Subgroups — #36466077

Order 22784.

226 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26019015	C₂	1	2	yes	maximal
2	#26018326	C₂	1	2	yes	maximal, complemented
3	#26018324	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151655	—	2	2	no	—
6	#19151351	C₂²	1	4	yes	—
7	#19151349	—	2	2	no	—
8	C₂²×C₁₆×C₈₉	C₄	1	1	yes	characteristic
9	C₂²×C₁₆×C₈₉	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₄×C₁₆×C₈₉	C₄	1	1	yes	characteristic
12	C₄×C₁₆×C₈₉	C₄	1	1	yes	characteristic
13	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₄⋊#10623494	—	2	4	no	—
16	C₂²⋊#10623494	—	4	2	no	—
17	#15667923	—	4	2	no	—
18	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₈₉	—	2	2	no	—
22	C₂×C₁₆×C₈₉	—	2	2	no	—
23	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
30	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×#10623494	—	4	4	no	—
32	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
35	C₂×C₈×C₈₉	—	2	2	no	—
36	C₂×C₈×C₈₉	—	2	1	no	—
37	C₂×C₈×C₈₉	—	2	1	no	—
38	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₈₉	D₈	1	1	yes	characteristic
44	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
45	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₈₉	—	2	1	no	—
47	C₁₆×C₈₉	—	2	1	no	—
48	C₁₆×C₈₉	M₁₆	1	2	yes	—
49	C₁₆×C₈₉	M₁₆	1	2	yes	—
50	C₂³×C₈₉	#107	1	1	yes	characteristic
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	2	1	no	—
53	C₂×C₄×C₈₉	—	2	1	no	—
54	C₂×C₄×C₈₉	—	2	2	no	—
55	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
61	C₈×C₈₉	—	2	1	no	—
62	C₈×C₈₉	—	2	1	no	—
63	C₈×C₈₉	#108	1	2	yes	—
64	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
65	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
66	#10623494	—	8	4	no	—
67	C₂²×C₈₉	—	2	1	no	—
68	C₂²×C₈₉	—	2	1	no	—
69	C₂²×C₈₉	#368	1	2	yes	—
70	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
71	C₄×C₈₉	—	2	1	no	—
72	C₄×C₈₉	—	2	1	no	—
73	C₄×C₈₉	—	2	1	no	—
74	C₄×C₈₉	—	2	1	no	—
75	C₄×C₈₉	#423	1	1	yes	characteristic
76	C₄×C₈₉	C₂×#108	1	1	yes	characteristic
77	C₄×C₈₉	#508	1	2	yes	—
78	#12024	—	89	1	no	maximal, Sylow
79	C₂×C₈₉	—	2	2	no	—
80	C₂×C₈₉	#1567	1	1	yes	characteristic
81	C₂×C₈₉	#2158	1	1	yes	characteristic
82	C₂×C₈₉	C₂×#368	1	1	yes	characteristic
83	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic
84	#1568	—	89	2	no	—
85	#1372	—	89	2	no	—
86	#1371	—	89	2	no	—
87	C₈₉	#12024	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	Dic₈₉	1	1	yes	characteristic
89	C₂²×C₁₆	Dic₈₉	1	1	yes	characteristic
90	#432	—	178	2	no	—
91	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
92	C₄×C₁₆	Dic₈₉	1	1	yes	characteristic
93	C₄×C₁₆	Dic₈₉	1	1	yes	characteristic
94	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
95	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
96	#335	—	89	4	no	—
97	#334	—	178	2	no	—
98	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
99	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
100	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
104	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
105	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
106	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
107	C₄⋊C₈	—	178	4	no	—
108	C₂²⋊C₈	—	356	2	no	—
109	#97	—	356	2	no	—
110	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
111	C₄×C₈	D₃₅₆	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
114	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
115	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
116	C₂²×C₄	Dic₃₅₆	1	1	yes	characteristic
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	356	4	no	—
121	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
122	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
123	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
124	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
125	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
126	C₄²	D₇₁₂	1	1	yes	characteristic
127	C₄²	Dic₃₅₆	1	1	yes	characteristic
128	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
129	C₁₆	—	2	1	no	—
130	C₁₆	—	2	1	no	—
131	C₁₆	#13083558	1	2	yes	—
132	C₁₆	#13083558	1	2	yes	—
133	C₂³	#15667946	1	1	yes	characteristic
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	#15667946	1	1	yes	characteristic
139	C₂×C₄	#15667946	1	1	yes	characteristic
140	C₂×C₄	#15667946	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
142	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
143	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	712	4	no	—
147	C₈	#15667948	1	2	yes	—
148	C₈	C₂×#13083558	1	1	yes	characteristic
149	C₈	C₂×#13083558	1	1	yes	characteristic
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#19151428	1	2	yes	—
153	C₂²	C₂×#15667946	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#19151622	1	1	yes	characteristic
159	C₄	C₂×#15667948	1	1	yes	characteristic
160	C₄	#19152030	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	#26019011	1	1	yes	characteristic
163	C₂	#26022036	1	1	yes	characteristic
164	C₂	C₂×#19151428	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×C₈₉	#107
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	#12024
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
2. #26018326	C₂	semidirect
3. #26018324	C₂	semidirect
87. C₈₉	#12024	semidirect

Maximal subgroups

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

subgroup	index
#26019015	2
#26018326	2
#26018324	2
C₂×C₄×C₁₆×C₈₉	2
#12024	89