Subgroups — #36469538

Order 22784.

418 conjugacy classes of subgroups.

The subgroup lattice has 294 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	C₂×#19151381	C₂	1	1	yes	maximal, characteristic
3	#26021848	C₂	1	1	yes	maximal, characteristic, complemented
4	#26020089	C₂	1	1	yes	maximal, characteristic
5	#26018340	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018334	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	C₂²	1	1	yes	characteristic
9	#19152014	—	2	2	no	—
10	#19152014	C₄	1	4	yes	complemented
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	C₂×#15667951	—	2	1	no	—
14	C₂×C₁₆×D₈₉	—	2	1	no	—
15	C₂×#15667940	C₄	1	1	yes	characteristic
16	#19151595	C₄	1	1	yes	characteristic
17	#19151553	C₂²	1	1	yes	characteristic
18	#19151381	C₄	1	4	yes	complemented
19	#19151373	—	2	1	no	—
20	C₁₆×Dic₈₉	—	2	1	no	—
21	#19151363	C₄	1	1	yes	characteristic
22	#19151358	C₄	1	1	yes	characteristic
23	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	#346×C₈₉	—	2	2	no	—
27	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
28	#15668018	—	2	1	no	—
29	#15668018	—	2	1	no	—
30	#15668018	C₂×C₄	1	2	yes	—
31	#15668018	C₂×C₄	1	2	yes	—
32	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
33	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
34	C₂×#13083557	—	2	1	no	—
35	C₂×C₈×D₈₉	—	2	1	no	—
36	#15667971	C₂×C₄	1	1	yes	characteristic
37	#15667952	—	2	1	no	—
38	#15667952	—	2	1	no	—
39	#15667952	C₂×C₄	1	2	yes	—
40	#15667952	C₂×C₄	1	2	yes	—
41	#15667951	—	2	4	no	—
42	C₂×#13083549	—	2	1	no	—
43	C₂×#13083549	—	2	1	no	—
44	C₁₆×D₈₉	—	2	4	no	—
45	#15667940	C₈	1	4	yes	—
46	#15667937	C₈	1	2	yes	—
47	#15667934	C₈	1	2	yes	—
48	#15667922	—	2	1	no	—
49	C₈×Dic₈₉	—	2	1	no	—
50	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₈₉	—	2	1	no	—
54	C₂×C₁₆×C₈₉	—	2	1	no	—
55	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
56	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
57	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
58	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
59	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
60	C₄×C₈×C₈₉	—	2	1	no	—
61	C₄×C₈×C₈₉	—	2	1	no	—
62	#13083574	—	2	1	no	—
63	#13083574	—	2	1	no	—
64	#13083574	C₄²	1	2	yes	—
65	#13083574	C₂×C₈	1	2	yes	—
66	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
67	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	#13083558	—	2	1	no	—
71	#13083558	—	2	1	no	—
72	#13083558	C₄²	1	2	yes	—
73	#13083558	C₂×C₈	1	2	yes	—
74	#13083557	—	2	2	no	—
75	#13083557	—	2	2	no	—
76	C₂×#10623494	—	2	1	no	—
77	C₂×#10623494	—	2	1	no	—
78	C₈×D₈₉	—	2	2	no	—
79	C₈×D₈₉	—	2	2	no	—
80	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
81	C₂²⋊Dic₈₉	—	2	1	no	—
82	C₄×Dic₈₉	—	2	1	no	—
83	#13083549	—	2	2	no	—
84	#13083549	—	2	2	no	—
85	C₂²×C₄×C₈₉	—	2	1	no	—
86	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	—	2	1	no	—
88	C₂×C₈×C₈₉	—	2	1	no	—
89	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
90	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
95	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
96	C₄²×C₈₉	—	2	1	no	—
97	C₄²×C₈₉	C₂×C₈	1	2	yes	—
98	C₁₆×C₈₉	—	2	2	no	—
99	C₁₆×C₈₉	C₂²⋊C₄	1	2	yes	—
100	C₁₆×C₈₉	C₂²⋊C₄	1	2	yes	—
101	C₂³×C₈₉	#131	1	1	yes	characteristic
102	C₂²×D₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	—	4	2	no	—
105	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
111	C₂²⋊D₈₉	—	2	2	no	—
112	C₂²⋊D₈₉	—	2	2	no	—
113	C₂×Dic₈₉	—	2	1	no	—
114	C₂×Dic₈₉	—	2	2	no	—
115	D₃₅₆	—	2	1	no	—
116	D₃₅₆	C₄×C₈	1	2	yes	—
117	C₄×D₈₉	—	2	2	no	—
118	C₄×D₈₉	—	2	2	no	—
119	Dic₁₇₈	—	2	1	no	—
120	Dic₁₇₈	C₄×C₈	1	2	yes	—
121	C₈×C₈₉	—	2	2	no	—
122	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₈×C₈₉	#117	1	1	yes	characteristic
124	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
125	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
126	#10623494	—	2	2	no	—
127	#10623494	—	2	2	no	—
128	C₂²×C₈₉	—	2	2	no	—
129	C₂²×C₈₉	#405	1	1	yes	characteristic
130	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
131	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
132	C₂×D₈₉	—	2	2	no	—
133	C₂×D₈₉	—	4	2	no	—
134	C₄×C₈₉	—	4	2	no	—
135	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
137	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
138	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
139	Dic₈₉	—	2	2	no	—
140	Dic₈₉	—	4	2	no	—
141	#12615	—	89	1	no	maximal, Sylow
142	C₂×C₈₉	—	2	1	no	—
143	C₂×C₈₉	—	2	1	no	—
144	C₂×C₈₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₈₉	#2171	1	2	yes	—
146	D₈₉	—	4	2	no	—
147	C₂×#504	—	89	1	no	—
148	C₂×#350	—	89	1	no	—
149	#2108	—	89	1	no	—
