Subgroups — #36448197

Order 22784.

454 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	#26020411	C₂	1	2	yes	maximal
2	#26020409	C₂	1	1	yes	maximal, characteristic
3	#26020379	C₂	1	2	yes	maximal
4	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667945	—	2	4	no	—
7	C₂×#15667944	—	2	2	no	—
8	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
10	#19151468	C₂²	1	2	yes	—
11	#19151360	—	2	4	no	—
12	C₈×#10623494	C₄	1	4	yes	complemented
13	C₂×#106×C₈₉	—	2	2	no	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	#384×C₈₉	C₂²	1	2	yes	—
16	C₂²×#10623494	D₄	1	2	yes	—
17	C₂²×#10623494	D₄	1	2	yes	—
18	C₂×C₄⋊Dic₈₉	—	2	2	no	—
19	C₂×C₄⋊Dic₈₉	—	2	2	no	—
20	C₂×C₄⋊Dic₈₉	—	2	2	no	—
21	#15667945	—	4	8	no	—
22	#15667944	—	4	4	no	—
23	C₄×#10623494	C₂×C₄	1	2	yes	—
24	C₄×#10623494	C₂×C₄	1	2	yes	—
25	C₄×#10623494	Q₈	1	2	yes	—
26	C₄×#10623494	Q₈	1	2	yes	—
27	C₂²×C₈×C₈₉	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	#106×C₈₉	—	4	4	no	—
33	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₈₉	Q₈	1	2	yes	—
35	C₂²×Dic₈₉	—	4	2	no	—
36	C₂×#10623494	—	2	4	no	—
37	C₂×#10623494	C₄⋊C₄	1	4	yes	—
38	C₂×#10623494	C₄⋊C₄	1	4	yes	—
39	C₂×#10623494	SD₁₆	1	8	yes	—
40	C₄⋊Dic₈₉	—	4	4	no	—
41	C₄⋊Dic₈₉	—	4	4	no	—
42	C₄⋊Dic₈₉	—	4	4	no	—
43	C₂²×C₄×C₈₉	—	4	2	no	—
44	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
48	C₂×C₈×C₈₉	D₈	1	4	yes	—
49	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
50	C₄⋊C₄×C₈₉	—	4	4	no	—
51	C₄⋊C₄×C₈₉	—	4	4	no	—
52	C₄⋊C₄×C₈₉	—	4	4	no	—
53	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
57	C₂³×C₈₉	#127	1	1	yes	characteristic
58	C₂×C₄×C₈₉	—	8	2	no	—
59	C₂×C₄×C₈₉	—	8	2	no	—
60	C₂×C₄×C₈₉	—	8	2	no	—
61	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
68	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
69	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
70	C₂×C₄×C₈₉	#135	1	4	yes	—
71	C₂×Dic₈₉	—	8	2	no	—
72	C₂×Dic₈₉	—	8	2	no	—
73	C₂×Dic₈₉	—	8	2	no	—
74	C₈×C₈₉	#107	1	8	yes	complemented
75	#10623494	—	2	4	no	—
76	#10623494	#106	1	8	yes	complemented
77	C₂²×C₈₉	#384	1	1	yes	characteristic
78	C₂²×C₈₉	#492	1	2	yes	—
79	C₂²×C₈₉	#493	1	1	yes	characteristic
80	C₂²×C₈₉	#494	1	1	yes	characteristic
81	C₂²×C₈₉	#495	1	2	yes	—
82	C₄×C₈₉	—	16	2	no	—
83	C₄×C₈₉	C₂×#106	1	2	yes	—
84	C₄×C₈₉	C₂×#107	1	2	yes	—
85	C₄×C₈₉	#427	1	2	yes	—
86	C₄×C₈₉	#498	1	2	yes	—
87	C₄×C₈₉	#499	1	2	yes	—
88	C₄×C₈₉	#500	1	2	yes	—
89	Dic₈₉	—	16	2	no	—
90	#8814	—	89	1	no	maximal, Sylow
91	C₂×C₈₉	#1706	1	2	yes	—
92	C₂×C₈₉	#1707	1	2	yes	—
93	C₂×C₈₉	#1844	1	1	yes	characteristic
94	C₂×C₈₉	#1845	1	1	yes	characteristic
95	C₂×C₈₉	#1846	1	1	yes	characteristic
96	#1845	—	89	2	no	—
97	#1844	—	89	1	no	—
98	#1844	D₈₉	1	1	yes	characteristic
99	#1834	—	89	2	no	—
100	C₂×C₈²	—	89	1	no	—
101	C₈₉	#8814	1	1	yes	characteristic, complemented, Sylow
102	C₂×#107	—	178	4	no	—
103	C₂×#106	—	2	2	no	—
104	C₂×#106	—	178	2	no	—
105	C₂×C₄×C₈	—	89	1	no	—
106	C₂×C₄×C₈	—	89	1	no	—
107	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
