Subgroups — #36581017

Order 22784.

664 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668012	C₂	1	1	yes	maximal, characteristic
2	#26021186	C₂	1	4	yes	maximal
3	#26020810	C₂	1	4	yes	maximal
4	#26020808	C₂	1	4	yes	maximal
5	#26020395	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#1839	C₂	1	1	yes	maximal, characteristic
7	C₂×#15668012	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
9	C₂×#15667941	C₂²	1	4	yes	—
10	C₂×C₈×Dic₈₉	C₂²	1	4	yes	—
11	#19151504	C₄	1	2	yes	—
12	#19151496	C₄	1	2	yes	—
13	#19151496	C₄	1	2	yes	—
14	#19151468	C₄	1	1	yes	characteristic
15	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
16	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
17	#19151353	C₂²	1	2	yes	—
18	#19151353	C₂²	1	2	yes	—
19	#19151353	C₂²	1	2	yes	—
20	#19151352	C₂²	1	8	yes	—
21	C₄³×C₈₉	C₂²	1	1	yes	characteristic
22	#336×C₈₉	C₂²	1	2	yes	—
23	#336×C₈₉	C₂²	1	2	yes	—
24	#336×C₈₉	C₂²	1	2	yes	—
25	C₂²×Dic₁₇₈	C₂×C₄	1	2	yes	—
26	C₂²×#10623494	C₂³	1	4	yes	—
27	#15668012	C₈	1	8	yes	complemented
28	C₄×Dic₁₇₈	—	4	2	no	—
29	C₄×Dic₁₇₈	—	4	2	no	—
30	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
35	C₂×#13083553	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₈₉	C₂³	1	4	yes	—
37	#15667941	—	4	4	no	—
38	C₈×Dic₈₉	—	4	4	no	—
39	#15667919	C₂×C₄	1	4	yes	—
40	#15667919	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
42	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₇₈	—	4	2	no	—
50	C₂×Dic₁₇₈	—	4	2	no	—
51	C₂×Dic₁₇₈	C₂×C₈	1	8	yes	—
52	C₂²×Dic₈₉	C₂²×C₄	1	4	yes	—
53	C₂²×Dic₈₉	C₂²×C₄	1	4	yes	—
54	C₂×#10623494	—	2	4	no	—
55	C₂×#10623494	—	2	4	no	—
56	C₂×#10623494	—	2	4	no	—
57	C₄⋊Dic₈₉	—	4	2	no	—
58	C₄⋊Dic₈₉	—	4	2	no	—
59	C₄⋊Dic₈₉	—	4	2	no	—
60	C₄⋊Dic₈₉	C₂×C₈	1	8	yes	—
61	C₄⋊Dic₈₉	C₂×C₈	1	8	yes	—
62	#13083553	—	4	4	no	—
63	C₄×Dic₈₉	—	2	4	no	—
64	C₄×Dic₈₉	—	2	4	no	—
65	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₈₉	—	2	4	no	—
73	C₂×C₈×C₈₉	—	2	4	no	—
74	C₂×C₈×C₈₉	—	2	4	no	—
75	C₄²×C₈₉	—	2	2	no	—
76	C₄²×C₈₉	—	2	2	no	—
77	C₄²×C₈₉	—	2	2	no	—
78	C₄²×C₈₉	—	4	1	no	—
79	C₄²×C₈₉	—	4	1	no	—
80	C₄²×C₈₉	—	4	1	no	—
81	C₄²×C₈₉	C₂×C₈	1	4	yes	—
82	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	—	2	1	no	—
84	C₂×C₄×C₈₉	—	2	1	no	—
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
102	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
103	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
104	C₂×Dic₈₉	—	2	4	no	—
105	C₂×Dic₈₉	—	2	4	no	—
106	C₂×Dic₈₉	—	2	4	no	—
107	C₂×Dic₈₉	—	2	4	no	—
108	C₂×Dic₈₉	—	2	4	no	—
109	C₂×Dic₈₉	C₂²×C₈	1	8	yes	—
110	Dic₁₇₈	—	4	8	no	—
111	C₈×C₈₉	—	4	4	no	—
112	#10623494	—	4	4	no	—
113	C₂²×C₈₉	#518	1	1	yes	characteristic
114	C₂²×C₈₉	#519	1	1	yes	characteristic
115	C₂²×C₈₉	#519	1	1	yes	characteristic
116	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
117	C₂²×C₈₉	#568	1	1	yes	characteristic
118	C₂²×C₈₉	#568	1	1	yes	characteristic
119	C₂²×C₈₉	#568	1	1	yes	characteristic
120	C₄×C₈₉	—	2	2	no	—
121	C₄×C₈₉	—	2	2	no	—
122	C₄×C₈₉	—	2	2	no	—
123	C₄×C₈₉	—	4	1	no	—
124	C₄×C₈₉	—	4	1	no	—
125	C₄×C₈₉	—	4	1	no	—
126	C₄×C₈₉	#563	1	4	yes	—
127	Dic₈₉	—	2	8	no	—
128	Dic₈₉	—	4	4	no	—
129	#23363	—	89	1	no	maximal, Sylow
130	C₂×C₈₉	#2381	1	1	yes	characteristic
131	C₂×C₈₉	#2972	1	1	yes	characteristic
132	C₂×C₈₉	#2980	1	1	yes	characteristic
133	C₂×C₈₉	#2982	1	1	yes	characteristic
134	C₂×C₈₉	#2982	1	1	yes	characteristic
135	C₂×C₈₉	#2987	1	1	yes	characteristic
136	C₂×C₈₉	#2987	1	1	yes	characteristic
137	C₄×C₄⋊Q₈	—	89	1	no	—
