Subgroups — #36441038

Order 22784.

304 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020369	C₂	1	1	yes	maximal, characteristic
2	#26020105	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10623494	—	2	1	no	—
5	C₂×C₄⋊#10623494	C₄	1	2	yes	—
6	C₂×#15667923	—	2	1	no	—
7	#19151353	—	2	1	no	—
8	#19151353	—	2	1	no	—
9	#19151353	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
11	C₂×#97×C₈₉	—	2	1	no	—
12	C₄³×C₈₉	C₂²	1	1	yes	characteristic
13	#336×C₈₉	C₄	1	2	yes	—
14	C₂²×#10623494	—	2	2	no	—
15	C₂²×#10623494	—	2	2	no	—
16	C₄⋊#10623494	—	2	4	no	—
17	C₄⋊#10623494	—	4	2	no	—
18	#15667923	—	4	1	no	—
19	#15667923	—	4	1	no	—
20	C₂²×C₈×C₈₉	—	2	2	no	—
21	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₈₉	D₄	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	C₄⋊C₈×C₈₉	—	2	4	no	—
29	#97×C₈₉	—	4	1	no	—
30	#97×C₈₉	—	4	1	no	—
31	C₂×#10623494	—	2	4	no	—
32	C₂×#10623494	—	4	2	no	—
33	C₂×#10623494	—	4	2	no	—
34	C₂×#10623494	—	4	2	no	—
35	C₂×#10623494	—	4	2	no	—
36	C₂×#10623494	—	4	2	no	—
37	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	—	2	4	no	—
45	C₂×C₈×C₈₉	—	4	2	no	—
46	C₂×C₈×C₈₉	—	4	2	no	—
47	C₄²×C₈₉	—	2	1	no	—
48	C₄²×C₈₉	—	2	1	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	—	4	1	no	—
54	C₄²×C₈₉	D₈	1	1	yes	characteristic
55	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
56	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
57	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
58	C₂³×C₈₉	#95	1	1	yes	characteristic
59	C₂×C₄×C₈₉	—	2	1	no	—
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	—	2	1	no	—
71	C₂×C₄×C₈₉	—	2	1	no	—
72	C₂×C₄×C₈₉	—	2	1	no	—
73	C₂×C₄×C₈₉	—	2	1	no	—
74	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
75	C₂×C₄×C₈₉	#101	1	1	yes	characteristic
76	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
78	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
79	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#104	1	2	yes	—
81	C₂×C₄×C₈₉	#104	1	2	yes	—
82	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
83	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
84	C₈×C₈₉	—	4	4	no	—
85	#10623494	—	4	4	no	—
86	#10623494	—	8	2	no	—
87	C₂²×C₈₉	#338	1	1	yes	characteristic
88	C₂²×C₈₉	#339	1	1	yes	characteristic
89	C₂²×C₈₉	#339	1	1	yes	characteristic
90	C₂²×C₈₉	#340	1	1	yes	characteristic
91	C₂²×C₈₉	#343	1	1	yes	characteristic
92	C₂²×C₈₉	#344	1	1	yes	characteristic
93	C₂²×C₈₉	#344	1	1	yes	characteristic
94	C₄×C₈₉	—	2	1	no	—
95	C₄×C₈₉	—	2	1	no	—
96	C₄×C₈₉	—	2	1	no	—
97	C₄×C₈₉	—	2	1	no	—
98	C₄×C₈₉	—	2	1	no	—
99	C₄×C₈₉	—	2	1	no	—
100	C₄×C₈₉	—	4	1	no	—
101	C₄×C₈₉	—	4	1	no	—
102	C₄×C₈₉	—	4	1	no	—
103	C₄×C₈₉	#329	1	2	yes	—
104	C₄×C₈₉	#330	1	2	yes	—
105	#7049	—	89	1	no	maximal, Sylow
106	C₂×C₈₉	#1291	1	1	yes	characteristic
107	C₂×C₈₉	#1291	1	1	yes	characteristic
108	C₂×C₈₉	#1292	1	1	yes	characteristic
109	C₂×C₈₉	#1295	1	1	yes	characteristic
110	C₂×C₈₉	#1303	1	1	yes	characteristic
111	C₂×C₈₉	#1303	1	1	yes	characteristic
112	C₂×C₈₉	#1307	1	1	yes	characteristic
113	#1831	—	89	1	no	—
114	#1767	—	89	1	no	—
115	#1767	D₈₉	1	1	yes	characteristic
116	C₈₉	#7049	1	1	yes	characteristic, complemented, Sylow
117	C₂×C₄⋊C₈	—	2	1	no	—
