Subgroups — #36708092

Order 22784.

444 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029184	C₂	1	1	yes	maximal, characteristic
2	#26022160	C₂	1	2	yes	maximal, complemented
3	#26022159	C₂	1	2	yes	maximal, complemented
4	#26022158	C₂	1	2	yes	maximal, complemented
5	#26022157	C₂	1	2	yes	maximal, complemented
6	#26022059	C₂	1	1	yes	maximal, characteristic
7	#26022056	C₂	1	2	yes	maximal
8	#26022052	C₂	1	2	yes	maximal
9	C₈₉×#2164	C₂	1	1	yes	maximal, characteristic
10	#19152151	C₄	1	1	yes	characteristic
11	#19151726	C₄	1	2	yes	—
12	#19151726	C₄	1	2	yes	—
13	C₈×Dic₁₇₈	C₂²	1	2	yes	—
14	C₈×Dic₁₇₈	C₂²	1	2	yes	—
15	#19151626	C₄	1	1	yes	characteristic
16	#19151625	C₄	1	1	yes	characteristic
17	#19151621	C₄	1	1	yes	characteristic
18	#19151553	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
20	C₄⋊#13083549	C₂²	1	2	yes	—
21	#19151416	C₂²	1	2	yes	—
22	#19151416	C₂²	1	2	yes	—
23	#19151416	C₂²	1	2	yes	—
24	#19151416	C₂²	1	2	yes	—
25	C₂²⋊#13083549	C₂²	1	2	yes	—
26	#19151374	C₂²	1	1	yes	characteristic
27	#19151373	C₂²	1	2	yes	—
28	#19151373	C₂²	1	2	yes	—
29	#19151373	C₂²	1	2	yes	—
30	C₄×#13083549	C₂²	1	1	yes	characteristic
31	C₁₆×Dic₈₉	C₂²	1	2	yes	—
32	#405×C₈₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
34	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
35	#346×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
38	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
39	#15668011	C₈	1	2	yes	—
40	#15668003	C₈	1	2	yes	—
41	#15667989	C₈	1	2	yes	—
42	#15667989	C₈	1	2	yes	—
43	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
45	#15667971	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×#13083549	C₂³	1	2	yes	—
48	C₂×#13083549	C₂³	1	2	yes	—
49	C₄⋊#10623494	C₂×C₄	1	2	yes	—
50	C₄⋊#10623494	C₂×C₄	1	2	yes	—
51	#15667941	C₂×C₄	1	2	yes	—
52	#15667941	C₂×C₄	1	2	yes	—
53	#15667941	C₂×C₄	1	2	yes	—
54	#15667941	C₂×C₄	1	2	yes	—
55	#15667922	C₂×C₄	1	2	yes	—
56	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
57	C₈×Dic₈₉	C₂³	1	2	yes	—
58	C₈×Dic₈₉	C₂³	1	2	yes	—
59	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
63	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
64	#97×C₈₉	C₂×C₄	1	1	yes	characteristic
65	#97×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
69	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
70	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
71	#13083558	—	4	1	no	—
72	C₂×#10623494	—	4	1	no	—
73	C₂×#10623494	C₂²×C₄	1	2	yes	—
74	C₂×#10623494	C₂²×C₄	1	2	yes	—
75	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
76	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
77	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
78	#13083553	C₂×C₈	1	2	yes	—
79	#13083553	C₂×C₈	1	2	yes	—
80	#13083553	C₂×C₈	1	2	yes	—
81	#13083553	C₂×C₈	1	2	yes	—
82	C₂²⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
83	C₂²⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
84	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
85	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
86	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
87	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
88	#13083549	—	2	2	no	—
89	#13083549	—	2	2	no	—
90	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	M₁₆×C₈₉	—	4	1	no	—
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
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	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
98	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
99	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₁₆×C₈₉	—	2	2	no	—
102	C₁₆×C₈₉	—	2	2	no	—
103	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
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	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₈₉	—	4	2	no	—
111	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
112	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
113	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
114	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
115	Dic₁₇₈	—	4	2	no	—
116	Dic₁₇₈	—	4	2	no	—
117	C₈×C₈₉	—	2	1	no	—
118	C₈×C₈₉	—	2	1	no	—
119	C₈×C₈₉	—	2	1	no	—
120	C₈×C₈₉	2₋¹⁺⁴	1	2	yes	—
121	#10623494	—	2	2	no	—
122	#10623494	—	2	2	no	—
123	C₂²×C₈₉	—	2	2	no	—
124	C₂²×C₈₉	—	2	1	no	—
125	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	2	2	no	—
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	#519	1	1	yes	characteristic
130	C₄×C₈₉	#568	1	1	yes	characteristic
131	Dic₈₉	—	2	2	no	—
132	Dic₈₉	—	2	2	no	—
133	Dic₈₉	—	2	2	no	—
