Subgroups — #34187554

Order 20224.

474 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23764324	C₂	1	2	yes	maximal
2	#23763267	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098314	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098309	C₂	1	2	yes	maximal
5	C₂×#500×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×#14579085	—	2	4	no	—
9	C₂×#14579071	—	2	1	no	—
10	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
11	#19098317	—	2	1	no	—
12	#19098314	—	2	2	no	—
13	#19098309	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
15	#500×C₇₉	—	2	2	no	—
16	#500×C₇₉	C₄	1	4	yes	complemented
17	C₂×#107×C₇₉	—	2	1	no	—
18	C₂×#107×C₇₉	—	2	1	no	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂²×C₇₉⋊C₈	—	2	1	no	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	C₂×C₄⋊Dic₇₉	D₄	1	2	yes	—
23	C₂×C₄×Dic₇₉	—	2	1	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
25	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
27	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
28	#14579085	—	4	8	no	—
29	#14579071	—	4	2	no	—
30	C₇₉⋊#95	—	2	1	no	—
31	C₂²×Q₈×C₇₉	—	2	2	no	—
32	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₇₉	—	2	2	no	—
35	C₄⋊Q₈×C₇₉	—	2	2	no	—
36	C₄⋊Q₈×C₇₉	—	2	2	no	—
37	C₄⋊Q₈×C₇₉	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
40	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
41	#107×C₇₉	—	2	4	no	—
42	#107×C₇₉	—	2	4	no	—
43	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₇₉	D₄	1	2	yes	—
45	C₂²×Dic₇₉	—	2	2	no	—
46	C₂×C₇₉⋊C₈	—	2	2	no	—
47	C₂×C₇₉⋊C₈	—	4	2	no	—
48	C₂×C₇₉⋊C₈	—	4	1	no	—
49	C₂×C₇₉⋊C₈	—	4	4	no	—
50	C₂×C₇₉⋊C₈	—	4	1	no	—
51	C₄⋊Dic₇₉	—	2	4	no	—
52	C₄×Dic₇₉	—	4	1	no	—
53	C₄×Dic₇₉	—	4	1	no	—
54	C₂×Q₈×C₇₉	—	2	4	no	—
55	C₂×Q₈×C₇₉	—	4	2	no	—
56	C₂×Q₈×C₇₉	—	4	4	no	—
57	C₂×Q₈×C₇₉	—	4	4	no	—
58	C₂²×C₄×C₇₉	—	4	2	no	—
59	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
62	C₂×C₈×C₇₉	—	2	1	no	—
63	C₂×C₈×C₇₉	—	2	1	no	—
64	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
67	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
68	C₄⋊C₄×C₇₉	—	2	4	no	—
69	C₄⋊C₄×C₇₉	—	2	4	no	—
70	C₄⋊C₄×C₇₉	—	4	2	no	—
71	C₄⋊C₄×C₇₉	—	4	2	no	—
72	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₇₉	#41	1	1	yes	characteristic
76	C₂³×C₇₉	#121	1	1	yes	characteristic
77	Q₈×C₇₉	—	4	8	no	—
78	Q₈×C₇₉	—	8	2	no	—
79	Q₈×C₇₉	—	8	2	no	—
80	C₂×C₄×C₇₉	—	2	2	no	—
81	C₂×C₄×C₇₉	—	4	4	no	—
82	C₂×C₄×C₇₉	—	8	4	no	—
83	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
87	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
88	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
89	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
92	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
93	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
94	C₂×Dic₇₉	—	4	2	no	—
95	C₂×Dic₇₉	—	4	2	no	—
96	C₂×Dic₇₉	—	4	2	no	—
97	C₈×C₇₉	—	2	2	no	—
98	C₈×C₇₉	#103	1	4	yes	complemented
99	C₇₉⋊C₈	—	4	2	no	—
100	C₇₉⋊C₈	—	8	2	no	—
101	C₂²×C₇₉	#390	1	1	yes	characteristic
102	C₂²×C₇₉	#461	1	1	yes	characteristic
103	C₂²×C₇₉	#462	1	1	yes	characteristic
104	C₂²×C₇₉	#469	1	2	yes	—
105	C₂²×C₇₉	#473	1	2	yes	—
106	C₄×C₇₉	—	2	1	no	—
107	C₄×C₇₉	—	2	1	no	—
108	C₄×C₇₉	—	8	4	no	—
109	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
110	C₄×C₇₉	#419	1	1	yes	characteristic
111	C₄×C₇₉	#442	1	2	yes	—
112	C₄×C₇₉	#461	1	1	yes	characteristic
113	C₄×C₇₉	#462	1	1	yes	characteristic
114	C₄×C₇₉	#496	1	2	yes	—
115	Dic₇₉	—	8	2	no	—
116	#9740	—	79	1	no	maximal, Sylow
117	C₂×C₇₉	#1693	1	2	yes	—
118	C₂×C₇₉	#1694	1	2	yes	—
119	C₂×C₇₉	#1972	1	1	yes	characteristic
120	C₂×C₇₉	#1979	1	1	yes	characteristic
121	C₂×C₇₉	#1981	1	1	yes	characteristic
122	C₂×#500	D₇₉	1	1	yes	characteristic
123	#1981	—	79	2	no	—
124	#1770	—	79	1	no	—
125	C₂×#334	—	79	1	no	—
126	C₂×#332	—	79	2	no	—
127	C₇₉	#9740	1	1	yes	characteristic, complemented, Sylow
128	C₂×C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
129	#500	—	2	2	no	—
