Subgroups — #36448098

Order 22784.

362 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020404	C₂	1	1	yes	maximal, characteristic
2	#26019998	C₂	1	1	yes	maximal, characteristic
3	#26018086	C₂	1	4	yes	maximal
4	C₈₉×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10623494	—	2	1	no	—
6	C₂×C₄⋊#10623494	—	2	1	no	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	C₂×C₄×#10623494	C₂²	1	2	yes	—
9	#19151353	C₄	1	2	yes	—
10	#19151353	C₄	1	2	yes	—
11	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
12	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
13	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
14	C₄³×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×#10623494	—	2	2	no	—
16	C₂²×#10623494	C₂×C₄	1	4	yes	—
17	C₄⋊#10623494	—	2	4	no	—
18	C₄⋊#10623494	—	4	2	no	—
19	C₄⋊#10623494	—	4	2	no	—
20	C₄×#10623494	—	2	2	no	—
21	C₄×#10623494	—	2	2	no	—
22	C₂²×C₈×C₈₉	—	2	2	no	—
23	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
28	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₈₉	—	2	4	no	—
31	C₄⋊C₈×C₈₉	—	4	2	no	—
32	C₄⋊C₈×C₈₉	—	4	2	no	—
33	C₂×#10623494	—	2	4	no	—
34	C₂×#10623494	—	2	4	no	—
35	C₂×#10623494	—	2	4	no	—
36	C₂×#10623494	—	4	2	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₂×#10623494	—	4	2	no	—
39	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₈₉	—	4	2	no	—
47	C₂×C₈×C₈₉	—	4	2	no	—
48	C₂×C₈×C₈₉	—	4	2	no	—
49	C₄²×C₈₉	—	2	2	no	—
50	C₄²×C₈₉	—	2	2	no	—
51	C₄²×C₈₉	—	4	1	no	—
52	C₄²×C₈₉	—	4	1	no	—
53	C₄²×C₈₉	D₈	1	2	yes	—
54	C₄²×C₈₉	SD₁₆	1	2	yes	—
55	C₄²×C₈₉	SD₁₆	1	2	yes	—
56	C₄²×C₈₉	Q₁₆	1	2	yes	—
57	C₄²×C₈₉	#41	1	2	yes	—
58	C₄²×C₈₉	#41	1	2	yes	—
59	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₈₉	—	2	1	no	—
61	C₂×C₄×C₈₉	—	2	1	no	—
62	C₂×C₄×C₈₉	—	2	1	no	—
63	C₂×C₄×C₈₉	—	2	1	no	—
64	C₂×C₄×C₈₉	—	2	1	no	—
65	C₂×C₄×C₈₉	—	2	1	no	—
66	C₂×C₄×C₈₉	—	2	1	no	—
67	C₂×C₄×C₈₉	—	2	1	no	—
68	C₂×C₄×C₈₉	—	2	1	no	—
69	C₂×C₄×C₈₉	—	2	1	no	—
70	C₂×C₄×C₈₉	#106	1	2	yes	—
71	C₂×C₄×C₈₉	#106	1	2	yes	—
72	C₂×C₄×C₈₉	#107	1	2	yes	—
73	C₂×C₄×C₈₉	#107	1	2	yes	—
74	C₂×C₄×C₈₉	#117	1	2	yes	—
75	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#125	1	2	yes	—
78	C₂×C₄×C₈₉	#131	1	2	yes	—
79	C₂×C₄×C₈₉	#131	1	2	yes	—
80	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
84	C₈×C₈₉	—	8	2	no	—
85	#10623494	—	4	4	no	—
86	#10623494	—	8	2	no	—
87	C₂²×C₈₉	#382	1	1	yes	characteristic
88	C₂²×C₈₉	#424	1	1	yes	characteristic
89	C₂²×C₈₉	#424	1	1	yes	characteristic
90	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
91	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
92	C₂²×C₈₉	#430	1	1	yes	characteristic
93	C₂²×C₈₉	#430	1	1	yes	characteristic
94	C₄×C₈₉	—	2	2	no	—
95	C₄×C₈₉	—	2	2	no	—
96	C₄×C₈₉	—	2	2	no	—
97	C₄×C₈₉	—	2	2	no	—
98	C₄×C₈₉	—	2	2	no	—
99	C₄×C₈₉	#433	1	4	yes	—
100	C₄×C₈₉	#486	1	1	yes	characteristic
101	C₄×C₈₉	#486	1	1	yes	characteristic
102	C₄×C₈₉	#487	1	1	yes	characteristic
103	C₄×C₈₉	#487	1	1	yes	characteristic
104	#8787	—	89	1	no	maximal, Sylow
105	C₂×C₈₉	#1570	1	1	yes	characteristic
106	C₂×C₈₉	#1570	1	1	yes	characteristic
107	C₂×C₈₉	#1571	1	1	yes	characteristic
108	C₂×C₈₉	#1571	1	1	yes	characteristic
109	C₂×C₈₉	#1832	1	1	yes	characteristic
110	C₂×C₈₉	#1834	1	1	yes	characteristic
111	C₂×C₈₉	#1838	1	1	yes	characteristic
112	#1842	—	89	1	no	—
113	#1842	D₈₉	1	1	yes	characteristic
114	#1744	—	89	1	no	—
115	#1277	—	89	4	no	—
116	C₈₉	#8787	1	1	yes	characteristic, complemented, Sylow
