Subgroups — #36472488

Order 22784.

274 conjugacy classes of subgroups.

The subgroup lattice has 192 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	C₂×#19151424	C₂	1	1	yes	maximal, characteristic
3	#26021037	C₂	1	2	yes	maximal
4	#26018361	C₂	1	2	yes	maximal
5	C₈₉×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667948	C₂²	1	2	yes	—
7	#19151641	—	2	2	no	—
8	C₂×#15667946	C₂²	1	2	yes	—
9	#19151428	C₄	1	4	yes	complemented
10	#19151424	—	2	1	no	—
11	#19151424	—	2	1	no	—
12	#19151366	—	2	2	no	—
13	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	#405×C₈₉	C₂²	1	1	yes	characteristic
16	#346×C₈₉	—	2	1	no	—
17	C₄×C₁₆×C₈₉	—	2	1	no	—
18	C₂×#13083558	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	#15667948	—	2	2	no	—
21	#15667948	C₂×C₄	1	4	yes	—
22	#15667946	—	2	2	no	—
23	#15667946	—	2	2	no	—
24	#15667945	—	4	2	no	—
25	#15667943	—	4	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	—	2	1	no	—
30	M₃₂×C₈₉	Q₈	1	2	yes	—
31	C₂×C₁₆×C₈₉	—	2	1	no	—
32	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₈₉	Q₈	1	2	yes	—
36	#97×C₈₉	—	2	1	no	—
37	C₄×C₈×C₈₉	—	2	1	no	—
38	C₂²×Dic₈₉	—	4	2	no	—
39	#13083558	—	2	4	no	—
40	#13083558	—	4	2	no	—
41	C₂×#10623494	—	4	2	no	—
42	C₄⋊Dic₈₉	—	4	2	no	—
43	C₄⋊Dic₈₉	—	4	2	no	—
44	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₈₉	—	2	1	no	—
46	M₁₆×C₈₉	Q₁₆	1	2	yes	—
47	C₂×C₈×C₈₉	—	2	1	no	—
48	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
53	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
55	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
56	C₂²⋊C₄×C₈₉	—	2	1	no	—
57	C₄²×C₈₉	—	2	1	no	—
58	C₁₆×C₈₉	—	2	2	no	—
59	C₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
60	C₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
61	C₂³×C₈₉	#135	1	1	yes	characteristic
62	C₂×C₄×C₈₉	—	2	1	no	—
63	C₂×C₄×C₈₉	—	2	1	no	—
64	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
70	C₂×Dic₈₉	—	8	2	no	—
71	C₂×Dic₈₉	—	16	2	no	—
72	C₈×C₈₉	—	2	2	no	—
73	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₈₉	#125	1	1	yes	characteristic
77	#10623494	—	4	4	no	—
78	C₂²×C₈₉	—	2	1	no	—
79	C₂²×C₈₉	—	2	1	no	—
80	C₂²×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
81	C₂²×C₈₉	#466	1	1	yes	characteristic
82	C₂²×C₈₉	#499	1	1	yes	characteristic
83	C₄×C₈₉	—	4	1	no	—
84	C₄×C₈₉	—	4	1	no	—
85	C₄×C₈₉	#389	1	1	yes	characteristic
86	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
87	C₄×C₈₉	#467	1	1	yes	characteristic
88	C₄×C₈₉	#500	1	1	yes	characteristic
89	Dic₈₉	—	16	2	no	—
90	#13171	—	89	1	no	maximal, Sylow
91	C₂×C₈₉	—	2	1	no	—
92	C₂×C₈₉	—	2	1	no	—
93	C₂×C₈₉	#1942	1	1	yes	characteristic
94	C₂×C₈₉	#2221	1	1	yes	characteristic
95	C₂×C₈₉	#2222	1	1	yes	characteristic
96	C₂×#368	—	89	1	no	—
97	C₂×#366	—	89	1	no	—
98	#2108	D₈₉	1	1	yes	characteristic
99	#1952	—	89	2	no	—
100	#1387	—	89	2	no	—
101	C₈₉	#13171	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
103	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
104	C₂×#108	—	89	2	no	—
105	#428	—	178	2	no	—
106	C₂×#107	—	89	2	no	—
107	#405	C₂×D₈₉	1	1	yes	characteristic
108	#368	—	89	4	no	—
109	#366	—	178	1	no	—
110	#366	—	178	1	no	—
111	#346	—	2	1	no	—
112	C₄×C₁₆	—	2	1	no	—
113	#343	—	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	D₃₅₆	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	178	2	no	—
119	M₃₂	—	2	1	no	—
120	M₃₂	Dic₁₇₈	1	2	yes	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
123	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
124	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
125	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
126	#108	—	89	4	no	—
127	#108	—	178	2	no	—
128	#107	—	356	2	no	—
129	#107	—	178	2	no	—
130	#107	—	178	2	no	—
131	#106	—	356	2	no	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₂²×C₄	—	356	2	no	—
135	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
136	M₁₆	—	2	1	no	—
137	M₁₆	—	356	2	no	—
138	M₁₆	—	178	4	no	—
139	M₁₆	Dic₃₅₆	1	2	yes	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	356	2	no	—
142	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₈₉	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	#13083574	1	1	yes	characteristic
147	C₂×C₈	#13083574	1	1	yes	characteristic
148	C₄⋊C₄	—	356	2	no	—
149	C₄⋊C₄	—	356	2	no	—
150	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
151	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
152	C₂²⋊C₄	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₁₆	—	2	2	no	—
155	C₁₆	#13083553	1	2	yes	—
156	C₁₆	#13083553	1	2	yes	—
157	C₂³	#15668034	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	1424	2	no	—
161	C₂×C₄	—	712	2	no	—
162	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
163	C₂×C₄	#15667986	1	1	yes	characteristic
164	C₂×C₄	#15668012	1	1	yes	characteristic
165	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
166	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
167	C₂×C₄	#15668034	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	356	4	no	—
170	C₈	C₂×#13083553	1	1	yes	characteristic
171	C₈	C₄×Dic₁₇₈	1	1	yes	characteristic
172	C₈	#15667991	1	1	yes	characteristic
173	C₈	#15668004	1	1	yes	characteristic
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×Dic₃₅₆	1	1	yes	characteristic
177	C₂²	#19151834	1	1	yes	characteristic
178	C₂²	#19152000	1	1	yes	characteristic
179	C₄	—	4	1	no	—
180	C₄	—	4	1	no	—
181	C₄	—	1424	2	no	—
182	C₄	#19151496	1	1	yes	characteristic
183	C₄	C₄×D₇₁₂	1	1	yes	characteristic
184	C₄	#19151839	1	1	yes	characteristic
185	C₄	#19152003	1	1	yes	characteristic
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#26020985	1	1	yes	characteristic
189	C₂	#26022372	1	1	yes	characteristic
190	C₂	#26022377	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151834
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2108	C₂
socle	C₂²×C₈₉	#466
prime radical	C₂³×C₈₉	#135
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2108	D₈₉
hypocenter	C₈₉	#13171
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
9. #19151428	C₄	semidirect
101. C₈₉	#13171	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
C₂×#19151424	2
#26021037	2
#26018361	2
C₈₉×#2108	2
#13171	89