Subgroups — #36470884

Order 22784.

266 conjugacy classes of subgroups.

The subgroup lattice has 188 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	1	yes	maximal, characteristic
2	#26022036	C₂	1	1	yes	maximal, characteristic
3	#26022025	C₂	1	2	yes	maximal
4	#26020376	C₂	1	2	yes	maximal
5	C₈₉×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667948	C₂²	1	2	yes	—
7	#19151636	C₂²	1	2	yes	—
8	#19151428	—	2	1	no	—
9	#19151428	—	2	1	no	—
10	#19151426	—	2	1	no	—
11	#19151424	—	2	1	no	—
12	#19151421	—	2	4	no	—
13	#19151369	—	2	2	no	—
14	#19151354	—	2	2	no	—
15	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	#405×C₈₉	C₂²	1	1	yes	characteristic
18	#346×C₈₉	C₄	1	2	yes	complemented
19	C₄×C₁₆×C₈₉	C₄	1	2	yes	complemented
20	C₂×#13083558	—	2	2	no	—
21	#15667971	—	2	2	no	—
22	#15667948	—	2	2	no	—
23	#15667948	—	2	2	no	—
24	#15667946	—	2	2	no	—
25	#15667944	—	2	2	no	—
26	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
28	#117×C₈₉	D₄	1	1	yes	characteristic
29	M₃₂×C₈₉	D₄	1	2	yes	—
30	M₃₂×C₈₉	Q₈	1	2	yes	—
31	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
37	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
38	#97×C₈₉	Q₈	1	2	yes	—
39	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
40	#13083558	—	4	2	no	—
41	#13083558	—	4	2	no	—
42	C₂×#10623494	—	4	2	no	—
43	C₄⋊Dic₈₉	—	4	2	no	—
44	C₂²⋊Dic₈₉	—	4	2	no	—
45	C₄×Dic₈₉	—	4	2	no	—
46	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₈₉	—	2	1	no	—
48	M₁₆×C₈₉	—	2	1	no	—
49	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
56	C₄⋊C₄×C₈₉	—	2	1	no	—
57	C₂²⋊C₄×C₈₉	—	2	1	no	—
58	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
59	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
60	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
61	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
62	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
63	C₁₆×C₈₉	C₄⋊C₄	1	4	yes	—
64	C₂³×C₈₉	#135	1	1	yes	characteristic
65	C₂×C₄×C₈₉	—	2	2	no	—
66	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
70	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
71	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
72	C₂×Dic₈₉	—	8	2	no	—
73	C₈×C₈₉	—	2	2	no	—
74	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
77	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
78	#10623494	—	8	2	no	—
79	C₂²×C₈₉	—	4	1	no	—
80	C₂²×C₈₉	#427	1	1	yes	characteristic
81	C₂²×C₈₉	#495	1	1	yes	characteristic
82	C₂²×C₈₉	#499	1	1	yes	characteristic
83	C₄×C₈₉	—	4	2	no	—
84	C₄×C₈₉	#384	1	1	yes	characteristic
85	C₄×C₈₉	#427	1	1	yes	characteristic
86	C₄×C₈₉	#495	1	1	yes	characteristic
87	C₄×C₈₉	#499	1	1	yes	characteristic
88	Dic₈₉	—	16	2	no	—
89	#12911	—	89	1	no	maximal, Sylow
90	C₂×C₈₉	—	2	1	no	—
91	C₂×C₈₉	—	2	1	no	—
92	C₂×C₈₉	—	2	1	no	—
93	C₂×C₈₉	#1846	1	1	yes	characteristic
94	C₂×#368	—	89	1	no	—
95	#2158	—	89	1	no	—
96	#2155	—	89	2	no	—
97	#2108	D₈₉	1	1	yes	characteristic
98	#1833	—	89	2	no	—
99	C₈₉	#12911	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
101	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
102	C₂×#108	—	89	2	no	—
103	#427	—	89	2	no	—
104	#405	C₂×D₈₉	1	1	yes	characteristic
105	#368	—	178	1	no	—
106	#368	—	178	1	no	—
107	#367	—	178	1	no	—
108	#366	—	178	1	no	—
109	#365	—	178	4	no	—
110	#346	Dic₈₉	1	2	yes	complemented
111	C₄×C₁₆	Dic₈₉	1	2	yes	complemented
112	#344	—	178	2	no	—
113	H(3,ℤ/4)	—	178	2	no	—
114	C₂×M₁₆	—	178	2	no	—
115	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
117	#117	—	178	2	no	—
118	#117	D₃₅₆	1	1	yes	characteristic
119	M₃₂	Dic₁₇₈	1	2	yes	—
120	M₃₂	D₃₅₆	1	2	yes	—
121	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
122	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
123	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
124	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
125	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
126	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
127	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
128	#108	—	178	2	no	—
129	#108	—	178	2	no	—
130	#107	—	178	2	no	—
131	#106	—	178	2	no	—
132	#97	Dic₁₇₈	1	2	yes	—
133	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
134	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
135	M₁₆	—	2	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	356	2	no	—
138	M₁₆	—	356	2	no	—
139	C₂×C₈	—	356	2	no	—
140	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
141	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
142	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
143	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
144	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
146	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
147	C₄⋊C₄	—	2	1	no	—
148	C₄⋊C₄	—	356	2	no	—
149	C₂²⋊C₄	—	2	1	no	—
150	C₂²⋊C₄	—	356	2	no	—
151	C₄²	—	356	2	no	—
152	C₄²	C₄⋊Dic₈₉	1	2	yes	—
153	C₁₆	C₄⋊Dic₈₉	1	1	yes	characteristic
154	C₁₆	C₄⋊Dic₈₉	1	1	yes	characteristic
155	C₁₆	C₄⋊Dic₈₉	1	1	yes	characteristic
156	C₁₆	C₄⋊Dic₈₉	1	1	yes	characteristic
157	C₁₆	C₄⋊Dic₈₉	1	4	yes	—
158	C₂³	#15668034	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	712	2	no	—
161	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
162	C₂×C₄	#15668009	1	1	yes	characteristic
163	C₂×C₄	#15668012	1	1	yes	characteristic
164	C₂×C₄	#15668034	1	1	yes	characteristic
165	C₂×C₄	#15668034	1	1	yes	characteristic
166	C₂×C₄	#15668034	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	712	2	no	—
169	C₈	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
170	C₈	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
171	C₈	#15668012	1	1	yes	characteristic
172	C₈	#15668012	1	1	yes	characteristic
173	C₂²	—	4	1	no	—
174	C₂²	#19151636	1	1	yes	characteristic
175	C₂²	#19151982	1	1	yes	characteristic
176	C₂²	#19152000	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	1424	2	no	—
179	C₄	#19151468	1	1	yes	characteristic
180	C₄	#19151636	1	1	yes	characteristic
181	C₄	#19151982	1	1	yes	characteristic
182	C₄	#19152000	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#26020416	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151636
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2108	C₂
socle	C₂×C₈₉	#1846
prime radical	C₂³×C₈₉	#135
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2108	D₈₉
hypocenter	C₈₉	#12911
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
18. #346×C₈₉	C₄	semidirect
19. C₄×C₁₆×C₈₉	C₄	semidirect
99. C₈₉	#12911	semidirect
110. #346	Dic₈₉	semidirect
111. C₄×C₁₆	Dic₈₉	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
#26022036	2
#26022025	2
#26020376	2
C₈₉×#2108	2
#12911	89