134	Dic₈₉	—	2	2	no	—
135	#33501	—	89	1	no	maximal, Sylow
136	C₂×C₈₉	—	4	1	no	—
137	C₂×C₈₉	#2982	1	1	yes	characteristic
138	C₂×C₈₉	#3407	1	2	yes	—
139	#2960	—	89	1	no	—
140	#2183	—	89	2	no	—
141	#2183	—	89	2	no	—
142	#2183	—	89	2	no	—
143	#2183	—	89	2	no	—
144	#2164	—	89	1	no	—
145	#2164	—	89	2	no	—
146	#2164	D₈₉	1	1	yes	characteristic
147	#2163	—	89	2	no	—
148	C₈₉	#33501	1	1	yes	characteristic, complemented, Sylow
149	#533	—	89	1	no	—
150	#446	—	89	2	no	—
151	#446	—	89	2	no	—
152	C₈×Q₈	—	89	2	no	—
153	C₈×Q₈	—	89	2	no	—
154	#424	—	89	1	no	—
155	#424	—	89	1	no	—
156	#423	—	89	1	no	—
157	#405	—	89	1	no	—
158	#405	C₂×D₈₉	1	1	yes	characteristic
159	C₂×C₄×C₈	—	89	1	no	—
160	C₄⋊C₁₆	—	89	2	no	—
161	C₄⋊C₁₆	—	89	2	no	—
162	C₄⋊C₁₆	—	89	2	no	—
163	C₄⋊C₁₆	—	89	2	no	—
164	C₄⋊C₁₆	—	89	2	no	—
165	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
166	C₂²⋊C₁₆	—	89	2	no	—
167	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
168	#346	—	89	2	no	—
169	#346	—	89	2	no	—
170	#346	—	89	2	no	—
171	#346	—	89	1	no	—
172	#346	C₂×D₈₉	1	1	yes	characteristic
173	C₄×C₁₆	—	89	2	no	—
174	C₄×C₁₆	—	89	1	no	—
175	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
176	C₂×M₁₆	—	89	1	no	—
177	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
178	C₂²×C₈	—	89	1	no	—
179	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
180	C₄⋊Q₈	—	89	2	no	—
181	#125	—	89	2	no	—
182	C₂²⋊Q₈	—	89	2	no	—
183	C₂²⋊Q₈	—	89	2	no	—
184	C₄×Q₈	—	89	2	no	—
185	C₄×Q₈	—	89	2	no	—
186	#117	—	89	1	no	—
187	#117	C₄×D₈₉	1	1	yes	characteristic
188	C₂×C₄²	—	89	1	no	—
189	C₂×C₁₆	—	89	2	no	—
190	C₂×C₁₆	—	89	2	no	—
191	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
192	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
193	C₄⋊C₈	—	89	2	no	—
194	C₄⋊C₈	—	89	2	no	—
195	C₄⋊C₈	—	89	2	no	—
196	C₄⋊C₈	—	89	2	no	—
197	C₄⋊C₈	—	89	2	no	—
198	C₄⋊C₈	—	89	2	no	—
199	#97	—	89	2	no	—
200	#97	C₄×D₈₉	1	1	yes	characteristic
201	#97	C₄×D₈₉	1	1	yes	characteristic
202	C₄×C₈	—	89	2	no	—
203	C₄×C₈	—	89	2	no	—
204	C₄×C₈	—	89	2	no	—
205	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
207	C₂×Q₈	—	89	2	no	—
208	C₂×Q₈	—	89	2	no	—
209	C₂²×C₄	—	89	2	no	—
210	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
211	M₁₆	—	4	1	no	—
212	M₁₆	—	356	1	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	89	2	no	—
215	C₂×C₈	—	89	2	no	—
216	C₂×C₈	—	356	1	no	—
217	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
218	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
219	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
221	C₄⋊C₄	—	89	2	no	—
222	C₄⋊C₄	—	89	2	no	—
223	C₄⋊C₄	—	89	2	no	—
224	C₄⋊C₄	—	89	2	no	—
225	C₄⋊C₄	—	89	2	no	—
226	C₄⋊C₄	—	89	2	no	—
227	C₄⋊C₄	—	89	2	no	—
228	C₄⋊C₄	C₈×D₈₉	1	2	yes	—
229	C₂²⋊C₄	—	89	1	no	—
230	C₂²⋊C₄	—	89	1	no	—
231	C₂²⋊C₄	C₈×D₈₉	1	2	yes	—
232	C₄²	—	89	1	no	—
233	C₄²	—	89	1	no	—
234	C₄²	—	89	2	no	—
235	C₄²	—	89	2	no	—
236	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
238	C₁₆	—	2	2	no	—
239	C₁₆	—	2	2	no	—
240	C₁₆	—	178	2	no	—
241	C₁₆	—	178	2	no	—
242	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
243	Q₈	—	356	2	no	—
244	Q₈	—	356	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	89	2	no	—
248	C₂×C₄	—	89	2	no	—
249	C₂×C₄	—	89	2	no	—
250	C₂×C₄	—	89	2	no	—
251	C₂×C₄	—	356	2	no	—
252	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
253	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
254	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
255	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	178	2	no	—
260	C₈	—	178	2	no	—
261	C₈	#15668059	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	178	2	no	—
269	C₄	—	178	2	no	—
270	C₄	—	178	2	no	—
271	C₄	—	178	2	no	—
272	C₄	#19152075	1	1	yes	characteristic
273	C₄	#19152364	1	1	yes	characteristic
274	C₂	—	4	1	no	—
275	C₂	#26029332	1	1	yes	characteristic
276	C₂	#26033353	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2164	C₂
prime radical	C₂³×C₈₉	C₂²×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2164	D₈₉
hypocenter	C₈₉	#33501
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2164

Complements

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

N	H ≅ G/N	type
2. #26022160	C₂	semidirect
3. #26022159	C₂	semidirect
4. #26022158	C₂	semidirect
5. #26022157	C₂	semidirect
148. C₈₉	#33501	semidirect

Maximal subgroups

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

subgroup	index
#26029184	2
#26022160	2
#26022159	2
#26022158	2
#26022157	2
#26022059	2
#26022056	2
#26022052	2
C₈₉×#2164	2
#33501	89