Subgroups — #34187549

Order 20224.

442 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23764349	C₂	1	2	yes	maximal
2	#23763267	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098314	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098312	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×#14579071	—	2	1	no	—
9	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
10	#19098325	—	2	4	no	—
11	#19098317	—	2	1	no	—
12	#19098314	—	2	2	no	—
13	#19098312	—	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	#14579071	—	4	2	no	—
29	C₇₉⋊#95	—	2	1	no	—
30	C₂²×Q₈×C₇₉	—	2	2	no	—
31	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
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	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
39	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
40	#107×C₇₉	—	2	4	no	—
41	#107×C₇₉	—	2	4	no	—
42	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₇₉	D₄	1	2	yes	—
44	C₂²×Dic₇₉	—	2	2	no	—
45	C₂×C₇₉⋊C₈	—	2	2	no	—
46	C₂×C₇₉⋊C₈	—	4	2	no	—
47	C₂×C₇₉⋊C₈	—	4	1	no	—
48	C₂×C₇₉⋊C₈	—	4	1	no	—
49	C₂×C₇₉⋊C₈	—	4	4	no	—
50	C₄⋊Dic₇₉	—	2	4	no	—
51	C₄×Dic₇₉	—	4	1	no	—
52	C₄×Dic₇₉	—	4	1	no	—
53	C₂×Q₈×C₇₉	—	2	4	no	—
54	C₂×Q₈×C₇₉	—	4	2	no	—
55	C₂×Q₈×C₇₉	—	8	4	no	—
56	C₂²×C₄×C₇₉	—	4	2	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	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
65	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
66	C₄⋊C₄×C₇₉	—	2	4	no	—
67	C₄⋊C₄×C₇₉	—	2	4	no	—
68	C₄⋊C₄×C₇₉	—	4	2	no	—
69	C₄⋊C₄×C₇₉	—	4	2	no	—
70	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₇₉	#41	1	1	yes	characteristic
74	C₂³×C₇₉	#121	1	1	yes	characteristic
75	Q₈×C₇₉	—	8	4	no	—
76	Q₈×C₇₉	—	8	4	no	—
77	C₂×C₄×C₇₉	—	2	2	no	—
78	C₂×C₄×C₇₉	—	4	4	no	—
79	C₂×C₄×C₇₉	—	8	4	no	—
80	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
84	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
85	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
86	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
89	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
90	C₂×C₄×C₇₉	#137	1	4	yes	—
91	C₂×Dic₇₉	—	4	2	no	—
92	C₂×Dic₇₉	—	4	2	no	—
93	C₂×Dic₇₉	—	4	2	no	—
94	C₈×C₇₉	—	2	2	no	—
95	C₈×C₇₉	#103	1	4	yes	—
96	C₇₉⋊C₈	—	4	2	no	—
97	C₇₉⋊C₈	—	8	2	no	—
98	C₂²×C₇₉	#390	1	1	yes	characteristic
99	C₂²×C₇₉	#461	1	1	yes	characteristic
100	C₂²×C₇₉	#462	1	1	yes	characteristic
101	C₂²×C₇₉	#470	1	2	yes	—
102	C₂²×C₇₉	#472	1	2	yes	—
103	C₄×C₇₉	—	2	1	no	—
104	C₄×C₇₉	—	2	1	no	—
105	C₄×C₇₉	—	8	4	no	—
106	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
107	C₄×C₇₉	#419	1	1	yes	characteristic
108	C₄×C₇₉	#441	1	2	yes	—
109	C₄×C₇₉	#461	1	1	yes	characteristic
110	C₄×C₇₉	#462	1	1	yes	characteristic
111	C₄×C₇₉	#497	1	2	yes	—
112	Dic₇₉	—	8	2	no	—
113	#9739	—	79	1	no	maximal, Sylow
114	C₂×C₇₉	#1695	1	2	yes	—
115	C₂×C₇₉	#1696	1	2	yes	—
116	C₂×C₇₉	#1971	1	1	yes	characteristic
117	C₂×C₇₉	#1978	1	1	yes	characteristic
118	C₂×C₇₉	#1981	1	1	yes	characteristic
119	C₂×#500	D₇₉	1	1	yes	characteristic
120	#1987	—	79	2	no	—
121	#1770	—	79	1	no	—
122	C₂×#334	—	79	1	no	—
123	C₂×#333	—	79	2	no	—
124	C₇₉	#9739	1	1	yes	characteristic, complemented, Sylow
125	C₂×C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
126	#500	—	2	2	no	—
127	#500	Dic₇₉	1	4	yes	complemented
128	C₂×#107	—	2	1	no	—
129	C₂×#107	—	2	1	no	—
130	C₂×C₄⋊C₈	—	79	1	no	—
131	C₂×C₄⋊C₈	—	79	2	no	—
132	C₂×#97	—	158	1	no	—
