Subgroups — #34179530

Order 20224.

398 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763269	C₂	1	2	yes	maximal
2	C₂×#19098314	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098284	C₂	1	1	yes	maximal, characteristic
4	#23761241	C₂	1	2	yes	maximal
5	C₄×#107×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14579072	—	2	2	no	—
8	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	#19098318	—	2	2	no	—
11	#19098314	C₄	1	4	yes	complemented
12	#19098284	C₄	1	4	yes	complemented
13	C₂×#107×C₇₉	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₇₉	C₂²	1	2	yes	—
16	#340×C₇₉	C₄	1	2	yes	—
17	C₂²×C₇₉⋊C₈	—	2	2	no	—
18	C₂²×C₇₉⋊C₈	—	2	1	no	—
19	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
20	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
22	#14579072	—	4	4	no	—
23	C₄×C₇₉⋊C₈	—	2	1	no	—
24	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₇₉⋊C₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₇₉	—	2	2	no	—
31	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
32	#107×C₇₉	—	2	4	no	—
33	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
35	#95×C₇₉	—	2	2	no	—
36	C₂×C₇₉⋊C₈	—	2	2	no	—
37	C₂×C₇₉⋊C₈	—	2	4	no	—
38	C₂×C₇₉⋊C₈	—	2	2	no	—
39	C₂×C₇₉⋊C₈	—	2	2	no	—
40	C₂×C₇₉⋊C₈	—	4	2	no	—
41	C₂×C₇₉⋊C₈	—	4	2	no	—
42	C₂×C₇₉⋊C₈	—	4	2	no	—
43	C₂×C₇₉⋊C₈	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₇₉	—	2	4	no	—
45	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
48	C₂×C₈×C₇₉	—	2	2	no	—
49	C₂×C₈×C₇₉	C₄²	1	2	yes	—
50	C₂×C₈×C₇₉	C₄²	1	2	yes	—
51	C₂×C₈×C₇₉	M₁₆	1	4	yes	—
52	C₄⋊C₄×C₇₉	—	2	4	no	—
53	C₄⋊C₄×C₇₉	—	2	4	no	—
54	C₄²×C₇₉	—	4	2	no	—
55	C₄²×C₇₉	—	4	2	no	—
56	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₇₉	#41	1	1	yes	characteristic
60	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₉	—	2	2	no	—
62	C₂×C₄×C₇₉	—	4	4	no	—
63	C₂×C₄×C₇₉	—	4	4	no	—
64	C₂×C₄×C₇₉	—	4	4	no	—
65	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
72	C₂×C₄×C₇₉	#131	1	2	yes	—
73	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₇₉	#137	1	2	yes	—
75	C₈×C₇₉	—	2	2	no	—
76	C₈×C₇₉	#97	1	4	yes	complemented
77	C₇₉⋊C₈	—	2	4	no	—
78	C₇₉⋊C₈	—	4	2	no	—
79	C₇₉⋊C₈	—	4	4	no	—
80	C₂²×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₇₉	#436	1	2	yes	—
82	C₂²×C₇₉	#441	1	1	yes	characteristic
83	C₂²×C₇₉	#442	1	1	yes	characteristic
84	C₂²×C₇₉	#444	1	2	yes	—
85	C₄×C₇₉	—	2	1	no	—
86	C₄×C₇₉	—	2	1	no	—
87	C₄×C₇₉	—	8	4	no	—
88	C₄×C₇₉	C₂×#97	1	1	yes	characteristic
89	C₄×C₇₉	#405	1	1	yes	characteristic
90	C₄×C₇₉	#428	1	2	yes	—
91	C₄×C₇₉	#440	1	2	yes	—
92	C₄×C₇₉	#446	1	2	yes	—
93	#8287	—	79	1	no	maximal, Sylow
94	C₂×C₇₉	#1593	1	2	yes	—
95	C₂×C₇₉	#1594	1	2	yes	—
96	C₂×C₇₉	#1770	1	1	yes	characteristic
97	C₂×C₇₉	#1773	1	1	yes	characteristic
98	C₂×C₇₉	#1776	1	1	yes	characteristic
99	C₄×#107	D₇₉	1	1	yes	characteristic
100	#1770	—	79	2	no	—
101	C₂×#334	—	79	1	no	—
102	C₂×#322	—	79	1	no	—
103	#1279	—	79	2	no	—
104	C₇₉	#8287	1	1	yes	characteristic, complemented, Sylow
105	C₂×#107	Dic₇₉	1	2	yes	—
106	C₂×C₄⋊C₈	—	79	2	no	—
107	C₂×#97	—	158	2	no	—
108	C₂×C₄×C₈	—	79	1	no	—
109	C₂×C₄×C₈	—	79	1	no	—
110	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
