Subgroups — #26034026

Order 11392.

1748 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	C₂×#15668058	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152267	C₂	1	4	yes	maximal, complemented
3	D₄×C₂²⋊D₈₉	C₂	1	12	yes	maximal, complemented
4	#19152165	C₂	1	3	yes	maximal, complemented
5	#19152162	C₂	1	3	yes	maximal, complemented
6	#19152144	C₂	1	3	yes	maximal, complemented
7	C₂×#15668008	C₂	1	3	yes	maximal, complemented
8	#41×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×H(5,𝔽₂)×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#15668058	—	2	8	no	—
11	C₂×#13083575	C₂²	1	2	yes	complemented
12	C₂×#13083574	C₂²	1	3	yes	complemented
13	C₂²×D₃₅₆	C₂²	1	3	yes	complemented
14	C₂²×C₂²⋊D₈₉	C₂²	1	12	yes	complemented
15	C₂×D₄×D₈₉	C₂²	1	6	yes	complemented
16	#15668010	C₂²	1	1	yes	characteristic, complemented
17	#15668008	C₂²	1	3	yes	complemented
18	#15668008	C₂²	1	6	yes	complemented
19	#15668008	C₂²	1	12	yes	complemented
20	#15668001	C₂²	1	3	yes	complemented
21	#15667999	C₂²	1	3	yes	complemented
22	#15667986	C₂²	1	12	yes	complemented
23	#15667984	C₂²	1	12	yes	complemented
24	#15667983	C₂²	1	12	yes	complemented
25	#15667981	C₂²	1	12	yes	complemented
26	#15667980	C₂²	1	12	yes	complemented
27	Q₈×Dic₈₉	C₂²	1	2	yes	complemented
28	D₄×Dic₈₉	C₂²	1	6	yes	complemented
29	C₄×C₂²⋊D₈₉	C₂²	1	12	yes	complemented
30	#15667971	C₂²	1	3	yes	complemented
31	C₂×C₄×Dic₈₉	C₂²	1	3	yes	complemented
32	H(5,𝔽₂)×C₈₉	—	2	8	no	—
33	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#41×C₈₉	C₂²	1	3	yes	complemented
35	C₂×#41×C₈₉	C₂²	1	2	yes	complemented
36	C₂²×D₄×C₈₉	C₂²	1	3	yes	complemented
37	C₂²×D₄×C₈₉	C₂²	1	6	yes	complemented
38	C₂³×D₈₉	C₂³	1	6	yes	—
39	#13083575	—	2	8	no	—
40	#13083574	—	2	12	no	—
41	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
42	C₂×D₃₅₆	—	2	12	no	—
43	C₂×D₃₅₆	C₂³	1	6	yes	—
44	C₂×D₃₅₆	C₂³	1	3	yes	—
45	C₂×C₂²⋊D₈₉	—	4	24	no	—
46	C₂×C₂²⋊D₈₉	C₂³	1	12	yes	—
47	C₂×C₂²⋊D₈₉	C₂³	1	6	yes	—
48	C₂×C₂²⋊D₈₉	C₂³	1	24	yes	—
49	D₄×D₈₉	—	2	24	no	—
50	C₂²×Dic₈₉	C₂³	1	6	yes	—
51	C₂×C₄×D₈₉	C₂³	1	6	yes	—
52	C₄⋊Dic₈₉	C₂³	1	6	yes	—
53	#13083553	C₂³	1	4	yes	—
54	C₂²⋊Dic₈₉	C₂³	1	6	yes	—
55	C₂²⋊Dic₈₉	C₂³	1	12	yes	—
56	#13083551	C₂³	1	12	yes	—
57	C₄×Dic₈₉	C₂³	1	3	yes	—
58	C₄×Dic₈₉	C₂³	1	6	yes	—
