Subgroups — #26034126

Order 11392.

1232 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	#19152334	C₂	1	4	yes	maximal, complemented
2	#19152268	C₂	1	4	yes	maximal, complemented
3	#19152242	C₂	1	8	yes	maximal, complemented
4	#19152202	C₂	1	4	yes	maximal, complemented
5	#19152165	C₂	1	2	yes	maximal, complemented
6	#19152155	C₂	1	1	yes	maximal, characteristic, complemented
7	#19152146	C₂	1	4	yes	maximal, complemented
8	#19152073	C₂	1	2	yes	maximal, complemented
9	C₄×#13083574	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#127×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13083574	C₂²	1	1	yes	characteristic
12	C₂×#13083573	C₂²	1	8	yes	complemented
13	#15668012	C₂²	1	2	yes	complemented
14	#15668008	C₂²	1	4	yes	complemented
15	#15668004	C₂²	1	2	yes	complemented
16	#15667999	C₂²	1	8	yes	complemented
17	#15667999	C₂²	1	4	yes	complemented
18	#15667993	C₂²	1	16	yes	complemented
19	#15667991	C₂²	1	4	yes	—
20	#15667984	C₂²	1	16	yes	complemented
21	#15667983	C₂²	1	8	yes	complemented
22	#15667983	C₂²	1	8	yes	complemented
23	#15667981	C₂²	1	8	yes	—
24	C₄×Dic₁₇₈	C₂²	1	4	yes	complemented
25	C₄×Dic₁₇₈	C₂²	1	2	yes	complemented
26	C₄×D₃₅₆	C₂²	1	2	yes	complemented
27	D₄×Dic₈₉	C₂²	1	8	yes	complemented
28	D₄×Dic₈₉	C₂²	1	8	yes	complemented
29	D₄×Dic₈₉	C₂²	1	8	yes	complemented
30	C₄×C₂²⋊D₈₉	C₂²	1	4	yes	—
31	#15667971	C₂²	1	2	yes	—
32	#15667971	C₂²	1	1	yes	characteristic
33	#15667969	C₂²	1	2	yes	complemented
34	C₂×C₂²⋊Dic₈₉	C₂²	1	8	yes	—
35	C₄²×D₈₉	C₂²	1	2	yes	complemented
36	C₂²×D₄×C₈₉	C₂²	1	2	yes	—
37	C₂²×D₄×C₈₉	C₂²	1	4	yes	—
38	#127×C₈₉	C₂²	1	2	yes	—
39	#127×C₈₉	C₂²	1	2	yes	—
40	#127×C₈₉	C₂²	1	4	yes	—
41	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
42	#13083574	—	2	4	no	—
43	#13083573	—	4	16	no	—
44	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₇₈	C₂³	1	4	yes	—
46	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₈₉	C₂³	1	2	yes	—
48	C₂×C₂²⋊D₈₉	C₂³	1	16	yes	—
49	C₂²×Dic₈₉	C₂³	1	16	yes	—
50	C₂×C₄×D₈₉	C₂³	1	2	yes	—
51	C₂×C₄×D₈₉	C₂³	1	4	yes	—
52	C₄⋊Dic₈₉	C₂³	1	2	yes	—
53	C₄⋊Dic₈₉	C₂³	1	4	yes	—
54	C₄⋊Dic₈₉	C₂³	1	4	yes	—
55	#13083553	C₂³	1	4	yes	—
56	#13083553	C₂³	1	8	yes	—
57	C₂²⋊Dic₈₉	C₂³	1	2	yes	—
58	C₂²⋊Dic₈₉	C₂³	1	4	yes	—
59	C₂²⋊Dic₈₉	C₂³	1	16	yes	—
60	C₂²⋊Dic₈₉	C₂³	1	16	yes	—
61	#13083551	C₂³	1	4	yes	—
62	C₄×Dic₈₉	C₂³	1	2	yes	—
63	C₄×Dic₈₉	C₂³	1	4	yes	—
64	C₄×Dic₈₉	C₂³	1	4	yes	—
65	C₂⁴×C₈₉	C₂³	1	4	yes	—
66	C₂×D₄×C₈₉	—	2	8	no	—
67	C₂×D₄×C₈₉	—	4	8	no	—
68	C₂×D₄×C₈₉	C₂³	1	4	yes	—
69	C₂×D₄×C₈₉	C₂³	1	4	yes	—
70	C₂×D₄×C₈₉	C₂³	1	8	yes	—
71	C₂×D₄×C₈₉	C₂³	1	8	yes	—
72	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
74	C₄²×C₈₉	C₂³	1	1	yes	characteristic
75	C₄²×C₈₉	C₂³	1	2	yes	—
76	C₄²×C₈₉	C₂³	1	1	yes	characteristic
77	C₂³×C₈₉	—	2	8	no	—
78	C₂³×C₈₉	—	4	8	no	—
79	C₂³×C₈₉	C₂⁴	1	8	yes	—
80	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
