Subgroups — #36525174

Order 22784.

590 conjugacy classes of subgroups.

The subgroup lattice has 242 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	#26019122	C₂	1	8	yes	maximal, complemented
4	#26018706	C₂	1	2	yes	maximal
5	C₂×#19151320	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic
7	#19152155	C₂²	1	2	yes	—
8	#19152003	C₄	1	1	yes	characteristic
9	#19151978	C₄	1	1	yes	characteristic
10	#19151949	C₄	1	2	yes	—
11	#19151839	C₄	1	4	yes	—
12	#19151725	—	2	2	no	—
13	#19151725	C₂²	1	4	yes	—
14	#19151722	—	2	2	no	—
15	#19151722	C₂²	1	4	yes	—
16	C₄×Dic₃₅₆	C₂²	1	4	yes	—
17	#19151626	—	2	2	no	—
18	#19151556	—	2	2	no	—
19	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
20	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
21	#19151328	C₂²	1	8	yes	—
22	#19151320	C₂²	1	4	yes	—
23	#423×C₈₉	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	#334×C₈₉	C₂²	1	4	yes	—
26	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
27	C₂²×#10623494	—	2	1	no	—
28	C₂²×#10623494	D₄	1	2	yes	—
29	#15668012	C₂×C₄	1	2	yes	—
30	#15668012	C₂×C₄	1	2	yes	—
31	#15668004	—	2	2	no	—
32	#15667991	—	2	2	no	—
33	#15667991	C₂×C₄	1	4	yes	—
34	C₄×Dic₁₇₈	—	2	2	no	—
35	C₄×Dic₁₇₈	C₂³	1	4	yes	—
36	#15667971	—	2	2	no	—
37	#15667946	C₂×C₄	1	4	yes	—
38	#15667941	—	2	4	no	—
39	#15667941	—	2	4	no	—
40	#15667937	C₂×C₄	1	8	yes	—
41	#15667922	—	2	4	no	—
42	C₄×#10623494	—	2	1	no	—
43	C₄×#10623494	D₄	1	1	yes	characteristic
44	C₄×#10623494	D₄	1	1	yes	characteristic
45	C₄×#10623494	C₂³	1	2	yes	—
46	C₄×#10623494	C₂³	1	2	yes	—
47	C₈×Dic₈₉	—	2	4	no	—
48	C₂×M₁₆×C₈₉	—	2	2	no	—
49	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₈₉	—	2	2	no	—
52	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
53	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₈₉	C₂³	1	2	yes	—
56	C₂×Dic₁₇₈	—	2	2	no	—
57	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
58	Dic₃₅₆	—	4	4	no	—
59	C₂×#10623494	—	2	4	no	—
60	C₂×#10623494	—	2	4	no	—
61	C₂×#10623494	—	2	2	no	—
62	C₂×#10623494	C₂×D₄	1	2	yes	—
63	C₂×#10623494	C₂×D₄	1	2	yes	—
64	C₄⋊Dic₈₉	—	2	2	no	—
65	C₄⋊Dic₈₉	—	2	2	no	—
66	C₄⋊Dic₈₉	—	2	2	no	—
67	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
68	#13083553	—	2	4	no	—
69	#13083553	—	2	4	no	—
70	C₂²⋊Dic₈₉	—	2	4	no	—
71	C₄×Dic₈₉	—	2	4	no	—
72	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
75	M₁₆×C₈₉	—	4	4	no	—
76	C₂×C₈×C₈₉	—	2	4	no	—
77	C₂×C₈×C₈₉	—	4	2	no	—
78	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
80	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₈₉	#41	1	1	yes	characteristic
83	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
84	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	—	2	2	no	—
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#137	1	4	yes	—
91	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
94	C₂×Dic₈₉	—	2	4	no	—
95	C₂×Dic₈₉	—	2	4	no	—
96	Dic₁₇₈	—	2	8	no	—
97	Dic₁₇₈	—	4	4	no	—
98	C₈×C₈₉	—	2	2	no	—
99	C₈×C₈₉	—	2	2	no	—
100	C₈×C₈₉	—	4	4	no	—
101	#10623494	—	2	4	no	—
102	#10623494	—	2	2	no	—
103	#10623494	—	2	2	no	—
104	C₂²×C₈₉	—	2	1	no	—
105	C₂²×C₈₉	—	2	1	no	—
106	C₂²×C₈₉	#441	1	2	yes	—
107	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
108	C₄×C₈₉	—	2	1	no	—
109	C₄×C₈₉	—	2	1	no	—
110	C₄×C₈₉	#441	1	2	yes	—
111	C₄×C₈₉	#443	1	2	yes	—
112	C₄×C₈₉	#568	1	2	yes	—
113	C₄×C₈₉	C₂×#137	1	2	yes	—
114	Dic₈₉	—	4	4	no	—
115	Dic₈₉	—	4	4	no	—
116	#17980	—	89	1	no	maximal, Sylow
117	C₂×C₈₉	—	2	2	no	—
118	C₂×C₈₉	#2932	1	1	yes	characteristic
119	C₂×C₈₉	C₂×#441	1	1	yes	characteristic
120	C₂×C₈₉	C₂×#443	1	1	yes	characteristic
121	#3170	—	89	1	no	—
122	#2960	—	89	2	no	—
123	#1587	—	89	8	no	—
124	#1566	D₈₉	1	1	yes	characteristic
