Subgroups — #34206842

Order 20224.

398 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579095	C₂	1	1	yes	maximal, characteristic
3	#23761491	C₂	1	4	yes	maximal
4	C₂×#500×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14579095	C₂²	1	2	yes	—
7	C₄×C₄⋊Dic₇₉	—	2	1	no	—
8	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
9	#19098326	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₇₉	—	2	1	no	—
11	#500×C₇₉	—	2	2	no	—
12	#500×C₇₉	C₄	1	4	yes	complemented
13	C₂×#107×C₇₉	—	2	1	no	—
14	C₂×#107×C₇₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇₉	—	2	2	no	—
17	C₂×C₄×Dic₇₉	—	4	1	no	—
18	C₂×C₇₉⋊C₁₆	—	2	2	no	—
19	C₂×C₇₉⋊C₁₆	—	2	4	no	—
20	#14579095	—	2	2	no	—
21	#14579095	Q₈	1	4	yes	complemented
22	C₇₉⋊#95	—	4	1	no	—
23	C₂²×Q₈×C₇₉	—	4	1	no	—
24	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
26	C₄⋊Q₈×C₇₉	—	2	4	no	—
27	C₄⋊Q₈×C₇₉	—	4	2	no	—
28	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
30	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
31	#107×C₇₉	—	2	4	no	—
32	#107×C₇₉	—	2	2	no	—
33	#107×C₇₉	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₉	D₄	1	2	yes	—
36	C₂²×Dic₇₉	—	4	2	no	—
37	C₄⋊Dic₇₉	—	2	4	no	—
38	C₄⋊Dic₇₉	—	4	2	no	—
39	C₄×Dic₇₉	—	8	2	no	—
40	C₇₉⋊C₁₆	—	4	2	no	—
41	C₂×Q₈×C₇₉	—	4	4	no	—
42	C₂×Q₈×C₇₉	—	8	2	no	—
43	C₂×Q₈×C₇₉	—	8	2	no	—
44	C₂²×C₄×C₇₉	—	4	2	no	—
45	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₉	—	2	2	no	—
49	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₇₉	#41	1	2	yes	—
54	C₄⋊C₄×C₇₉	—	2	4	no	—
55	C₄⋊C₄×C₇₉	—	4	4	no	—
56	C₄⋊C₄×C₇₉	—	4	2	no	—
57	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₇₉	SD₁₆	1	2	yes	—
59	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
60	Q₈×C₇₉	—	8	8	no	—
61	C₂×C₄×C₇₉	—	2	2	no	—
62	C₂×C₄×C₇₉	—	4	4	no	—
63	C₂×C₄×C₇₉	—	8	2	no	—
64	C₂×C₄×C₇₉	—	8	2	no	—
65	C₂×C₄×C₇₉	#102	1	2	yes	—
66	C₂×C₄×C₇₉	#102	1	2	yes	—
67	C₂×C₄×C₇₉	D₁₆	1	2	yes	—
68	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
69	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₉	#137	1	2	yes	—
73	C₂×Dic₇₉	—	4	4	no	—
74	C₂×Dic₇₉	—	8	2	no	—
75	C₂×Dic₇₉	—	8	2	no	—
76	C₈×C₇₉	—	2	2	no	—
77	C₈×C₇₉	C₄×Q₈	1	2	yes	—
78	C₈×C₇₉	#124	1	1	yes	characteristic
79	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
80	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
81	C₂²×C₇₉	#474	1	2	yes	—
82	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
83	C₂²×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
84	C₂²×C₇₉	#510	1	2	yes	—
85	C₄×C₇₉	—	2	2	no	—
86	C₄×C₇₉	—	8	4	no	—
87	C₄×C₇₉	#357	1	4	yes	—
88	C₄×C₇₉	#391	1	1	yes	characteristic
89	C₄×C₇₉	#419	1	1	yes	characteristic
90	C₄×C₇₉	#475	1	2	yes	—
91	Dic₇₉	—	8	4	no	—
92	#13275	—	79	1	no	maximal, Sylow
93	C₂×C₇₉	#1988	1	1	yes	characteristic
94	C₂×C₇₉	C₂×#357	1	1	yes	characteristic
95	C₂×C₇₉	#2141	1	1	yes	characteristic
96	C₂×C₇₉	#2224	1	2	yes	—
97	C₂×C₇₉	#2225	1	2	yes	—
98	C₂×#500	D₇₉	1	1	yes	characteristic
99	C₂×C₄⋊C₁₆	—	79	1	no	—
100	C₄×#107	—	79	1	no	—
101	#1380	—	79	4	no	—
102	C₇₉	#13275	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	—	2	1	no	—