81	C₂²×D₈₉	C₂⁴	1	2	yes	—
82	D₄×C₈₉	—	2	8	no	—
83	D₄×C₈₉	—	2	8	no	—
84	D₄×C₈₉	—	2	16	no	—
85	D₄×C₈₉	—	4	8	no	—
86	C₂×C₄×C₈₉	—	2	4	no	—
87	C₂×C₄×C₈₉	#41	1	4	yes	complemented
88	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
89	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
90	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
92	C₂²⋊D₈₉	—	2	4	no	—
93	C₂²⋊D₈₉	—	4	16	no	—
94	C₂×Dic₈₉	—	4	16	no	—
95	C₂×Dic₈₉	C₂⁴	1	4	yes	—
96	C₂×Dic₈₉	C₂⁴	1	2	yes	—
97	C₂×Dic₈₉	C₂⁴	1	8	yes	—
98	D₃₅₆	—	2	2	no	—
99	C₄×D₈₉	—	2	4	no	—
100	C₄×D₈₉	—	2	8	no	—
101	Dic₁₇₈	—	2	2	no	—
102	Dic₁₇₈	—	2	8	no	—
103	C₂²×C₈₉	—	2	2	no	—
104	C₂²×C₈₉	—	2	8	no	—
105	C₂²×C₈₉	—	2	16	no	—
106	C₂²×C₈₉	—	8	8	no	—
107	C₂²×C₈₉	C₂×#41	1	2	yes	—
108	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
109	C₂×D₈₉	—	2	2	no	—
110	C₂×D₈₉	—	2	4	no	—
111	C₄×C₈₉	—	2	2	no	—
112	C₄×C₈₉	C₂×#41	1	2	yes	—
113	C₄×C₈₉	C₂×#41	1	2	yes	—
114	C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
115	Dic₈₉	—	2	2	no	—
116	Dic₈₉	—	2	8	no	—
117	Dic₈₉	—	2	4	no	—
118	C₂×C₈₉	—	2	2	no	—
119	C₂×C₈₉	—	4	8	no	—
120	C₂×C₈₉	C₂²×#41	1	1	yes	characteristic
121	C₂×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
122	D₈₉	—	4	2	no	—
123	#3508	—	89	1	no	maximal, Sylow
124	C₈₉	#3508	1	1	yes	characteristic, complemented, Sylow
125	#559	—	89	4	no	—
126	#550	—	89	4	no	—
127	#546	—	89	8	no	—
128	#540	—	89	4	no	—
129	#535	—	89	2	no	—
130	#533	—	89	1	no	—
131	#532	—	89	4	no	—
132	C₂×#127	D₈₉	1	1	yes	characteristic, complemented
133	#518	—	89	2	no	—
134	C₄×#41	—	89	1	no	—
135	C₂×#41	—	89	8	no	—
136	C₂×#41	—	89	1	no	—
137	C₂²×D₄	C₂×D₈₉	1	2	yes	—
138	C₂²×D₄	C₂×D₈₉	1	4	yes	—
139	C₄⋊Q₈	—	89	2	no	—
140	#127	—	89	4	no	—
141	#127	C₂×D₈₉	1	2	yes	—
142	#127	C₂×D₈₉	1	2	yes	—
143	#127	C₂×D₈₉	1	4	yes	—
144	#125	—	89	2	no	—
145	#124	—	89	8	no	—
146	#124	—	89	4	no	—
147	#123	—	89	16	no	—
148	C₂²⋊Q₈	—	89	4	no	—
149	#121	—	89	8	no	—
150	#121	—	89	8	no	—
151	#121	—	89	16	no	—
152	#120	—	89	8	no	—
153	C₄×Q₈	—	89	2	no	—
154	C₄×Q₈	—	89	4	no	—
155	C₄×D₄	—	89	8	no	—
156	C₄×D₄	—	89	8	no	—
157	C₄×D₄	—	89	8	no	—
158	C₄×D₄	—	89	4	no	—
159	C₄×D₄	—	89	2	no	—
160	#117	—	89	1	no	—
161	#117	—	89	2	no	—
162	#117	—	89	2	no	—
163	C₂×C₂²⋊C₄	—	89	8	no	—
164	C₂×C₄²	—	89	2	no	—
165	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
166	C₂⁴	C₂²×D₈₉	1	4	yes	—
167	#41	—	356	16	no	—
168	#41	—	178	4	no	—
169	C₂×Q₈	—	89	1	no	—
170	C₂×Q₈	—	89	4	no	—
171	C₂×D₄	—	2	8	no	—
172	C₂×D₄	—	4	8	no	—
173	C₂×D₄	—	89	16	no	—
174	C₂×D₄	—	89	2	no	—
175	C₂×D₄	—	89	1	no	—
176	C₂×D₄	C₂²×D₈₉	1	4	yes	—
177	C₂×D₄	C₂²×D₈₉	1	4	yes	—
178	C₂×D₄	C₂²×D₈₉	1	8	yes	—
179	C₂×D₄	C₂²×D₈₉	1	8	yes	—
180	C₂²×C₄	—	89	16	no	—
181	C₂²×C₄	—	89	2	no	—
182	C₂²×C₄	—	89	4	no	—
183	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
184	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
185	C₄⋊C₄	—	89	2	no	—
186	C₄⋊C₄	—	89	4	no	—