150	C₂×#346	D₈₉	1	1	yes	characteristic, complemented
151	#1764	—	89	1	no	—
152	#1379	—	89	1	no	—
153	#1376	—	89	1	no	—
154	C₈₉	#12615	1	1	yes	characteristic, complemented, Sylow
155	C₂×#131	—	89	1	no	—
156	#504	—	89	4	no	—
157	#504	—	178	2	no	—
158	C₂×M₃₂	—	89	1	no	—
159	C₂×M₃₂	—	89	1	no	—
160	C₂×M₃₂	—	178	1	no	—
161	C₂²×C₁₆	—	178	1	no	—
162	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
163	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
164	C₂×#104	—	89	1	no	—
165	#416	—	89	1	no	—
166	#405	—	89	1	no	—
167	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
168	#350	—	89	4	no	—
169	#346	—	2	2	no	—
170	#346	—	178	1	no	—
171	C₄×C₁₆	—	178	1	no	—
172	#341	—	89	1	no	—
173	#339	—	89	1	no	—
174	C₂×#41	—	89	1	no	—
175	#131	—	89	2	no	—
176	#131	—	89	2	no	—
177	#131	—	178	1	no	—
178	#131	—	178	1	no	—
179	C₂×M₁₆	—	89	1	no	—
180	C₂×M₁₆	—	89	1	no	—
181	C₂×M₁₆	—	178	1	no	—
182	C₂²×C₈	—	178	1	no	—
183	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
185	#117	—	89	1	no	—
186	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
187	M₃₂	—	89	2	no	—
188	M₃₂	—	89	2	no	—
189	M₃₂	—	178	1	no	—
190	M₃₂	—	178	1	no	—
191	M₃₂	—	178	4	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	178	1	no	—
195	C₂×C₁₆	—	178	1	no	—
196	C₂×C₁₆	—	178	4	no	—
197	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
198	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
199	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
200	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
201	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
202	#104	—	89	4	no	—
203	#103	—	89	2	no	—
204	#102	—	89	2	no	—
205	#97	—	178	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	178	1	no	—
209	#41	—	89	2	no	—
210	#41	—	89	2	no	—
211	#41	—	178	1	no	—
212	#41	—	178	1	no	—
213	C₂×Q₈	—	89	1	no	—
214	C₂×D₄	—	89	1	no	—
215	C₂×D₄	—	178	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	178	1	no	—
218	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
219	M₁₆	—	89	2	no	—
220	M₁₆	—	89	2	no	—
221	M₁₆	—	178	1	no	—
222	M₁₆	—	178	1	no	—
223	M₁₆	—	178	2	no	—
224	M₁₆	—	178	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	178	1	no	—
228	C₂×C₈	—	178	1	no	—
229	C₂×C₈	—	178	2	no	—
230	C₂×C₈	—	178	2	no	—
231	C₂×C₈	C₈×D₈₉	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
234	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
235	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
236	C₂×C₈	#13083574	1	1	yes	characteristic
237	C₂×C₈	#13083574	1	1	yes	characteristic
238	C₄⋊C₄	—	89	2	no	—
239	C₂²⋊C₄	—	178	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	178	1	no	—
242	C₄²	C₈×D₈₉	1	2	yes	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	178	2	no	—
245	C₁₆	—	178	2	no	—
246	C₁₆	#13083551	1	2	yes	—
247	C₁₆	#13083551	1	2	yes	—
248	C₂³	—	178	1	no	—
249	C₂³	#15668018	1	1	yes	characteristic
250	Q₈	—	89	2	no	—
251	Q₈	—	178	1	no	—
252	D₄	—	89	2	no	—
253	D₄	—	178	2	no	—
254	D₄	—	178	2	no	—
255	D₄	—	178	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	178	2	no	—
259	C₂×C₄	—	178	1	no	—
260	C₂×C₄	—	178	2	no	—
261	C₂×C₄	—	178	2	no	—
262	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
264	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
266	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
267	C₂×C₄	#15668018	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	178	2	no	—
270	C₈	—	178	2	no	—
271	C₈	C₂×#13083551	1	1	yes	characteristic
272	C₈	#15667969	1	1	yes	characteristic
273	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
274	C₈	C₄×D₃₅₆	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	356	2	no	—
277	C₂²	—	178	2	no	—
278	C₂²	#19151552	1	1	yes	characteristic
279	C₂²	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
280	C₂²	C₈×D₃₅₆	1	1	yes	characteristic
281	C₄	—	4	2	no	—
282	C₄	—	356	2	no	—
283	C₄	—	178	2	no	—
284	C₄	C₄×#13083551	1	1	yes	characteristic
285	C₄	C₄×C₈×D₈₉	1	1	yes	characteristic
286	C₄	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
287	C₄	C₈×D₃₅₆	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	356	2	no	—
291	C₂	C₈×#13083551	1	1	yes	characteristic
292	C₂	#26022095	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle	C₂²×C₈₉	C₈×D₄
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	#12615
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #26021848	C₂	semidirect
5. #26018340	C₂	semidirect
6. #26018334	C₂	semidirect
7. C₂×#346×C₈₉	C₂	semidirect
10. #19152014	C₄	semidirect
18. #19151381	C₄	semidirect
150. C₂×#346	D₈₉	semidirect
154. C₈₉	#12615	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
C₂×#19151381	2
#26021848	2
#26020089	2
#26018340	2
#26018334	2
C₂×#346×C₈₉	2
#12615	89