Subgroups — #36563065

Order 22784.

1006 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030698	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151716	C₂	1	4	yes	maximal, complemented
3	#26028965	C₂	1	2	yes	maximal, complemented
4	#26021145	C₂	1	4	yes	maximal, complemented
5	#26021084	C₂	1	2	yes	maximal, complemented
6	#26020427	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	C₂²	1	4	yes	—
9	C₂×#15668018	C₂²	1	4	yes	—
10	#19152148	C₂²	1	2	yes	—
11	#19151992	C₄	1	4	yes	complemented
12	#19151987	C₄	1	4	yes	complemented
13	#19151716	—	2	16	no	—
14	#19151677	—	2	4	no	—
15	C₈×D₃₅₆	—	2	4	no	—
16	C₂×#15667948	C₂²	1	4	yes	—
17	#19151622	C₂²	1	2	yes	—
18	C₂×#15667940	C₂²	1	8	yes	—
19	#19151564	C₂²	1	4	yes	—
20	C₂×#97×C₈₉	C₂²	1	2	yes	—
21	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#376×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#13083574	C₂³	1	4	yes	—
26	#15668034	—	2	8	no	—
27	#15668034	—	2	8	no	—
28	C₂×Dic₃₅₆	C₂×C₄	1	4	yes	—
29	C₂×#13083563	C₂×C₄	1	8	yes	—
30	C₂×D₇₁₂	C₂×C₄	1	4	yes	—
31	#15668018	—	2	16	no	—
32	C₂×#13083558	C₂³	1	4	yes	—
33	C₂×#13083557	—	2	4	no	—
34	C₂×C₈×D₈₉	—	2	4	no	—
35	#15668012	C₂×C₄	1	2	yes	—
36	#15668009	C₂×C₄	1	2	yes	—
37	#15668000	C₂×C₄	1	4	yes	—
38	#15667986	—	2	4	no	—
39	C₄×D₃₅₆	—	2	4	no	—
40	#15667948	—	2	8	no	—
41	C₄⋊#10623494	—	2	4	no	—
42	#15667940	—	2	16	no	—
43	#15667924	—	2	8	no	—
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
46	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	#97×C₈₉	—	2	4	no	—
50	#97×C₈₉	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₈₉	D₄	1	2	yes	—
52	C₄×C₈×C₈₉	D₄	1	2	yes	—
53	C₄×C₈×C₈₉	D₄	1	8	yes	complemented
54	#95×C₈₉	—	2	2	no	—
55	#13083574	—	2	8	no	—
56	#13083574	—	2	8	no	—
57	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
58	C₂×D₃₅₆	—	2	4	no	—
59	C₂×D₃₅₆	C₂²×C₄	1	4	yes	—
60	C₂×C₂²⋊D₈₉	—	2	4	no	—
61	C₂×C₄×D₈₉	—	2	4	no	—
62	Dic₃₅₆	—	2	8	no	—
63	#13083563	—	2	16	no	—
64	D₇₁₂	—	2	8	no	—
65	#13083558	—	2	8	no	—
66	#13083557	—	4	8	no	—
67	C₂×#10623494	—	2	4	no	—
68	C₈×D₈₉	—	4	8	no	—
69	C₄⋊Dic₈₉	—	2	4	no	—
70	#13083551	—	2	8	no	—
71	C₂²×C₄×C₈₉	—	2	2	no	—
72	C₂²×C₄×C₈₉	#41	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₈₉	—	2	2	no	—
75	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₈₉	#41	1	4	yes	—
82	C₄²×C₈₉	—	2	2	no	—
83	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
84	C₄²×C₈₉	C₂×D₄	1	4	yes	—
85	C₄²×C₈₉	C₂×D₄	1	2	yes	—
86	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
87	C₂²×D₈₉	—	2	4	no	—
88	C₂×C₄×C₈₉	—	2	4	no	—
89	C₂×C₄×C₈₉	—	4	2	no	—
90	C₂×C₄×C₈₉	—	4	4	no	—
91	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
94	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
96	C₂²⋊D₈₉	—	4	8	no	—
97	C₂×Dic₈₉	—	2	4	no	—
98	D₃₅₆	—	2	8	no	—
99	D₃₅₆	—	4	8	no	—
100	C₄×D₈₉	—	4	4	no	—
101	C₄×D₈₉	—	4	4	no	—
102	Dic₁₇₈	—	2	8	no	—
103	C₈×C₈₉	—	2	2	no	—
104	C₈×C₈₉	C₄×D₄	1	8	yes	—
105	C₈×C₈₉	#127	1	2	yes	—
106	C₈×C₈₉	#127	1	2	yes	—
107	#10623494	—	4	4	no	—
108	C₂²×C₈₉	—	2	2	no	—
109	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
110	C₂²×C₈₉	#532	1	2	yes	—
111	C₂×D₈₉	—	4	4	no	—
112	C₂×D₈₉	—	4	8	no	—
113	C₄×C₈₉	—	2	4	no	—
114	C₄×C₈₉	—	4	4	no	—
115	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
116	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
117	C₄×C₈₉	#532	1	1	yes	characteristic
118	Dic₈₉	—	4	4	no	—
119	#21708	—	89	1	no	maximal, Sylow
120	C₂×C₈₉	—	2	2	no	—
121	C₂×C₈₉	C₄×#127	1	1	yes	characteristic
122	C₂×C₈₉	#2955	1	2	yes	—
123	D₈₉	—	8	4	no	—
124	#3150	—	89	1	no	—
125	C₂×#443	—	89	4	no	—
126	#2931	—	89	2	no	—
127	#1975	—	89	4	no	—
128	#1963	—	89	2	no	—
129	#1849	—	89	1	no	—
130	#1726	D₈₉	1	1	yes	characteristic, complemented
131	C₈₉	#21708	1	1	yes	characteristic, complemented, Sylow
132	C₂×#135	—	89	4	no	—
133	C₂×#131	—	89	4	no	—
134	#532	—	89	2	no	—
135	#497	—	89	4	no	—
136	#496	—	89	4	no	—
137	#443	—	178	16	no	—
