Subgroups — #36520148

Order 22784.

814 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029218	C₂	1	2	yes	maximal
2	C₄×#15668018	C₂	1	1	yes	maximal, characteristic
3	#26018554	C₂	1	2	yes	maximal, complemented
4	#26018546	C₂	1	2	yes	maximal, complemented
5	#26018541	C₂	1	2	yes	maximal, complemented
6	#26018535	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#13083557	C₂	1	4	yes	maximal, complemented
8	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668018	C₄	1	2	yes	—
10	C₄×#13083574	C₂²	1	1	yes	characteristic
11	#19151728	C₄	1	4	yes	—
12	#19151677	C₄	1	4	yes	—
13	#19151669	C₄	1	8	yes	—
14	#19151651	C₂²	1	2	yes	—
15	#19151649	C₂²	1	4	yes	—
16	#19151553	C₄	1	2	yes	—
17	#19151552	C₄	1	4	yes	—
18	C₄×#13083558	C₂²	1	1	yes	characteristic
19	C₄×#13083557	C₂²	1	2	yes	—
20	C₄×#13083557	C₂²	1	4	yes	—
21	C₄×C₈×D₈₉	C₂²	1	2	yes	—
22	#19151320	C₂²	1	4	yes	—
23	#19151319	C₂²	1	4	yes	—
24	#19151319	C₂²	1	2	yes	—
25	C₈×#10623494	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
28	C₈²×C₈₉	C₂²	1	2	yes	—
29	C₈²×C₈₉	C₂²	1	1	yes	characteristic
30	C₈²×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
32	#15668018	—	2	8	no	—
33	C₂×#13083558	C₂×C₄	1	2	yes	—
34	C₂×#13083557	C₂×C₄	1	8	yes	—
35	C₂×#13083557	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
38	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
39	C₄×C₂²⋊D₈₉	C₂×C₄	1	4	yes	—
40	#15667971	C₂×C₄	1	1	yes	characteristic
41	#15667969	C₂×C₄	1	2	yes	—
42	C₄²×D₈₉	C₂³	1	2	yes	—
43	C₄⋊#10623494	C₂×C₄	1	4	yes	—
44	#15667941	C₂×C₄	1	8	yes	—
45	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
46	#15667924	C₂×C₄	1	8	yes	—
47	#15667923	C₂×C₄	1	2	yes	—
48	#15667922	C₂×C₄	1	8	yes	—
49	#15667922	C₂×C₄	1	4	yes	—
50	C₄×#10623494	C₂³	1	4	yes	—
51	C₄×#10623494	C₂³	1	2	yes	—
52	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₈₉	C₂³	1	2	yes	—
59	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₈₉	C₂³	1	2	yes	—
61	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	#13083574	—	2	4	no	—
63	C₂×Dic₁₇₈	C₄²	1	1	yes	characteristic
64	C₂×D₃₅₆	C₄²	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₉	C₄²	1	2	yes	—
66	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
67	C₂×C₄×D₈₉	C₂²×C₄	1	4	yes	—
68	#13083558	—	2	4	no	—
69	#13083557	—	2	8	no	—
70	#13083557	—	2	16	no	—
71	C₂×#10623494	C₂²×C₄	1	4	yes	—
72	C₂×#10623494	C₂²×C₄	1	8	yes	—
73	C₈×D₈₉	—	2	8	no	—
74	C₄⋊Dic₈₉	C₄²	1	2	yes	—
75	#13083553	C₄²	1	4	yes	—
76	C₂²⋊Dic₈₉	C₄²	1	2	yes	—
77	#13083551	C₄²	1	4	yes	—
78	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
79	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₈₉	—	2	8	no	—
83	C₂×C₈×C₈₉	M₁₆	1	8	yes	complemented
84	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
88	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