130	#500	Dic₇₉	1	4	yes	complemented
131	C₂×#107	—	2	1	no	—
132	C₂×#107	—	2	1	no	—
133	C₂×C₄⋊C₈	—	79	1	no	—
134	C₂×C₄⋊C₈	—	79	2	no	—
135	C₂×#103	—	158	4	no	—
136	C₂×#97	—	158	1	no	—
137	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
138	C₄×C₄⋊C₄	—	79	1	no	—
139	#336	—	158	1	no	—
140	#334	—	158	2	no	—
141	#332	—	158	4	no	—
142	C₂²×Q₈	—	2	2	no	—
143	C₂²×C₈	—	158	2	no	—
144	C₂²×C₈	—	158	1	no	—
145	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	C₂×Dic₇₉	1	4	yes	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	79	2	no	—
154	C₂×C₄²	—	158	1	no	—
155	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
156	#107	—	2	4	no	—
157	#107	—	2	4	no	—
158	C₄⋊C₈	—	79	2	no	—
159	C₄⋊C₈	—	158	2	no	—
160	C₄⋊C₈	—	158	2	no	—
161	C₄⋊C₈	—	158	1	no	—
162	#103	—	316	8	no	—
163	#97	—	316	2	no	—
164	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
165	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
166	#95	—	158	1	no	—
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	4	4	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	—	158	2	no	—
173	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
175	C₂²×C₄	#11929623	1	1	yes	characteristic
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	316	1	no	—
179	C₂×C₈	—	316	1	no	—
180	C₂×C₈	—	316	2	no	—
181	C₂×C₈	—	316	4	no	—
182	C₂×C₈	—	158	2	no	—
183	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
184	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
185	C₂×C₈	#11929611	1	2	yes	—
186	C₂×C₈	#11929614	1	2	yes	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	158	4	no	—
192	C₄²	—	316	1	no	—
193	C₄²	—	316	1	no	—
194	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
195	C₄²	D₄×D₇₉	1	1	yes	characteristic
196	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
197	C₄²	#11929623	1	1	yes	characteristic
198	C₂³	#14579132	1	1	yes	characteristic
199	Q₈	—	4	8	no	—
200	Q₈	—	8	2	no	—
201	Q₈	—	8	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	—	316	2	no	—
206	C₂×C₄	—	316	2	no	—
207	C₂×C₄	—	316	2	no	—
208	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
209	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
210	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
211	C₂×C₄	#14579132	1	1	yes	characteristic
212	C₂×C₄	#14579148	1	1	yes	characteristic
213	C₂×C₄	#14579157	1	1	yes	characteristic
214	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
215	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
216	C₂×C₄	#14579185	1	4	yes	—
217	C₂×C₄	#14579192	1	1	yes	characteristic
218	C₂×C₄	#14579192	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	316	2	no	—
221	C₈	—	632	2	no	—
222	C₈	#14579085	1	4	yes	complemented
223	C₂²	#19098466	1	1	yes	characteristic
224	C₂²	#19098765	1	1	yes	characteristic
225	C₂²	#19098770	1	1	yes	characteristic
226	C₂²	#19098813	1	2	yes	—
227	C₂²	#19098835	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	8	4	no	—
231	C₄	—	632	2	no	—
232	C₄	C₂×#14579085	1	1	yes	characteristic
233	C₄	#19098572	1	1	yes	characteristic
234	C₄	#19098675	1	2	yes	—
235	C₄	#19098765	1	1	yes	characteristic
236	C₄	#19098770	1	1	yes	characteristic
237	C₄	#19098951	1	2	yes	—
238	C₂	#23762910	1	2	yes	—
239	C₂	#23762915	1	2	yes	—
240	C₂	#23764277	1	1	yes	characteristic
241	C₂	#23764316	1	1	yes	characteristic
242	C₂	#23764324	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579132
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#500×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	#121
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#500	D₇₉
hypocenter	C₇₉	#9740
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
16. #500×C₇₉	C₄	semidirect
98. C₈×C₇₉	#103	semidirect
127. C₇₉	#9740	semidirect
130. #500	Dic₇₉	semidirect
222. C₈	#14579085	semidirect

Maximal subgroups

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

subgroup	index
#23764324	2
#23763267	2
C₂×#19098314	2
C₂×#19098309	2
C₂×#500×C₇₉	2
#9740	79