59	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
60	C₂⁴×C₈₉	C₂³	1	6	yes	—
61	#41×C₈₉	—	2	8	no	—
62	#41×C₈₉	—	2	12	no	—
63	#41×C₈₉	D₄	1	8	yes	complemented
64	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₈₉	—	2	24	no	—
67	C₂×D₄×C₈₉	—	2	12	no	—
68	C₂×D₄×C₈₉	C₂³	1	6	yes	—
69	C₂×D₄×C₈₉	C₂³	1	6	yes	—
70	C₂×D₄×C₈₉	C₂³	1	3	yes	—
71	C₂×D₄×C₈₉	C₂³	1	3	yes	—
72	C₂²×C₄×C₈₉	C₂³	1	6	yes	—
73	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
74	C₂³×C₈₉	—	2	12	no	—
75	C₂³×C₈₉	—	4	12	no	—
76	C₂³×C₈₉	C₂⁴	1	6	yes	—
77	C₂³×C₈₉	C₂⁴	1	3	yes	—
78	C₂²×D₈₉	—	2	12	no	—
79	C₂²×D₈₉	—	4	12	no	—
80	C₂²×D₈₉	C₂⁴	1	6	yes	—
81	Q₈×C₈₉	—	2	2	no	—
82	Q₈×C₈₉	C₂×D₄	1	4	yes	complemented
83	D₄×C₈₉	—	2	12	no	—
84	D₄×C₈₉	—	2	6	no	—
85	D₄×C₈₉	—	2	12	no	—
86	D₄×C₈₉	C₂×D₄	1	12	yes	complemented
87	C₂×C₄×C₈₉	—	2	12	no	—
88	C₂×C₄×C₈₉	C₂×D₄	1	12	yes	complemented
89	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
90	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂⁴	1	3	yes	—
92	C₂²⋊D₈₉	—	2	12	no	—
93	C₂²⋊D₈₉	—	2	24	no	—
94	C₂²⋊D₈₉	—	4	24	no	—
95	C₂×Dic₈₉	—	2	12	no	—
96	C₂×Dic₈₉	C₂⁴	1	6	yes	—
97	C₂×Dic₈₉	C₂⁴	1	2	yes	—
98	C₂×Dic₈₉	C₂⁴	1	2	yes	—
99	D₃₅₆	—	2	12	no	—
100	D₃₅₆	—	2	6	no	—
101	C₄×D₈₉	—	2	12	no	—
102	Dic₁₇₈	—	2	2	no	—
103	C₂²×C₈₉	—	2	6	no	—
104	C₂²×C₈₉	—	2	6	no	—
105	C₂²×C₈₉	—	2	12	no	—
106	C₂²×C₈₉	—	8	12	no	—
107	C₂²×C₈₉	C₂²×D₄	1	6	yes	—
108	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
109	C₂×D₈₉	—	2	6	no	—
110	C₂×D₈₉	—	2	12	no	—
111	C₂×D₈₉	—	8	12	no	—
112	C₄×C₈₉	—	2	2	no	—
113	C₄×C₈₉	C₂²×D₄	1	2	yes	—
114	C₄×C₈₉	C₂²×D₄	1	6	yes	—
115	Dic₈₉	—	2	2	no	—
116	Dic₈₉	—	2	6	no	—
117	Dic₈₉	H(5,𝔽₂)	1	4	yes	complemented
118	C₂×C₈₉	—	2	6	no	—
119	C₂×C₈₉	—	4	6	no	—
120	C₂×C₈₉	C₂³×D₄	1	1	yes	characteristic
121	C₂×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
122	D₈₉	—	4	6	no	—
123	#3498	—	89	1	no	maximal, Sylow
124	C₈₉	#3498	1	1	yes	characteristic, complemented, Sylow
125	C₂×H(5,𝔽₂)	—	89	1	no	—
126	C₂×H(5,𝔽₂)	D₈₉	1	1	yes	characteristic, complemented
127	#550	—	89	4	no	—
128	D₄²	—	89	12	no	—
129	#535	—	89	3	no	—
130	#535	—	89	3	no	—
131	#532	—	89	3	no	—
132	C₂×#127	—	89	3	no	—
133	C₄×#41	—	89	1	no	—
134	H(5,𝔽₂)	—	2	8	no	—
135	H(5,𝔽₂)	—	178	8	no	—
136	C₂×#41	—	89	2	no	—
137	C₂×#41	—	89	3	no	—
138	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