138	#436	—	178	4	no	—
139	C₈×D₄	—	178	4	no	—
140	C₂×#108	—	89	4	no	—
141	#423	—	89	2	no	—
142	C₂×#104	—	89	8	no	—
143	#408	—	89	4	no	—
144	C₂×#97	C₂×D₈₉	1	2	yes	—
145	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
147	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
148	#376	C₂×D₈₉	1	1	yes	characteristic
149	C₂×#41	—	89	4	no	—
150	#135	—	178	8	no	—
151	#135	—	178	8	no	—
152	C₂×Q₁₆	—	89	4	no	—
153	C₂×SD₁₆	—	89	8	no	—
154	C₂×D₈	—	89	4	no	—
155	#131	—	178	16	no	—
156	C₂×M₁₆	—	89	4	no	—
157	C₂×M₁₆	—	178	4	no	—
158	C₂²×C₈	—	178	4	no	—
159	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
161	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
162	C₄⋊Q₈	—	89	2	no	—
163	#127	—	89	2	no	—
164	#124	—	89	4	no	—
165	#121	—	178	4	no	—
166	C₄×D₄	—	178	4	no	—
167	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
168	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
169	#108	—	178	8	no	—
170	C₄⋊C₈	—	178	4	no	—
171	#104	—	178	16	no	—
172	C₂²⋊C₈	—	178	8	no	—
173	#97	—	2	4	no	—
174	#97	C₄×D₈₉	1	4	yes	—
175	C₄×C₈	D₃₅₆	1	2	yes	—
176	C₄×C₈	D₃₅₆	1	2	yes	—
177	C₄×C₈	D₃₅₆	1	8	yes	complemented
178	#95	—	2	2	no	—
179	#41	—	178	8	no	—
180	#41	—	178	8	no	—
181	C₂×Q₈	—	89	4	no	—
182	C₂×D₄	—	89	4	no	—
183	C₂×D₄	—	178	4	no	—
184	C₂×D₄	—	178	4	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	178	4	no	—
187	C₂²×C₄	#13083574	1	2	yes	—
188	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
189	Q₁₆	—	178	8	no	—
190	SD₁₆	—	178	16	no	—
191	D₈	—	178	8	no	—
192	M₁₆	—	356	8	no	—
193	M₁₆	—	178	8	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	356	8	no	—
196	C₂×C₈	—	178	4	no	—
197	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
198	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
199	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
200	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
201	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
202	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
203	C₂×C₈	#13083574	1	4	yes	—
204	C₄⋊C₄	—	178	4	no	—
205	C₂²⋊C₄	—	178	8	no	—
206	C₄²	—	2	2	no	—
207	C₄²	C₂×C₄×D₈₉	1	2	yes	—
208	C₄²	C₂×D₃₅₆	1	4	yes	—
209	C₄²	C₂×D₃₅₆	1	2	yes	—
210	C₂³	—	178	4	no	—
211	C₂³	C₂×#13083574	1	1	yes	characteristic
212	Q₈	—	178	8	no	—
213	D₄	—	356	8	no	—
214	D₄	—	356	8	no	—
215	D₄	—	178	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	356	4	no	—
220	C₂×C₄	—	356	4	no	—
221	C₂×C₄	—	178	4	no	—
222	C₂×C₄	C₄×D₃₅₆	1	4	yes	—
223	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
224	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₃₅₆	1	2	yes	—
226	C₂×C₄	C₂×#13083574	1	2	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	356	4	no	—
229	C₈	C₄×D₃₅₆	1	8	yes	—
230	C₈	#15668009	1	2	yes	—
231	C₈	#15668009	1	2	yes	—
232	C₂²	—	2	2	no	—
233	C₂²	—	356	8	no	—
234	C₂²	—	356	4	no	—
235	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
236	C₂²	#19152148	1	2	yes	—
237	C₄	—	2	4	no	—
238	C₄	—	4	4	no	—
239	C₄	—	356	4	no	—
240	C₄	C₂×C₄×D₃₅₆	1	2	yes	—
241	C₄	C₂×#15668009	1	1	yes	characteristic
242	C₄	#19152148	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	712	4	no	—
245	C₂	C₄×#15668009	1	1	yes	characteristic
246	C₂	#26029156	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#1726	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#26030698	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1726	D₈₉
hypocenter	C₈₉	#21708
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
2. C₂×#19151716	C₂	semidirect
3. #26028965	C₂	semidirect
4. #26021145	C₂	semidirect
5. #26021084	C₂	semidirect
6. #26020427	C₂	semidirect
7. C₈₉×#1726	C₂	semidirect
11. #19151992	C₄	semidirect
12. #19151987	C₄	semidirect
53. C₄×C₈×C₈₉	D₄	semidirect
130. #1726	D₈₉	semidirect
131. C₈₉	#21708	semidirect
177. C₄×C₈	D₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
#26030698	2
C₂×#19151716	2
#26028965	2
#26021145	2
#26021084	2
#26020427	2
C₈₉×#1726	2
#21708	89