125	#1493	—	89	2	no	—
126	C₂×#322	—	89	1	no	—
127	C₈₉	#17980	1	1	yes	characteristic, complemented, Sylow
128	#533	—	89	2	no	—
129	#500	—	89	1	no	—
130	#494	—	89	1	no	—
131	#487	—	89	2	no	—
132	#467	—	89	4	no	—
133	#446	—	89	4	no	—
134	#446	—	178	2	no	—
135	C₈×Q₈	—	89	4	no	—
136	C₈×Q₈	—	178	2	no	—
137	C₄×Q₁₆	—	89	4	no	—
138	#424	—	178	2	no	—
139	#423	C₂×D₈₉	1	2	yes	—
140	#405	—	178	2	no	—
141	C₂×C₄×C₈	—	89	1	no	—
142	C₂×C₄×C₈	—	89	1	no	—
143	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
144	#334	C₂×D₈₉	1	4	yes	—
145	Q₈⋊C₈	—	89	8	no	—
146	#322	—	89	4	no	—
147	C₂×Q₁₆	—	89	4	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂²×C₈	—	89	2	no	—
150	C₂²×C₈	—	178	1	no	—
151	C₂²×C₈	C₄×D₈₉	1	2	yes	—
152	C₄⋊Q₈	—	89	2	no	—
153	C₄⋊Q₈	—	89	2	no	—
154	#125	—	178	2	no	—
155	C₂²⋊Q₈	—	89	4	no	—
156	C₂²⋊Q₈	—	178	2	no	—
157	C₄×Q₈	—	89	4	no	—
158	C₄×Q₈	—	178	2	no	—
159	#117	—	178	2	no	—
160	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
161	#107	—	89	4	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	178	4	no	—
164	C₄⋊C₈	—	178	4	no	—
165	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
166	#103	—	89	8	no	—
167	#97	—	178	4	no	—
168	C₄×C₈	—	89	1	no	—
169	C₄×C₈	—	89	1	no	—
170	C₄×C₈	—	89	2	no	—
171	C₄×C₈	—	89	2	no	—
172	C₄×C₈	—	178	4	no	—
173	C₄×C₈	—	178	1	no	—
174	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
175	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
176	C₄×C₈	C₂²×D₈₉	1	2	yes	—
177	C₂×Q₈	—	89	4	no	—
178	C₂×Q₈	—	178	2	no	—
179	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
180	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
181	C₂²×C₄	#13083575	1	1	yes	characteristic
182	Q₁₆	—	356	4	no	—
183	M₁₆	—	4	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	89	2	no	—
187	C₂×C₈	—	89	2	no	—
188	C₂×C₈	—	178	4	no	—
189	C₂×C₈	—	178	4	no	—
190	C₂×C₈	—	178	2	no	—
191	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
192	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
193	C₄⋊C₄	—	89	4	no	—
194	C₄⋊C₄	—	178	2	no	—
195	C₄⋊C₄	—	178	2	no	—
196	C₄⋊C₄	—	178	4	no	—
197	C₄⋊C₄	—	178	2	no	—
198	C₄⋊C₄	—	178	4	no	—
199	C₂²⋊C₄	—	178	4	no	—
200	C₄²	—	178	4	no	—
201	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
202	C₄²	D₄×D₈₉	1	1	yes	characteristic
203	C₄²	#13083575	1	1	yes	characteristic
204	C₄²	C₂³×D₈₉	1	1	yes	characteristic
205	C₂³	#15667974	1	1	yes	characteristic
206	Q₈	—	356	4	no	—
207	Q₈	—	178	8	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	178	4	no	—
211	C₂×C₄	—	178	4	no	—
212	C₂×C₄	#15667974	1	1	yes	characteristic
213	C₂×C₄	#15667974	1	1	yes	characteristic
214	C₂×C₄	#15667974	1	1	yes	characteristic
215	C₂×C₄	#15668041	1	4	yes	—
216	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
217	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
218	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	4	4	no	—
222	C₈	—	178	4	no	—
223	C₈	—	178	2	no	—
224	C₈	—	178	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	#19151704	1	2	yes	—
228	C₂²	C₂×#15667974	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	356	4	no	—
232	C₄	—	356	4	no	—
233	C₄	#19151704	1	2	yes	—
234	C₄	#19151714	1	2	yes	—
235	C₄	#19152364	1	2	yes	—
236	C₄	C₂×#15668041	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	#26028969	1	1	yes	characteristic
239	C₂	C₂×#19151704	1	1	yes	characteristic
240	C₂	C₂×#19151714	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	#17980
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. #26019122	C₂	semidirect
127. C₈₉	#17980	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
#26019122	2
#26018706	2
C₂×#19151320	2
C₈₉×#1566	2
#17980	89