138	#1984	—	89	4	no	—
139	#1916	—	89	4	no	—
140	#1916	—	89	4	no	—
141	#1839	—	89	1	no	—
142	#1839	D₈₉	1	1	yes	characteristic
143	C₈₉	#23363	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄⋊Q₈	—	89	1	no	—
145	C₂×C₄×Q₈	—	89	2	no	—
146	C₂×C₄⋊C₈	—	89	4	no	—
147	C₂×C₄×C₈	—	89	4	no	—
148	#391	—	89	2	no	—
149	#389	—	89	2	no	—
150	#389	—	89	2	no	—
151	#384	—	89	1	no	—
152	C₄×C₄⋊C₄	—	89	2	no	—
153	C₄×C₄⋊C₄	—	89	2	no	—
154	C₄³	C₂×D₈₉	1	1	yes	characteristic
155	#336	—	89	8	no	—
156	#336	—	89	2	no	—
157	#336	—	89	2	no	—
158	#336	—	89	2	no	—
159	#336	C₂×D₈₉	1	2	yes	—
160	#336	C₂×D₈₉	1	2	yes	—
161	#336	C₂×D₈₉	1	2	yes	—
162	C₂²×Q₈	—	89	2	no	—
163	C₂²×C₈	—	89	4	no	—
164	C₂²×C₈	C₂²×D₈₉	1	4	yes	—
165	C₄⋊Q₈	—	89	8	no	—
166	C₄×Q₈	—	356	2	no	—
167	C₄×Q₈	—	356	2	no	—
168	C₂×C₄⋊C₄	—	89	2	no	—
169	C₂×C₄⋊C₄	—	89	2	no	—
170	C₂×C₄⋊C₄	—	89	2	no	—
171	C₂×C₄⋊C₄	—	89	4	no	—
172	C₂×C₄⋊C₄	—	89	2	no	—
173	C₂×C₄⋊C₄	—	89	2	no	—
174	C₂×C₄²	—	89	4	no	—
175	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
176	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
182	C₄⋊C₈	—	356	4	no	—
183	C₄×C₈	—	356	4	no	—
184	#95	—	89	4	no	—
185	#95	—	89	4	no	—
186	C₂×Q₈	—	89	8	no	—
187	C₂×Q₈	—	356	2	no	—
188	C₂×Q₈	—	356	2	no	—
189	C₂²×C₄	—	89	4	no	—
190	C₂²×C₄	—	89	4	no	—
191	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
197	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	178	4	no	—
202	C₂×C₈	—	178	4	no	—
203	C₂×C₈	—	178	4	no	—
204	C₄⋊C₄	—	89	8	no	—
205	C₄⋊C₄	—	89	8	no	—
206	C₄⋊C₄	—	356	4	no	—
207	C₄⋊C₄	—	356	2	no	—
208	C₄⋊C₄	—	356	2	no	—
209	C₄⋊C₄	—	356	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	4	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	178	4	no	—
217	C₄²	—	178	4	no	—
218	C₄²	C₈×D₈₉	1	4	yes	—
219	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
220	Q₈	—	356	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	89	8	no	—
237	C₂×C₄	—	178	4	no	—
238	C₂×C₄	—	178	4	no	—
239	C₂×C₄	—	178	4	no	—
240	C₂×C₄	—	178	4	no	—
241	C₂×C₄	—	178	4	no	—
242	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
243	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
244	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
245	C₂×C₄	#15668058	1	2	yes	—
246	C₂×C₄	#15668059	1	2	yes	—
247	C₂×C₄	#15668059	1	2	yes	—
248	C₈	—	4	4	no	—
249	C₈	—	356	4	no	—
250	C₂²	#19152072	1	1	yes	characteristic
251	C₂²	#19152075	1	1	yes	characteristic
252	C₂²	#19152075	1	1	yes	characteristic
253	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
254	C₂²	#19152364	1	1	yes	characteristic
255	C₂²	#19152364	1	1	yes	characteristic
256	C₂²	#19152364	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	356	4	no	—
264	C₄	—	178	8	no	—
265	C₄	#19152350	1	4	yes	—
266	C₂	#26023432	1	1	yes	characteristic
267	C₂	#26029275	1	1	yes	characteristic
268	C₂	#26029322	1	1	yes	characteristic
269	C₂	#26029332	1	1	yes	characteristic
270	C₂	#26029332	1	1	yes	characteristic
271	C₂	#26029357	1	1	yes	characteristic
272	C₂	#26029357	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#1839	C₂
socle, prime radical	C₂³×C₈₉	C₂³×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1839	D₈₉
hypocenter	C₈₉	#23363
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #1839

Complements

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

N	H ≅ G/N	type
27. #15668012	C₈	semidirect
143. C₈₉	#23363	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668012	2
#26021186	2
#26020810	2
#26020808	2
#26020395	2
C₈₉×#1839	2
#23363	89