Subgroups — #36581033

Order 22784.

636 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26021193	C₂	1	2	yes	maximal
2	#26021189	C₂	1	2	yes	maximal
3	#26020818	C₂	1	4	yes	maximal
4	#26020811	C₂	1	4	yes	maximal
5	#26020395	C₂	1	2	yes	maximal
6	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	C₂×#15667923	C₂²	1	2	yes	—
10	C₂×C₄×#10623494	C₂²	1	2	yes	—
11	#19151353	C₂²	1	4	yes	—
12	#19151353	C₂²	1	2	yes	—
13	#19151353	C₂²	1	4	yes	—
14	#19151353	C₂²	1	2	yes	—
15	#19151353	C₂²	1	4	yes	—
16	#19151353	C₂²	1	4	yes	—
17	C₂×C₄⋊Q₈×C₈₉	C₄	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
20	#391×C₈₉	C₄	1	1	yes	characteristic
21	#391×C₈₉	C₄	1	1	yes	characteristic
22	#389×C₈₉	C₄	1	4	yes	—
23	#384×C₈₉	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
25	C₄³×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×#10623494	C₂³	1	4	yes	—
27	C₂²×#10623494	C₂³	1	4	yes	—
28	C₄⋊#10623494	—	4	2	no	—
29	C₄⋊#10623494	—	4	2	no	—
30	#15667923	—	4	2	no	—
31	C₄×#10623494	—	4	2	no	—
32	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₈₉	—	2	4	no	—
35	C₄×Q₈×C₈₉	—	4	1	no	—
36	C₄×Q₈×C₈₉	—	4	1	no	—
37	C₄×Q₈×C₈₉	—	4	1	no	—
38	C₄×Q₈×C₈₉	—	4	1	no	—
39	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
49	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
51	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
52	#95×C₈₉	C₂×C₄	1	4	yes	—
53	#95×C₈₉	C₂×C₄	1	4	yes	—
54	C₂×#10623494	—	2	4	no	—
55	C₂×#10623494	—	2	4	no	—
56	C₂×#10623494	—	2	8	no	—
57	C₂×#10623494	—	2	8	no	—
58	C₂×Q₈×C₈₉	—	2	2	no	—
59	C₂×Q₈×C₈₉	—	4	1	no	—
60	C₂×Q₈×C₈₉	—	4	1	no	—
61	C₂×Q₈×C₈₉	—	4	1	no	—
62	C₂×Q₈×C₈₉	—	4	1	no	—
63	C₂×Q₈×C₈₉	M₁₆	1	4	yes	—
64	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₈₉	—	2	8	no	—
74	C₄⋊C₄×C₈₉	—	2	8	no	—
75	C₄⋊C₄×C₈₉	—	4	1	no	—
76	C₄⋊C₄×C₈₉	—	4	1	no	—
77	C₄⋊C₄×C₈₉	—	4	2	no	—
78	C₄⋊C₄×C₈₉	—	4	2	no	—
79	C₄²×C₈₉	—	2	2	no	—
80	C₄²×C₈₉	—	2	4	no	—
81	C₄²×C₈₉	—	2	1	no	—
82	C₄²×C₈₉	—	2	1	no	—
83	C₄²×C₈₉	—	2	2	no	—
84	C₄²×C₈₉	—	2	2	no	—
85	C₄²×C₈₉	—	2	2	no	—
86	C₄²×C₈₉	—	2	2	no	—
87	C₄²×C₈₉	—	2	4	no	—
88	C₄²×C₈₉	—	4	1	no	—
89	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
90	Q₈×C₈₉	—	4	4	no	—
91	Q₈×C₈₉	—	4	4	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	—	2	2	no	—
101	C₂×C₄×C₈₉	—	2	2	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	2	4	no	—
107	C₂×C₄×C₈₉	—	2	4	no	—
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
112	C₂×C₄×C₈₉	#131	1	4	yes	—
113	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₄×C₈₉	2₋¹⁺⁴	1	4	yes	—
116	#10623494	—	4	4	no	—
117	#10623494	—	4	4	no	—
118	C₂²×C₈₉	#518	1	1	yes	characteristic
119	C₂²×C₈₉	#519	1	2	yes	—
120	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
121	C₂²×C₈₉	C₂×#131	1	2	yes	—
122	C₂²×C₈₉	#568	1	1	yes	characteristic
123	C₄×C₈₉	—	2	4	no	—
124	C₄×C₈₉	—	2	2	no	—
125	C₄×C₈₉	—	2	4	no	—
126	C₄×C₈₉	—	4	2	no	—
127	C₄×C₈₉	—	4	1	no	—
128	C₄×C₈₉	—	4	2	no	—
129	C₄×C₈₉	—	4	2	no	—
130	C₄×C₈₉	—	4	2	no	—
131	C₄×C₈₉	#563	1	2	yes	—
132	C₄×C₈₉	#563	1	2	yes	—
133	#23365	—	89	1	no	maximal, Sylow
134	C₂×C₈₉	#2381	1	1	yes	characteristic
135	C₂×C₈₉	#2973	1	1	yes	characteristic
136	C₂×C₈₉	#2979	1	1	yes	characteristic
137	C₂×C₈₉	#2984	1	2	yes	—
138	C₂×C₈₉	#2985	1	2	yes	—
139	C₄×C₄⋊Q₈	D₈₉	1	1	yes	characteristic
140	#1985	—	89	2	no	—
141	#1984	—	89	2	no	—
142	#1917	—	89	4	no	—
143	#1916	—	89	4	no	—
144	#1839	—	89	2	no	—
145	C₈₉	#23365	1	1	yes	characteristic, complemented, Sylow
146	C₂×C₄⋊Q₈	Dic₈₉	1	1	yes	characteristic
147	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
