Subgroups — #36708093

Order 22784.

444 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029186	C₂	1	1	yes	maximal, characteristic
2	#26022159	C₂	1	4	yes	maximal, complemented
3	#26022158	C₂	1	4	yes	maximal, complemented
4	#26022056	C₂	1	4	yes	maximal
5	#26022055	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#2163	C₂	1	1	yes	maximal, characteristic
7	#19152155	C₄	1	1	yes	characteristic
8	#19151728	C₄	1	2	yes	—
9	#19151728	C₄	1	2	yes	—
10	C₈×Dic₁₇₈	C₂²	1	2	yes	—
11	C₈×Dic₁₇₈	C₂²	1	2	yes	—
12	#19151624	C₄	1	2	yes	—
13	#19151622	C₄	1	1	yes	characteristic
14	#19151553	C₂²	1	2	yes	—
15	C₄⋊#13083549	C₂²	1	2	yes	—
16	#19151416	C₂²	1	8	yes	—
17	C₂²⋊#13083549	C₂²	1	2	yes	—
18	#19151374	C₂²	1	2	yes	—
19	#19151373	C₂²	1	4	yes	—
20	C₁₆×Dic₈₉	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
23	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
24	#346×C₈₉	C₂²	1	2	yes	—
25	C₂×#13083558	C₂×C₄	1	2	yes	—
26	#15668012	C₈	1	1	yes	characteristic
27	#15668012	C₈	1	1	yes	characteristic
28	#15668004	C₈	1	2	yes	—
29	#15667991	C₈	1	2	yes	—
30	#15667991	C₈	1	2	yes	—
31	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
32	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
33	#15667971	C₂×C₄	1	2	yes	—
34	C₂×#13083549	C₂³	1	4	yes	—
35	C₄⋊#10623494	C₂×C₄	1	2	yes	—
36	C₄⋊#10623494	C₂×C₄	1	2	yes	—
37	#15667941	C₂×C₄	1	4	yes	—
38	#15667941	C₂×C₄	1	4	yes	—
39	#15667922	C₂×C₄	1	4	yes	—
40	C₈×Dic₈₉	C₂³	1	4	yes	—
41	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
45	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
50	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
51	#13083558	—	4	2	no	—
52	C₂×#10623494	C₂²×C₄	1	4	yes	—
53	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
54	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
55	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
56	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
57	#13083553	C₂×C₈	1	4	yes	—
58	#13083553	C₂×C₈	1	4	yes	—
59	C₂²⋊Dic₈₉	C₂×C₈	1	4	yes	—
60	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
61	#13083549	—	2	4	no	—
62	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	—	2	2	no	—
66	C₂×C₈×C₈₉	—	4	1	no	—
67	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
71	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₁₆×C₈₉	—	2	4	no	—
76	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₈₉	—	2	1	no	—
78	C₂×C₄×C₈₉	—	2	1	no	—
79	C₂×C₄×C₈₉	—	4	1	no	—
80	C₂×C₄×C₈₉	—	4	1	no	—
81	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
87	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
88	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
89	Dic₁₇₈	—	4	2	no	—
90	Dic₁₇₈	—	4	2	no	—
91	C₈×C₈₉	—	2	1	no	—
92	C₈×C₈₉	—	2	1	no	—
93	C₈×C₈₉	—	2	1	no	—
94	C₈×C₈₉	2₋¹⁺⁴	1	2	yes	—
95	#10623494	—	2	4	no	—
96	C₂²×C₈₉	—	2	1	no	—
97	C₂²×C₈₉	—	2	1	no	—
98	C₂²×C₈₉	—	2	1	no	—
99	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
100	C₄×C₈₉	—	2	1	no	—
101	C₄×C₈₉	—	2	1	no	—
102	C₄×C₈₉	—	2	1	no	—
103	C₄×C₈₉	—	2	1	no	—
104	C₄×C₈₉	—	2	1	no	—
105	C₄×C₈₉	#519	1	1	yes	characteristic
106	C₄×C₈₉	#568	1	1	yes	characteristic
107	Dic₈₉	—	2	4	no	—
108	Dic₈₉	—	2	4	no	—
109	#33501	—	89	1	no	maximal, Sylow
110	C₂×C₈₉	—	4	1	no	—
111	C₂×C₈₉	#2982	1	1	yes	characteristic
112	C₂×C₈₉	#3407	1	1	yes	characteristic
113	C₂×C₈₉	#3407	1	1	yes	characteristic
114	#2960	—	89	1	no	—
115	#2183	—	89	4	no	—
116	#2183	—	89	4	no	—
117	#2164	—	89	4	no	—
118	#2163	—	89	1	no	—
