Subgroups — #36463656

Order 22784.

438 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26021863	C₂	1	2	yes	maximal, complemented
2	#26018192	C₂	1	2	yes	maximal, complemented
3	#26018176	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667952	—	2	2	no	—
6	C₂²⋊#13083549	—	2	2	no	—
7	C₂²⋊#13083549	C₂²	1	4	yes	—
8	C₂³×C₈×C₈₉	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₈₉	C₄	1	1	yes	characteristic
11	C₈×D₄×C₈₉	—	2	2	no	—
12	C₈×D₄×C₈₉	—	2	2	no	—
13	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₈₉	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
17	#15667952	—	4	4	no	—
18	C₂×#13083549	—	2	4	no	—
19	C₂²×D₄×C₈₉	C₈	1	1	yes	characteristic
20	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₈₉	—	2	1	no	—
23	C₂²×C₈×C₈₉	—	2	1	no	—
24	C₂²×C₈×C₈₉	—	4	2	no	—
25	C₂²×C₈×C₈₉	—	4	2	no	—
26	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	D₄	1	2	yes	—
30	C₂²×C₈×C₈₉	D₄	1	2	yes	—
31	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
32	C₄×D₄×C₈₉	—	2	2	no	—
33	C₄×D₄×C₈₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₈₉	C₈	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₈₉	C₈	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₈₉	C₈	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₄⋊C₈×C₈₉	—	2	2	no	—
39	C₂²⋊C₈×C₈₉	—	2	2	no	—
40	C₂²⋊C₈×C₈₉	—	2	2	no	—
41	C₄×C₈×C₈₉	—	2	1	no	—
42	C₄×C₈×C₈₉	—	2	1	no	—
43	#13083549	—	4	4	no	—
44	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
45	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
46	C₂×D₄×C₈₉	—	2	2	no	—
47	C₂×D₄×C₈₉	—	2	2	no	—
48	C₂×D₄×C₈₉	—	2	2	no	—
49	C₂²×C₄×C₈₉	—	2	1	no	—
50	C₂²×C₄×C₈₉	—	2	1	no	—
51	C₂²×C₄×C₈₉	—	4	1	no	—
52	C₂²×C₄×C₈₉	—	4	1	no	—
53	C₂²×C₄×C₈₉	—	4	1	no	—
54	C₂²×C₄×C₈₉	—	4	1	no	—
55	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
61	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	—	2	2	no	—
63	C₂×C₈×C₈₉	—	2	4	no	—
64	C₂×C₈×C₈₉	—	2	1	no	—
65	C₂×C₈×C₈₉	—	2	1	no	—
66	C₂×C₈×C₈₉	—	4	2	no	—
67	C₂×C₈×C₈₉	—	4	1	no	—
68	C₂×C₈×C₈₉	—	4	2	no	—
69	C₂×C₈×C₈₉	—	4	2	no	—
70	C₂×C₈×C₈₉	—	4	1	no	—
71	C₂×C₈×C₈₉	—	4	1	no	—
72	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
75	C₄⋊C₄×C₈₉	—	2	2	no	—
76	C₂²⋊C₄×C₈₉	—	2	2	no	—
77	C₂²⋊C₄×C₈₉	—	2	2	no	—
78	C₄²×C₈₉	—	2	1	no	—
79	C₄²×C₈₉	—	2	1	no	—
80	C₂³×C₈₉	—	2	1	no	—
81	C₂³×C₈₉	—	2	1	no	—
82	C₂³×C₈₉	—	2	1	no	—
83	C₂³×C₈₉	—	2	1	no	—
84	C₂³×C₈₉	—	4	1	no	—
85	C₂³×C₈₉	—	4	1	no	—
86	C₂³×C₈₉	—	4	1	no	—
87	C₂³×C₈₉	—	4	1	no	—
88	C₂³×C₈₉	C₂²⋊C₈	1	2	yes	—
89	C₂³×C₈₉	M₃₂	1	2	yes	complemented
90	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
91	D₄×C₈₉	—	2	8	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	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	1	no	—
100	C₂×C₄×C₈₉	—	4	1	no	—
101	C₂×C₄×C₈₉	—	4	2	no	—
102	C₂×C₄×C₈₉	—	4	2	no	—
103	C₂×C₄×C₈₉	—	4	1	no	—
104	C₂×C₄×C₈₉	—	4	1	no	—
105	C₂×C₄×C₈₉	—	4	1	no	—
106	C₂×C₄×C₈₉	—	4	1	no	—
107	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
108	C₂×C₄×C₈₉	M₃₂	1	2	yes	complemented
109	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
111	C₈×C₈₉	—	2	1	no	—
112	C₈×C₈₉	—	2	2	no	—
113	C₈×C₈₉	—	4	1	no	—
114	C₈×C₈₉	—	4	1	no	—
115	C₈×C₈₉	#99	1	2	yes	—
116	C₂²×C₈₉	—	2	1	no	—
117	C₂²×C₈₉	—	2	1	no	—
118	C₂²×C₈₉	—	2	2	no	—
119	C₂²×C₈₉	—	2	2	no	—
120	C₂²×C₈₉	—	2	2	no	—
121	C₂²×C₈₉	—	2	2	no	—
122	C₂²×C₈₉	—	4	2	no	—
123	C₂²×C₈₉	—	4	2	no	—
124	C₂²×C₈₉	—	4	2	no	—
125	C₂²×C₈₉	—	4	2	no	—
126	C₂²×C₈₉	—	4	1	no	—
127	C₂²×C₈₉	—	4	1	no	—
128	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₂²×C₈₉	C₂×M₃₂	1	1	yes	characteristic
130	C₂²×C₈₉	C₂×M₃₂	1	1	yes	characteristic
131	C₄×C₈₉	—	2	2	no	—
132	C₄×C₈₉	—	2	1	no	—
133	C₄×C₈₉	—	2	2	no	—
134	C₄×C₈₉	—	4	1	no	—
135	C₄×C₈₉	—	4	1	no	—
136	C₄×C₈₉	C₂×#99	1	1	yes	characteristic
137	C₄×C₈₉	#413	1	1	yes	characteristic
138	#11466	—	89	1	no	maximal, Sylow
139	C₂×C₈₉	—	2	1	no	—
140	C₂×C₈₉	—	2	1	no	—
141	C₂×C₈₉	—	4	2	no	—
142	C₂×C₈₉	—	4	2	no	—
143	C₂×C₈₉	#1459	1	1	yes	characteristic
