Subgroups — #36447876

Order 22784.

522 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018566	C₂	1	2	yes	maximal
2	#26018085	C₂	1	2	yes	maximal, complemented
3	#26018081	C₂	1	2	yes	maximal, complemented
4	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151651	C₂²	1	4	yes	—
6	#19151324	—	2	2	no	—
7	#19151322	—	2	2	no	—
8	C₂×#117×C₈₉	C₄	1	1	yes	characteristic
9	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₈₉	C₄	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₈₉	—	2	2	no	—
12	C₄×C₂²⋊C₄×C₈₉	—	2	2	no	—
13	#376×C₈₉	C₄	1	1	yes	characteristic
14	#376×C₈₉	C₂²	1	1	yes	characteristic
15	C₄³×C₈₉	C₄	1	1	yes	characteristic
16	C₄³×C₈₉	C₂²	1	1	yes	characteristic
17	C₄⋊#10623494	—	2	4	no	—
18	C₂²⋊#10623494	—	2	4	no	—
19	C₄×#10623494	—	2	4	no	—
20	C₂³×C₄×C₈₉	—	2	1	no	—
21	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
22	#117×C₈₉	—	2	2	no	—
23	#117×C₈₉	—	2	2	no	—
24	#117×C₈₉	—	2	2	no	—
25	#117×C₈₉	C₈	1	4	yes	complemented
26	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
28	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
29	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
30	C₂×C₄²×C₈₉	—	2	1	no	—
31	C₂×C₄²×C₈₉	—	2	2	no	—
32	C₂×C₄²×C₈₉	—	2	2	no	—
33	C₂×C₄²×C₈₉	—	2	2	no	—
34	C₂×C₄²×C₈₉	—	2	2	no	—
35	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
41	#95×C₈₉	—	2	2	no	—
42	#95×C₈₉	—	2	2	no	—
43	C₂×#10623494	—	4	4	no	—
44	C₂⁴×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	—	2	1	no	—
46	C₂²×C₄×C₈₉	—	2	1	no	—
47	C₂²×C₄×C₈₉	—	2	1	no	—
48	C₂²×C₄×C₈₉	—	2	1	no	—
49	C₂²×C₄×C₈₉	—	2	1	no	—
50	C₂²×C₄×C₈₉	—	2	4	no	—
51	C₂²×C₄×C₈₉	—	2	1	no	—
52	C₂²×C₄×C₈₉	—	2	1	no	—
53	C₂²×C₄×C₈₉	—	2	1	no	—
54	C₂²×C₄×C₈₉	—	4	4	no	—
55	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
59	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
61	C₄⋊C₄×C₈₉	—	2	4	no	—
62	C₄⋊C₄×C₈₉	—	2	4	no	—
63	C₂²⋊C₄×C₈₉	—	2	4	no	—
64	C₂²⋊C₄×C₈₉	—	2	4	no	—
65	C₄²×C₈₉	—	2	4	no	—
66	C₄²×C₈₉	—	2	1	no	—
67	C₄²×C₈₉	—	2	1	no	—
68	C₄²×C₈₉	—	2	1	no	—
69	C₄²×C₈₉	—	2	1	no	—
70	C₄²×C₈₉	—	2	1	no	—
71	C₄²×C₈₉	—	2	1	no	—
72	C₄²×C₈₉	—	4	4	no	—
73	C₄²×C₈₉	—	4	4	no	—
74	C₄²×C₈₉	C₂×C₈	1	2	yes	—
75	C₄²×C₈₉	C₂×C₈	1	2	yes	—
76	C₄²×C₈₉	M₁₆	1	2	yes	—
77	C₄²×C₈₉	M₁₆	1	2	yes	—
78	C₄²×C₈₉	#41	1	2	yes	—
79	C₄²×C₈₉	#41	1	2	yes	—
80	C₂³×C₈₉	—	2	1	no	—
81	C₂³×C₈₉	—	2	1	no	—
82	C₂³×C₈₉	—	4	2	no	—
83	C₂³×C₈₉	C₂²⋊C₈	1	2	yes	—
84	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	—	2	4	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	2	no	—
89	C₂×C₄×C₈₉	—	2	2	no	—
90	C₂×C₄×C₈₉	—	2	2	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	4	2	no	—
98	C₂×C₄×C₈₉	—	4	8	no	—
99	C₂×C₄×C₈₉	—	4	2	no	—
100	C₂×C₄×C₈₉	—	4	2	no	—
101	C₂×C₄×C₈₉	—	4	2	no	—
102	C₂×C₄×C₈₉	—	4	2	no	—
103	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#123	1	4	yes	—
107	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
111	#10623494	—	8	4	no	—
112	C₂²×C₈₉	—	2	2	no	—
113	C₂²×C₈₉	—	2	2	no	—
114	C₂²×C₈₉	—	4	2	no	—
115	C₂²×C₈₉	—	4	2	no	—
116	C₂²×C₈₉	—	4	2	no	—
117	C₂²×C₈₉	#381	1	1	yes	characteristic
118	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₈₉	#407	1	1	yes	characteristic
120	C₄×C₈₉	—	2	4	no	—
121	C₄×C₈₉	—	2	2	no	—
122	C₄×C₈₉	—	2	4	no	—
123	C₄×C₈₉	—	4	2	no	—
124	C₄×C₈₉	—	4	2	no	—
125	C₄×C₈₉	—	4	2	no	—
126	C₄×C₈₉	—	4	2	no	—
127	C₄×C₈₉	#431	1	1	yes	characteristic
128	C₄×C₈₉	#431	1	1	yes	characteristic
129	C₄×C₈₉	#432	1	1	yes	characteristic
130	C₄×C₈₉	#432	1	1	yes	characteristic
131	#8731	—	89	1	no	maximal, Sylow