108	#384	—	89	2	no	—
109	#384	C₂×D₈₉	1	2	yes	—
110	#340	—	178	4	no	—
111	C₈²	—	89	4	no	—
112	C₂²×C₈	—	89	2	no	—
113	C₂²×C₈	—	89	2	no	—
114	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	178	2	no	—
119	C₂×C₄⋊C₄	—	178	2	no	—
120	C₂×C₄⋊C₄	—	178	2	no	—
121	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
122	#107	—	356	8	no	—
123	#106	—	4	4	no	—
124	#106	—	356	4	no	—
125	C₄×C₈	—	89	2	no	—
126	C₄×C₈	—	89	2	no	—
127	C₄×C₈	—	89	2	no	—
128	C₄×C₈	—	89	2	no	—
129	C₄×C₈	Dic₁₇₈	1	2	yes	—
130	C₄×C₈	C₄×D₈₉	1	2	yes	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	356	2	no	—
133	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
134	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
136	C₂×C₈	—	89	8	no	—
137	C₂×C₈	—	89	4	no	—
138	C₂×C₈	—	89	4	no	—
139	C₂×C₈	—	178	4	no	—
140	C₂×C₈	#13083553	1	4	yes	—
141	C₂×C₈	#13083559	1	4	yes	—
142	C₂×C₈	#13083564	1	4	yes	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	356	4	no	—
147	C₄⋊C₄	—	356	4	no	—
148	C₄⋊C₄	—	356	4	no	—
149	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
150	C₄²	Q₈×D₈₉	1	1	yes	characteristic
151	C₄²	Q₈×D₈₉	1	1	yes	characteristic
152	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
153	C₂³	#15668008	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	712	2	no	—
158	C₂×C₄	—	712	2	no	—
159	C₂×C₄	—	712	2	no	—
160	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
161	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
162	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
163	C₂×C₄	#15668010	1	1	yes	characteristic
164	C₂×C₄	#15668011	1	1	yes	characteristic
165	C₂×C₄	#15668011	1	1	yes	characteristic
166	C₂×C₄	C₂×#13083559	1	2	yes	—
167	C₂×C₄	SD₁₆×D₈₉	1	4	yes	—
168	C₂×C₄	C₂×#13083564	1	2	yes	—
169	C₂×C₄	#15668033	1	4	yes	—
170	C₈	—	89	8	no	—
171	C₈	—	178	4	no	—
172	C₈	#15667945	1	8	yes	complemented
173	C₂²	#19151467	1	1	yes	characteristic
174	C₂²	#19151969	1	2	yes	—
175	C₂²	#19151974	1	1	yes	characteristic
176	C₂²	#19151977	1	1	yes	characteristic
177	C₂²	#19151979	1	2	yes	—
178	C₄	—	16	2	no	—
179	C₄	—	1424	2	no	—
180	C₄	#106×D₈₉	1	2	yes	—
181	C₄	C₂×#15667945	1	2	yes	—
182	C₄	#19151633	1	2	yes	—
183	C₄	#19151995	1	2	yes	—
184	C₄	#19151998	1	2	yes	—
185	C₄	#19152001	1	2	yes	—
186	C₂	#26019842	1	2	yes	—
187	C₂	#26019847	1	2	yes	—
188	C₂	#26020407	1	1	yes	characteristic
189	C₂	#26020411	1	1	yes	characteristic
190	C₂	#26020413	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15668008
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1844	C₂
socle, prime radical	C₂³×C₈₉	#127
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1844	D₈₉
hypocenter	C₈₉	#8814
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
12. C₈×#10623494	C₄	semidirect
74. C₈×C₈₉	#107	semidirect
76. #10623494	#106	semidirect
101. C₈₉	#8814	semidirect
172. C₈	#15667945	semidirect

Maximal subgroups

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

subgroup	index
#26020411	2
#26020409	2
#26020379	2
C₂×C₈×#10623494	2
C₈₉×#1844	2
#8814	89