Subgroups — #36445129

Order 22784.

420 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028894	C₂	1	1	yes	maximal, characteristic
2	#26018635	C₂	1	1	yes	maximal, characteristic
3	#26018635	C₂	1	1	yes	maximal, characteristic
4	#26018535	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018089	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018089	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13083574	C₂²	1	1	yes	characteristic
9	#19151657	—	2	2	no	—
10	#19151651	C₂²	1	1	yes	characteristic
11	#19151651	C₂²	1	1	yes	characteristic
12	#19151649	—	2	1	no	—
13	#19151649	—	2	1	no	—
14	#19151624	C₄	1	2	yes	—
15	#19151564	C₄	1	2	yes	—
16	C₄×#13083558	C₂²	1	1	yes	characteristic
17	#19151330	C₄	1	2	yes	—
18	#19151330	C₄	1	2	yes	—
19	#19151327	C₄	1	2	yes	—
20	#19151327	C₄	1	2	yes	—
21	#19151319	—	2	1	no	—
22	C₈×#10623494	—	2	1	no	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	#322×C₈₉	—	2	2	no	—
27	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
28	C₂×#13083558	C₂×C₄	1	2	yes	—
29	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
31	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
32	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₉	—	2	1	no	—
34	C₄×C₂²⋊D₈₉	—	2	1	no	—
35	#15667971	C₂×C₄	1	1	yes	characteristic
36	#15667969	—	2	1	no	—
37	C₄²×D₈₉	—	2	1	no	—
38	C₄⋊#10623494	C₂×C₄	1	2	yes	—
39	C₄⋊#10623494	C₂×C₄	1	2	yes	—
40	#15667941	—	2	2	no	—
41	#15667941	—	2	2	no	—
42	C₂²⋊#10623494	—	2	1	no	—
43	C₂²⋊#10623494	—	2	1	no	—
44	#15667924	—	2	2	no	—
45	#15667924	—	2	2	no	—
46	#15667923	—	2	1	no	—
47	C₄×#10623494	—	2	1	no	—
48	C₄×#10623494	—	2	1	no	—
49	C₄×#10623494	—	2	1	no	—
50	C₂²×C₈×C₈₉	—	2	1	no	—
51	C₂²×C₈×C₈₉	—	2	1	no	—
52	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₈₉	—	2	1	no	—
55	C₄×C₈×C₈₉	—	2	1	no	—
56	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
59	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₈₉	D₄	1	2	yes	—
61	#13083574	—	2	4	no	—
62	C₂×Dic₁₇₈	C₄²	1	1	yes	characteristic
63	C₂×D₃₅₆	C₄²	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	#13083558	—	2	4	no	—
69	C₂×#10623494	—	2	2	no	—
70	C₂×#10623494	—	2	2	no	—
71	C₂×#10623494	—	2	2	no	—
72	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
73	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
74	#13083553	—	2	1	no	—
75	#13083553	—	2	1	no	—
76	C₂²⋊Dic₈₉	—	2	1	no	—
77	#13083551	—	2	1	no	—
78	#13083551	—	2	1	no	—
79	C₄×Dic₈₉	—	2	1	no	—
80	C₄×Dic₈₉	—	2	1	no	—
81	C₄×Dic₈₉	—	2	1	no	—
82	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
85	C₂×C₈×C₈₉	—	2	4	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	2	4	no	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	4	2	no	—
90	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
91	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂³×C₈₉	#117	1	1	yes	characteristic
97	C₂²×D₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	2	2	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#131	1	4	yes	—
107	C₂²⋊D₈₉	—	4	2	no	—
108	C₂×Dic₈₉	—	2	1	no	—
109	C₂×Dic₈₉	—	2	1	no	—
110	C₂×Dic₈₉	—	2	1	no	—
111	D₃₅₆	—	2	2	no	—
112	C₄×D₈₉	—	4	2	no	—
113	C₄×D₈₉	—	4	2	no	—
114	C₄×D₈₉	—	4	2	no	—
115	Dic₁₇₈	—	2	2	no	—
116	C₈×C₈₉	—	2	4	no	—
117	C₈×C₈₉	—	2	2	no	—
118	C₈×C₈₉	—	2	2	no	—
119	C₈×C₈₉	—	4	2	no	—
120	#10623494	—	4	2	no	—
121	#10623494	—	4	2	no	—
122	#10623494	—	4	2	no	—
123	C₂²×C₈₉	—	2	1	no	—
124	C₂²×C₈₉	—	2	1	no	—
125	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₈₉	#433	1	2	yes	—
127	C₂×D₈₉	—	4	1	no	—
128	C₂×D₈₉	—	4	1	no	—
129	C₂×D₈₉	—	4	1	no	—
130	C₄×C₈₉	—	2	1	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	#405	1	2	yes	—
133	C₄×C₈₉	#408	1	2	yes	—
134	C₄×C₈₉	#443	1	2	yes	—
135	C₄×C₈₉	#444	1	2	yes	—
136	Dic₈₉	—	4	1	no	—
137	Dic₈₉	—	4	1	no	—
138	Dic₈₉	—	4	1	no	—
139	#8189	—	89	1	no	maximal, Sylow
140	C₂×C₈₉	—	2	2	no	—
141	C₂×C₈₉	#1750	1	1	yes	characteristic
142	C₂×C₈₉	#1764	1	1	yes	characteristic
143	C₂×C₈₉	#1765	1	1	yes	characteristic
144	D₈₉	—	8	1	no	—
145	#2922	—	89	1	no	—
146	#1476	—	89	1	no	—
147	#1476	—	89	1	no	—
148	#1450	—	89	1	no	—
149	C₂×#322	D₈₉	1	1	yes	characteristic, complemented
