Subgroups — #36445451

Order 22784.

444 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018914	C₂	1	1	yes	maximal, characteristic
2	#26018844	C₂	1	1	yes	maximal, characteristic
3	#26018541	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018141	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018141	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2931	C₂	1	1	yes	maximal, characteristic
8	#19151651	C₂²	1	1	yes	characteristic
9	#19151651	C₂²	1	1	yes	characteristic
10	C₄×#13083558	C₂²	1	1	yes	characteristic
11	C₄×#13083558	C₂²	1	1	yes	characteristic
12	#19151347	C₄	1	2	yes	—
13	#19151342	C₄	1	2	yes	—
14	#19151338	C₄	1	4	yes	—
15	#19151320	—	2	1	no	—
16	#19151320	—	2	1	no	—
17	#19151319	—	2	1	no	—
18	C₈×#10623494	—	2	1	no	—
19	C₂×#131×C₈₉	C₄	1	2	yes	—
20	#532×C₈₉	C₂²	1	1	yes	characteristic
21	#436×C₈₉	—	2	1	no	—
22	#436×C₈₉	—	2	1	no	—
23	C₈×D₄×C₈₉	—	2	1	no	—
24	C₈×D₄×C₈₉	—	2	1	no	—
25	#423×C₈₉	C₂²	1	1	yes	characteristic
26	#408×C₈₉	C₄	1	2	yes	—
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#13083558	—	2	1	no	—
29	C₂×#13083558	—	2	1	no	—
30	C₄⋊#10623494	C₂×C₄	1	2	yes	—
31	C₄⋊#10623494	C₂×C₄	1	2	yes	—
32	C₂²⋊#10623494	—	2	1	no	—
33	C₂²⋊#10623494	—	2	1	no	—
34	#15667923	C₂×C₄	1	2	yes	—
35	#15667923	C₂×C₄	1	2	yes	—
36	C₄×#10623494	—	2	1	no	—
37	C₄×#10623494	—	2	1	no	—
38	C₄×#10623494	—	2	1	no	—
39	C₄×#10623494	—	2	1	no	—
40	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
41	#131×C₈₉	—	4	4	no	—
42	C₂×M₁₆×C₈₉	—	2	2	no	—
43	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₈₉	—	2	2	no	—
45	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
48	#127×C₈₉	C₂×C₄	1	1	yes	characteristic
49	#124×C₈₉	C₂×C₄	1	2	yes	—
50	#121×C₈₉	—	2	1	no	—
51	#121×C₈₉	—	2	1	no	—
52	C₄×D₄×C₈₉	—	2	1	no	—
53	C₄×D₄×C₈₉	—	2	1	no	—
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₈₉	—	2	1	no	—
56	C₄⋊C₈×C₈₉	—	2	1	no	—
57	C₂²⋊C₈×C₈₉	—	2	2	no	—
58	C₂²⋊C₈×C₈₉	—	2	2	no	—
59	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
63	#13083558	—	2	4	no	—
64	#13083558	—	4	2	no	—
65	C₂×#10623494	—	2	2	no	—
66	C₂×#10623494	—	2	2	no	—
67	C₂×#10623494	—	2	2	no	—
68	C₂×#10623494	—	2	2	no	—
69	#41×C₈₉	—	4	2	no	—
70	#41×C₈₉	—	4	2	no	—
71	C₂×Q₈×C₈₉	C₄²	1	2	yes	—
72	C₂×D₄×C₈₉	—	2	2	no	—
73	C₂×D₄×C₈₉	—	2	1	no	—
74	C₂×D₄×C₈₉	—	2	1	no	—
75	C₂×D₄×C₈₉	C₄²	1	2	yes	—
76	C₂²×C₄×C₈₉	—	2	2	no	—
77	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
80	M₁₆×C₈₉	—	4	4	no	—
81	M₁₆×C₈₉	—	4	2	no	—
82	C₂×C₈×C₈₉	—	2	2	no	—
83	C₂×C₈×C₈₉	—	2	2	no	—
84	C₂×C₈×C₈₉	—	4	1	no	—
85	C₂×C₈×C₈₉	—	4	4	no	—
86	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₈₉	—	2	1	no	—
91	C₄⋊C₄×C₈₉	—	2	1	no	—
92	C₂²⋊C₄×C₈₉	—	2	2	no	—
93	C₂²⋊C₄×C₈₉	—	2	2	no	—
94	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₈₉	—	2	2	no	—
99	C₂³×C₈₉	#117	1	1	yes	characteristic
100	Q₈×C₈₉	—	4	2	no	—
101	D₄×C₈₉	—	4	2	no	—
102	D₄×C₈₉	—	4	4	no	—
103	D₄×C₈₉	—	4	2	no	—
104	D₄×C₈₉	—	4	2	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	4	2	no	—
111	C₂×C₄×C₈₉	—	4	2	no	—
112	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	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	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
118	C₈×C₈₉	—	2	1	no	—
119	C₈×C₈₉	—	2	1	no	—
120	C₈×C₈₉	—	2	1	no	—
121	C₈×C₈₉	—	4	2	no	—
122	C₈×C₈₉	#100	1	2	yes	—
123	#10623494	—	2	4	no	—
124	#10623494	—	4	2	no	—
125	#10623494	—	4	2	no	—
126	#10623494	—	4	2	no	—
127	C₂²×C₈₉	—	2	1	no	—
128	C₂²×C₈₉	—	2	1	no	—
129	C₂²×C₈₉	—	2	1	no	—
130	C₂²×C₈₉	—	4	2	no	—
131	C₂²×C₈₉	—	4	2	no	—
132	C₂²×C₈₉	—	4	2	no	—
133	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	—	4	2	no	—
138	C₄×C₈₉	C₂×#100	1	1	yes	characteristic
139	C₄×C₈₉	#413	1	1	yes	characteristic
140	C₄×C₈₉	#443	1	2	yes	—
141	C₄×C₈₉	#444	1	2	yes	—
142	#8245	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	—	4	1	no	—
144	C₂×C₈₉	—	8	2	no	—
145	C₂×C₈₉	C₄×#100	1	1	yes	characteristic
146	C₂×C₈₉	#1764	1	1	yes	characteristic