90	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
91	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
92	C₂²×D₈₉	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₈₉	—	2	4	no	—
94	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
95	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
97	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
99	C₂²⋊D₈₉	—	2	4	no	—
100	C₂×Dic₈₉	C₂×C₄²	1	4	yes	—
101	C₂×Dic₈₉	C₂×C₄²	1	2	yes	—
102	D₃₅₆	—	2	2	no	—
103	C₄×D₈₉	—	2	4	no	—
104	C₄×D₈₉	—	2	8	no	—
105	Dic₁₇₈	—	2	2	no	—
106	C₈×C₈₉	—	2	4	no	—
107	C₈×C₈₉	C₂×M₁₆	1	4	yes	—
108	C₈×C₈₉	C₂×M₁₆	1	4	yes	—
109	C₈×C₈₉	#131	1	8	yes	complemented
110	#10623494	—	2	4	no	—
111	#10623494	—	2	8	no	—
112	C₂²×C₈₉	—	2	2	no	—
113	C₂²×C₈₉	C₄×M₁₆	1	2	yes	—
114	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
115	C₂×D₈₉	—	2	2	no	—
116	C₂×D₈₉	—	2	4	no	—
117	C₄×C₈₉	—	2	2	no	—
118	C₄×C₈₉	C₄×M₁₆	1	2	yes	—
119	C₄×C₈₉	C₂²×M₁₆	1	2	yes	—
120	C₄×C₈₉	C₂×#131	1	4	yes	—
121	Dic₈₉	—	2	2	no	—
122	Dic₈₉	—	2	4	no	—
123	#17335	—	89	1	no	maximal, Sylow
124	C₂×C₈₉	—	2	2	no	—
125	C₂×C₈₉	C₂×C₄×M₁₆	1	1	yes	characteristic
126	C₂×C₈₉	C₄×#131	1	2	yes	—
127	D₈₉	—	4	2	no	—
128	#2965	—	89	2	no	—
129	C₄×#131	—	89	1	no	—
130	#1455	—	89	2	no	—
131	#1453	—	89	2	no	—
132	#1451	—	89	2	no	—
133	#1450	—	89	1	no	—
134	C₈×M₁₆	—	89	4	no	—
135	C₂×C₈²	D₈₉	1	1	yes	characteristic, complemented
136	C₈₉	#17335	1	1	yes	characteristic, complemented, Sylow
137	C₂×#131	—	89	2	no	—
138	C₄×#41	—	89	1	no	—
139	#446	—	89	4	no	—
140	#436	—	89	4	no	—
141	#435	—	89	8	no	—
142	#431	—	89	2	no	—
143	#431	—	89	4	no	—
144	#405	—	89	2	no	—
145	#405	—	89	4	no	—
146	C₄×M₁₆	—	89	1	no	—
147	C₄×M₁₆	—	89	2	no	—
148	C₄×M₁₆	—	89	4	no	—
149	C₂×C₄×C₈	—	89	2	no	—
150	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
151	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
152	#322	—	89	4	no	—
153	#322	—	89	4	no	—
154	#322	—	89	2	no	—
155	C₈²	—	89	2	no	—
156	C₈²	C₂×D₈₉	1	1	yes	characteristic
157	C₈²	C₂×D₈₉	1	1	yes	characteristic
158	C₈²	C₂×D₈₉	1	2	yes	—
159	C₂×#41	—	89	1	no	—
160	#131	—	178	8	no	—
161	C₂×M₁₆	—	89	2	no	—
162	C₂×M₁₆	—	89	4	no	—
163	C₂×M₁₆	—	89	8	no	—
164	C₂²×C₈	—	89	4	no	—
165	C₂²×C₈	C₄×D₈₉	1	4	yes	—
166	C₂²×C₈	C₄×D₈₉	1	2	yes	—
167	C₄×Q₈	—	89	2	no	—
168	C₄×D₄	—	89	4	no	—
169	C₄×D₄	—	89	2	no	—
170	#117	—	89	1	no	—
171	#117	—	89	2	no	—
172	C₂×C₄²	—	89	2	no	—
173	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
174	C₄⋊C₈	—	89	8	no	—
175	C₄⋊C₈	—	89	4	no	—
176	C₂²⋊C₈	—	89	4	no	—
177	C₂²⋊C₈	—	89	8	no	—
178	#97	—	89	4	no	—
179	#97	—	89	8	no	—
180	#97	—	89	2	no	—
181	C₄×C₈	—	89	4	no	—
182	C₄×C₈	—	89	4	no	—
183	C₄×C₈	—	89	2	no	—
184	C₄×C₈	C₄×D₈₉	1	2	yes	—
185	C₄×C₈	C₄×D₈₉	1	4	yes	—
186	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₈₉	1	2	yes	—