118	C₂×C₄⋊C₈	—	89	2	no	—
119	C₂×C₄⋊C₈	—	178	1	no	—
120	C₂×#97	—	2	1	no	—
121	C₂×#97	—	178	1	no	—
122	C₄³	C₂×D₈₉	1	1	yes	characteristic
123	#336	—	89	2	no	—
124	#336	—	178	1	no	—
125	#336	—	178	1	no	—
126	#336	Dic₈₉	1	2	yes	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	178	2	no	—
129	C₂²×C₈	—	178	2	no	—
130	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
132	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
133	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
134	C₂×C₄²	C₂×Dic₈₉	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	—	356	2	no	—
139	C₄⋊C₈	—	178	4	no	—
140	#97	—	4	1	no	—
141	#97	—	4	1	no	—
142	#97	—	356	1	no	—
143	#97	—	356	1	no	—
144	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
145	C₂²×C₄	#13083551	1	1	yes	characteristic
146	C₂²×C₄	#13083551	1	1	yes	characteristic
147	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
148	C₂²×C₄	#13083553	1	1	yes	characteristic
149	C₂²×C₄	#13083553	1	1	yes	characteristic
150	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	356	2	no	—
155	C₂×C₈	—	356	2	no	—
156	C₂×C₈	—	356	2	no	—
157	C₂×C₈	—	356	2	no	—
158	C₂×C₈	—	356	2	no	—
159	C₂×C₈	—	178	4	no	—
160	C₄²	—	2	1	no	—
161	C₄²	—	2	1	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	4	1	no	—
165	C₄²	—	4	1	no	—
166	C₄²	—	4	1	no	—
167	C₄²	#13083559	1	1	yes	characteristic
168	C₄²	#13083561	1	1	yes	characteristic
169	C₄²	#13083562	1	1	yes	characteristic
170	C₄²	#13083564	1	1	yes	characteristic
171	C₂³	#15667919	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	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	#15667928	1	1	yes	characteristic
188	C₂×C₄	#15667930	1	1	yes	characteristic
189	C₂×C₄	#15667932	1	1	yes	characteristic
190	C₂×C₄	#15667933	1	1	yes	characteristic
191	C₂×C₄	#15667935	1	1	yes	characteristic
192	C₂×C₄	#15667936	1	1	yes	characteristic
193	C₂×C₄	#15667938	1	2	yes	—
194	C₂×C₄	#15667940	1	2	yes	—
195	C₂×C₄	#15667943	1	1	yes	characteristic
196	C₂×C₄	#15667945	1	1	yes	characteristic
197	C₈	—	4	4	no	—
198	C₈	—	356	4	no	—
199	C₈	—	712	2	no	—
200	C₂²	#19151356	1	1	yes	characteristic
201	C₂²	#19151358	1	1	yes	characteristic
202	C₂²	#19151359	1	1	yes	characteristic
203	C₂²	#19151360	1	1	yes	characteristic
204	C₂²	#19151366	1	1	yes	characteristic
205	C₂²	#19151368	1	1	yes	characteristic
206	C₂²	#19151369	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	4	1	no	—
214	C₄	—	4	1	no	—
215	C₄	—	4	1	no	—
216	C₄	#19151337	1	2	yes	—
217	C₄	#19151339	1	2	yes	—
218	C₂	#26018124	1	1	yes	characteristic
219	C₂	#26018125	1	1	yes	characteristic
220	C₂	#26018126	1	1	yes	characteristic
221	C₂	#26018134	1	1	yes	characteristic
222	C₂	#26018153	1	1	yes	characteristic
223	C₂	#26018155	1	1	yes	characteristic
224	C₂	#26018162	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667919
commutator	C₂²×C₄×C₈₉	C₄²
Frattini	C₄³	C₂×D₈₉
Fitting	C₈₉×#1767	C₂
socle, prime radical	C₂³×C₈₉	#95
cosocle	C₄³×C₈₉	C₂²
hypercenter	#1767	D₈₉
hypocenter	C₈₉	#7049
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
116. C₈₉	#7049	semidirect

Maximal subgroups

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

subgroup	index
#26020369	2
#26020105	2
C₈₉×#1767	2
#7049	89