147	C₂×C₈₉	#1765	1	1	yes	characteristic
148	#2931	D₈₉	1	1	yes	characteristic
149	#1543	—	89	1	no	—
150	#1525	—	89	1	no	—
151	#1451	—	89	1	no	—
152	C₈×M₁₆	—	89	1	no	—
153	#1299	—	89	1	no	—
154	#1299	—	89	1	no	—
155	C₈₉	#8245	1	1	yes	characteristic, complemented, Sylow
156	C₂×#131	Dic₈₉	1	2	yes	—
157	#532	C₂×D₈₉	1	1	yes	characteristic
158	#436	—	2	1	no	—
159	#436	—	2	1	no	—
160	C₈×D₄	—	2	1	no	—
161	C₈×D₄	—	2	1	no	—
162	#431	—	89	1	no	—
163	#431	—	89	1	no	—
164	#423	C₂×D₈₉	1	1	yes	characteristic
165	#408	Dic₈₉	1	2	yes	—
166	C₄×M₁₆	—	89	1	no	—
167	C₄×M₁₆	—	89	1	no	—
168	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
169	#333	—	89	2	no	—
170	#331	—	89	2	no	—
171	#329	—	89	4	no	—
172	#322	—	178	1	no	—
173	#322	—	178	1	no	—
174	#322	—	178	1	no	—
175	C₈²	—	178	1	no	—
176	C₂×#41	C₂×Dic₈₉	1	2	yes	—
177	#131	—	4	4	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	178	1	no	—
180	C₂×M₁₆	—	178	1	no	—
181	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
184	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
185	C₄⋊Q₈	C₄×D₈₉	1	1	yes	characteristic
186	#127	C₄×D₈₉	1	1	yes	characteristic
187	#124	C₄×D₈₉	1	2	yes	—
188	#121	—	2	1	no	—
189	#121	—	2	1	no	—
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	—	2	1	no	—
192	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	89	2	no	—
196	C₄⋊C₈	—	89	2	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	C₂²⋊C₈	—	2	2	no	—
199	C₂²⋊C₈	—	178	1	no	—
200	C₂²⋊C₈	—	178	1	no	—
201	#97	—	89	2	no	—
202	#97	—	89	2	no	—
203	C₄×C₈	—	178	1	no	—
204	C₄×C₈	—	178	1	no	—
205	C₄×C₈	—	178	1	no	—
206	C₄×C₈	—	178	1	no	—
207	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
208	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
209	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
210	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
211	#41	—	4	2	no	—
212	#41	—	4	2	no	—
213	C₂×Q₈	C₄×Dic₈₉	1	2	yes	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	C₄×Dic₈₉	1	2	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
220	C₂²×C₄	#13083574	1	1	yes	characteristic
221	C₂²×C₄	#13083574	1	1	yes	characteristic
222	M₁₆	—	4	4	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	356	2	no	—
225	M₁₆	—	178	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	178	2	no	—
231	C₂×C₈	—	178	2	no	—
232	C₂×C₈	—	178	2	no	—
233	C₂×C₈	—	178	2	no	—
234	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
237	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
243	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
244	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
245	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
246	C₂³	—	2	2	no	—
247	C₂³	#15667971	1	1	yes	characteristic
248	Q₈	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
261	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
262	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
264	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	356	2	no	—
271	C₈	—	356	2	no	—
272	C₈	—	356	2	no	—
273	C₈	—	178	4	no	—
274	C₈	#15667929	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	2	no	—
286	C₄	C₂×#15667929	1	1	yes	characteristic
287	C₄	#19151582	1	1	yes	characteristic
288	C₄	#19151713	1	2	yes	—
289	C₄	#19151718	1	2	yes	—
290	C₂	—	4	1	no	—
291	C₂	—	8	2	no	—
292	C₂	C₄×#15667929	1	1	yes	characteristic
293	C₂	#26020087	1	1	yes	characteristic
294	C₂	#26020094	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#2931	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	#532×C₈₉	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#2931	D₈₉
hypocenter	C₈₉	#8245
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
3. #26018541	C₂	semidirect
4. C₈×#13083558	C₂	semidirect
5. #26018141	C₂	semidirect
6. #26018141	C₂	semidirect
155. C₈₉	#8245	semidirect

Maximal subgroups

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

subgroup	index
#26018914	2
#26018844	2
#26018541	2
C₈×#13083558	2
#26018141	2
C₈₉×#2931	2
#8245	89