Subgroups — #36562779

Order 22784.

618 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020976	C₂	1	2	yes	maximal
2	#26020818	C₂	1	4	yes	maximal
3	#26020372	C₂	1	1	yes	maximal, characteristic
4	#26020127	C₂	1	4	yes	maximal
5	#26020121	C₂	1	2	yes	maximal
6	#26019939	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#2402	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
10	C₂×#15667923	C₂²	1	4	yes	—
11	C₂×#15667923	C₂²	1	1	yes	characteristic
12	C₂×#15667923	C₂²	1	2	yes	—
13	C₂×C₄×#10623494	C₂²	1	2	yes	—
14	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
15	#19151353	C₂²	1	4	yes	—
16	#19151353	C₂²	1	4	yes	—
17	#19151353	C₂²	1	4	yes	—
18	#400×C₈₉	C₄	1	4	yes	—
19	#398×C₈₉	C₄	1	4	yes	—
20	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
22	#376×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×#10623494	C₂³	1	4	yes	—
24	C₂²×#10623494	C₂³	1	1	yes	characteristic
25	C₂²×#10623494	C₂³	1	2	yes	—
26	C₂²×#10623494	C₂³	1	1	yes	characteristic
27	C₄⋊#10623494	—	2	8	no	—
28	C₄⋊#10623494	—	2	4	no	—
29	#15667923	—	2	4	no	—
30	#15667923	—	2	8	no	—
31	#15667923	—	2	2	no	—
32	C₄×#10623494	—	2	4	no	—
33	C₄×#10623494	Q₈	1	4	yes	—
34	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
38	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	#95×C₈₉	C₂×C₄	1	4	yes	—
41	#95×C₈₉	C₂×C₄	1	2	yes	—
42	#95×C₈₉	C₂×C₄	1	4	yes	—
43	#95×C₈₉	C₂×C₄	1	4	yes	—
44	#95×C₈₉	C₂×C₄	1	2	yes	—
45	C₂×#10623494	—	2	2	no	—
46	C₂×#10623494	—	2	2	no	—
47	C₂×#10623494	—	2	2	no	—
48	C₂×#10623494	—	2	4	no	—
49	C₂×#10623494	—	2	4	no	—
50	C₂×#10623494	—	2	4	no	—
51	C₂×#10623494	C₂×Q₈	1	2	yes	—
52	C₂×#10623494	C₂×Q₈	1	2	yes	—
53	C₂×#10623494	#41	1	4	yes	—
54	C₂×#10623494	#41	1	4	yes	—
55	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
60	C₄⋊C₄×C₈₉	—	4	4	no	—
61	C₄⋊C₄×C₈₉	—	4	4	no	—
62	C₄⋊C₄×C₈₉	—	4	4	no	—
63	C₄²×C₈₉	—	2	2	no	—
64	C₄²×C₈₉	—	2	4	no	—
65	C₄²×C₈₉	—	2	2	no	—
66	C₄²×C₈₉	—	2	4	no	—
67	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
68	C₄²×C₈₉	#41	1	2	yes	—
69	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₉	—	2	4	no	—
71	C₂×C₄×C₈₉	—	2	4	no	—
72	C₂×C₄×C₈₉	—	2	4	no	—
73	C₂×C₄×C₈₉	—	2	4	no	—
74	C₂×C₄×C₈₉	—	2	4	no	—
75	C₂×C₄×C₈₉	—	2	4	no	—
76	C₂×C₄×C₈₉	—	2	4	no	—
77	C₂×C₄×C₈₉	—	2	4	no	—
78	C₂×C₄×C₈₉	—	2	4	no	—
79	C₂×C₄×C₈₉	—	2	4	no	—
80	C₂×C₄×C₈₉	C₄×Q₈	1	4	yes	—
81	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
83	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
84	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
85	#10623494	—	2	4	no	—
86	#10623494	—	2	2	no	—
87	#10623494	—	4	4	no	—
88	#10623494	#125	1	4	yes	—
89	C₂²×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
90	C₂²×C₈₉	C₄×#41	1	2	yes	—
91	C₂²×C₈₉	#568	1	4	yes	—
92	C₄×C₈₉	—	2	4	no	—
93	C₄×C₈₉	—	4	4	no	—
94	C₄×C₈₉	—	4	4	no	—
95	C₄×C₈₉	—	4	4	no	—
96	C₄×C₈₉	C₂×#125	1	1	yes	characteristic
97	C₄×C₈₉	#529	1	2	yes	—
98	C₄×C₈₉	#533	1	1	yes	characteristic
99	#21677	—	89	1	no	maximal, Sylow
100	C₂×C₈₉	C₄×#125	1	1	yes	characteristic
101	C₂×C₈₉	#2961	1	2	yes	—
102	C₂×C₈₉	#2967	1	4	yes	—
103	#2402	D₈₉	1	1	yes	characteristic
104	#1940	—	89	2	no	—
105	#1917	—	89	4	no	—
106	#1832	—	89	1	no	—
107	#1771	—	89	4	no	—