187	C₄⋊C₄	—	89	4	no	—
188	C₄⋊C₄	—	89	4	no	—
189	C₄⋊C₄	—	89	8	no	—
190	C₂²⋊C₄	—	89	2	no	—
191	C₂²⋊C₄	—	89	4	no	—
192	C₂²⋊C₄	—	89	16	no	—
193	C₂²⋊C₄	—	89	16	no	—
194	C₂²⋊C₄	—	89	4	no	—
195	C₄²	—	89	2	no	—
196	C₄²	—	89	4	no	—
197	C₄²	—	89	4	no	—
198	C₄²	C₂²×D₈₉	1	1	yes	characteristic
199	C₄²	C₂²×D₈₉	1	1	yes	characteristic
200	C₄²	C₂²×D₈₉	1	2	yes	—
201	C₂³	—	2	8	no	—
202	C₂³	—	4	8	no	—
203	C₂³	—	89	2	no	—
204	C₂³	C₂³×D₈₉	1	8	yes	—
205	C₂³	C₂³×D₈₉	1	1	yes	characteristic
206	Q₈	—	178	2	no	—
207	Q₈	—	178	8	no	—
208	D₄	—	2	8	no	—
209	D₄	—	2	8	no	—
210	D₄	—	2	16	no	—
211	D₄	—	4	8	no	—
212	D₄	—	356	16	no	—
213	D₄	—	178	4	no	—
214	D₄	—	178	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	89	2	no	—
217	C₂×C₄	—	89	4	no	—
218	C₂×C₄	—	89	8	no	—
219	C₂×C₄	—	356	16	no	—
220	C₂×C₄	—	178	4	no	—
221	C₂×C₄	—	178	8	no	—
222	C₂×C₄	#13083573	1	4	yes	complemented
223	C₂×C₄	C₂³×D₈₉	1	2	yes	—
224	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
225	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₈₉	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	8	no	—
229	C₂²	—	2	16	no	—
230	C₂²	—	8	8	no	—
231	C₂²	—	178	2	no	—
232	C₂²	—	178	4	no	—
233	C₂²	C₂×#13083573	1	2	yes	—
234	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	178	2	no	—
237	C₄	—	178	4	no	—
238	C₄	—	178	8	no	—
239	C₄	C₂×#13083573	1	2	yes	—
240	C₄	C₂×#13083573	1	2	yes	—
241	C₄	#15668057	1	4	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	4	8	no	—
244	C₂	—	356	2	no	—
245	C₂	C₂²×#13083573	1	1	yes	characteristic
246	C₂	C₂×#15668057	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	C₂×#127×C₈₉	C₂
hypercenter	C₂×#127	D₈₉
hypocenter	C₈₉	#3508
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. #19152334	C₂	semidirect
2. #19152268	C₂	semidirect
3. #19152242	C₂	semidirect
4. #19152202	C₂	semidirect
5. #19152165	C₂	semidirect
6. #19152155	C₂	semidirect
7. #19152146	C₂	semidirect
8. #19152073	C₂	semidirect
9. C₄×#13083574	C₂	semidirect
10. C₂×#127×C₈₉	C₂	semidirect
12. C₂×#13083573	C₂²	semidirect
13. #15668012	C₂²	semidirect
14. #15668008	C₂²	semidirect
15. #15668004	C₂²	semidirect
16. #15667999	C₂²	semidirect
17. #15667999	C₂²	semidirect
18. #15667993	C₂²	semidirect
20. #15667984	C₂²	semidirect
21. #15667983	C₂²	semidirect
22. #15667983	C₂²	semidirect
24. C₄×Dic₁₇₈	C₂²	semidirect
25. C₄×Dic₁₇₈	C₂²	semidirect
26. C₄×D₃₅₆	C₂²	semidirect
27. D₄×Dic₈₉	C₂²	semidirect
28. D₄×Dic₈₉	C₂²	semidirect
29. D₄×Dic₈₉	C₂²	semidirect
33. #15667969	C₂²	semidirect
35. C₄²×D₈₉	C₂²	semidirect
87. C₂×C₄×C₈₉	#41	semidirect
124. C₈₉	#3508	semidirect
132. C₂×#127	D₈₉	semidirect
222. C₂×C₄	#13083573	semidirect

Maximal subgroups

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

subgroup	index
#19152334	2
#19152268	2
#19152242	2
#19152202	2
#19152165	2
#19152155	2
#19152146	2
#19152073	2
C₄×#13083574	2
C₂×#127×C₈₉	2
#3508	89