Subgroups — #34179510

Order 20224.

414 conjugacy classes of subgroups.

The subgroup lattice has 202 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×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#23761241	C₂	1	2	yes	maximal
5	C₄×#106×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	C₈×C₇₉⋊C₈	C₄	1	4	yes	complemented
13	C₂×#106×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	1	no	—
18	C₂²×C₇₉⋊C₈	—	2	2	no	—
19	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
21	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
23	#14579072	—	4	4	no	—
24	C₄×C₇₉⋊C₈	—	2	1	no	—
25	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₇₉⋊C₈	Q₈	1	2	yes	—
28	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₇₉	—	2	2	no	—
32	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
33	#106×C₇₉	—	2	4	no	—
34	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
36	#95×C₇₉	—	2	2	no	—
37	C₂×C₇₉⋊C₈	—	2	4	no	—
38	C₂×C₇₉⋊C₈	—	2	2	no	—
39	C₂×C₇₉⋊C₈	—	2	4	no	—
40	C₂×C₇₉⋊C₈	—	2	2	no	—
41	C₂×C₇₉⋊C₈	—	4	2	no	—
42	C₂×C₇₉⋊C₈	—	4	2	no	—
43	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₇₉⋊C₈	SD₁₆	1	4	yes	—
46	C₂²×C₄×C₇₉	—	2	4	no	—
47	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
50	C₂×C₈×C₇₉	—	2	2	no	—
51	C₂×C₈×C₇₉	C₄²	1	2	yes	—
52	C₂×C₈×C₇₉	C₄²	1	2	yes	—
53	C₂×C₈×C₇₉	M₁₆	1	4	yes	—
54	C₄⋊C₄×C₇₉	—	2	4	no	—
55	C₄⋊C₄×C₇₉	—	2	4	no	—
56	C₄²×C₇₉	—	4	2	no	—
57	C₄²×C₇₉	—	4	2	no	—
58	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₇₉	#41	1	1	yes	characteristic
62	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₉	—	2	2	no	—
64	C₂×C₄×C₇₉	—	4	4	no	—
65	C₂×C₄×C₇₉	—	4	4	no	—
66	C₂×C₄×C₇₉	—	4	4	no	—
67	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₇₉	#131	1	2	yes	—
75	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₇₉	#135	1	2	yes	—
77	C₈×C₇₉	—	2	2	no	—
78	C₈×C₇₉	#97	1	4	yes	complemented
79	C₇₉⋊C₈	—	2	4	no	—
80	C₇₉⋊C₈	—	2	2	no	—
81	C₇₉⋊C₈	—	4	4	no	—
82	C₇₉⋊C₈	#106	1	4	yes	complemented
83	C₂²×C₇₉	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₇₉	#436	1	2	yes	—
85	C₂²×C₇₉	C₄×SD₁₆	1	2	yes	—
86	C₂²×C₇₉	#443	1	2	yes	—
87	C₄×C₇₉	—	2	1	no	—
88	C₄×C₇₉	—	2	1	no	—
89	C₄×C₇₉	—	8	4	no	—
90	C₄×C₇₉	C₂×#97	1	1	yes	characteristic
91	C₄×C₇₉	#405	1	1	yes	characteristic
92	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
93	C₄×C₇₉	#427	1	1	yes	characteristic
94	C₄×C₇₉	C₄×SD₁₆	1	2	yes	—
95	C₄×C₇₉	#446	1	2	yes	—
96	#8283	—	79	1	no	maximal, Sylow
97	C₂×C₇₉	#1590	1	4	yes	—
98	C₂×C₇₉	#1770	1	1	yes	characteristic
99	C₂×C₇₉	#1772	1	1	yes	characteristic
100	C₂×C₇₉	C₄×#106	1	1	yes	characteristic
101	C₄×#106	D₇₉	1	1	yes	characteristic
102	#1770	—	79	2	no	—
103	C₂×#334	—	79	1	no	—
104	C₂×C₈²	—	79	1	no	—
105	#1279	—	79	2	no	—
106	C₇₉	#8283	1	1	yes	characteristic, complemented, Sylow
107	C₂×#106	Dic₇₉	1	2	yes	—
108	C₂×C₄⋊C₈	—	79	2	no	—
109	C₂×#97	—	158	2	no	—
110	C₂×C₄×C₈	—	79	1	no	—
111	C₂×C₄×C₈	—	79	1	no	—
112	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
113	C₄×C₄⋊C₄	C₂×D₇₉	1	2	yes	—
