44	C₂²×Dic₇₉	—	4	2	no	—
45	C₂×C₇₉⋊C₈	—	4	1	no	—
46	C₂×C₇₉⋊C₈	—	4	1	no	—
47	C₂×C₇₉⋊C₈	—	4	1	no	—
48	C₄⋊Dic₇₉	—	2	2	no	—
49	C₄⋊Dic₇₉	—	4	2	no	—
50	C₄⋊Dic₇₉	—	4	2	no	—
51	C₄⋊Dic₇₉	—	4	2	no	—
52	C₄⋊Dic₇₉	—	4	2	no	—
53	C₄⋊Dic₇₉	—	8	2	no	—
54	Dic₇₉⋊C₄	—	4	2	no	—
55	C₄×Dic₇₉	—	4	1	no	—
56	C₄×Dic₇₉	—	4	1	no	—
57	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
60	C₂×C₈×C₇₉	—	2	1	no	—
61	C₂×C₈×C₇₉	—	2	1	no	—
62	C₂×C₈×C₇₉	—	2	1	no	—
63	C₂×C₈×C₇₉	—	2	1	no	—
64	C₂×C₈×C₇₉	—	4	2	no	—
65	C₂×C₈×C₇₉	—	4	2	no	—
66	C₂×C₈×C₇₉	—	4	2	no	—
67	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
68	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
69	C₄²×C₇₉	#41	1	2	yes	—
70	C₄²×C₇₉	#41	1	2	yes	—
71	C₂³×C₇₉	#121	1	1	yes	characteristic
72	C₂×C₄×C₇₉	—	2	1	no	—
73	C₂×C₄×C₇₉	—	2	1	no	—
74	C₂×C₄×C₇₉	#123	1	2	yes	—
75	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
76	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
77	C₂×C₄×C₇₉	#126	1	2	yes	—
78	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
82	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
85	C₂×Dic₇₉	—	2	2	no	—
86	C₂×Dic₇₉	—	4	1	no	—
87	C₂×Dic₇₉	—	4	1	no	—
88	C₂×Dic₇₉	—	4	1	no	—
89	C₂×Dic₇₉	—	4	1	no	—
90	C₂×Dic₇₉	—	4	1	no	—
91	C₂×Dic₇₉	—	8	2	no	—
92	C₂×Dic₇₉	—	8	2	no	—
93	C₂×Dic₇₉	—	8	2	no	—
94	Dic₁₅₈	—	2	4	no	—
95	Dic₁₅₈	—	4	2	no	—
96	C₈×C₇₉	—	2	2	no	—
97	C₈×C₇₉	—	2	2	no	—
98	C₈×C₇₉	—	8	2	no	—
99	C₇₉⋊C₈	—	8	1	no	—
100	C₂²×C₇₉	#397	1	1	yes	characteristic
101	C₂²×C₇₉	#460	1	1	yes	characteristic
102	C₂²×C₇₉	#462	1	1	yes	characteristic
103	C₂²×C₇₉	#467	1	1	yes	characteristic
104	C₂²×C₇₉	#469	1	1	yes	characteristic
105	C₂²×C₇₉	#472	1	1	yes	characteristic
106	C₂²×C₇₉	#473	1	1	yes	characteristic
107	C₄×C₇₉	—	2	2	no	—
108	C₄×C₇₉	#482	1	2	yes	—
109	C₄×C₇₉	#483	1	1	yes	characteristic
110	C₄×C₇₉	#484	1	2	yes	—
111	C₄×C₇₉	#484	1	1	yes	characteristic
112	C₄×C₇₉	#487	1	1	yes	characteristic
113	C₄×C₇₉	#489	1	1	yes	characteristic
114	Dic₇₉	—	4	2	no	—
115	Dic₇₉	—	8	1	no	—
116	Dic₇₉	—	16	2	no	—
117	#10618	—	79	1	no	maximal, Sylow
118	C₂×C₇₉	#1670	1	1	yes	characteristic
119	C₂×C₇₉	#1676	1	1	yes	characteristic
120	C₂×C₇₉	#1688	1	1	yes	characteristic
121	C₂×C₇₉	#1694	1	1	yes	characteristic
122	C₂×C₇₉	#2072	1	1	yes	characteristic
123	C₂×C₇₉	#2076	1	1	yes	characteristic
124	C₂×C₇₉	#2078	1	1	yes	characteristic
125	#1984	—	79	1	no	—
126	#1845	—	79	1	no	—
127	#1803	—	79	2	no	—
128	#1296	—	79	2	no	—
129	#1277	D₇₉	1	1	yes	characteristic
130	C₇₉	#10618	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₄×Q₈	—	79	1	no	—
132	C₂×#107	—	158	1	no	—
133	C₂×#107	—	158	1	no	—
134	C₂×C₄⋊C₈	—	79	1	no	—
135	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
136	C₂×#103	—	158	2	no	—
137	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
138	#384	—	79	2	no	—
139	#340	—	158	2	no	—
140	#336	—	158	1	no	—
141	#336	—	158	1	no	—
142	C₂²×Q₈	—	79	1	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	158	1	no	—
145	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
146	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
147	C₄×Q₈	—	158	2	no	—
148	C₄×Q₈	—	158	2	no	—
149	C₂×C₄⋊C₄	—	79	1	no	—
150	C₂×C₄⋊C₄	—	158	2	no	—
