Subgroups — #26034045

Order 11392.

1540 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668059	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15668058	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152260	C₂	1	4	yes	maximal, complemented
4	#19152243	C₂	1	12	yes	maximal, complemented
5	#19152174	C₂	1	3	yes	maximal, complemented
6	#19152167	C₂	1	3	yes	maximal, complemented
7	#19152148	C₂	1	3	yes	maximal, complemented
8	C₂×#15668000	C₂	1	3	yes	maximal, complemented
9	C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#15668059	—	2	8	no	—
11	#15668058	—	2	8	no	—
12	C₂×#13083575	C₂²	1	2	yes	—
13	C₂×#13083574	C₂²	1	3	yes	—
14	C₂×#13083574	C₂²	1	3	yes	complemented
15	C₂×#13083573	C₂²	1	6	yes	complemented
16	C₂²×Dic₁₇₈	C₂²	1	3	yes	complemented
17	C₂×Q₈×D₈₉	C₂²	1	2	yes	complemented
18	C₂²×D₃₅₆	C₂²	1	3	yes	—
19	C₂×D₄×D₈₉	C₂²	1	6	yes	—
20	#15668012	C₂²	1	3	yes	complemented
21	#15668009	C₂²	1	3	yes	complemented
22	#15668000	C₂²	1	1	yes	characteristic, complemented
23	#15668000	C₂²	1	3	yes	complemented
24	#15668000	C₂²	1	6	yes	complemented
25	#15668000	C₂²	1	12	yes	complemented
26	#15667996	C₂²	1	12	yes	complemented
27	#15667992	C₂²	1	12	yes	complemented
28	#15667987	C₂²	1	12	yes	complemented
29	#15667986	C₂²	1	12	yes	complemented
30	#15667982	C₂²	1	12	yes	complemented
31	C₄×D₃₅₆	C₂²	1	12	yes	complemented
32	C₂×#13083551	C₂²	1	12	yes	complemented
33	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₈₉	C₂²	1	2	yes	complemented
35	C₄×D₄×C₈₉	C₂²	1	6	yes	complemented
36	#117×C₈₉	C₂²	1	3	yes	complemented
37	C₂×C₄²×C₈₉	C₂²	1	3	yes	complemented
38	C₂³×D₈₉	C₂³	1	6	yes	—
39	#13083575	—	2	8	no	—
40	#13083574	—	2	12	no	—
41	#13083574	—	2	12	no	—
42	#13083573	—	2	24	no	—
43	C₂×Dic₁₇₈	—	2	12	no	—
44	C₂×Dic₁₇₈	C₂³	1	6	yes	—
45	C₂×Dic₁₇₈	C₂³	1	3	yes	—
46	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
47	Q₈×D₈₉	—	2	8	no	—
48	C₂×D₃₅₆	—	2	12	no	—
49	C₂×D₃₅₆	C₂³	1	6	yes	—
50	C₂×D₃₅₆	C₂³	1	3	yes	—
51	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
52	C₂×D₃₅₆	C₂³	1	12	yes	—
53	C₂×C₂²⋊D₈₉	C₂³	1	6	yes	—
54	C₂×C₂²⋊D₈₉	C₂³	1	6	yes	—
55	D₄×D₈₉	—	2	24	no	—
56	C₂²×Dic₈₉	C₂³	1	6	yes	—
57	C₂×C₄×D₈₉	C₂³	1	6	yes	—
58	C₂×C₄×D₈₉	C₂³	1	6	yes	—
59	C₄⋊Dic₈₉	C₂³	1	12	yes	—
60	#13083551	C₂³	1	4	yes	—
61	#13083551	C₂³	1	12	yes	—
62	#13083551	C₂³	1	24	yes	—
63	#41×C₈₉	D₄	1	8	yes	complemented
64	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₈₉	C₂³	1	3	yes	—
66	C₂²×C₄×C₈₉	C₂³	1	6	yes	—
67	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
68	C₄⋊C₄×C₈₉	C₂³	1	6	yes	—
69	C₂²⋊C₄×C₈₉	C₂³	1	6	yes	—
70	C₄²×C₈₉	C₂³	1	6	yes	—
71	C₄²×C₈₉	C₂³	1	1	yes	characteristic
72	C₄²×C₈₉	C₂³	1	3	yes	—
73	C₂³×C₈₉	C₂⁴	1	3	yes	—
74	C₂²×D₈₉	—	2	12	no	—
75	C₂²×D₈₉	—	4	12	no	—
76	C₂²×D₈₉	C₂⁴	1	6	yes	—
77	C₂²×D₈₉	C₂⁴	1	2	yes	—
78	Q₈×C₈₉	C₂×D₄	1	4	yes	—
