Subgroups — #36470877

Order 22784.

314 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151428	C₂	1	4	yes	maximal
2	#26020424	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667948	—	2	2	no	—
5	C₂×#15667948	C₂²	1	4	yes	—
6	#19151622	—	2	2	no	—
7	#19151428	—	2	8	no	—
8	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₈₉	C₄	1	2	yes	—
11	C₄×C₁₆×C₈₉	C₄	1	2	yes	—
12	C₂×#13083558	—	2	4	no	—
13	#15667948	—	2	8	no	—
14	#15667948	—	4	4	no	—
15	C₄⋊#10623494	—	4	4	no	—
16	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
20	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
21	C₂×C₁₆×C₈₉	Q₈	1	4	yes	—
22	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
25	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
26	#13083558	—	4	4	no	—
27	#13083558	—	4	4	no	—
28	C₂×#10623494	—	4	4	no	—
29	C₂²×C₄×C₈₉	Q₁₆	1	2	yes	—
30	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₈₉	D₈	1	2	yes	—
34	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
35	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
37	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
38	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₈₉	D₈	1	2	yes	—
42	C₁₆×C₈₉	C₄⋊C₄	1	8	yes	—
43	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₄×C₈₉	—	2	4	no	—
45	C₂×C₄×C₈₉	#107	1	2	yes	—
46	C₂×C₄×C₈₉	#107	1	2	yes	—
47	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
53	C₈×C₈₉	#107	1	2	yes	—
54	C₈×C₈₉	#107	1	2	yes	—
55	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
56	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
57	#10623494	—	8	4	no	—
58	C₂²×C₈₉	—	2	2	no	—
59	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
60	C₂²×C₈₉	#494	1	1	yes	characteristic
61	C₂²×C₈₉	#500	1	1	yes	characteristic
62	C₄×C₈₉	—	2	2	no	—
63	C₄×C₈₉	#368	1	4	yes	complemented
64	C₄×C₈₉	#384	1	1	yes	characteristic
65	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
66	C₄×C₈₉	#493	1	1	yes	characteristic
67	C₄×C₈₉	#500	1	1	yes	characteristic
68	#12909	—	89	1	no	maximal, Sylow
69	C₂×C₈₉	—	2	1	no	—
70	C₂×C₈₉	—	2	1	no	—
71	C₂×C₈₉	#1845	1	1	yes	characteristic
72	C₂×C₈₉	C₂×#368	1	2	yes	—
73	C₂×#368	—	89	4	no	—
74	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic
75	#1848	—	89	2	no	—
76	C₈₉	#12909	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
78	C₂×#108	—	89	4	no	—
79	C₂×#108	—	178	2	no	—
80	#423	—	178	2	no	—
81	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
82	#368	—	178	8	no	—
83	C₄×C₁₆	Dic₈₉	1	2	yes	—
84	C₄×C₁₆	Dic₈₉	1	2	yes	—
85	C₂×M₁₆	—	178	4	no	—
86	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
87	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
88	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
89	C₂×C₁₆	Dic₁₇₈	1	4	yes	—
90	C₂×C₁₆	D₃₅₆	1	4	yes	—
91	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	—
92	#108	—	356	4	no	—
93	#108	—	178	8	no	—
94	C₄⋊C₈	—	356	4	no	—
95	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
96	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
97	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
98	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
99	C₂²×C₄	Dic₃₅₆	1	2	yes	—
100	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
101	M₁₆	—	356	4	no	—
102	M₁₆	—	356	4	no	—
103	C₂×C₈	—	356	4	no	—
104	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
105	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
106	C₂×C₈	D₇₁₂	1	2	yes	—
107	C₂×C₈	Dic₃₅₆	1	2	yes	—
108	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
109	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
110	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
111	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
112	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
113	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
114	C₄²	D₇₁₂	1	2	yes	—
115	C₁₆	C₄⋊Dic₈₉	1	8	yes	—
116	C₂³	C₂×Dic₃₅₆	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	#15667946	1	2	yes	—
119	C₂×C₄	#15667946	1	2	yes	—
120	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
121	C₂×C₄	#15668009	1	1	yes	characteristic
122	C₂×C₄	#15668012	1	1	yes	characteristic
123	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
124	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
126	C₈	—	712	4	no	—
127	C₈	#15667946	1	2	yes	—
128	C₈	#15667946	1	2	yes	—
129	C₈	C₂×C₄⋊Dic₈₉	1	2	yes	—
130	C₈	#15668012	1	2	yes	—
131	C₂²	—	2	2	no	—
132	C₂²	C₂×#15667946	1	1	yes	characteristic
133	C₂²	#19151978	1	1	yes	characteristic
134	C₂²	#19152003	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	#19151428	1	4	yes	complemented
137	C₄	#19151468	1	1	yes	characteristic
138	C₄	C₂×#15667946	1	1	yes	characteristic
139	C₄	#19151975	1	1	yes	characteristic
140	C₄	#19152003	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#26020412	1	1	yes	characteristic
144	C₂	C₂×#19151428	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×C₈₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	#12909
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
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
63. C₄×C₈₉	#368	semidirect
76. C₈₉	#12909	semidirect
136. C₄	#19151428	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151428	2
#26020424	2
C₂×C₄×C₁₆×C₈₉	2
#12909	89