Subgroups — #34181117

Order 20224.

478 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763380	C₂	1	2	yes	maximal
2	C₂×#19098314	C₂	1	2	yes	maximal, complemented
3	#23761235	C₂	1	2	yes	maximal, complemented
4	C₂×#431×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	4	yes	—
6	C₂×C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
7	#19098318	—	2	2	no	—
8	#19098314	—	2	4	no	—
9	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
10	#431×C₇₉	—	2	4	no	—
11	C₂×C₄⋊C₈×C₇₉	C₄	1	2	yes	—
12	C₂×C₂²⋊C₈×C₇₉	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂²×C₇₉⋊C₈	—	2	4	no	—
16	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
17	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
18	C₂²⋊(C₇₉⋊C₈)	—	4	4	no	—
19	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₇₉	Q₈	1	1	yes	characteristic
22	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₇₉	—	2	2	no	—
25	C₂×C₄²×C₇₉	—	2	2	no	—
26	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₉	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₇₉	—	2	4	no	—
31	C₂²⋊C₈×C₇₉	—	2	4	no	—
32	C₄×C₈×C₇₉	—	2	2	no	—
33	C₄×C₈×C₇₉	D₄	1	4	yes	—
34	C₂×C₇₉⋊C₈	—	4	4	no	—
35	C₂×C₇₉⋊C₈	—	4	4	no	—
36	C₂×C₇₉⋊C₈	—	4	4	no	—
37	C₂⁴×C₇₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₉	—	2	2	no	—
39	C₂²×C₄×C₇₉	—	2	2	no	—
40	C₂²×C₄×C₇₉	—	2	1	no	—
41	C₂²×C₄×C₇₉	—	2	1	no	—
42	C₂²×C₄×C₇₉	—	2	2	no	—
43	C₂²×C₄×C₇₉	—	2	4	no	—
44	C₂²×C₄×C₇₉	—	4	2	no	—
45	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	SD₁₆	1	4	yes	—
49	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₇₉	—	2	2	no	—
53	C₂×C₈×C₇₉	—	2	4	no	—
54	C₂×C₈×C₇₉	—	2	4	no	—
55	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₇₉	—	2	1	no	—
57	C₄²×C₇₉	—	2	1	no	—
58	C₄²×C₇₉	—	2	2	no	—
59	C₄²×C₇₉	—	2	2	no	—
60	C₄²×C₇₉	C₂×D₄	1	2	yes	—
61	C₄²×C₇₉	#41	1	2	yes	—
62	C₂³×C₇₉	—	2	1	no	—
63	C₂³×C₇₉	—	2	2	no	—
64	C₂³×C₇₉	—	2	1	no	—
65	C₂³×C₇₉	—	2	2	no	—
66	C₂³×C₇₉	#106	1	2	yes	—
67	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₉	—	2	1	no	—
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	—	2	8	no	—
72	C₂×C₄×C₇₉	—	2	1	no	—
73	C₂×C₄×C₇₉	—	2	1	no	—
74	C₂×C₄×C₇₉	—	2	2	no	—
75	C₂×C₄×C₇₉	—	2	2	no	—
76	C₂×C₄×C₇₉	—	4	4	no	—
77	C₂×C₄×C₇₉	—	4	2	no	—
78	C₂×C₄×C₇₉	—	4	2	no	—
79	C₂×C₄×C₇₉	#106	1	2	yes	—
80	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
82	C₂×C₄×C₇₉	#121	1	2	yes	—
83	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
84	C₂×C₄×C₇₉	#131	1	4	yes	—
85	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
86	C₂×C₄×C₇₉	#135	1	2	yes	—
87	C₈×C₇₉	—	2	4	no	—
88	C₈×C₇₉	—	4	2	no	—
89	C₇₉⋊C₈	—	8	4	no	—
90	C₂²×C₇₉	—	2	2	no	—
91	C₂²×C₇₉	—	2	4	no	—
92	C₂²×C₇₉	—	4	1	no	—
93	C₂²×C₇₉	—	4	1	no	—
94	C₂²×C₇₉	—	4	2	no	—
95	C₂²×C₇₉	#380	1	1	yes	characteristic
96	C₂²×C₇₉	#424	1	2	yes	—
97	C₂²×C₇₉	C₂×#106	1	1	yes	characteristic
98	C₂²×C₇₉	#427	1	1	yes	characteristic
99	C₂²×C₇₉	#430	1	2	yes	—
100	C₄×C₇₉	—	2	2	no	—
101	C₄×C₇₉	—	2	2	no	—
102	C₄×C₇₉	—	4	1	no	—
103	C₄×C₇₉	—	4	1	no	—
104	C₄×C₇₉	#408	1	2	yes	—
105	C₄×C₇₉	#433	1	2	yes	—
106	C₄×C₇₉	#465	1	4	yes	—
107	#8598	—	79	1	no	maximal, Sylow
108	C₂×C₇₉	—	4	2	no	—
109	C₂×C₇₉	#1570	1	2	yes	—
110	C₂×C₇₉	#1572	1	2	yes	—
111	C₂×C₇₉	#1798	1	1	yes	characteristic
112	C₂×C₇₉	#1807	1	1	yes	characteristic
113	C₂×C₇₉	#1811	1	1	yes	characteristic