79	D₄×C₈₉	C₂×D₄	1	12	yes	—
80	C₂×C₄×C₈₉	—	4	6	no	—
81	C₂×C₄×C₈₉	C₂×D₄	1	12	yes	—
82	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
83	C₂×C₄×C₈₉	C₂⁴	1	6	yes	—
84	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂⁴	1	3	yes	—
86	C₂²⋊D₈₉	—	2	12	no	—
87	C₂²⋊D₈₉	—	2	12	no	—
88	C₂×Dic₈₉	—	2	12	no	—
89	C₂×Dic₈₉	C₂⁴	1	6	yes	—
90	C₂×Dic₈₉	C₂⁴	1	2	yes	—
91	D₃₅₆	—	2	12	no	—
92	D₃₅₆	—	2	6	no	—
93	D₃₅₆	—	2	2	no	—
94	D₃₅₆	—	4	12	no	—
95	C₄×D₈₉	—	2	12	no	—
96	C₄×D₈₉	—	2	12	no	—
97	Dic₁₇₈	—	2	12	no	—
98	Dic₁₇₈	—	2	6	no	—
99	Dic₁₇₈	—	2	2	no	—
100	C₂²×C₈₉	—	2	6	no	—
101	C₂²×C₈₉	C₂²×D₄	1	6	yes	—
102	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
103	C₂×D₈₉	—	2	2	no	—
104	C₂×D₈₉	—	2	6	no	—
105	C₂×D₈₉	—	2	4	no	—
106	C₂×D₈₉	—	2	12	no	—
107	C₂×D₈₉	—	8	12	no	—
108	C₄×C₈₉	—	2	6	no	—
109	C₄×C₈₉	—	2	2	no	—
110	C₄×C₈₉	C₂²×D₄	1	2	yes	—
111	C₄×C₈₉	C₂²×D₄	1	6	yes	—
112	Dic₈₉	—	2	6	no	—
113	Dic₈₉	—	2	2	no	—
114	C₂×C₈₉	—	2	6	no	—
115	C₂×C₈₉	C₂³×D₄	1	1	yes	characteristic
116	C₂×C₈₉	#585	1	2	yes	—
117	D₈₉	—	4	2	no	—
118	D₈₉	—	4	6	no	—
119	#3500	—	89	1	no	maximal, Sylow
120	C₈₉	#3500	1	1	yes	characteristic, complemented, Sylow
121	C₂×2₋¹⁺⁴	—	89	1	no	—
122	C₂×H(5,𝔽₂)	—	89	1	no	—
123	#548	—	89	4	no	—
124	#546	—	89	12	no	—
125	#536	—	89	3	no	—
126	#535	—	89	3	no	—
127	#532	—	89	3	no	—
128	C₂×#124	—	89	3	no	—
129	C₄×#41	D₈₉	1	1	yes	characteristic, complemented
130	2₋¹⁺⁴	—	178	8	no	—
131	H(5,𝔽₂)	—	178	8	no	—
132	C₂×#41	—	89	2	no	—
133	C₂×#41	—	89	6	no	—
134	C₂×#41	—	89	3	no	—
135	C₂×#41	—	89	3	no	—
136	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
137	C₂²×Q₈	—	89	2	no	—
138	C₂²×Q₈	—	89	3	no	—
139	C₂²×D₄	—	89	6	no	—
140	C₂²×D₄	—	89	3	no	—
141	C₄⋊Q₈	—	89	3	no	—
142	#127	—	89	3	no	—
143	#124	—	89	3	no	—
144	#124	—	89	6	no	—
145	#124	—	89	12	no	—
146	#124	—	89	1	no	—
147	#123	—	89	12	no	—
148	C₂²⋊Q₈	—	89	12	no	—
149	#121	—	89	12	no	—
150	#121	—	89	12	no	—
151	#120	—	89	12	no	—
152	C₄×Q₈	C₂×D₈₉	1	2	yes	complemented
153	C₄×D₄	—	89	12	no	—
154	C₄×D₄	C₂×D₈₉	1	6	yes	complemented
155	#117	C₂×D₈₉	1	3	yes	complemented
156	C₂×C₂²⋊C₄	—	89	12	no	—
157	C₂×C₄²	C₂×D₈₉	1	3	yes	complemented
158	C₂⁴	—	89	6	no	—
159	#41	—	178	8	no	—
160	#41	—	178	24	no	—
161	#41	—	178	12	no	—
162	#41	—	178	12	no	—
163	#41	D₃₅₆	1	8	yes	complemented
164	C₂×Q₈	—	89	6	no	—
165	C₂×Q₈	—	89	3	no	—
166	C₂×Q₈	—	89	1	no	—
167	C₂×Q₈	—	178	8	no	—
168	C₂×Q₈	—	178	12	no	—
169	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
170	C₂×D₄	—	89	6	no	—
171	C₂×D₄	—	89	6	no	—
172	C₂×D₄	—	89	12	no	—
173	C₂×D₄	—	89	6	no	—
174	C₂×D₄	—	89	3	no	—
175	C₂×D₄	—	89	1	no	—
176	C₂×D₄	—	178	24	no	—
177	C₂×D₄	—	178	12	no	—
178	C₂×D₄	C₂²×D₈₉	1	3	yes	—
179	C₂²×C₄	—	89	6	no	—
180	C₂²×C₄	—	89	6	no	—
181	C₂²×C₄	—	89	6	no	—
182	C₂²×C₄	C₂²×D₈₉	1	6	yes	—
183	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
