Subgroups — #34203149

Order 20224.

606 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098601	C₂	1	2	yes	maximal
2	#23761492	C₂	1	4	yes	maximal, complemented
3	C₂³×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579120	—	2	2	no	—
5	#19098601	—	2	4	no	—
6	#19098601	C₄	1	8	yes	complemented
7	C₂×#14579095	C₂²	1	4	yes	—
8	C₂×#14579093	—	2	2	no	—
9	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₇₉	—	2	4	no	—
11	C₂²×C₁₆×C₇₉	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
13	#14579120	—	2	8	no	—
14	#14579120	—	4	4	no	—
15	C₂×C₄⋊Dic₇₉	—	2	4	no	—
16	C₂×C₂²⋊Dic₇₉	—	4	2	no	—
17	C₂×C₄×Dic₇₉	—	4	2	no	—
18	#14579095	—	2	4	no	—
19	#14579095	C₂×C₄	1	8	yes	—
20	#14579093	—	2	8	no	—
21	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₇₉	D₄	1	4	yes	—
25	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
27	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₇₉	—	2	8	no	—
29	C₂×C₁₆×C₇₉	—	2	4	no	—
30	C₂×C₁₆×C₇₉	C₂×C₄	1	4	yes	—
31	C₂²×Dic₇₉	—	4	4	no	—
32	C₄⋊Dic₇₉	—	2	8	no	—
33	C₄⋊Dic₇₉	—	4	4	no	—
34	C₂²⋊Dic₇₉	—	4	8	no	—
35	C₄×Dic₇₉	—	4	8	no	—
36	C₂⁴×C₇₉	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	SD₁₆	1	4	yes	—
41	C₂²×C₄×C₇₉	Q₁₆	1	2	yes	—
42	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₄²	1	4	yes	—
44	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
46	C₂×C₈×C₇₉	C₄⋊C₄	1	8	yes	—
47	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
52	C₁₆×C₇₉	—	2	4	no	—
53	C₂³×C₇₉	—	2	4	no	—
54	C₂³×C₇₉	#102	1	4	yes	—
55	C₂³×C₇₉	#107	1	2	yes	—
56	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₇₉	#95	1	4	yes	—
58	C₂×C₄×C₇₉	#102	1	4	yes	—
59	C₂×C₄×C₇₉	#103	1	8	yes	—
60	C₂×C₄×C₇₉	#106	1	4	yes	—
61	C₂×C₄×C₇₉	#107	1	2	yes	—
62	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
65	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×Dic₇₉	—	4	8	no	—
67	C₂×Dic₇₉	—	8	8	no	—
68	C₈×C₇₉	#95	1	4	yes	—
69	C₈×C₇₉	C₂×C₄²	1	1	yes	characteristic
70	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₈×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
72	C₂²×C₇₉	—	2	8	no	—
73	C₂²×C₇₉	—	2	4	no	—
74	C₂²×C₇₉	#340	1	4	yes	—
75	C₂²×C₇₉	C₂×#102	1	2	yes	—
76	C₂²×C₇₉	C₂×#107	1	1	yes	characteristic
77	C₂²×C₇₉	#508	1	4	yes	—
78	C₄×C₇₉	#340	1	4	yes	—
79	C₄×C₇₉	C₂×#95	1	1	yes	characteristic
80	C₄×C₇₉	C₂×#103	1	2	yes	—
81	C₄×C₇₉	C₂×#106	1	1	yes	characteristic
82	Dic₇₉	—	8	8	no	—
83	#12556	—	79	1	no	maximal, Sylow
84	C₂×C₇₉	—	2	4	no	—
85	C₂×C₇₉	C₂×#340	1	1	yes	characteristic
86	C₂×C₇₉	#2138	1	4	yes	—
87	C₂×C₇₉	#2158	1	2	yes	—
88	C₂³×C₁₆	D₇₉	1	1	yes	characteristic
89	C₂×#427	—	79	2	no	—
90	#1380	—	79	4	no	—
91	C₇₉	#12556	1	1	yes	characteristic, complemented, Sylow
92	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
93	C₂×#117	—	158	2	no	—
94	C₂²×C₁₆	—	2	4	no	—
95	C₂²×C₁₆	Dic₇₉	1	4	yes	complemented
96	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
97	#427	—	79	8	no	—
98	#427	—	158	4	no	—
99	C₂×#107	—	79	4	no	—
100	C₂×#106	—	158	2	no	—
101	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
102	C₂²×C₈	Dic₁₅₈	1	2	yes	—
