Subgroups — #36707075

Order 22784.

818 conjugacy classes of subgroups.

The subgroup lattice has 284 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	2	yes	maximal
2	C₄×#15668018	C₂	1	1	yes	maximal, characteristic
3	#26021850	C₂	1	4	yes	maximal, complemented
4	#26021848	C₂	1	2	yes	maximal, complemented
5	C₄×#15667952	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021845	C₂	1	2	yes	maximal, complemented
7	#346×D₈₉	C₂	1	2	yes	maximal, complemented
8	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668018	C₄	1	1	yes	characteristic
10	C₂×#15668018	C₂²	1	1	yes	characteristic
11	C₄×#13083574	C₄	1	1	yes	characteristic
12	#19152014	C₄	1	16	yes	complemented
13	C₂×#15667952	C₂²	1	2	yes	—
14	C₂×#15667951	C₂²	1	4	yes	—
15	C₂×C₁₆×D₈₉	C₂²	1	4	yes	—
16	#19151553	C₄	1	1	yes	characteristic
17	#19151553	C₂²	1	1	yes	characteristic
18	#19151552	C₄	1	2	yes	—
19	#19151552	C₂²	1	2	yes	—
20	C₄×#13083558	C₄	1	1	yes	characteristic
21	C₄×#13083557	C₄	1	2	yes	—
22	C₄×C₈×D₈₉	C₂²	1	2	yes	—
23	#19151374	C₂²	1	2	yes	—
24	#19151373	C₂²	1	4	yes	—
25	C₄×#13083549	C₂²	1	2	yes	—
26	C₁₆×Dic₈₉	C₂²	1	4	yes	—
27	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	#346×C₈₉	C₂²	1	1	yes	characteristic
30	#346×C₈₉	C₂²	1	1	yes	characteristic
31	#346×C₈₉	C₂²	1	2	yes	—
32	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
33	#15668018	C₈	1	8	yes	—
34	#15668018	C₂×C₄	1	8	yes	—
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×#13083557	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₈₉	C₂³	1	2	yes	—
41	C₄×Dic₁₇₈	C₈	1	2	yes	—
42	C₄×D₃₅₆	C₈	1	2	yes	—
43	C₄×C₂²⋊D₈₉	C₈	1	4	yes	—
44	#15667971	C₂×C₄	1	1	yes	characteristic
45	#15667969	C₂×C₄	1	2	yes	—
46	C₄²×D₈₉	C₂×C₄	1	2	yes	—
47	#15667952	C₂×C₄	1	8	yes	complemented
48	#15667951	C₂×C₄	1	16	yes	complemented
49	C₂×#13083549	C₂³	1	4	yes	—
50	C₁₆×D₈₉	C₂×C₄	1	16	yes	complemented
51	#15667923	C₂×C₄	1	2	yes	—
52	#15667922	C₂×C₄	1	2	yes	—
53	#15667922	C₂×C₄	1	2	yes	—
54	C₄×#10623494	C₂×C₄	1	2	yes	—
55	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
56	C₈×Dic₈₉	C₂³	1	2	yes	—
57	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₂×C₁₆×C₈₉	C₂×C₄	1	8	yes	complemented
61	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
62	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
63	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	#13083574	C₄²	1	4	yes	—
68	#13083574	C₂×C₈	1	4	yes	—
69	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
70	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
71	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₈₉	C₂×C₈	1	4	yes	—
73	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
74	#13083558	C₄²	1	4	yes	—
75	#13083558	C₂×C₈	1	4	yes	—
76	#13083557	C₄²	1	8	yes	—
77	#13083557	C₂×C₈	1	8	yes	—
78	C₂×#10623494	C₂²×C₄	1	2	yes	—
79	C₂×#10623494	C₂²×C₄	1	2	yes	—
80	C₈×D₈₉	C₂×C₈	1	8	yes	—
81	C₈×D₈₉	C₂²×C₄	1	8	yes	—
82	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
83	#13083553	C₂×C₈	1	4	yes	—
84	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
85	#13083551	C₂×C₈	1	4	yes	—
86	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
87	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
