Subgroups — #36563072

Order 22784.

1006 conjugacy classes of subgroups.

The subgroup lattice has 242 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×#19151721	C₂	1	4	yes	maximal, complemented
3	#26028965	C₂	1	2	yes	maximal, complemented
4	#26021140	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	#19151721	—	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	1	yes	characteristic
21	C₂×#97×C₈₉	C₂²	1	2	yes	—
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	C₂×C₄	1	4	yes	—
50	#97×C₈₉	D₄	1	8	yes	complemented
51	C₄×C₈×C₈₉	—	2	4	no	—
52	C₄×C₈×C₈₉	D₄	1	4	yes	—
53	#95×C₈₉	—	2	2	no	—
54	#13083574	—	2	8	no	—
55	#13083574	—	2	8	no	—
56	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
57	C₂×D₃₅₆	—	2	4	no	—
58	C₂×D₃₅₆	C₂²×C₄	1	4	yes	—
59	C₂×C₂²⋊D₈₉	—	2	4	no	—
60	C₂×C₄×D₈₉	—	2	4	no	—
61	Dic₃₅₆	—	2	8	no	—
62	#13083563	—	2	16	no	—
63	D₇₁₂	—	2	8	no	—
64	#13083558	—	2	8	no	—
65	#13083557	—	4	8	no	—
66	C₂×#10623494	—	2	4	no	—
67	C₈×D₈₉	—	4	8	no	—
68	C₄⋊Dic₈₉	—	2	4	no	—
69	#13083551	—	2	8	no	—
70	C₂²×C₄×C₈₉	—	2	2	no	—
71	C₂²×C₄×C₈₉	#41	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₈₉	—	2	2	no	—
74	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₈₉	#41	1	4	yes	—
80	C₄²×C₈₉	—	2	2	no	—
81	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
82	C₄²×C₈₉	C₂×D₄	1	4	yes	—
83	C₄²×C₈₉	C₂×D₄	1	2	yes	—
84	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
85	C₂²×D₈₉	—	2	4	no	—
86	C₂×C₄×C₈₉	—	2	4	no	—
87	C₂×C₄×C₈₉	—	4	2	no	—
88	C₂×C₄×C₈₉	—	4	4	no	—
89	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
90	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
94	C₂²⋊D₈₉	—	4	8	no	—
95	C₂×Dic₈₉	—	2	4	no	—
96	D₃₅₆	—	2	8	no	—
97	D₃₅₆	—	4	8	no	—
98	C₄×D₈₉	—	4	4	no	—
99	C₄×D₈₉	—	4	4	no	—
100	Dic₁₇₈	—	2	8	no	—
101	C₈×C₈₉	—	2	2	no	—
102	C₈×C₈₉	C₄×D₄	1	8	yes	—
103	C₈×C₈₉	#127	1	4	yes	—
104	#10623494	—	4	4	no	—
105	C₂²×C₈₉	—	2	2	no	—
106	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
107	C₂²×C₈₉	#532	1	2	yes	—
108	C₂×D₈₉	—	4	4	no	—
109	C₂×D₈₉	—	4	8	no	—
110	C₄×C₈₉	—	2	4	no	—
111	C₄×C₈₉	—	4	4	no	—
112	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
113	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
114	C₄×C₈₉	#532	1	1	yes	characteristic
115	Dic₈₉	—	4	4	no	—
116	#21709	—	89	1	no	maximal, Sylow
117	C₂×C₈₉	—	2	2	no	—
118	C₂×C₈₉	C₄×#127	1	1	yes	characteristic
119	C₂×C₈₉	#2955	1	2	yes	—
120	D₈₉	—	8	4	no	—
121	#3150	—	89	1	no	—
122	C₂×#444	—	89	4	no	—
123	#2931	—	89	2	no	—
124	#1974	—	89	4	no	—
125	#1963	—	89	2	no	—
126	#1849	—	89	1	no	—
127	#1726	D₈₉	1	1	yes	characteristic, complemented
128	C₈₉	#21709	1	1	yes	characteristic, complemented, Sylow
129	C₂×#135	—	89	4	no	—
130	C₂×#131	—	89	4	no	—
131	#532	—	89	2	no	—
132	#497	—	89	4	no	—
133	#496	—	89	4	no	—
134	#444	—	178	16	no	—
135	#436	—	178	4	no	—
136	C₈×D₄	—	178	4	no	—
137	C₂×#108	—	89	4	no	—
138	#423	—	89	2	no	—
139	C₂×#104	—	89	8	no	—
140	#408	—	89	4	no	—
141	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
142	C₂×#97	C₂×D₈₉	1	2	yes	—
143	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
144	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
145	#376	C₂×D₈₉	1	1	yes	characteristic
146	C₂×#41	—	89	4	no	—
147	#135	—	178	8	no	—
148	#135	—	178	8	no	—
149	C₂×Q₁₆	—	89	4	no	—