188	C₄×C₈	C₂²×D₈₉	1	2	yes	—
189	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
190	#41	—	178	4	no	—
191	C₂×Q₈	—	89	1	no	—
192	C₂×D₄	—	89	2	no	—
193	C₂×D₄	—	89	1	no	—
194	C₂²×C₄	—	89	4	no	—
195	C₂²×C₄	—	89	2	no	—
196	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
198	M₁₆	—	178	4	no	—
199	M₁₆	—	178	8	no	—
200	M₁₆	—	178	16	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	89	4	no	—
203	C₂×C₈	—	89	8	no	—
204	C₂×C₈	—	178	8	no	—
205	C₂×C₈	#13083557	1	8	yes	complemented
206	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
207	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
208	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
209	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
210	C₄⋊C₄	—	89	4	no	—
211	C₄⋊C₄	—	89	2	no	—
212	C₂²⋊C₄	—	89	2	no	—
213	C₂²⋊C₄	—	89	4	no	—
214	C₄²	—	89	2	no	—
215	C₄²	—	89	4	no	—
216	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
217	C₄²	C₂×C₄×D₈₉	1	2	yes	—
218	C₄²	C₂³×D₈₉	1	1	yes	characteristic
219	C₂³	—	89	2	no	—
220	C₂³	C₄²×D₈₉	1	1	yes	characteristic
221	Q₈	—	178	2	no	—
222	D₄	—	178	4	no	—
223	D₄	—	178	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	89	4	no	—
226	C₂×C₄	—	89	2	no	—
227	C₂×C₄	—	178	4	no	—
228	C₂×C₄	—	178	8	no	—
229	C₂×C₄	C₄²×D₈₉	1	2	yes	—
230	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
231	C₂×C₄	C₂×#13083557	1	4	yes	—
232	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₄×D₈₉	1	2	yes	—
234	C₈	—	2	4	no	—
235	C₈	—	178	4	no	—
236	C₈	—	178	8	no	—
237	C₈	C₂×#13083557	1	4	yes	—
238	C₈	C₂×#13083557	1	4	yes	—
239	C₈	#15668018	1	8	yes	complemented
240	C₂²	—	2	2	no	—
241	C₂²	—	178	2	no	—
242	C₂²	—	178	4	no	—
243	C₂²	C₄×#13083557	1	2	yes	—
244	C₂²	C₂×C₄²×D₈₉	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	178	2	no	—
247	C₄	—	178	4	no	—
248	C₄	C₄×#13083557	1	2	yes	—
249	C₄	C₂²×#13083557	1	2	yes	—
250	C₄	C₂×#15668018	1	4	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	356	2	no	—
253	C₂	C₂×C₄×#13083557	1	1	yes	characteristic
254	C₂	C₄×#15668018	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₈₉
commutator, socle	C₂²×C₈₉	C₂²×C₄²
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	#17335
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #26018554	C₂	semidirect
4. #26018546	C₂	semidirect
5. #26018541	C₂	semidirect
6. #26018535	C₂	semidirect
7. C₈×#13083557	C₂	semidirect
8. C₂×C₈²×C₈₉	C₂	semidirect
83. C₂×C₈×C₈₉	M₁₆	semidirect
109. C₈×C₈₉	#131	semidirect
135. C₂×C₈²	D₈₉	semidirect
136. C₈₉	#17335	semidirect
205. C₂×C₈	#13083557	semidirect
239. C₈	#15668018	semidirect

Maximal subgroups

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

subgroup	index
#26029218	2
C₄×#15668018	2
#26018554	2
#26018546	2
#26018541	2
#26018535	2
C₈×#13083557	2
C₂×C₈²×C₈₉	2
#17335	89