88	#13083549	C₂²×C₄	1	8	yes	—
89	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
90	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
92	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
97	C₄²×C₈₉	C₂×C₈	1	2	yes	—
98	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
101	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
102	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₈₉	—	4	2	no	—
105	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
111	C₂²⋊D₈₉	C₄×C₈	1	8	yes	—
112	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
113	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
114	D₃₅₆	C₄×C₈	1	4	yes	—
115	C₄×D₈₉	—	4	4	no	—
116	C₄×D₈₉	C₂×C₄²	1	4	yes	—
117	C₄×D₈₉	C₂²×C₈	1	4	yes	—
118	Dic₁₇₈	C₄×C₈	1	4	yes	—
119	C₈×C₈₉	C₂×C₄²	1	2	yes	—
120	C₈×C₈₉	C₂²×C₈	1	2	yes	—
121	C₈×C₈₉	C₂²×C₈	1	2	yes	—
122	C₈×C₈₉	C₂³×C₄	1	2	yes	—
123	#10623494	C₂×C₄²	1	4	yes	—
124	#10623494	C₂²×C₈	1	4	yes	—
125	C₂²×C₈₉	—	2	2	no	—
126	C₂²×C₈₉	C₂×C₄×C₈	1	2	yes	—
127	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
128	C₂×D₈₉	—	2	4	no	—
129	C₂×D₈₉	C₂×C₄×C₈	1	4	yes	—
130	C₄×C₈₉	—	2	2	no	—
131	C₄×C₈₉	—	2	2	no	—
132	C₄×C₈₉	C₂×C₄×C₈	1	2	yes	—
133	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
134	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
135	Dic₈₉	—	2	4	no	—
136	Dic₈₉	C₂×C₄×C₈	1	4	yes	—
137	#33326	—	89	1	no	maximal, Sylow
138	C₂×C₈₉	—	2	2	no	—
139	C₂×C₈₉	C₂²×C₄×C₈	1	1	yes	characteristic
140	C₂×C₈₉	#3407	1	2	yes	—
141	D₈₉	—	2	4	no	—
142	C₂×#504	—	89	2	no	—
143	C₄×#131	—	89	1	no	—
144	#2108	—	89	2	no	—
145	#2108	—	89	4	no	—
146	C₄×M₃₂	—	89	1	no	—
147	C₄×M₃₂	—	89	2	no	—
148	C₂×#346	—	89	2	no	—
149	C₂×#346	D₈₉	1	1	yes	characteristic, complemented
150	C₈₉	#33326	1	1	yes	characteristic, complemented, Sylow
151	C₂×#131	—	89	1	no	—
152	C₂×#131	—	89	1	no	—
153	C₄×#41	—	89	1	no	—
154	#504	—	89	16	no	—
155	C₂×M₃₂	—	89	2	no	—
156	C₂×M₃₂	—	89	4	no	—
157	C₂²×C₁₆	—	89	4	no	—
158	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
159	#405	—	89	1	no	—
160	#405	—	89	1	no	—
161	#405	—	89	2	no	—
162	#405	—	89	2	no	—
163	C₄×M₁₆	—	89	1	no	—
164	C₄×M₁₆	—	89	2	no	—
165	C₂×C₄×C₈	—	89	2	no	—
166	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
167	#346	—	89	4	no	—
168	#346	—	89	2	no	—
169	#346	C₂×D₈₉	1	1	yes	characteristic
170	#346	C₂×D₈₉	1	1	yes	characteristic
171	#346	C₂×D₈₉	1	2	yes	—
172	C₄×C₁₆	—	89	4	no	—
173	C₄×C₁₆	—	89	2	no	—
174	C₂×#41	—	89	1	no	—
175	#131	—	89	8	no	—
176	#131	—	89	8	no	—
177	C₂×M₁₆	—	89	1	no	—
178	C₂×M₁₆	—	89	1	no	—
179	C₂×M₁₆	—	89	2	no	—
180	C₂×M₁₆	—	89	2	no	—
181	C₂²×C₈	—	89	2	no	—
182	C₂²×C₈	—	89	2	no	—
183	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
185	C₄×Q₈	—	89	2	no	—
186	C₄×D₄	—	89	4	no	—
187	C₄×D₄	—	89	2	no	—
188	#117	—	89	1	no	—
189	#117	—	89	2	no	—
190	C₂×C₄²	—	89	2	no	—
191	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
192	M₃₂	—	89	8	no	—
193	M₃₂	—	89	16	no	—
194	C₂×C₁₆	—	89	4	no	—
195	C₂×C₁₆	—	89	16	no	—
196	C₂×C₁₆	C₄×D₈₉	1	8	yes	complemented
197	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
198	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
199	#97	—	89	2	no	—
