Subgroups — #36469525

Order 22784.

410 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152014	C₂	1	1	yes	maximal, characteristic
2	#26021872	C₂	1	1	yes	maximal, characteristic
3	#26021848	C₂	1	1	yes	maximal, characteristic, complemented
4	#26020053	C₂	1	1	yes	maximal, characteristic
5	#26018339	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018338	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	C₂²	1	1	yes	characteristic
9	#19152014	—	2	1	no	—
10	#19152014	—	2	1	no	—
11	#19152014	—	2	1	no	—
12	#19152014	—	2	1	no	—
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	C₂×#15667952	C₂²	1	1	yes	characteristic
15	C₂×#15667951	C₄	1	2	yes	complemented
16	C₂×C₁₆×D₈₉	C₄	1	2	yes	complemented
17	#19151580	C₄	1	1	yes	characteristic
18	#19151571	C₄	1	1	yes	characteristic
19	#19151553	C₂²	1	1	yes	characteristic
20	#19151379	C₄	1	4	yes	complemented
21	#19151373	—	2	1	no	—
22	C₁₆×Dic₈₉	—	2	1	no	—
23	#19151370	C₄	1	1	yes	characteristic
24	#19151368	C₄	1	1	yes	characteristic
25	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	#405×C₈₉	C₂²	1	1	yes	characteristic
28	#346×C₈₉	—	2	1	no	—
29	C₄×C₁₆×C₈₉	—	2	1	no	—
30	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
31	#15668018	—	2	1	no	—
32	#15668018	—	2	1	no	—
33	#15668018	—	2	1	no	—
34	#15668018	—	2	1	no	—
35	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
36	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
37	C₂×#13083557	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
39	#15667971	C₂×C₄	1	1	yes	characteristic
40	#15667952	—	2	1	no	—
41	#15667952	—	2	1	no	—
42	#15667952	—	2	1	no	—
43	#15667952	C₂×C₄	1	2	yes	—
44	#15667951	—	2	2	no	—
45	#15667951	—	2	2	no	—
46	C₂×#13083549	—	2	1	no	—
47	C₂×#13083549	C₂×C₄	1	2	yes	—
48	C₁₆×D₈₉	—	2	2	no	—
49	C₁₆×D₈₉	—	2	2	no	—
50	#15667930	C₈	1	2	yes	—
51	#15667928	C₈	1	2	yes	—
52	#15667926	C₈	1	4	yes	—
53	#15667922	—	2	1	no	—
54	C₈×Dic₈₉	—	2	1	no	—
55	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
58	M₃₂×C₈₉	—	2	1	no	—
59	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
60	C₂×C₁₆×C₈₉	—	2	1	no	—
61	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
63	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
64	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
65	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
66	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
67	#97×C₈₉	—	2	1	no	—
68	C₄×C₈×C₈₉	—	2	1	no	—
69	#13083574	—	2	1	no	—
70	#13083574	—	2	1	no	—
71	#13083574	—	2	1	no	—
72	#13083574	—	2	1	no	—
73	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
74	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
75	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
76	C₂×C₄×D₈₉	C₄²	1	2	yes	—
77	#13083558	—	2	1	no	—
78	#13083558	—	2	1	no	—
79	#13083558	—	2	1	no	—
80	#13083558	C₂×C₈	1	2	yes	—
81	#13083557	—	2	2	no	—
82	#13083557	—	2	2	no	—
83	C₂×#10623494	—	2	1	no	—
84	C₂×#10623494	C₄²	1	2	yes	—
85	C₈×D₈₉	—	2	2	no	—
86	C₈×D₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
89	C₄×Dic₈₉	—	2	1	no	—
90	#13083549	—	2	2	no	—
91	#13083549	—	2	2	no	—
92	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	M₁₆×C₈₉	—	2	1	no	—
94	M₁₆×C₈₉	C₂×C₈	1	2	yes	—
95	C₂×C₈×C₈₉	—	2	1	no	—
96	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
101	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	—	2	1	no	—
103	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
104	C₄²×C₈₉	—	2	1	no	—
105	C₁₆×C₈₉	—	2	2	no	—
106	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