104	C₂²×C₁₆	—	79	2	no	—
105	#500	—	2	2	no	—
106	#500	Dic₇₉	1	4	yes	complemented
107	C₂×#107	—	2	1	no	—
108	C₂×#107	—	79	2	no	—
109	C₂×#107	C₂×D₇₉	1	2	yes	—
110	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	158	1	no	—
112	C₄⋊C₁₆	—	158	2	no	—
113	#340	—	158	1	no	—
114	C₂²×Q₈	—	4	1	no	—
115	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
117	C₄⋊Q₈	—	2	4	no	—
118	C₄⋊Q₈	—	4	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄⋊C₄	—	158	2	no	—
122	C₂×C₄²	—	316	1	no	—
123	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
124	C₂×C₁₆	—	158	2	no	—
125	C₂×C₁₆	—	158	4	no	—
126	#107	—	2	4	no	—
127	#107	—	2	2	no	—
128	#107	—	79	4	no	—
129	#107	—	158	2	no	—
130	#107	C₂×Dic₇₉	1	4	yes	—
131	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
132	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
133	#95	—	316	1	no	—
134	C₂×Q₈	—	4	4	no	—
135	C₂×Q₈	—	8	2	no	—
136	C₂×Q₈	—	8	2	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	316	2	no	—
139	C₂²×C₄	#11929609	1	1	yes	characteristic
140	C₂²×C₄	#11929609	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
144	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
145	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
146	C₂×C₈	Q₈×D₇₉	1	2	yes	—
147	C₂×C₈	#11929625	1	2	yes	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	316	2	no	—
152	C₄⋊C₄	—	158	4	no	—
153	C₄²	—	632	2	no	—
154	C₄²	C₂²⋊Dic₇₉	1	2	yes	—
155	C₄²	#11929612	1	2	yes	—
156	C₁₆	—	316	2	no	—
157	C₂³	C₂×#11929609	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	316	4	no	—
164	C₂×C₄	—	632	2	no	—
165	C₂×C₄	—	632	2	no	—
166	C₂×C₄	#14579082	1	2	yes	—
167	C₂×C₄	#14579082	1	2	yes	—
168	C₂×C₄	#14579102	1	2	yes	—
169	C₂×C₄	#14579105	1	2	yes	—
170	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
171	C₂×C₄	#14579137	1	2	yes	—
172	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
173	C₂×C₄	#14579192	1	2	yes	—
174	C₈	—	2	2	no	—
175	C₈	Q₈×Dic₇₉	1	2	yes	—
176	C₈	#14579150	1	1	yes	characteristic
177	C₈	#14579159	1	1	yes	characteristic
178	C₂²	C₂×#14579082	1	1	yes	characteristic
179	C₂²	#19098840	1	2	yes	—
180	C₂²	C₂×#14579102	1	1	yes	characteristic
181	C₂²	C₂×#14579105	1	1	yes	characteristic
182	C₂²	#19099004	1	2	yes	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	632	4	no	—
186	C₄	#19098364	1	4	yes	—
187	C₄	#19098470	1	1	yes	characteristic
188	C₄	#19098572	1	1	yes	characteristic
189	C₄	#19098845	1	2	yes	—
190	C₂	#23764353	1	1	yes	characteristic
191	C₂	C₂×#19098364	1	1	yes	characteristic
192	C₂	#23765120	1	1	yes	characteristic
193	C₂	#23765534	1	2	yes	—
194	C₂	#23765539	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#500×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#500	D₇₉
hypocenter	C₇₉	#13275
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
12. #500×C₇₉	C₄	semidirect
21. #14579095	Q₈	semidirect
102. C₇₉	#13275	semidirect
106. #500	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₉⋊C₁₆)	2
C₄×#14579095	2
#23761491	2
C₂×#500×C₇₉	2
#13275	79