139	C₂×#41	C₂×D₈₉	1	3	yes	complemented
140	C₂×#41	C₂×D₈₉	1	2	yes	complemented
141	C₂²×D₄	—	89	12	no	—
142	C₂²×D₄	—	89	6	no	—
143	C₂²×D₄	—	89	3	no	—
144	C₂²×D₄	C₂×D₈₉	1	3	yes	complemented
145	C₂²×D₄	C₂×D₈₉	1	6	yes	complemented
146	C₄⋊Q₈	—	89	1	no	—
147	#127	—	89	3	no	—
148	#127	—	89	6	no	—
149	#127	—	89	12	no	—
150	#124	—	89	3	no	—
151	#124	—	89	3	no	—
152	#121	—	89	12	no	—
153	#121	—	89	12	no	—
154	#121	—	89	12	no	—
155	#120	—	89	12	no	—
156	#120	—	89	12	no	—
157	C₄×Q₈	—	89	2	no	—
158	C₄×D₄	—	89	6	no	—
159	C₄×D₄	—	89	12	no	—
160	#117	—	89	3	no	—
161	C₂×C₄²	—	89	3	no	—
162	C₂⁴	—	89	6	no	—
163	C₂⁴	C₂²×D₈₉	1	6	yes	—
164	#41	—	2	8	no	—
165	#41	—	2	12	no	—
166	#41	—	178	8	no	—
167	#41	—	178	12	no	—
168	#41	C₂²⋊D₈₉	1	8	yes	complemented
169	C₂×Q₈	—	89	1	no	—
170	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
171	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
172	C₂×D₄	—	2	24	no	—
173	C₂×D₄	—	2	12	no	—
174	C₂×D₄	—	89	24	no	—
175	C₂×D₄	—	89	12	no	—
176	C₂×D₄	—	89	6	no	—
177	C₂×D₄	—	89	6	no	—
178	C₂×D₄	—	89	3	no	—
179	C₂×D₄	—	356	24	no	—
180	C₂×D₄	—	178	24	no	—
181	C₂×D₄	—	178	12	no	—
182	C₂×D₄	C₂²×D₈₉	1	6	yes	—
183	C₂×D₄	C₂²×D₈₉	1	6	yes	—
184	C₂×D₄	C₂²×D₈₉	1	3	yes	—
185	C₂×D₄	C₂²×D₈₉	1	3	yes	—
186	C₂²×C₄	—	89	6	no	—
187	C₂²×C₄	—	89	6	no	—
188	C₂²×C₄	C₂²×D₈₉	1	6	yes	—
189	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
190	C₄⋊C₄	—	89	6	no	—
191	C₄⋊C₄	—	89	4	no	—
192	C₂²⋊C₄	—	89	6	no	—
193	C₂²⋊C₄	—	89	12	no	—
194	C₂²⋊C₄	—	89	12	no	—
195	C₄²	—	89	1	no	—
196	C₄²	—	89	3	no	—
197	C₄²	—	89	6	no	—
198	C₂³	—	2	12	no	—
199	C₂³	—	4	12	no	—
200	C₂³	—	89	6	no	—
201	C₂³	—	356	12	no	—
202	C₂³	—	178	12	no	—
203	C₂³	C₂³×D₈₉	1	6	yes	—
204	C₂³	C₂³×D₈₉	1	3	yes	—
205	Q₈	—	2	2	no	—
206	Q₈	—	178	2	no	—
207	Q₈	C₂×C₂²⋊D₈₉	1	4	yes	complemented
208	D₄	—	2	12	no	—
209	D₄	—	2	12	no	—
210	D₄	—	2	6	no	—
211	D₄	—	356	24	no	—
212	D₄	—	178	24	no	—
213	D₄	—	178	12	no	—
214	D₄	—	178	12	no	—
215	D₄	—	178	6	no	—
216	D₄	C₂×C₂²⋊D₈₉	1	12	yes	complemented
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	89	2	no	—
219	C₂×C₄	—	89	6	no	—
220	C₂×C₄	—	89	2	no	—
221	C₂×C₄	—	178	12	no	—
222	C₂×C₄	—	178	12	no	—
223	C₂×C₄	C₂×C₂²⋊D₈₉	1	12	yes	complemented
224	C₂×C₄	C₂³×D₈₉	1	2	yes	—
225	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₈₉	1	3	yes	—