111	C₄×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
112	#340	Dic₇₉	1	2	yes	—
113	#336	—	158	2	no	—
114	#334	—	79	4	no	—
115	#322	—	79	4	no	—
116	C₂²×C₈	—	79	2	no	—
117	C₂²×C₈	—	158	2	no	—
118	C₂²×C₈	—	158	1	no	—
119	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
120	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
122	C₂×C₄²	—	2	2	no	—
123	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
124	#107	—	2	4	no	—
125	C₄⋊C₈	—	79	4	no	—
126	C₄⋊C₈	—	158	2	no	—
127	#97	—	316	4	no	—
128	C₄×C₈	—	79	2	no	—
129	C₄×C₈	—	79	2	no	—
130	C₄×C₈	—	79	2	no	—
131	C₄×C₈	—	158	1	no	—
132	C₄×C₈	C₄×D₇₉	1	2	yes	—
133	C₄×C₈	C₄×D₇₉	1	2	yes	—
134	#95	—	2	2	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
137	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
138	C₂²×C₄	#11929623	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	316	2	no	—
141	C₂×C₈	—	316	2	no	—
142	C₂×C₈	—	316	2	no	—
143	C₂×C₈	—	79	4	no	—
144	C₂×C₈	—	158	2	no	—
145	C₂×C₈	—	158	4	no	—
146	C₂×C₈	—	158	2	no	—
147	C₂×C₈	—	158	2	no	—
148	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
149	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
150	C₂×C₈	#11929608	1	4	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄²	—	4	2	no	—
154	C₄²	—	4	2	no	—
155	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
156	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
157	C₄²	Q₈×D₇₉	1	1	yes	characteristic
158	C₄²	#11929625	1	1	yes	characteristic
159	C₂³	D₄×Dic₇₉	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
165	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
166	C₂×C₄	#14579119	1	1	yes	characteristic
167	C₂×C₄	#14579123	1	1	yes	characteristic
168	C₂×C₄	#14579126	1	1	yes	characteristic
169	C₂×C₄	Q₈×Dic₇₉	1	1	yes	characteristic
170	C₂×C₄	C₂×#11929608	1	2	yes	—
171	C₂×C₄	#14579168	1	2	yes	—
172	C₂×C₄	#14579185	1	2	yes	—
173	C₂×C₄	#14579193	1	2	yes	—
174	C₈	—	2	2	no	—
175	C₈	—	316	2	no	—
176	C₈	—	316	4	no	—
177	C₈	—	158	4	no	—
178	C₈	#14579072	1	4	yes	complemented
179	C₂²	C₄⋊C₄×Dic₇₉	1	1	yes	characteristic
180	C₂²	#19098641	1	2	yes	—
181	C₂²	#19098670	1	1	yes	characteristic
182	C₂²	#19098675	1	1	yes	characteristic
183	C₂²	#19098685	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	8	4	no	—
187	C₄	C₂×#14579072	1	1	yes	characteristic
188	C₄	#19098519	1	1	yes	characteristic
189	C₄	#19098604	1	2	yes	—
190	C₄	#19098662	1	2	yes	—
191	C₄	#19098694	1	2	yes	—
192	C₂	#23762309	1	2	yes	—
193	C₂	#23762314	1	2	yes	—
194	C₂	#23763269	1	1	yes	characteristic
195	C₂	#23763286	1	1	yes	characteristic
196	C₂	#23763299	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×#107×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₄×D₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×#107	D₇₉
hypocenter	C₇₉	#8287
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. #19098314	C₄	semidirect
12. #19098284	C₄	semidirect
76. C₈×C₇₉	#97	semidirect
104. C₇₉	#8287	semidirect
178. C₈	#14579072	semidirect

Maximal subgroups

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

subgroup	index
#23763269	2
C₂×#19098314	2
C₂×#19098284	2
#23761241	2
C₄×#107×C₇₉	2
#8287	79