133	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
134	C₄×C₄⋊C₄	—	79	1	no	—
135	#340	—	158	4	no	—
136	#336	—	158	1	no	—
137	#334	—	158	2	no	—
138	#333	—	158	4	no	—
139	C₂²×Q₈	—	2	2	no	—
140	C₂²×C₈	—	158	2	no	—
141	C₂²×C₈	—	158	1	no	—
142	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
143	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	C₂×Dic₇₉	1	4	yes	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	79	2	no	—
151	C₂×C₄²	—	158	1	no	—
152	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
153	#107	—	2	4	no	—
154	#107	—	2	4	no	—
155	C₄⋊C₈	—	79	2	no	—
156	C₄⋊C₈	—	158	2	no	—
157	C₄⋊C₈	—	158	2	no	—
158	C₄⋊C₈	—	158	1	no	—
159	#97	—	316	2	no	—
160	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
161	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
162	#95	—	158	1	no	—
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂×Q₈	—	8	4	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	—	158	2	no	—
168	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
170	C₂²×C₄	#11929623	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	316	1	no	—
174	C₂×C₈	—	316	1	no	—
175	C₂×C₈	—	316	2	no	—
176	C₂×C₈	—	316	4	no	—
177	C₂×C₈	—	158	2	no	—
178	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
179	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
180	C₂×C₈	#11929611	1	2	yes	—
181	C₂×C₈	#11929614	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	158	4	no	—
187	C₄²	—	316	1	no	—
188	C₄²	—	316	1	no	—
189	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
190	C₄²	D₄×D₇₉	1	1	yes	characteristic
191	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
192	C₄²	#11929623	1	1	yes	characteristic
193	C₂³	#14579132	1	1	yes	characteristic
194	Q₈	—	8	4	no	—
195	Q₈	—	8	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	8	4	no	—
199	C₂×C₄	—	316	2	no	—
200	C₂×C₄	—	316	2	no	—
201	C₂×C₄	—	316	2	no	—
202	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
203	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
204	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
205	C₂×C₄	#14579132	1	1	yes	characteristic
206	C₂×C₄	#14579148	1	1	yes	characteristic
207	C₂×C₄	#14579157	1	1	yes	characteristic
208	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
209	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
210	C₂×C₄	#14579192	1	1	yes	characteristic
211	C₂×C₄	#14579192	1	1	yes	characteristic
212	C₂×C₄	#14579193	1	4	yes	—
213	C₈	—	2	2	no	—
214	C₈	—	316	2	no	—
215	C₈	—	632	2	no	—
216	C₈	#14579085	1	4	yes	—
217	C₂²	#19098466	1	1	yes	characteristic
218	C₂²	#19098765	1	1	yes	characteristic
219	C₂²	#19098770	1	1	yes	characteristic
220	C₂²	#19098818	1	2	yes	—
221	C₂²	#19098830	1	2	yes	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	8	4	no	—
225	C₄	—	632	2	no	—
226	C₄	C₂×#14579085	1	1	yes	characteristic
227	C₄	#19098572	1	1	yes	characteristic
228	C₄	#19098670	1	2	yes	—
229	C₄	#19098765	1	1	yes	characteristic
230	C₄	#19098770	1	1	yes	characteristic
231	C₄	#19098956	1	2	yes	—
232	C₂	#23762920	1	2	yes	—
233	C₂	#23762925	1	2	yes	—
234	C₂	#23764272	1	1	yes	characteristic
235	C₂	#23764311	1	1	yes	characteristic
236	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	#9739
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
124. C₇₉	#9739	semidirect
127. #500	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23764349	2
#23763267	2
C₂×#19098314	2
C₂×#19098312	2
C₂×#500×C₇₉	2
#9739	79