150	#1278	—	89	1	no	—
151	#1278	—	89	1	no	—
152	C₈₉	#8189	1	1	yes	characteristic, complemented, Sylow
153	C₄×#41	—	89	1	no	—
154	#433	—	178	2	no	—
155	#431	—	89	1	no	—
156	#431	—	89	1	no	—
157	#431	—	178	1	no	—
158	#431	—	178	1	no	—
159	#424	—	89	2	no	—
160	#408	—	89	2	no	—
161	C₄×M₁₆	—	89	1	no	—
162	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
165	Q₈⋊C₈	—	89	2	no	—
166	Q₈⋊C₈	—	89	2	no	—
167	D₄⋊C₈	—	89	2	no	—
168	D₄⋊C₈	—	89	2	no	—
169	#322	—	2	2	no	—
170	#322	—	178	1	no	—
171	C₈²	—	178	1	no	—
172	C₂×#41	—	89	1	no	—
173	C₂×M₁₆	—	89	2	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	C₄×D₈₉	1	2	yes	—
177	C₄×Q₈	—	89	1	no	—
178	C₄×Q₈	—	89	1	no	—
179	C₄×D₄	—	89	1	no	—
180	C₄×D₄	—	89	1	no	—
181	C₄×D₄	—	178	1	no	—
182	C₄×D₄	—	178	1	no	—
183	#117	—	89	1	no	—
184	#117	—	178	1	no	—
185	C₂×C₄²	—	178	1	no	—
186	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
187	C₄⋊C₈	—	89	2	no	—
188	C₄⋊C₈	—	89	2	no	—
189	C₄⋊C₈	—	178	2	no	—
190	C₄⋊C₈	—	178	2	no	—
191	C₂²⋊C₈	—	178	1	no	—
192	C₂²⋊C₈	—	178	1	no	—
193	C₂²⋊C₈	—	178	2	no	—
194	C₂²⋊C₈	—	178	2	no	—
195	#97	—	178	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	178	1	no	—
199	C₄×C₈	—	178	1	no	—
200	C₄×C₈	—	178	1	no	—
201	C₄×C₈	C₄×D₈₉	1	2	yes	—
202	C₄×C₈	C₄×D₈₉	1	2	yes	—
203	C₄×C₈	D₃₅₆	1	1	yes	characteristic
204	C₄×C₈	D₃₅₆	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
206	#41	—	178	4	no	—
207	C₂×Q₈	—	89	1	no	—
208	C₂×D₄	—	89	1	no	—
209	C₂×D₄	—	178	1	no	—
210	C₂²×C₄	—	178	1	no	—
211	C₂²×C₄	—	178	1	no	—
212	C₂²×C₄	—	178	1	no	—
213	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
214	C₂²×C₄	#13083574	1	1	yes	characteristic
215	C₂²×C₄	#13083574	1	1	yes	characteristic
216	M₁₆	—	178	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	178	2	no	—
223	C₂×C₈	—	178	2	no	—
224	C₂×C₈	—	178	2	no	—
225	C₂×C₈	#13083551	1	2	yes	—
226	C₂×C₈	#13083551	1	2	yes	—
227	C₄⋊C₄	—	89	1	no	—
228	C₄⋊C₄	—	89	1	no	—
229	C₄⋊C₄	—	178	1	no	—
230	C₄⋊C₄	—	178	1	no	—
231	C₂²⋊C₄	—	178	1	no	—
232	C₂²⋊C₄	—	178	1	no	—
233	C₂²⋊C₄	—	178	1	no	—
234	C₄²	—	178	1	no	—
235	C₄²	—	178	1	no	—
236	C₄²	—	178	1	no	—
237	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
238	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
239	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
240	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
241	C₂³	—	178	1	no	—
242	C₂³	#15667969	1	1	yes	characteristic
243	Q₈	—	178	2	no	—
244	D₄	—	356	2	no	—
245	D₄	—	178	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	356	2	no	—
249	C₂×C₄	—	356	2	no	—
250	C₂×C₄	—	356	2	no	—
251	C₂×C₄	—	178	1	no	—
252	C₂×C₄	—	178	1	no	—
253	C₂×C₄	—	178	1	no	—
254	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
255	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
256	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
257	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
258	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
259	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
260	C₂×C₄	#15668020	1	4	yes	—
261	C₈	—	2	4	no	—
262	C₈	—	2	2	no	—
263	C₈	—	2	2	no	—
264	C₈	—	4	2	no	—
265	C₈	—	356	2	no	—
266	C₈	—	356	2	no	—
267	C₈	—	356	2	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	356	1	no	—
271	C₂²	—	356	1	no	—
272	C₂²	—	356	1	no	—
273	C₂²	C₄×#13083551	1	1	yes	characteristic
274	C₂²	#19151660	1	2	yes	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	356	1	no	—
278	C₄	—	356	1	no	—
279	C₄	—	356	1	no	—
280	C₄	#19151555	1	2	yes	—
281	C₄	#19151566	1	2	yes	—
282	C₄	#19151716	1	2	yes	—
283	C₄	#19151721	1	2	yes	—
284	C₂	—	2	2	no	—
285	C₂	—	712	1	no	—
286	C₂	#26020021	1	1	yes	characteristic
287	C₂	#26020089	1	1	yes	characteristic
288	C₂	#26020096	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#8189
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. #26018535	C₂	semidirect
5. #26018089	C₂	semidirect
6. #26018089	C₂	semidirect
7. C₂×#322×C₈₉	C₂	semidirect
149. C₂×#322	D₈₉	semidirect
152. C₈₉	#8189	semidirect

Maximal subgroups

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

subgroup	index
#26028894	2
#26018635	2
#26018535	2
#26018089	2
C₂×#322×C₈₉	2
#8189	89