151	C₂×C₄⋊C₄	—	158	2	no	—
152	C₂×C₄⋊C₄	—	158	1	no	—
153	C₂×C₄⋊C₄	—	158	2	no	—
154	C₂×C₄²	—	158	1	no	—
155	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
156	#107	—	316	2	no	—
157	#107	—	316	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	158	2	no	—
161	#103	—	316	4	no	—
162	C₄×C₈	—	2	2	no	—
163	C₂×Q₈	—	79	2	no	—
164	C₂×Q₈	—	158	2	no	—
165	C₂×Q₈	—	158	2	no	—
166	C₂×Q₈	—	158	1	no	—
167	C₂²×C₄	—	316	2	no	—
168	C₂²×C₄	—	158	1	no	—
169	C₂²×C₄	—	158	1	no	—
170	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
172	C₂²×C₄	#11929625	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	316	1	no	—
181	C₂×C₈	—	316	1	no	—
182	C₂×C₈	—	316	1	no	—
183	C₂×C₈	#11929613	1	2	yes	—
184	C₂×C₈	Dic₃₁₆	1	2	yes	—
185	C₄⋊C₄	—	316	2	no	—
186	C₄⋊C₄	—	316	2	no	—
187	C₄⋊C₄	—	316	2	no	—
188	C₄⋊C₄	—	316	2	no	—
189	C₄⋊C₄	—	316	2	no	—
190	C₄⋊C₄	—	632	2	no	—
191	C₄⋊C₄	—	158	2	no	—
192	C₄²	—	316	1	no	—
193	C₄²	—	316	1	no	—
194	C₄²	#11929623	1	2	yes	—
195	C₄²	#11929624	1	2	yes	—
196	C₂³	#14579136	1	1	yes	characteristic
197	Q₈	—	316	2	no	—
198	Q₈	—	158	4	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	316	1	no	—
202	C₂×C₄	—	316	1	no	—
203	C₂×C₄	—	316	1	no	—
204	C₂×C₄	—	316	1	no	—
205	C₂×C₄	—	316	1	no	—
206	C₂×C₄	—	632	2	no	—
207	C₂×C₄	—	632	2	no	—
208	C₂×C₄	—	632	2	no	—
209	C₂×C₄	—	158	2	no	—
210	C₂×C₄	#14579143	1	2	yes	—
211	C₂×C₄	#14579145	1	1	yes	characteristic
212	C₂×C₄	#14579149	1	1	yes	characteristic
213	C₂×C₄	#14579156	1	2	yes	—
214	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
215	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
216	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
217	C₂×C₄	#14579180	1	1	yes	characteristic
218	C₂×C₄	#14579188	1	2	yes	—
219	C₂×C₄	#14579189	1	1	yes	characteristic
220	C₂×C₄	#14579194	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	8	2	no	—
224	C₈	—	632	1	no	—
225	C₂²	#19098490	1	1	yes	characteristic
226	C₂²	#19098764	1	1	yes	characteristic
227	C₂²	#19098774	1	1	yes	characteristic
228	C₂²	#19098805	1	1	yes	characteristic
229	C₂²	#19098817	1	1	yes	characteristic
230	C₂²	#19098834	1	1	yes	characteristic
231	C₂²	#19098839	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	316	2	no	—
234	C₄	—	632	1	no	—
235	C₄	—	1264	2	no	—
236	C₄	#19098885	1	2	yes	—
237	C₄	#19098894	1	1	yes	characteristic
238	C₄	#19098897	1	2	yes	—
239	C₄	#19098899	1	1	yes	characteristic
240	C₄	#19098914	1	1	yes	characteristic
241	C₄	#19098923	1	1	yes	characteristic
242	C₂	#23762771	1	1	yes	characteristic
243	C₂	#23762805	1	1	yes	characteristic
244	C₂	#23762885	1	1	yes	characteristic
245	C₂	#23762919	1	1	yes	characteristic
246	C₂	#23764810	1	1	yes	characteristic
247	C₂	#23764830	1	1	yes	characteristic
248	C₂	#23764842	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579136
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1277	C₂
socle, prime radical	C₂³×C₇₉	#121
hypercenter	#1277	D₇₉
hypocenter	C₇₉	#10618
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
130. C₇₉	#10618	semidirect

Maximal subgroups

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

subgroup	index
#23764336	2
#23763560	2
#23763406	2
#23761284	2
C₇₉×#1277	2
#10618	79