117	C₂×C₄⋊C₈	—	2	1	no	—
118	C₂×C₄⋊C₈	—	2	1	no	—
119	C₂×C₄⋊C₈	—	89	2	no	—
120	C₂×C₄⋊C₈	—	178	1	no	—
121	C₂×C₄⋊C₈	—	178	1	no	—
122	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
123	C₂×C₄×C₈	—	89	2	no	—
124	C₄³	C₂×D₈₉	1	1	yes	characteristic
125	#336	—	89	2	no	—
126	#336	—	89	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	89	4	no	—
129	C₂²×C₈	—	178	2	no	—
130	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
131	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
132	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
133	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
135	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
136	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
137	C₄⋊C₈	—	2	4	no	—
138	C₄⋊C₈	—	4	2	no	—
139	C₄⋊C₈	—	4	2	no	—
140	C₄⋊C₈	—	356	2	no	—
141	C₄⋊C₈	—	356	2	no	—
142	C₄⋊C₈	—	178	4	no	—
143	C₄×C₈	—	178	2	no	—
144	C₄×C₈	—	178	2	no	—
145	C₂²×C₄	#13083553	1	1	yes	characteristic
146	C₂²×C₄	#13083553	1	1	yes	characteristic
147	C₂²×C₄	#13083553	1	1	yes	characteristic
148	C₂²×C₄	#13083553	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
151	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	356	2	no	—
156	C₂×C₈	—	356	2	no	—
157	C₂×C₈	—	356	2	no	—
158	C₂×C₈	—	178	4	no	—
159	C₂×C₈	—	178	4	no	—
160	C₂×C₈	—	178	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	4	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	#13083559	1	2	yes	—
166	C₄²	#13083561	1	2	yes	—
167	C₄²	#13083562	1	2	yes	—
168	C₄²	#13083564	1	2	yes	—
169	C₄²	#13083573	1	2	yes	—
170	C₄²	#13083575	1	2	yes	—
171	C₂³	C₂×#13083553	1	1	yes	characteristic
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	#15667943	1	2	yes	—
183	C₂×C₄	#15667943	1	2	yes	—
184	C₂×C₄	#15667945	1	2	yes	—
185	C₂×C₄	#15667945	1	2	yes	—
186	C₂×C₄	#15667970	1	2	yes	—
187	C₂×C₄	#15667999	1	1	yes	characteristic
188	C₂×C₄	#15668001	1	1	yes	characteristic
189	C₂×C₄	#15668003	1	2	yes	—
190	C₂×C₄	#15668018	1	2	yes	—
191	C₂×C₄	#15668020	1	2	yes	—
192	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
193	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
194	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
195	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
196	C₈	—	8	2	no	—
197	C₈	—	356	4	no	—
198	C₈	—	712	2	no	—
199	C₂²	#19151463	1	1	yes	characteristic
200	C₂²	#19151624	1	1	yes	characteristic
201	C₂²	#19151626	1	1	yes	characteristic
202	C₂²	C₂×#15667943	1	1	yes	characteristic
203	C₂²	C₂×#15667945	1	1	yes	characteristic
204	C₂²	#19151646	1	1	yes	characteristic
205	C₂²	#19151646	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	#19151658	1	4	yes	—
212	C₄	#19151944	1	1	yes	characteristic
213	C₄	#19151946	1	1	yes	characteristic
214	C₄	#19151948	1	1	yes	characteristic
215	C₄	#19151950	1	1	yes	characteristic
216	C₂	#26019022	1	1	yes	characteristic
217	C₂	#26019024	1	1	yes	characteristic
218	C₂	#26019026	1	1	yes	characteristic
219	C₂	#26019028	1	1	yes	characteristic
220	C₂	#26020371	1	1	yes	characteristic
221	C₂	#26020379	1	1	yes	characteristic
222	C₂	#26020392	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1842	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1842	D₈₉
hypocenter	C₈₉	#8787
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
116. C₈₉	#8787	semidirect

Maximal subgroups

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

subgroup	index
#26020404	2
#26019998	2
#26018086	2
C₈₉×#1842	2
#8787	89