114	#340	Dic₇₉	1	2	yes	—
115	#336	—	158	2	no	—
116	#334	—	79	4	no	—
117	C₈²	—	79	4	no	—
118	C₂²×C₈	—	79	1	no	—
119	C₂²×C₈	—	79	1	no	—
120	C₂²×C₈	—	158	2	no	—
121	C₂²×C₈	—	158	1	no	—
122	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
123	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂×Dic₇₉	1	4	yes	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
127	#106	—	2	4	no	—
128	C₄⋊C₈	—	79	4	no	—
129	C₄⋊C₈	—	158	2	no	—
130	#97	—	316	4	no	—
131	C₄×C₈	—	79	2	no	—
132	C₄×C₈	—	79	2	no	—
133	C₄×C₈	—	79	2	no	—
134	C₄×C₈	—	158	1	no	—
135	C₄×C₈	C₄×D₇₉	1	2	yes	—
136	C₄×C₈	C₄×D₇₉	1	2	yes	—
137	#95	—	2	2	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
140	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
141	C₂²×C₄	#11929623	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	316	2	no	—
144	C₂×C₈	—	316	2	no	—
145	C₂×C₈	—	79	2	no	—
146	C₂×C₈	—	79	4	no	—
147	C₂×C₈	—	79	2	no	—
148	C₂×C₈	—	158	4	no	—
149	C₂×C₈	—	158	4	no	—
150	C₂×C₈	—	158	2	no	—
151	C₂×C₈	—	158	2	no	—
152	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
153	C₂×C₈	C₄×Dic₇₉	1	2	yes	—
154	C₂×C₈	#11929608	1	4	yes	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄²	—	4	2	no	—
158	C₄²	—	4	2	no	—
159	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
161	C₄²	Q₈×D₇₉	1	1	yes	characteristic
162	C₄²	#11929625	1	1	yes	characteristic
163	C₂³	D₄×Dic₇₉	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
169	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
170	C₂×C₄	#14579119	1	1	yes	characteristic
171	C₂×C₄	#14579123	1	1	yes	characteristic
172	C₂×C₄	#14579126	1	1	yes	characteristic
173	C₂×C₄	Q₈×Dic₇₉	1	1	yes	characteristic
174	C₂×C₄	C₂×#11929608	1	2	yes	—
175	C₂×C₄	#14579168	1	2	yes	—
176	C₂×C₄	SD₁₆×D₇₉	1	2	yes	—
177	C₂×C₄	#14579182	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	316	4	no	—
180	C₈	—	79	4	no	—
181	C₈	—	158	4	no	—
182	C₈	—	158	2	no	—
183	C₈	#14579072	1	4	yes	complemented
184	C₂²	C₄⋊C₄×Dic₇₉	1	1	yes	characteristic
185	C₂²	#19098641	1	2	yes	—
186	C₂²	SD₁₆×Dic₇₉	1	2	yes	—
187	C₂²	#19098680	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	8	4	no	—
191	C₄	C₂×#14579072	1	1	yes	characteristic
192	C₄	#19098519	1	1	yes	characteristic
193	C₄	#106×D₇₉	1	1	yes	characteristic
194	C₄	#19098598	1	1	yes	characteristic
195	C₄	#19098650	1	2	yes	—
196	C₄	#19098694	1	2	yes	—
197	C₂	#23762291	1	4	yes	—
198	C₂	#23763269	1	1	yes	characteristic
199	C₂	#23763281	1	1	yes	characteristic
200	C₂	#106×Dic₇₉	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×#106×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₄×D₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×#106	D₇₉
hypocenter	C₇₉	#8283
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
11. #19098314	C₄	semidirect
12. C₈×C₇₉⋊C₈	C₄	semidirect
78. C₈×C₇₉	#97	semidirect
82. C₇₉⋊C₈	#106	semidirect
106. C₇₉	#8283	semidirect
183. 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×C₈×C₇₉⋊C₈	2
#23761241	2
C₄×#106×C₇₉	2
#8283	79