Subgroups — #36453655

Order 22784.

486 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152003	C₂	1	1	yes	maximal, characteristic
2	#26021174	C₂	1	2	yes	maximal
3	C₈×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151345	C₂	1	2	yes	maximal
5	C₂×#335×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668012	C₂²	1	2	yes	—
7	#19152003	—	2	2	no	—
8	#19152003	C₄	1	4	yes	complemented
9	C₂×#15667946	—	2	1	no	—
10	C₂×#15667946	—	2	1	no	—
11	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	C₂×#15667937	—	2	4	no	—
13	C₂×C₈×Dic₈₉	—	2	1	no	—
14	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
15	#19151352	—	2	1	no	—
16	#19151345	—	2	4	no	—
17	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#335×C₈₉	—	2	2	no	—
20	C₂²×Dic₁₇₈	—	2	2	no	—
21	C₂²×#10623494	—	2	1	no	—
22	#15668012	—	2	2	no	—
23	#15668012	—	2	2	no	—
24	#15668012	—	2	2	no	—
25	#15668012	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₈₉	—	2	2	no	—
27	C₂×C₄⋊Dic₈₉	—	2	2	no	—
28	C₂×C₄⋊Dic₈₉	D₄	1	2	yes	—
29	C₂×C₄×Dic₈₉	—	2	1	no	—
30	#15667946	—	2	4	no	—
31	#15667946	—	2	4	no	—
32	C₄⋊#10623494	—	2	1	no	—
33	C₄⋊#10623494	D₄	1	2	yes	—
34	#15667937	—	4	8	no	—
35	C₈×Dic₈₉	—	4	2	no	—
36	#15667919	—	2	1	no	—
37	C₂²×C₈×C₈₉	—	2	2	no	—
38	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₈₉	—	2	2	no	—
42	C₄⋊C₈×C₈₉	—	2	2	no	—
43	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₈₉	D₄	1	2	yes	—
45	C₂×Dic₁₇₈	—	2	4	no	—
46	C₂×Dic₁₇₈	—	4	4	no	—
47	C₂×Dic₁₇₈	—	4	4	no	—
48	C₂×Dic₁₇₈	—	4	2	no	—
49	C₂²×Dic₈₉	—	2	2	no	—
50	C₂²×Dic₈₉	—	4	2	no	—
51	C₂×#10623494	—	2	2	no	—
52	C₂×#10623494	—	4	1	no	—
53	C₂×#10623494	—	4	1	no	—
54	C₄⋊Dic₈₉	—	2	2	no	—
55	C₄⋊Dic₈₉	—	2	4	no	—
56	C₄⋊Dic₈₉	—	2	4	no	—
57	C₄⋊Dic₈₉	—	4	2	no	—
58	C₄⋊Dic₈₉	—	4	2	no	—
59	C₄⋊Dic₈₉	D₈	1	4	yes	—
60	C₄×Dic₈₉	—	4	1	no	—
61	C₄×Dic₈₉	—	4	1	no	—
62	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
65	C₂×C₈×C₈₉	—	2	1	no	—
66	C₂×C₈×C₈₉	—	2	1	no	—
67	C₂×C₈×C₈₉	—	4	2	no	—
68	C₂×C₈×C₈₉	—	4	4	no	—
69	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
72	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
73	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₈₉	#41	1	1	yes	characteristic
77	C₂³×C₈₉	#121	1	1	yes	characteristic
78	C₂×C₄×C₈₉	—	2	2	no	—
79	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
83	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
84	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#135	1	2	yes	—
89	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
90	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
91	C₂×Dic₈₉	—	2	4	no	—
92	C₂×Dic₈₉	—	4	2	no	—
93	C₂×Dic₈₉	—	4	2	no	—
94	C₂×Dic₈₉	—	4	4	no	—
95	C₂×Dic₈₉	—	8	4	no	—
96	Dic₁₇₈	—	4	8	no	—
97	Dic₁₇₈	—	8	2	no	—
98	Dic₁₇₈	—	8	2	no	—
99	C₈×C₈₉	—	2	2	no	—
100	C₈×C₈₉	—	8	2	no	—
101	C₈×C₈₉	#103	1	4	yes	complemented
102	#10623494	—	4	2	no	—
103	C₂²×C₈₉	#390	1	1	yes	characteristic
104	C₂²×C₈₉	#461	1	1	yes	characteristic
105	C₂²×C₈₉	#462	1	1	yes	characteristic
106	C₂²×C₈₉	#466	1	2	yes	—
107	C₂²×C₈₉	#473	1	2	yes	—
108	C₄×C₈₉	—	2	1	no	—
109	C₄×C₈₉	—	2	1	no	—
110	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
111	C₄×C₈₉	#419	1	1	yes	characteristic
112	C₄×C₈₉	C₄×D₈	1	2	yes	—
113	C₄×C₈₉	#461	1	1	yes	characteristic
114	C₄×C₈₉	#462	1	1	yes	characteristic
115	C₄×C₈₉	#493	1	1	yes	characteristic
116	C₄×C₈₉	#495	1	1	yes	characteristic
117	Dic₈₉	—	4	4	no	—
118	Dic₈₉	—	8	4	no	—
119	#9731	—	89	1	no	maximal, Sylow
120	C₂×C₈₉	#1673	1	2	yes	—
121	C₂×C₈₉	#1674	1	2	yes	—
122	C₂×C₈₉	#1970	1	1	yes	characteristic
123	C₂×C₈₉	#1979	1	1	yes	characteristic
124	C₂×C₈₉	#1981	1	1	yes	characteristic
125	C₂×#500	—	89	1	no	—
126	#1981	—	89	2	no	—
127	C₈×C₄⋊C₄	—	89	1	no	—
128	C₂×#335	D₈₉	1	1	yes	characteristic
129	C₂×#332	—	89	2	no	—
130	C₈₉	#9731	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₄⋊Q₈	—	89	2	no	—
132	#500	—	89	4	no	—
133	#500	—	178	2	no	—
134	C₂×#107	—	178	1	no	—
135	C₂×#107	—	178	1	no	—
136	C₂×C₄⋊C₈	—	89	1	no	—