107	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
108	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
109	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
110	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
111	C₂²×D₈₉	C₄×C₈	1	2	yes	—
112	C₂×C₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
119	C₂²⋊D₈₉	—	2	2	no	—
120	C₂²⋊D₈₉	—	2	2	no	—
121	C₂×Dic₈₉	—	2	2	no	—
122	C₂×Dic₈₉	C₄×C₈	1	2	yes	—
123	D₃₅₆	—	2	1	no	—
124	D₃₅₆	—	2	1	no	—
125	C₄×D₈₉	—	2	2	no	—
126	C₄×D₈₉	—	2	2	no	—
127	Dic₁₇₈	—	2	1	no	—
128	Dic₁₇₈	—	2	1	no	—
129	C₈×C₈₉	—	2	2	no	—
130	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₈×C₈₉	#117	1	1	yes	characteristic
132	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
133	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
134	#10623494	—	2	2	no	—
135	#10623494	—	2	2	no	—
136	C₂²×C₈₉	—	4	1	no	—
137	C₂²×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
138	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
139	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
140	C₂×D₈₉	—	2	2	no	—
141	C₂×D₈₉	—	4	2	no	—
142	C₄×C₈₉	—	4	2	no	—
143	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₈₉	#405	1	1	yes	characteristic
145	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
146	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
147	Dic₈₉	—	2	2	no	—
148	Dic₈₉	—	4	2	no	—
149	#12613	—	89	1	no	maximal, Sylow
150	C₂×C₈₉	—	2	1	no	—
151	C₂×C₈₉	—	2	1	no	—
152	C₂×C₈₉	—	2	1	no	—
153	C₂×C₈₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
154	D₈₉	—	4	2	no	—
155	C₂×#504	—	89	1	no	—
156	#2115	—	89	1	no	—
157	#2108	—	89	1	no	—
158	#2108	D₈₉	1	1	yes	characteristic, complemented
159	#1757	—	89	1	no	—
160	#1378	—	89	1	no	—
161	#1378	—	89	1	no	—
162	C₈₉	#12613	1	1	yes	characteristic, complemented, Sylow
163	C₂×#131	—	89	1	no	—
164	#504	—	178	1	no	—
165	#504	—	178	1	no	—
166	#504	—	178	1	no	—
167	#504	—	178	1	no	—
168	C₂×M₃₂	—	89	1	no	—
169	C₂×M₃₂	—	89	2	no	—
170	C₂×M₃₂	—	89	1	no	—
171	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
172	C₂²×C₁₆	—	89	2	no	—
173	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
174	#413	—	89	1	no	—
175	#410	—	89	1	no	—
176	#405	—	89	1	no	—
177	#405	C₂×D₈₉	1	1	yes	characteristic
178	#349	—	89	4	no	—
179	#346	—	2	1	no	—
180	#346	—	178	1	no	—
181	C₄×C₁₆	—	2	1	no	—
182	C₄×C₁₆	—	178	1	no	—
183	#344	—	89	1	no	—
184	#344	—	89	1	no	—
185	C₂×#41	—	89	1	no	—
186	#131	—	178	1	no	—
187	#131	—	178	1	no	—
188	#131	—	178	1	no	—
189	#131	—	178	1	no	—
190	C₂×M₁₆	—	89	1	no	—
191	C₂×M₁₆	—	89	2	no	—
192	C₂×M₁₆	—	89	1	no	—
193	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
194	C₂²×C₈	—	89	2	no	—
195	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
196	#117	—	89	1	no	—
197	#117	C₄×D₈₉	1	1	yes	characteristic
198	M₃₂	—	2	1	no	—
199	M₃₂	—	89	2	no	—
200	M₃₂	—	178	1	no	—
201	M₃₂	—	178	1	no	—
202	M₃₂	—	178	1	no	—
203	M₃₂	—	178	2	no	—
204	M₃₂	—	178	2	no	—
205	M₃₂	C₄×D₈₉	1	2	yes	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	89	2	no	—
208	C₂×C₁₆	—	178	1	no	—
209	C₂×C₁₆	—	178	2	no	—
210	C₂×C₁₆	—	178	2	no	—
211	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
212	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
213	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
214	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
215	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
216	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
217	#101	—	89	2	no	—