200	#97	—	89	2	no	—
201	#97	—	89	2	no	—
202	C₄×C₈	—	89	2	no	—
203	C₄×C₈	—	89	2	no	—
204	C₄×C₈	—	89	2	no	—
205	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
206	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
209	#41	—	89	4	no	—
210	#41	—	89	4	no	—
211	C₂×Q₈	—	89	1	no	—
212	C₂×D₄	—	89	2	no	—
213	C₂×D₄	—	89	1	no	—
214	C₂²×C₄	—	89	2	no	—
215	C₂²×C₄	—	89	4	no	—
216	C₂²×C₄	C₈×D₈₉	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
218	M₁₆	—	89	4	no	—
219	M₁₆	—	89	4	no	—
220	M₁₆	—	89	8	no	—
221	M₁₆	—	89	8	no	—
222	C₂×C₈	—	89	2	no	—
223	C₂×C₈	—	89	2	no	—
224	C₂×C₈	—	89	8	no	—
225	C₂×C₈	—	89	8	no	—
226	C₂×C₈	C₈×D₈₉	1	4	yes	—
227	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
230	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
232	C₄⋊C₄	—	89	4	no	—
233	C₄⋊C₄	—	89	2	no	—
234	C₂²⋊C₄	—	89	2	no	—
235	C₂²⋊C₄	—	89	4	no	—
236	C₄²	—	89	4	no	—
237	C₄²	—	89	2	no	—
238	C₄²	C₈×D₈₉	1	2	yes	—
239	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
240	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
241	C₁₆	—	89	8	no	—
242	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
243	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
244	C₂³	—	89	2	no	—
245	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
246	Q₈	—	89	4	no	—
247	D₄	—	89	8	no	—
248	D₄	—	89	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	89	2	no	—
251	C₂×C₄	—	89	4	no	—
252	C₂×C₄	—	89	4	no	—
253	C₂×C₄	—	89	4	no	—
254	C₂×C₄	—	356	4	no	—
255	C₂×C₄	C₄²×D₈₉	1	2	yes	—
256	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
257	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
258	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
259	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
260	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
261	C₈	—	89	4	no	—
262	C₈	—	89	4	no	—
263	C₈	C₄²×D₈₉	1	2	yes	—
264	C₈	C₂×C₈×D₈₉	1	2	yes	—
265	C₈	C₂×C₈×D₈₉	1	2	yes	—
266	C₈	C₂²×C₄×D₈₉	1	2	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	89	4	no	—
269	C₂²	—	178	4	no	—
270	C₂²	C₄×C₈×D₈₉	1	2	yes	—
271	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	89	4	no	—
275	C₄	—	178	4	no	—
276	C₄	C₄×C₈×D₈₉	1	2	yes	—
277	C₄	C₂×C₄²×D₈₉	1	1	yes	characteristic
278	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	178	4	no	—
281	C₂	C₂×C₄×C₈×D₈₉	1	1	yes	characteristic
282	C₂	#26033354	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator	C₂×C₈₉	C₂²×C₄×C₈
Fitting	C₂×#346×C₈₉	C₂
socle	C₂²×C₈₉	C₂³×C₈
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	#33326
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
3. #26021850	C₂	semidirect
4. #26021848	C₂	semidirect
5. C₄×#15667952	C₂	semidirect
6. #26021845	C₂	semidirect
7. #346×D₈₉	C₂	semidirect
8. C₂×#346×C₈₉	C₂	semidirect
12. #19152014	C₄	semidirect
47. #15667952	C₂×C₄	semidirect
48. #15667951	C₂×C₄	semidirect
50. C₁₆×D₈₉	C₂×C₄	semidirect
60. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
149. C₂×#346	D₈₉	semidirect
150. C₈₉	#33326	semidirect
196. C₂×C₁₆	C₄×D₈₉	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
C₄×#15668018	2
#26021850	2
#26021848	2
C₄×#15667952	2
#26021845	2
#346×D₈₉	2
C₂×#346×C₈₉	2
#33326	89