114	C₂×#431	D₇₉	1	1	yes	characteristic
115	#1797	—	79	2	no	—
116	C₂×#334	—	79	2	no	—
117	#1277	—	79	2	no	—
118	C₇₉	#8598	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
120	#431	—	2	4	no	—
121	C₂×C₄⋊C₈	—	79	4	no	—
122	C₂×C₄⋊C₈	Dic₇₉	1	2	yes	—
123	C₂×C₂²⋊C₈	—	158	2	no	—
124	C₂×C₂²⋊C₈	Dic₇₉	1	2	yes	—
125	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
126	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
127	#336	—	158	2	no	—
128	#334	—	158	4	no	—
129	C₂³×C₄	Dic₁₅₈	1	1	yes	characteristic
130	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
131	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
132	C₂²×C₈	—	158	4	no	—
133	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
134	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	—	2	2	no	—
137	C₂×C₄²	Dic₁₅₈	1	1	yes	characteristic
138	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
139	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
140	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	158	4	no	—
143	C₄⋊C₈	—	158	4	no	—
144	C₂²⋊C₈	—	2	4	no	—
145	C₂²⋊C₈	—	316	4	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	C₂²⋊D₇₉	1	4	yes	—
148	C₂⁴	C₄⋊Dic₇₉	1	1	yes	characteristic
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
157	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
158	C₂²×C₄	C₄⋊Dic₇₉	1	1	yes	characteristic
159	C₂²×C₄	#11929613	1	4	yes	—
160	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
162	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	316	4	no	—
167	C₂×C₈	—	316	4	no	—
168	C₂×C₈	—	316	4	no	—
169	C₂×C₈	C₂²⋊Dic₇₉	1	4	yes	—
170	C₄²	—	2	1	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
175	C₄²	#11929624	1	2	yes	—
176	C₂³	—	2	1	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	1	no	—
179	C₂³	—	2	2	no	—
180	C₂³	#14579093	1	2	yes	—
181	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	#14579093	1	2	yes	—
194	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
195	C₂×C₄	#14579120	1	1	yes	characteristic
196	C₂×C₄	#14579135	1	2	yes	—
197	C₂×C₄	#14579140	1	2	yes	—
198	C₂×C₄	#14579168	1	4	yes	—
199	C₂×C₄	C₂×#11929613	1	2	yes	—
200	C₂×C₄	#14579183	1	2	yes	—
201	C₈	—	2	4	no	—
202	C₈	—	4	2	no	—
203	C₈	—	632	4	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	1	no	—
208	C₂²	—	4	1	no	—
209	C₂²	#19098423	1	1	yes	characteristic
210	C₂²	#19098590	1	2	yes	—
211	C₂²	C₂×#14579093	1	1	yes	characteristic
212	C₂²	#19098601	1	1	yes	characteristic
213	C₂²	#19098612	1	2	yes	—
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	1	no	—
218	C₄	#19098530	1	2	yes	—
219	C₄	#19098624	1	2	yes	—
220	C₄	#19098792	1	4	yes	—
221	C₂	—	4	2	no	—
222	C₂	#23762171	1	2	yes	—
223	C₂	#23762180	1	2	yes	—
224	C₂	#23763384	1	1	yes	characteristic
225	C₂	#23763425	1	1	yes	characteristic
226	C₂	#23763442	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#431×C₇₉	C₂
socle	C₂³×C₇₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₉	C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#431	D₇₉
hypocenter	C₇₉	#8598
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. C₂×#19098314	C₂	semidirect
3. #23761235	C₂	semidirect
118. C₇₉	#8598	semidirect

Maximal subgroups

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

subgroup	index
#23763380	2
C₂×#19098314	2
#23761235	2
C₂×#431×C₇₉	2
#8598	79