184	C₄⋊C₄	—	89	12	no	—
185	C₄⋊C₄	C₂²×D₈₉	1	6	yes	—
186	C₂²⋊C₄	—	89	12	no	—
187	C₂²⋊C₄	—	89	24	no	—
188	C₂²⋊C₄	—	89	4	no	—
189	C₂²⋊C₄	C₂²×D₈₉	1	6	yes	—
190	C₄²	C₂²×D₈₉	1	1	yes	characteristic
191	C₄²	C₂²×D₈₉	1	3	yes	—
192	C₄²	C₂²×D₈₉	1	6	yes	—
193	C₂³	—	89	6	no	—
194	C₂³	—	89	2	no	—
195	C₂³	—	356	12	no	—
196	C₂³	—	178	12	no	—
197	C₂³	C₂³×D₈₉	1	3	yes	—
198	Q₈	—	178	12	no	—
199	Q₈	—	178	6	no	—
200	Q₈	—	178	2	no	—
201	Q₈	C₂×D₃₅₆	1	4	yes	—
202	D₄	—	356	12	no	—
203	D₄	—	178	12	no	—
204	D₄	—	178	12	no	—
205	D₄	—	178	12	no	—
206	D₄	—	178	6	no	—
207	D₄	—	178	2	no	—
208	D₄	C₂×D₃₅₆	1	12	yes	—
209	C₂×C₄	—	4	6	no	—
210	C₂×C₄	—	89	6	no	—
211	C₂×C₄	—	89	2	no	—
212	C₂×C₄	—	178	12	no	—
213	C₂×C₄	—	178	12	no	—
214	C₂×C₄	—	178	12	no	—
215	C₂×C₄	C₂×D₃₅₆	1	12	yes	—
216	C₂×C₄	C₂³×D₈₉	1	2	yes	—
217	C₂×C₄	C₂³×D₈₉	1	6	yes	—
218	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
219	C₂×C₄	C₂³×D₈₉	1	3	yes	—
220	C₂²	—	2	6	no	—
221	C₂²	—	178	2	no	—
222	C₂²	—	178	6	no	—
223	C₂²	—	178	4	no	—
224	C₂²	—	178	12	no	—
225	C₂²	—	712	12	no	—
226	C₂²	C₂²×D₃₅₆	1	6	yes	—
227	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	6	no	—
230	C₄	—	178	2	no	—
231	C₄	—	178	6	no	—
232	C₄	C₂²×D₃₅₆	1	6	yes	—
233	C₄	C₂²×D₃₅₆	1	2	yes	—
234	C₂	—	2	6	no	—
235	C₂	—	356	2	no	—
236	C₂	—	356	6	no	—
237	C₂	C₂³×D₃₅₆	1	1	yes	characteristic
238	C₂	#19152434	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	C₄×#41×C₈₉	C₂
hypercenter	C₄×#41	D₈₉
hypocenter	C₈₉	#3500
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#15668059	C₂	semidirect
2. C₂×#15668058	C₂	semidirect
3. #19152260	C₂	semidirect
4. #19152243	C₂	semidirect
5. #19152174	C₂	semidirect
6. #19152167	C₂	semidirect
7. #19152148	C₂	semidirect
8. C₂×#15668000	C₂	semidirect
9. C₄×#41×C₈₉	C₂	semidirect
14. C₂×#13083574	C₂²	semidirect
15. C₂×#13083573	C₂²	semidirect
16. C₂²×Dic₁₇₈	C₂²	semidirect
17. C₂×Q₈×D₈₉	C₂²	semidirect
20. #15668012	C₂²	semidirect
21. #15668009	C₂²	semidirect
22. #15668000	C₂²	semidirect
23. #15668000	C₂²	semidirect
24. #15668000	C₂²	semidirect
25. #15668000	C₂²	semidirect
26. #15667996	C₂²	semidirect
27. #15667992	C₂²	semidirect
28. #15667987	C₂²	semidirect
29. #15667986	C₂²	semidirect
30. #15667982	C₂²	semidirect
31. C₄×D₃₅₆	C₂²	semidirect
32. C₂×#13083551	C₂²	semidirect
34. C₄×Q₈×C₈₉	C₂²	semidirect
35. C₄×D₄×C₈₉	C₂²	semidirect
36. #117×C₈₉	C₂²	semidirect
37. C₂×C₄²×C₈₉	C₂²	semidirect
63. #41×C₈₉	D₄	semidirect
120. C₈₉	#3500	semidirect
129. C₄×#41	D₈₉	semidirect
152. C₄×Q₈	C₂×D₈₉	semidirect
154. C₄×D₄	C₂×D₈₉	semidirect
155. #117	C₂×D₈₉	semidirect
157. C₂×C₄²	C₂×D₈₉	semidirect
163. #41	D₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668059	2
C₂×#15668058	2
#19152260	2
#19152243	2
#19152174	2
#19152167	2
#19152148	2
C₂×#15668000	2
C₄×#41×C₈₉	2
#3500	89