218	#100	—	89	2	no	—
219	#99	—	89	4	no	—
220	#97	—	2	1	no	—
221	#97	—	178	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	178	1	no	—
224	#41	—	178	1	no	—
225	#41	—	178	1	no	—
226	#41	—	178	1	no	—
227	#41	—	178	1	no	—
228	C₂×Q₈	—	89	1	no	—
229	C₂×D₄	—	89	2	no	—
230	C₂×D₄	—	89	1	no	—
231	C₂²×C₄	—	89	2	no	—
232	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
233	M₁₆	—	2	1	no	—
234	M₁₆	—	89	2	no	—
235	M₁₆	—	178	1	no	—
236	M₁₆	—	178	1	no	—
237	M₁₆	—	178	1	no	—
238	M₁₆	—	178	2	no	—
239	M₁₆	—	178	2	no	—
240	M₁₆	C₈×D₈₉	1	2	yes	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	89	2	no	—
243	C₂×C₈	—	178	1	no	—
244	C₂×C₈	—	178	2	no	—
245	C₂×C₈	—	178	2	no	—
246	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
249	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
250	C₂×C₈	#13083574	1	1	yes	characteristic
251	C₂×C₈	#13083574	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	178	1	no	—
254	C₂²⋊C₄	—	89	2	no	—
255	C₂²⋊C₄	C₈×D₈₉	1	2	yes	—
256	C₄²	—	2	1	no	—
257	C₄²	—	178	1	no	—
258	C₁₆	—	2	2	no	—
259	C₁₆	—	178	2	no	—
260	C₁₆	—	178	2	no	—
261	C₁₆	#13083551	1	1	yes	characteristic
262	C₁₆	#13083551	1	1	yes	characteristic
263	C₁₆	#13083551	1	1	yes	characteristic
264	C₁₆	#13083551	1	1	yes	characteristic
265	C₂³	—	89	2	no	—
266	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
267	Q₈	—	178	1	no	—
268	Q₈	—	178	1	no	—
269	D₄	—	178	2	no	—
270	D₄	—	178	2	no	—
271	D₄	—	178	1	no	—
272	D₄	—	178	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	89	2	no	—
275	C₂×C₄	—	178	2	no	—
276	C₂×C₄	—	178	2	no	—
277	C₂×C₄	—	178	2	no	—
278	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
279	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
280	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
281	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
282	C₂×C₄	#15668018	1	1	yes	characteristic
283	C₂×C₄	#15668018	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	178	2	no	—
286	C₈	—	178	2	no	—
287	C₈	C₂×#13083551	1	1	yes	characteristic
288	C₈	#15667969	1	1	yes	characteristic
289	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
290	C₈	C₄×D₃₅₆	1	1	yes	characteristic
291	C₂²	—	4	1	no	—
292	C₂²	—	356	2	no	—
293	C₂²	—	178	2	no	—
294	C₂²	C₄×C₈×D₈₉	1	1	yes	characteristic
295	C₂²	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
296	C₂²	C₈×D₃₅₆	1	1	yes	characteristic
297	C₄	—	4	2	no	—
298	C₄	—	356	2	no	—
299	C₄	—	178	2	no	—
300	C₄	C₄×#13083551	1	1	yes	characteristic
301	C₄	#19151552	1	1	yes	characteristic
302	C₄	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
303	C₄	C₈×D₃₅₆	1	1	yes	characteristic
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	356	2	no	—
308	C₂	C₈×#13083551	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#13083551
commutator	C₂²×C₈₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2108	C₂
socle	C₂×C₈₉	C₈×C₂²⋊C₄
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2108	D₈₉
hypocenter	C₈₉	#12613
LCS	G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ #2108

Complements

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

N	H ≅ G/N	type
3. #26021848	C₂	semidirect
5. #26018339	C₂	semidirect
6. #26018338	C₂	semidirect
7. C₈₉×#2108	C₂	semidirect
15. C₂×#15667951	C₄	semidirect
16. C₂×C₁₆×D₈₉	C₄	semidirect
20. #19151379	C₄	semidirect
158. #2108	D₈₉	semidirect
162. C₈₉	#12613	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152014	2
#26021872	2
#26021848	2
#26020053	2
#26018339	2
#26018338	2
C₈₉×#2108	2
#12613	89