108	#1770	—	89	2	no	—
109	#1726	—	89	1	no	—
110	C₈₉	#21677	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₄⋊C₈	—	89	2	no	—
112	C₂×C₄⋊C₈	—	89	4	no	—
113	C₂×#97	—	89	2	no	—
114	C₂×#97	—	89	1	no	—
115	C₂×#97	—	89	4	no	—
116	C₂×C₄×C₈	—	89	1	no	—
117	C₂×C₄×C₈	—	89	2	no	—
118	#400	Dic₈₉	1	4	yes	—
119	#398	Dic₈₉	1	4	yes	—
120	C₄×C₄⋊C₄	C₂×D₈₉	1	4	yes	—
121	C₄×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
122	#376	C₂×D₈₉	1	1	yes	characteristic
123	#336	—	89	4	no	—
124	#336	—	89	4	no	—
125	#336	—	89	4	no	—
126	C₂²×C₈	—	89	1	no	—
127	C₂²×C₈	—	89	1	no	—
128	C₂²×C₈	—	89	2	no	—
129	C₂²×C₈	—	89	4	no	—
130	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
131	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
132	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
133	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
134	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
135	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
136	C₄⋊C₈	—	178	8	no	—
137	C₄⋊C₈	—	178	4	no	—
138	#97	—	178	8	no	—
139	#97	—	178	4	no	—
140	#97	—	178	2	no	—
141	C₄×C₈	—	89	4	no	—
142	C₄×C₈	—	178	4	no	—
143	#95	C₂×Dic₈₉	1	4	yes	—
144	#95	C₂×Dic₈₉	1	2	yes	—
145	#95	C₂×Dic₈₉	1	4	yes	—
146	#95	C₂×Dic₈₉	1	4	yes	—
147	#95	C₂×Dic₈₉	1	2	yes	—
148	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
149	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
150	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
151	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
152	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
153	C₂×C₈	—	89	2	no	—
154	C₂×C₈	—	89	4	no	—
155	C₂×C₈	—	89	2	no	—
156	C₂×C₈	—	89	4	no	—
157	C₂×C₈	—	178	4	no	—
158	C₂×C₈	—	178	4	no	—
159	C₂×C₈	—	178	4	no	—
160	C₂×C₈	—	178	2	no	—
161	C₂×C₈	—	178	2	no	—
162	C₂×C₈	—	178	2	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	4	no	—
169	C₄²	—	2	4	no	—
170	C₄²	Q₈×D₈₉	1	2	yes	—
171	C₄²	#13083575	1	2	yes	—
172	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	Q₈×Dic₈₉	1	4	yes	—
184	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
185	C₂×C₄	#41×D₈₉	1	2	yes	—
186	C₂×C₄	#41×D₈₉	1	2	yes	—
187	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
188	C₈	—	89	4	no	—
189	C₈	—	356	4	no	—
190	C₈	—	178	4	no	—
191	C₈	—	178	2	no	—
192	C₂²	C₂×Q₈×Dic₈₉	1	1	yes	characteristic
193	C₂²	#41×Dic₈₉	1	2	yes	—
194	C₂²	#19152365	1	4	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	4	4	no	—
197	C₄	—	4	4	no	—
198	C₄	—	4	4	no	—
199	C₄	#125×D₈₉	1	1	yes	characteristic
200	C₄	#19152133	1	2	yes	—
201	C₄	#19152154	1	1	yes	characteristic
202	C₂	#125×Dic₈₉	1	1	yes	characteristic
203	C₂	#26029193	1	2	yes	—
204	C₂	#26029239	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂³×C₄
Fitting	C₈₉×#2402	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#2402	D₈₉
hypocenter	C₈₉	#21677
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #2402

Complements

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

N	H ≅ G/N	type
110. C₈₉	#21677	semidirect

Maximal subgroups

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

subgroup	index
#26020976	2
#26020818	2
#26020372	2
#26020127	2
#26020121	2
#26019939	2
C₈₉×#2402	2
#21677	89