103	C₂²×C₈	C₄×D₇₉	1	4	yes	—
104	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
106	C₂²×C₈	C₂²⋊D₇₉	1	4	yes	—
107	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
108	#117	—	316	4	no	—
109	#117	—	158	8	no	—
110	C₂×C₄⋊C₄	—	158	4	no	—
111	C₂×C₂²⋊C₄	—	316	2	no	—
112	C₂×C₄²	—	316	2	no	—
113	C₂×C₁₆	—	2	8	no	—
114	C₂×C₁₆	—	2	4	no	—
115	C₂×C₁₆	C₂×Dic₇₉	1	4	yes	—
116	#107	—	79	8	no	—
117	#107	—	158	4	no	—
118	#106	—	158	8	no	—
119	C₂⁴	D₆₃₂	1	1	yes	characteristic
120	C₂²×C₄	—	316	4	no	—
121	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	4	yes	—
122	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
123	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
124	C₂²×C₄	#11929613	1	4	yes	—
125	C₂²×C₄	Dic₃₁₆	1	2	yes	—
126	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
127	C₂×C₈	C₄×Dic₇₉	1	4	yes	—
128	C₂×C₈	(C₂×D₇₉)⋊C₄	1	4	yes	—
129	C₂×C₈	C₂²⋊Dic₇₉	1	4	yes	—
130	C₂×C₈	Dic₇₉⋊C₄	1	8	yes	—
131	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
132	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
133	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
134	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
135	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
136	C₄⋊C₄	—	316	4	no	—
137	C₄⋊C₄	—	158	8	no	—
138	C₂²⋊C₄	—	316	8	no	—
139	C₄²	—	316	8	no	—
140	C₁₆	—	2	4	no	—
141	C₂³	—	2	4	no	—
142	C₂³	#14579083	1	4	yes	—
143	C₂³	#14579095	1	2	yes	—
144	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
145	C₂×C₄	—	316	8	no	—
146	C₂×C₄	—	632	8	no	—
147	C₂×C₄	C₇₉⋊#95	1	4	yes	—
148	C₂×C₄	#14579083	1	4	yes	—
149	C₂×C₄	#14579086	1	8	yes	—
150	C₂×C₄	#14579093	1	4	yes	—
151	C₂×C₄	#14579095	1	2	yes	—
152	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	2	yes	—
153	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
154	C₂×C₄	C₂×#11929613	1	2	yes	—
155	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
156	C₈	C₇₉⋊#95	1	4	yes	—
157	C₈	C₂×C₄×Dic₇₉	1	1	yes	characteristic
158	C₈	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
159	C₈	C₂×Dic₇₉⋊C₄	1	2	yes	—
160	C₂²	—	2	8	no	—
161	C₂²	—	2	4	no	—
162	C₂²	#19098326	1	4	yes	—
163	C₂²	C₂×#14579083	1	2	yes	—
164	C₂²	C₂×#14579095	1	1	yes	characteristic
165	C₂²	#19098995	1	4	yes	—
166	C₄	—	632	8	no	—
167	C₄	#19098326	1	4	yes	—
168	C₄	C₂×C₇₉⋊#95	1	1	yes	characteristic
169	C₄	C₂×#14579086	1	2	yes	—
170	C₄	C₂×#14579093	1	1	yes	characteristic
171	C₂	—	2	4	no	—
172	C₂	C₂×#19098326	1	1	yes	characteristic
173	C₂	#23765108	1	4	yes	—
174	C₂	#23765184	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₆₃₂
commutator	C₈×C₇₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂³×C₁₆×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂⁴×C₇₉	D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂³×C₁₆	D₇₉
hypocenter	C₇₉	#12556
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. #23761492	C₂	semidirect
6. #19098601	C₄	semidirect
11. C₂²×C₁₆×C₇₉	C₄	semidirect
91. C₇₉	#12556	semidirect
95. C₂²×C₁₆	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098601	2
#23761492	2
C₂³×C₁₆×C₇₉	2
#12556	79