119	#2163	D₈₉	1	1	yes	characteristic
120	C₈₉	#33501	1	1	yes	characteristic, complemented, Sylow
121	#533	—	89	1	no	—
122	#446	—	89	2	no	—
123	#446	—	89	2	no	—
124	C₈×Q₈	—	89	2	no	—
125	C₈×Q₈	—	89	2	no	—
126	#424	—	89	2	no	—
127	#423	—	89	1	no	—
128	#405	—	89	2	no	—
129	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
130	C₄⋊C₁₆	—	89	8	no	—
131	C₄⋊C₁₆	—	89	2	no	—
132	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
133	C₂²⋊C₁₆	—	89	2	no	—
134	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
135	#346	—	89	4	no	—
136	#346	—	89	2	no	—
137	#346	C₂×D₈₉	1	2	yes	—
138	C₄×C₁₆	—	89	4	no	—
139	C₂×M₁₆	—	89	2	no	—
140	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
142	C₄⋊Q₈	—	89	1	no	—
143	C₄⋊Q₈	—	89	1	no	—
144	#125	—	89	2	no	—
145	C₂²⋊Q₈	—	89	2	no	—
146	C₂²⋊Q₈	—	89	2	no	—
147	C₄×Q₈	—	89	2	no	—
148	C₄×Q₈	—	89	2	no	—
149	#117	—	89	2	no	—
150	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
151	C₂×C₁₆	—	89	4	no	—
152	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
153	C₄⋊C₈	—	89	4	no	—
154	C₄⋊C₈	—	89	4	no	—
155	C₄⋊C₈	—	89	2	no	—
156	C₄⋊C₈	—	89	2	no	—
157	#97	—	89	4	no	—
158	C₄×C₈	—	89	4	no	—
159	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
160	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
163	C₂×Q₈	—	89	2	no	—
164	C₂×Q₈	—	89	2	no	—
165	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
166	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
168	M₁₆	—	356	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	89	4	no	—
172	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
173	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
174	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
175	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
176	C₄⋊C₄	—	89	2	no	—
177	C₄⋊C₄	—	89	2	no	—
178	C₄⋊C₄	—	89	4	no	—
179	C₄⋊C₄	—	89	4	no	—
180	C₄⋊C₄	—	89	2	no	—
181	C₄⋊C₄	—	89	2	no	—
182	C₂²⋊C₄	—	89	4	no	—
183	C₄²	—	89	4	no	—
184	C₄²	C₈×D₈₉	1	1	yes	characteristic
185	C₄²	C₈×D₈₉	1	1	yes	characteristic
186	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
187	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
188	C₁₆	—	2	4	no	—
189	C₁₆	—	178	4	no	—
190	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
191	Q₈	—	356	2	no	—
192	Q₈	—	356	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	89	4	no	—
198	C₂×C₄	—	89	4	no	—
199	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
200	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
201	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
202	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
203	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	178	4	no	—
209	C₈	#15668059	1	2	yes	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	178	4	no	—
220	C₄	—	178	4	no	—
221	C₄	#19152075	1	1	yes	characteristic
222	C₄	#19152364	1	1	yes	characteristic
223	C₂	—	4	1	no	—
224	C₂	#26029332	1	1	yes	characteristic
225	C₂	#26033354	1	1	yes	characteristic
226	C₂	#26033354	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2163	C₂
prime radical	C₂³×C₈₉	C₂²×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2163	D₈₉
hypocenter	C₈₉	#33501
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. #26022159	C₂	semidirect
3. #26022158	C₂	semidirect
120. C₈₉	#33501	semidirect

Maximal subgroups

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

subgroup	index
#26029186	2
#26022159	2
#26022158	2
#26022056	2
#26022055	2
C₈₉×#2163	2
#33501	89