Subgroups — #36524986

Order 22784.

598 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030838	C₂	1	1	yes	maximal, characteristic
2	#26029182	C₂	1	2	yes	maximal
3	#26019160	C₂	1	4	yes	maximal, complemented
4	#26019070	C₂	1	4	yes	maximal, complemented
5	#26018706	C₂	1	2	yes	maximal
6	C₂×#19151319	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic
8	#19152155	C₂²	1	2	yes	—
9	#19152003	C₄	1	1	yes	characteristic
10	#19151978	C₄	1	1	yes	characteristic
11	#19151949	C₄	1	2	yes	—
12	#19151839	C₄	1	4	yes	—
13	#19151725	—	2	2	no	—
14	#19151725	C₂²	1	4	yes	—
15	#19151722	—	2	2	no	—
16	#19151722	C₂²	1	4	yes	—
17	C₄×Dic₃₅₆	C₂²	1	4	yes	—
18	#19151626	—	2	2	no	—
19	#19151556	—	2	2	no	—
20	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
21	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
22	#19151328	C₂²	1	8	yes	—
23	#19151319	C₂²	1	2	yes	—
24	#19151319	C₂²	1	2	yes	—
25	#423×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	#334×C₈₉	C₂²	1	4	yes	—
28	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
29	C₂²×#10623494	—	2	1	no	—
30	C₂²×#10623494	D₄	1	2	yes	—
31	#15668012	C₂×C₄	1	2	yes	—
32	#15668012	C₂×C₄	1	2	yes	—
33	#15668004	—	2	2	no	—
34	#15667991	—	2	2	no	—
35	#15667991	C₂×C₄	1	4	yes	—
36	C₄×Dic₁₇₈	—	2	2	no	—
37	C₄×Dic₁₇₈	C₂³	1	4	yes	—
38	#15667971	—	2	2	no	—
39	#15667946	C₂×C₄	1	4	yes	—
40	#15667941	—	2	4	no	—
41	#15667941	—	2	4	no	—
42	#15667937	C₂×C₄	1	8	yes	—
43	#15667922	—	2	4	no	—
44	C₄×#10623494	—	2	1	no	—
45	C₄×#10623494	D₄	1	2	yes	—
46	C₄×#10623494	C₂³	1	2	yes	—
47	C₄×#10623494	C₂³	1	1	yes	characteristic
48	C₄×#10623494	C₂³	1	1	yes	characteristic
49	C₈×Dic₈₉	—	2	4	no	—
50	C₂×M₁₆×C₈₉	—	2	2	no	—
51	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₈₉	—	2	2	no	—
54	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
55	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₈₉	C₂³	1	2	yes	—
58	C₂×Dic₁₇₈	—	2	2	no	—
59	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
60	Dic₃₅₆	—	2	8	no	—
61	C₂×#10623494	—	2	4	no	—
62	C₂×#10623494	—	2	2	no	—
63	C₂×#10623494	—	4	2	no	—
64	C₂×#10623494	C₂×D₄	1	2	yes	—
65	C₂×#10623494	C₂×D₄	1	2	yes	—
66	C₄⋊Dic₈₉	—	2	2	no	—
67	C₄⋊Dic₈₉	—	2	2	no	—
68	C₄⋊Dic₈₉	—	2	2	no	—
69	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
70	#13083553	—	2	4	no	—
71	#13083553	—	2	4	no	—
72	C₂²⋊Dic₈₉	—	2	4	no	—
73	C₄×Dic₈₉	—	2	4	no	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
77	M₁₆×C₈₉	—	4	4	no	—
78	C₂×C₈×C₈₉	—	2	4	no	—
79	C₂×C₈×C₈₉	—	2	4	no	—
80	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
82	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₈₉	#41	1	1	yes	characteristic
85	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
86	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	—	2	2	no	—
89	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₈₉	#137	1	4	yes	—
93	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
96	C₂×Dic₈₉	—	2	4	no	—
97	C₂×Dic₈₉	—	2	4	no	—
98	Dic₁₇₈	—	2	8	no	—
99	Dic₁₇₈	—	4	4	no	—
100	C₈×C₈₉	—	2	2	no	—
101	C₈×C₈₉	—	4	4	no	—
102	C₈×C₈₉	#131	1	4	yes	—
103	#10623494	—	2	2	no	—
104	#10623494	—	2	2	no	—
105	#10623494	—	4	2	no	—
106	C₂²×C₈₉	—	2	1	no	—
107	C₂²×C₈₉	—	2	1	no	—
108	C₂²×C₈₉	#441	1	2	yes	—
109	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₈₉	—	2	1	no	—
111	C₄×C₈₉	—	2	1	no	—
112	C₄×C₈₉	#441	1	2	yes	—
113	C₄×C₈₉	#444	1	2	yes	—
114	C₄×C₈₉	C₂×#131	1	2	yes	—
115	C₄×C₈₉	C₂×#137	1	2	yes	—
116	Dic₈₉	—	4	4	no	—
117	Dic₈₉	—	4	4	no	—
118	#17961	—	89	1	no	maximal, Sylow
119	C₂×C₈₉	—	2	2	no	—
120	C₂×C₈₉	#2931	1	1	yes	characteristic
121	C₂×C₈₉	C₂×#441	1	1	yes	characteristic
122	C₂×C₈₉	C₂×#444	1	1	yes	characteristic
123	#3170	—	89	1	no	—
124	#2960	—	89	2	no	—
125	#1593	—	89	4	no	—
126	#1579	—	89	4	no	—
127	#1566	D₈₉	1	1	yes	characteristic