150	C₂×SD₁₆	—	89	8	no	—
151	C₂×D₈	—	89	4	no	—
152	#131	—	178	16	no	—
153	C₂×M₁₆	—	89	4	no	—
154	C₂×M₁₆	—	178	4	no	—
155	C₂²×C₈	—	178	4	no	—
156	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
158	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
159	C₄⋊Q₈	—	89	2	no	—
160	#127	—	89	2	no	—
161	#124	—	89	4	no	—
162	#121	—	178	4	no	—
163	C₄×D₄	—	178	4	no	—
164	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
165	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
166	#108	—	178	8	no	—
167	C₄⋊C₈	—	178	4	no	—
168	#104	—	178	16	no	—
169	C₂²⋊C₈	—	178	8	no	—
170	#97	C₄×D₈₉	1	4	yes	—
171	#97	D₃₅₆	1	8	yes	complemented
172	C₄×C₈	—	2	4	no	—
173	C₄×C₈	D₃₅₆	1	4	yes	—
174	#95	—	2	2	no	—
175	#41	—	178	8	no	—
176	#41	—	178	8	no	—
177	C₂×Q₈	—	89	4	no	—
178	C₂×D₄	—	89	4	no	—
179	C₂×D₄	—	178	4	no	—
180	C₂×D₄	—	178	4	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	178	4	no	—
183	C₂²×C₄	#13083574	1	2	yes	—
184	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
185	Q₁₆	—	178	8	no	—
186	SD₁₆	—	178	16	no	—
187	D₈	—	178	8	no	—
188	M₁₆	—	356	8	no	—
189	M₁₆	—	178	8	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	356	8	no	—
192	C₂×C₈	—	178	4	no	—
193	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
194	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
195	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
196	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
197	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
198	C₂×C₈	#13083574	1	4	yes	—
199	C₄⋊C₄	—	178	4	no	—
200	C₂²⋊C₄	—	178	8	no	—
201	C₄²	—	2	2	no	—
202	C₄²	C₂×C₄×D₈₉	1	2	yes	—
203	C₄²	C₂×D₃₅₆	1	4	yes	—
204	C₄²	C₂×D₃₅₆	1	2	yes	—
205	C₂³	—	178	4	no	—
206	C₂³	C₂×#13083574	1	1	yes	characteristic
207	Q₈	—	178	8	no	—
208	D₄	—	356	8	no	—
209	D₄	—	356	8	no	—
210	D₄	—	178	8	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	356	4	no	—
215	C₂×C₄	—	356	4	no	—
216	C₂×C₄	—	178	4	no	—
217	C₂×C₄	C₄×D₃₅₆	1	4	yes	—
218	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
219	C₂×C₄	C₂²×D₃₅₆	1	2	yes	—
220	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
221	C₂×C₄	C₂×#13083574	1	2	yes	—
222	C₈	—	2	2	no	—
223	C₈	—	356	4	no	—
224	C₈	C₄×D₃₅₆	1	8	yes	—
225	C₈	#15668009	1	4	yes	—
226	C₂²	—	2	2	no	—
227	C₂²	—	356	8	no	—
228	C₂²	—	356	4	no	—
229	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
230	C₂²	#19152148	1	2	yes	—
231	C₄	—	2	4	no	—
232	C₄	—	4	4	no	—
233	C₄	—	356	4	no	—
234	C₄	C₂×C₄×D₃₅₆	1	2	yes	—
235	C₄	C₂×#15668009	1	1	yes	characteristic
236	C₄	#19152148	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	—	712	4	no	—
239	C₂	C₄×#15668009	1	1	yes	characteristic
240	C₂	#26029156	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×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	#21709
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
2. C₂×#19151721	C₂	semidirect
3. #26028965	C₂	semidirect
4. #26021140	C₂	semidirect
5. #26021084	C₂	semidirect
6. #26020427	C₂	semidirect
7. C₈₉×#1726	C₂	semidirect
11. #19151992	C₄	semidirect
12. #19151987	C₄	semidirect
50. #97×C₈₉	D₄	semidirect
127. #1726	D₈₉	semidirect
128. C₈₉	#21709	semidirect
171. #97	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×#19151721	2
#26028965	2
#26021140	2
#26021084	2
#26020427	2
C₈₉×#1726	2
#21709	89