227	C₂²	—	2	6	no	—
228	C₂²	—	2	6	no	—
229	C₂²	—	2	12	no	—
230	C₂²	—	8	12	no	—
231	C₂²	—	178	6	no	—
232	C₂²	—	178	12	no	—
233	C₂²	—	712	12	no	—
234	C₂²	C₂²×C₂²⋊D₈₉	1	6	yes	—
235	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	89	4	no	—
238	C₄	—	178	2	no	—
239	C₄	—	178	6	no	—
240	C₄	C₂²×C₂²⋊D₈₉	1	6	yes	—
241	C₄	C₂²×C₂²⋊D₈₉	1	2	yes	—
242	C₂	—	2	6	no	—
243	C₂	—	4	6	no	—
244	C₂	—	356	6	no	—
245	C₂	C₂³×C₂²⋊D₈₉	1	1	yes	characteristic
246	C₂	H(5,𝔽₂)×D₈₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	C₂×H(5,𝔽₂)×C₈₉	C₂
hypercenter	C₂×H(5,𝔽₂)	D₈₉
hypocenter	C₈₉	#3498
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#15668058	C₂	semidirect
2. #19152267	C₂	semidirect
3. D₄×C₂²⋊D₈₉	C₂	semidirect
4. #19152165	C₂	semidirect
5. #19152162	C₂	semidirect
6. #19152144	C₂	semidirect
7. C₂×#15668008	C₂	semidirect
8. #41×Dic₈₉	C₂	semidirect
9. C₂×H(5,𝔽₂)×C₈₉	C₂	semidirect
11. C₂×#13083575	C₂²	semidirect
12. C₂×#13083574	C₂²	semidirect
13. C₂²×D₃₅₆	C₂²	semidirect
14. C₂²×C₂²⋊D₈₉	C₂²	semidirect
15. C₂×D₄×D₈₉	C₂²	semidirect
16. #15668010	C₂²	semidirect
17. #15668008	C₂²	semidirect
18. #15668008	C₂²	semidirect
19. #15668008	C₂²	semidirect
20. #15668001	C₂²	semidirect
21. #15667999	C₂²	semidirect
22. #15667986	C₂²	semidirect
23. #15667984	C₂²	semidirect
24. #15667983	C₂²	semidirect
25. #15667981	C₂²	semidirect
26. #15667980	C₂²	semidirect
27. Q₈×Dic₈₉	C₂²	semidirect
28. D₄×Dic₈₉	C₂²	semidirect
29. C₄×C₂²⋊D₈₉	C₂²	semidirect
30. #15667971	C₂²	semidirect
31. C₂×C₄×Dic₈₉	C₂²	semidirect
34. C₂×#41×C₈₉	C₂²	semidirect
35. C₂×#41×C₈₉	C₂²	semidirect
36. C₂²×D₄×C₈₉	C₂²	semidirect
37. C₂²×D₄×C₈₉	C₂²	semidirect
63. #41×C₈₉	D₄	semidirect
82. Q₈×C₈₉	C₂×D₄	semidirect
86. D₄×C₈₉	C₂×D₄	semidirect
88. C₂×C₄×C₈₉	C₂×D₄	semidirect
117. Dic₈₉	H(5,𝔽₂)	semidirect
124. C₈₉	#3498	semidirect
126. C₂×H(5,𝔽₂)	D₈₉	semidirect
139. C₂×#41	C₂×D₈₉	semidirect
140. C₂×#41	C₂×D₈₉	semidirect
144. C₂²×D₄	C₂×D₈₉	semidirect
145. C₂²×D₄	C₂×D₈₉	semidirect
168. #41	C₂²⋊D₈₉	semidirect
207. Q₈	C₂×C₂²⋊D₈₉	semidirect
216. D₄	C₂×C₂²⋊D₈₉	semidirect
223. C₂×C₄	C₂×C₂²⋊D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668058	2
#19152267	2
D₄×C₂²⋊D₈₉	2
#19152165	2
#19152162	2
#19152144	2
C₂×#15668008	2
#41×Dic₈₉	2
C₂×H(5,𝔽₂)×C₈₉	2
#3498	89