128	#1493	—	89	2	no	—
129	C₂×#322	—	89	1	no	—
130	C₈₉	#17961	1	1	yes	characteristic, complemented, Sylow
131	#533	—	89	2	no	—
132	#500	—	89	1	no	—
133	#494	—	89	1	no	—
134	#487	—	89	2	no	—
135	#467	—	89	4	no	—
136	#446	—	89	4	no	—
137	#446	—	178	2	no	—
138	C₈×Q₈	—	89	4	no	—
139	C₈×Q₈	—	178	2	no	—
140	C₄×Q₁₆	—	89	4	no	—
141	#424	—	178	2	no	—
142	#423	C₂×D₈₉	1	2	yes	—
143	#405	—	178	2	no	—
144	C₂×C₄×C₈	—	89	1	no	—
145	C₂×C₄×C₈	—	89	1	no	—
146	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
147	#334	C₂×D₈₉	1	4	yes	—
148	Q₈⋊C₈	—	89	8	no	—
149	#322	—	89	2	no	—
150	#322	—	89	2	no	—
151	C₂×Q₁₆	—	89	4	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂²×C₈	—	89	2	no	—
154	C₂²×C₈	—	178	1	no	—
155	C₂²×C₈	C₄×D₈₉	1	2	yes	—
156	C₄⋊Q₈	—	89	2	no	—
157	C₄⋊Q₈	—	89	2	no	—
158	#125	—	178	2	no	—
159	C₂²⋊Q₈	—	89	4	no	—
160	C₂²⋊Q₈	—	178	2	no	—
161	C₄×Q₈	—	89	4	no	—
162	C₄×Q₈	—	178	2	no	—
163	#117	—	178	2	no	—
164	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
165	#107	—	89	4	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	178	4	no	—
168	C₄⋊C₈	—	178	4	no	—
169	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
170	#103	—	89	8	no	—
171	#97	—	178	4	no	—
172	C₄×C₈	—	89	2	no	—
173	C₄×C₈	—	89	1	no	—
174	C₄×C₈	—	89	1	no	—
175	C₄×C₈	—	89	2	no	—
176	C₄×C₈	—	178	4	no	—
177	C₄×C₈	—	178	1	no	—
178	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
179	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₈₉	1	2	yes	—
181	C₂×Q₈	—	89	4	no	—
182	C₂×Q₈	—	178	2	no	—
183	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
184	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
185	C₂²×C₄	#13083575	1	1	yes	characteristic
186	Q₁₆	—	178	8	no	—
187	M₁₆	—	4	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	89	2	no	—
191	C₂×C₈	—	89	2	no	—
192	C₂×C₈	—	356	2	no	—
193	C₂×C₈	—	178	2	no	—
194	C₂×C₈	—	178	4	no	—
195	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
196	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
197	C₄⋊C₄	—	89	4	no	—
198	C₄⋊C₄	—	178	2	no	—
199	C₄⋊C₄	—	178	2	no	—
200	C₄⋊C₄	—	178	4	no	—
201	C₄⋊C₄	—	178	2	no	—
202	C₄⋊C₄	—	178	4	no	—
203	C₂²⋊C₄	—	178	4	no	—
204	C₄²	—	178	4	no	—
205	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
206	C₄²	D₄×D₈₉	1	1	yes	characteristic
207	C₄²	#13083575	1	1	yes	characteristic
208	C₄²	C₂³×D₈₉	1	1	yes	characteristic
209	C₂³	#15667974	1	1	yes	characteristic
210	Q₈	—	356	4	no	—
211	Q₈	—	178	8	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	178	4	no	—
215	C₂×C₄	—	178	4	no	—
216	C₂×C₄	#15667974	1	1	yes	characteristic
217	C₂×C₄	#15667974	1	1	yes	characteristic
218	C₂×C₄	#15667974	1	1	yes	characteristic
219	C₂×C₄	#15668041	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
222	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	4	4	no	—
225	C₈	—	356	2	no	—
226	C₈	—	178	2	no	—
227	C₈	—	178	2	no	—
228	C₈	#15668020	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#19151704	1	2	yes	—
232	C₂²	C₂×#15667974	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	356	4	no	—
236	C₄	—	356	4	no	—
237	C₄	#19151704	1	2	yes	—
238	C₄	#19151719	1	2	yes	—
239	C₄	C₂×#15668020	1	2	yes	—
240	C₄	C₂×#15668041	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#26028963	1	1	yes	characteristic
243	C₂	C₂×#19151704	1	1	yes	characteristic
244	C₂	C₂×#19151719	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1566	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1566	D₈₉
hypocenter	C₈₉	#17961
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. #26019160	C₂	semidirect
4. #26019070	C₂	semidirect
130. C₈₉	#17961	semidirect

Maximal subgroups

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

subgroup	index
#26030838	2
#26029182	2
#26019160	2
#26019070	2
#26018706	2
C₂×#19151319	2
C₈₉×#1566	2
#17961	89