144	C₂×C₈₉	#2112	1	1	yes	characteristic
145	C₂×C₈₉	#2115	1	1	yes	characteristic
146	C₂×C₈×D₄	D₈₉	1	1	yes	characteristic
147	#2112	—	89	2	no	—
148	#1322	—	89	2	no	—
149	#1314	—	89	2	no	—
150	C₈₉	#11466	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₈	Dic₈₉	1	1	yes	characteristic
152	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
153	C₂×C₄×D₄	Dic₈₉	1	1	yes	characteristic
154	C₂×M₃₂	—	178	2	no	—
155	C₈×D₄	—	2	2	no	—
156	C₈×D₄	—	2	2	no	—
157	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
158	C₂×C₂²⋊C₈	Dic₈₉	1	1	yes	characteristic
159	C₂×C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
160	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
161	C₂²⋊C₁₆	—	89	4	no	—
162	C₂²⋊C₁₆	—	178	2	no	—
163	C₂²×D₄	#10623494	1	1	yes	characteristic
164	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
165	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	4	2	no	—
169	C₂²×C₈	—	4	2	no	—
170	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
171	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
172	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
173	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
174	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
175	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
176	C₄×D₄	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₂×C₄⋊C₄	#10623494	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	#10623494	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	#10623494	1	1	yes	characteristic
181	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
182	M₃₂	—	356	4	no	—
183	C₂×C₁₆	—	178	4	no	—
184	C₄⋊C₈	—	2	2	no	—
185	C₂²⋊C₈	—	2	2	no	—
186	C₂²⋊C₈	—	2	2	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	2	1	no	—
189	C₂⁴	C₂×#10623494	1	1	yes	characteristic
190	C₂⁴	C₂×#10623494	1	1	yes	characteristic
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
201	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
202	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
203	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
204	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
205	C₂²×C₄	#13083558	1	2	yes	—
206	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
219	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₁₆	—	356	4	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	4	1	no	—
231	C₂³	—	4	1	no	—
232	C₂³	—	4	1	no	—
233	C₂³	—	4	1	no	—
234	C₂³	C₂²⋊#10623494	1	2	yes	—
235	C₂³	#15667952	1	2	yes	complemented
236	C₂³	C₂²×#10623494	1	1	yes	characteristic
237	D₄	—	2	8	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	C₂²⋊#10623494	1	2	yes	—
254	C₂×C₄	#15667952	1	2	yes	complemented
255	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
256	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₈	#15667927	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	C₂×C₂²⋊#10623494	1	1	yes	characteristic
275	C₂²	C₂×#15667952	1	1	yes	characteristic
276	C₂²	C₂×#15667952	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	C₂×#15667927	1	1	yes	characteristic
283	C₄	#19151582	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	4	2	no	—
287	C₂	—	4	2	no	—
288	C₂	#26018570	1	1	yes	characteristic
289	C₂	#26021863	1	1	yes	characteristic
290	C₂	#26021873	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₈₉
commutator	C₂³×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×C₈×D₄×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₈₉	C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₈×D₄	D₈₉
hypocenter	C₈₉	#11466
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #26021863	C₂	semidirect
2. #26018192	C₂	semidirect
3. #26018176	C₂	semidirect
89. C₂³×C₈₉	M₃₂	semidirect
108. C₂×C₄×C₈₉	M₃₂	semidirect
150. C₈₉	#11466	semidirect
235. C₂³	#15667952	semidirect
254. C₂×C₄	#15667952	semidirect

Maximal subgroups

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

subgroup	index
#26021863	2
#26018192	2
#26018176	2
C₂×C₈×D₄×C₈₉	2
#11466	89