137	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
138	C₂×#103	—	178	4	no	—
139	C₂×C₄×C₈	—	178	1	no	—
140	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
141	C₄×C₄⋊C₄	—	89	1	no	—
142	#336	—	178	1	no	—
143	#335	—	2	2	no	—
144	#332	—	178	4	no	—
145	C₂²×Q₈	—	178	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	178	1	no	—
148	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
149	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
150	C₄⋊Q₈	—	89	4	no	—
151	C₄⋊Q₈	—	178	2	no	—
152	C₄⋊Q₈	—	178	2	no	—
153	C₄⋊Q₈	—	178	2	no	—
154	C₂×C₄⋊C₄	—	89	2	no	—
155	C₂×C₄⋊C₄	—	178	2	no	—
156	C₂×C₄⋊C₄	—	178	2	no	—
157	C₂×C₄²	—	178	1	no	—
158	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
159	#107	—	178	4	no	—
160	#107	—	178	4	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	89	2	no	—
164	C₄⋊C₈	—	178	1	no	—
165	#103	—	356	8	no	—
166	C₄×C₈	—	356	2	no	—
167	C₄×C₈	C₄×D₈₉	1	2	yes	—
168	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
169	#95	—	178	1	no	—
170	C₂×Q₈	—	356	2	no	—
171	C₂×Q₈	—	356	4	no	—
172	C₂×Q₈	—	356	4	no	—
173	C₂×Q₈	—	178	4	no	—
174	C₂²×C₄	—	356	2	no	—
175	C₂²×C₄	—	178	2	no	—
176	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
178	C₂²×C₄	#13083575	1	1	yes	characteristic
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	4	no	—
183	C₂×C₈	—	356	1	no	—
184	C₂×C₈	—	356	1	no	—
185	C₂×C₈	—	178	2	no	—
186	C₂×C₈	#13083551	1	2	yes	—
187	C₂×C₈	#13083551	1	2	yes	—
188	C₂×C₈	#13083563	1	2	yes	—
189	C₂×C₈	Dic₃₅₆	1	2	yes	—
190	C₄⋊C₄	—	89	4	no	—
191	C₄⋊C₄	—	356	2	no	—
192	C₄⋊C₄	—	356	2	no	—
193	C₄⋊C₄	—	178	4	no	—
194	C₄⋊C₄	—	178	4	no	—
195	C₄⋊C₄	—	178	2	no	—
196	C₄²	—	356	1	no	—
197	C₄²	—	356	1	no	—
198	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
199	C₄²	D₄×D₈₉	1	1	yes	characteristic
200	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
201	C₄²	#13083575	1	1	yes	characteristic
202	C₂³	#15667987	1	1	yes	characteristic
203	Q₈	—	356	8	no	—
204	Q₈	—	712	2	no	—
205	Q₈	—	712	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	356	2	no	—
208	C₂×C₄	—	356	2	no	—
209	C₂×C₄	—	356	4	no	—
210	C₂×C₄	—	178	4	no	—
211	C₂×C₄	—	712	4	no	—
212	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
213	C₂×C₄	#15667974	1	1	yes	characteristic
214	C₂×C₄	#15667974	1	1	yes	characteristic
215	C₂×C₄	#15667987	1	1	yes	characteristic
216	C₂×C₄	#15668001	1	1	yes	characteristic
217	C₂×C₄	#15668008	1	1	yes	characteristic
218	C₂×C₄	D₈×D₈₉	1	2	yes	—
219	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
220	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
221	C₂×C₄	#15668035	1	2	yes	—
222	C₂×C₄	#15668045	1	1	yes	characteristic
223	C₂×C₄	#15668045	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	8	2	no	—
226	C₈	—	356	2	no	—
227	C₈	#15667937	1	4	yes	complemented
228	C₂²	#19151499	1	1	yes	characteristic
229	C₂²	#19151804	1	1	yes	characteristic
230	C₂²	#19151809	1	1	yes	characteristic
231	C₂²	#19151835	1	2	yes	—
232	C₂²	#19151874	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	356	4	no	—
236	C₄	—	712	4	no	—
237	C₄	C₂×#15667937	1	1	yes	characteristic
238	C₄	#19151608	1	1	yes	characteristic
239	C₄	#19151680	1	2	yes	—
240	C₄	#19151804	1	1	yes	characteristic
241	C₄	#19151809	1	1	yes	characteristic
242	C₄	#19151973	1	1	yes	characteristic
243	C₄	#19151983	1	1	yes	characteristic
244	C₂	#26019642	1	2	yes	—
245	C₂	#26019647	1	2	yes	—
246	C₂	#26021116	1	1	yes	characteristic
247	C₂	#26021165	1	1	yes	characteristic
248	C₂	#26021174	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667987
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#335×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#121
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#335	D₈₉
hypocenter	C₈₉	#9731
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
8. #19152003	C₄	semidirect
101. C₈×C₈₉	#103	semidirect
130. C₈₉	#9731	semidirect
227. C₈	#15667937	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152003	2
#26021174	2
C₈×C₄⋊Dic₈₉	2
C₂×#19151345	2
C₂×#335×C₈₉	2
#9731	89