148	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
149	C₂×C₄⋊C₈	—	89	2	no	—
150	C₂×C₄⋊C₈	—	89	2	no	—
151	C₂×#97	—	89	2	no	—
152	C₂×C₄×C₈	—	89	2	no	—
153	#391	Dic₈₉	1	1	yes	characteristic
154	#391	Dic₈₉	1	1	yes	characteristic
155	#389	Dic₈₉	1	4	yes	—
156	#384	Dic₈₉	1	1	yes	characteristic
157	C₄×C₄⋊C₄	C₂×D₈₉	1	4	yes	—
158	C₄³	C₂×D₈₉	1	1	yes	characteristic
159	#336	—	89	2	no	—
160	#336	—	89	4	no	—
161	#336	—	89	2	no	—
162	#336	—	89	4	no	—
163	#336	—	89	4	no	—
164	#336	—	89	4	no	—
165	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
166	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
167	C₂²×C₈	—	89	4	no	—
168	C₂²×C₈	—	89	4	no	—
169	C₄⋊Q₈	—	2	4	no	—
170	C₄×Q₈	—	4	1	no	—
171	C₄×Q₈	—	4	1	no	—
172	C₄×Q₈	—	4	1	no	—
173	C₄×Q₈	—	4	1	no	—
174	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
175	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
176	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
178	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
180	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
181	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
182	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
185	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
186	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
187	C₄⋊C₈	—	356	2	no	—
188	C₄⋊C₈	—	356	2	no	—
189	#97	—	356	2	no	—
190	C₄×C₈	—	356	2	no	—
191	#95	C₂×Dic₈₉	1	4	yes	—
192	#95	C₂×Dic₈₉	1	4	yes	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂×Q₈	—	4	1	no	—
196	C₂×Q₈	—	4	1	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	#13083558	1	4	yes	—
199	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
201	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
202	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
205	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
207	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
208	C₂×C₈	—	178	4	no	—
209	C₂×C₈	—	178	4	no	—
210	C₂×C₈	—	178	8	no	—
211	C₂×C₈	—	178	8	no	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	2	8	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	4	no	—
227	C₄²	—	4	1	no	—
228	C₂³	C₂³×Dic₈₉	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	Q₈	—	4	4	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	C₂×#13083558	1	2	yes	—
250	C₂×C₄	C₂×#13083558	1	4	yes	—
251	C₂×C₄	#15668019	1	4	yes	—
252	C₂×C₄	#15668057	1	1	yes	characteristic
253	C₂×C₄	#15668057	1	1	yes	characteristic
254	C₂×C₄	#15668060	1	4	yes	—
255	C₈	—	356	4	no	—
256	C₈	—	356	4	no	—
257	C₂²	#19152073	1	1	yes	characteristic
258	C₂²	#19152076	1	2	yes	—
259	C₂²	C₂²×#13083558	1	1	yes	characteristic
260	C₂²	C₂×#15668019	1	2	yes	—
261	C₂²	#19152365	1	1	yes	characteristic
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	4	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	2	no	—
270	C₄	#19152347	1	2	yes	—
271	C₄	#19152347	1	2	yes	—
272	C₂	#26023433	1	1	yes	characteristic
273	C₂	#26029281	1	1	yes	characteristic
274	C₂	#26029319	1	1	yes	characteristic
275	C₂	#26029341	1	2	yes	—
276	C₂	#26029347	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle, prime radical	C₂³×C₈₉	C₂³×C₄
Fitting	C₄×C₄⋊Q₈×C₈₉	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#23365
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
145. C₈₉	#23365	semidirect

Maximal subgroups

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

subgroup	index
#26021193	2
#26021189	2
#26020818	2
#26020811	2
#26020395	2
C₄×C₄⋊Q₈×C₈₉	2
#23365	89