132	C₂×C₈₉	—	2	1	no	—
133	C₂×C₈₉	—	2	1	no	—
134	C₂×C₈₉	—	4	2	no	—
135	C₂×C₈₉	#1817	1	1	yes	characteristic
136	C₂×C₈₉	#1827	1	2	yes	—
137	C₄×#117	D₈₉	1	1	yes	characteristic
138	#1458	—	89	2	no	—
139	#1276	—	89	2	no	—
140	#1274	—	89	2	no	—
141	C₈₉	#8731	1	1	yes	characteristic, complemented, Sylow
142	C₂×#117	Dic₈₉	1	1	yes	characteristic
143	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
144	C₂²×C₄²	Dic₈₉	1	1	yes	characteristic
145	#431	—	89	4	no	—
146	C₄×C₄⋊C₄	—	2	2	no	—
147	C₄×C₂²⋊C₄	—	2	2	no	—
148	#376	Dic₈₉	1	1	yes	characteristic
149	#376	C₂×D₈₉	1	1	yes	characteristic
150	C₄³	Dic₈₉	1	1	yes	characteristic
151	C₄³	C₂×D₈₉	1	1	yes	characteristic
152	#324	—	178	2	no	—
153	#323	—	178	2	no	—
154	C₂³×C₄	—	2	1	no	—
155	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
156	#117	—	2	2	no	—
157	#117	—	2	2	no	—
158	#117	—	2	2	no	—
159	#117	#10623494	1	4	yes	complemented
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
162	C₂×C₂²⋊C₄	—	2	1	no	—
163	C₂×C₂²⋊C₄	C₂²⋊D₈₉	1	2	yes	—
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
170	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
171	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
172	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
175	C₄⋊C₈	—	178	4	no	—
176	C₂²⋊C₈	—	178	4	no	—
177	C₄×C₈	—	178	4	no	—
178	#95	—	2	2	no	—
179	#95	—	2	2	no	—
180	C₂⁴	C₂²⋊Dic₈₉	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
193	C₂²×C₄	C₂×#10623494	1	2	yes	—
194	C₂²×C₄	#13083558	1	2	yes	—
195	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
197	C₂×C₈	—	356	4	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	2	4	no	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	4	4	no	—
210	C₄²	—	4	4	no	—
211	C₄²	C₂×#10623494	1	2	yes	—
212	C₄²	C₂×#10623494	1	2	yes	—
213	C₄²	#13083558	1	2	yes	—
214	C₄²	#13083558	1	2	yes	—
215	C₄²	#13083574	1	2	yes	—
216	C₄²	#13083574	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	4	2	no	—
220	C₂³	C₂²⋊#10623494	1	2	yes	—
221	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	C₂²⋊#10623494	1	2	yes	—
241	C₂×C₄	#15667971	1	1	yes	characteristic
242	C₂×C₄	#15667971	1	1	yes	characteristic
243	C₂×C₄	#15667995	1	4	yes	—
244	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
245	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
246	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
247	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
248	C₈	—	712	4	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	#19151461	1	1	yes	characteristic
255	C₂²	C₂×C₂²⋊#10623494	1	1	yes	characteristic
256	C₂²	#19151562	1	1	yes	characteristic
257	C₄	—	2	4	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	#19151651	1	1	yes	characteristic
265	C₄	#19151651	1	1	yes	characteristic
266	C₄	#19151655	1	1	yes	characteristic
267	C₄	#19151655	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	4	2	no	—
271	C₂	#26020320	1	1	yes	characteristic
272	C₂	#26020356	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15667971
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×#117×C₈₉	C₂
socle	C₂²×C₈₉	#381
prime radical	C₂⁴×C₈₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×#117	D₈₉
hypocenter	C₈₉	#8731
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #26018085	C₂	semidirect
3. #26018081	C₂	semidirect
25. #117×C₈₉	C₈	semidirect
141. C₈₉	#8731	semidirect
159. #117	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26018566	2
#26018085	2
#26018081	2
C₄×#117×C₈₉	2
#8731	89