Subgroups — #34443761

Order 20224.

490 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773914	C₂	1	2	yes	maximal
2	#23765682	C₂	1	2	yes	maximal, complemented
3	#23765679	C₂	1	4	yes	maximal, complemented
4	#23765675	C₂	1	2	yes	maximal, complemented
5	#23765184	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
7	#19099120	—	2	2	no	—
8	#19098968	C₂²	1	4	yes	—
9	#19098965	—	2	2	no	—
10	#19098965	C₂²	1	4	yes	—
11	#19098961	—	2	2	no	—
12	#19098601	—	2	2	no	—
13	#19098601	C₂²	1	4	yes	—
14	#19098391	C₂²	1	8	yes	—
15	#19098389	C₂²	1	8	yes	—
16	C₂²×C₁₆×C₇₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₇₉	C₂²	1	2	yes	—
20	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
21	#14579161	—	2	4	no	—
22	#14579153	—	2	4	no	—
23	#14579140	—	4	4	no	—
24	C₄×Dic₁₅₈	—	4	4	no	—
25	#14579120	—	2	4	no	—
26	#14579095	—	2	4	no	—
27	#14579095	C₂³	1	8	yes	—
28	#14579093	—	2	4	no	—
29	#14579093	—	2	4	no	—
30	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₇₉	Q₈	1	8	yes	complemented
34	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
35	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
36	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
37	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
38	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
40	C₂×Dic₁₅₈	—	4	4	no	—
41	C₄⋊Dic₇₉	—	2	8	no	—
42	C₄⋊Dic₇₉	—	4	4	no	—
43	Dic₇₉⋊C₄	—	4	8	no	—
44	C₂²⋊Dic₇₉	—	4	4	no	—
45	C₄×Dic₇₉	—	4	4	no	—
46	C₂²×C₄×C₇₉	D₈	1	2	yes	—
47	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₉	Q₁₆	1	4	yes	—
49	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
53	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
54	C₄²×C₇₉	D₈	1	2	yes	—
55	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₁₆×C₇₉	C₂×Q₈	1	4	yes	—
58	C₁₆×C₇₉	C₂×Q₈	1	4	yes	—
59	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₉	—	2	4	no	—
61	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
62	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
66	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
67	C₂×Dic₇₉	—	4	8	no	—
68	Dic₁₅₈	—	8	8	no	—
69	C₈×C₇₉	C₄⋊Q₈	1	2	yes	—
70	C₈×C₇₉	C₂×Q₁₆	1	2	yes	—
71	C₈×C₇₉	C₂×Q₁₆	1	2	yes	—
72	C₈×C₇₉	C₂²×Q₈	1	2	yes	—
73	C₂²×C₇₉	—	2	2	no	—
74	C₂²×C₇₉	#500	1	2	yes	—
75	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
76	C₄×C₇₉	—	2	2	no	—
77	C₄×C₇₉	#500	1	2	yes	—
78	C₄×C₇₉	#508	1	4	yes	—
79	C₄×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
80	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
81	Dic₇₉	—	8	8	no	—
82	#33726	—	79	1	no	maximal, Sylow
83	C₂×C₇₉	—	2	2	no	—
84	C₂×C₇₉	C₂×#500	1	1	yes	characteristic
85	C₂×C₇₉	C₂×#508	1	2	yes	—
86	#3160	—	79	2	no	—
87	#2254	—	79	2	no	—
88	#2253	—	79	4	no	—
89	#2252	—	79	2	no	—
90	#2158	—	79	4	no	—
91	C₂×C₄×C₁₆	D₇₉	1	1	yes	characteristic
92	C₇₉	#33726	1	1	yes	characteristic, complemented, Sylow
93	#533	—	158	2	no	—
94	C₂²×C₁₆	C₂×D₇₉	1	2	yes	—
95	#500	—	79	4	no	—
96	#499	—	79	4	no	—
97	#499	—	158	2	no	—
98	#498	—	158	2	no	—
99	#427	—	79	4	no	—
100	#427	—	158	2	no	—
101	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
102	#367	—	79	8	no	—
103	#366	—	79	8	no	—
104	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
105	C₄×C₁₆	C₂×D₇₉	1	2	yes	—
106	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
107	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
109	C₄⋊Q₈	—	158	4	no	—
110	#125	—	158	4	no	—
111	C₂²⋊Q₈	—	316	4	no	—
112	C₄×Q₈	—	316	4	no	—
113	#117	—	158	4	no	—
114	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
115	C₂×C₁₆	Dic₁₅₈	1	8	yes	complemented
116	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
117	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
118	#107	—	79	8	no	—
119	#107	—	158	4	no	—
120	#106	—	158	4	no	—
121	#106	—	158	4	no	—
122	C₄×C₈	D₃₁₆	1	1	yes	characteristic
123	C₄×C₈	D₃₁₆	1	1	yes	characteristic
124	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
125	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
126	C₂×Q₈	—	316	4	no	—
127	C₂²×C₄	D₆₃₂	1	2	yes	—
128	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
129	C₂×C₈	Dic₃₁₆	1	4	yes	—
130	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
131	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
132	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
133	C₂×C₈	C₂×Dic₁₅₈	1	4	yes	—
134	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
135	C₄⋊C₄	—	316	8	no	—
136	C₄⋊C₄	—	316	4	no	—
137	C₄⋊C₄	—	158	8	no	—
138	C₂²⋊C₄	—	316	4	no	—
139	C₄²	—	316	4	no	—
140	C₄²	D₆₃₂	1	2	yes	—
141	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
142	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
143	C₁₆	C₂×Dic₁₅₈	1	4	yes	—
144	C₁₆	C₂×Dic₁₅₈	1	4	yes	—
145	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
146	Q₈	—	632	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	316	8	no	—
149	C₂×C₄	#14579161	1	2	yes	—
150	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
151	C₂×C₄	C₂×D₆₃₂	1	2	yes	—
152	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
153	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
154	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
155	C₈	#14579161	1	2	yes	—
156	C₈	C₂×Dic₃₁₆	1	2	yes	—
157	C₈	C₂×Dic₃₁₆	1	2	yes	—
158	C₈	C₂²×Dic₁₅₈	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	#19098968	1	2	yes	—
161	C₂²	C₂²×D₆₃₂	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	632	8	no	—
164	C₄	#19098968	1	2	yes	—
165	C₄	#19098995	1	4	yes	—
166	C₄	C₂×#14579161	1	1	yes	characteristic
167	C₄	C₂²×Dic₃₁₆	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×#19098968	1	1	yes	characteristic
170	C₂	C₂×#19098995	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₃₂
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×C₄×C₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂³×C₇₉	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₇₉
hypocenter	C₇₉	#33726
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #23765682	C₂	semidirect
3. #23765679	C₂	semidirect
4. #23765675	C₂	semidirect
33. C₂×C₁₆×C₇₉	Q₈	semidirect
92. C₇₉	#33726	semidirect
115. C₂×C₁₆	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23773914	2
#23765682	2
#23765679	2
#23765675	2
#23765184	2
C₂×C₄×C₁₆×C₇₉	2
#33726	79