Subgroups — #36470898

Order 22784.

420 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151418	C₂	1	2	yes	maximal, complemented
2	#26021863	C₂	1	1	yes	maximal, characteristic
3	C₈×C₂²⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	#26018333	C₂	1	2	yes	maximal, complemented
5	C₂²×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	C₂²	1	1	yes	characteristic
7	C₂×#15667952	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊#10623494	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
10	C₄×C₂²⋊Dic₈₉	C₄	1	1	yes	characteristic
11	#19151418	—	2	4	no	—
12	C₂²⋊#13083549	—	2	1	no	—
13	C₂²⋊#13083549	—	2	1	no	—
14	#19151352	C₄	1	2	yes	—
15	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₈₉	—	2	1	no	—
17	C₂×M₃₂×C₈₉	—	2	4	no	—
18	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
19	C₂²×C₁₆×C₈₉	—	2	1	no	—
20	C₂²×#10623494	C₂×C₄	1	2	yes	—
21	C₂×C₂²⋊Dic₈₉	—	2	1	no	—
22	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
23	#15667952	—	2	2	no	—
24	#15667952	—	2	2	no	—
25	C₂×#13083549	—	2	1	no	—
26	C₂×#13083549	—	2	1	no	—
27	C₂²⋊#10623494	C₈	1	2	yes	—
28	C₂²⋊#10623494	C₈	1	2	yes	—
29	C₈×Dic₈₉	—	2	2	no	—
30	C₈×Dic₈₉	—	2	2	no	—
31	#15667919	—	2	1	no	—
32	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	—	2	2	no	—
34	C₂²×C₈×C₈₉	—	2	2	no	—
35	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
41	M₃₂×C₈₉	—	2	4	no	—
42	M₃₂×C₈₉	—	2	2	no	—
43	M₃₂×C₈₉	D₄	1	4	yes	—
44	C₂×C₁₆×C₈₉	—	2	4	no	—
45	C₂×C₁₆×C₈₉	—	2	2	no	—
46	C₂²×Dic₈₉	—	2	2	no	—
47	C₂×#10623494	—	2	2	no	—
48	C₂×#10623494	—	2	2	no	—
49	C₂×#10623494	C₂×C₈	1	4	yes	—
50	C₂²⋊Dic₈₉	—	2	4	no	—
51	C₄×Dic₈₉	—	2	2	no	—
52	C₄×Dic₈₉	—	2	2	no	—
53	#13083549	—	4	1	no	—
54	#13083549	—	4	1	no	—
55	C₂⁴×C₈₉	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	—	2	2	no	—
57	C₂²×C₄×C₈₉	—	2	2	no	—
58	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	—	2	2	no	—
65	C₂×C₈×C₈₉	—	2	1	no	—
66	C₂×C₈×C₈₉	—	2	1	no	—
67	C₂×C₈×C₈₉	—	2	2	no	—
68	C₂×C₈×C₈₉	—	2	2	no	—
69	C₂×C₈×C₈₉	—	2	1	no	—
70	C₂×C₈×C₈₉	—	2	1	no	—
71	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₈×C₈₉	#41	1	2	yes	—
77	C₁₆×C₈₉	—	2	4	no	—
78	C₂³×C₈₉	—	2	2	no	—
79	C₂³×C₈₉	—	2	2	no	—
80	C₂³×C₈₉	—	4	1	no	—
81	C₂³×C₈₉	C₄⋊C₈	1	1	yes	characteristic
82	C₂³×C₈₉	C₄⋊C₈	1	1	yes	characteristic
83	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₈₉	—	2	2	no	—
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	2	no	—
90	C₂×C₄×C₈₉	—	2	2	no	—
91	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
92	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₈₉	#131	1	2	yes	—
97	C₂×Dic₈₉	—	2	4	no	—
98	C₂×Dic₈₉	—	4	2	no	—
99	C₂×Dic₈₉	—	4	2	no	—
100	C₈×C₈₉	—	2	1	no	—
101	C₈×C₈₉	—	2	1	no	—
102	C₈×C₈₉	#120	1	1	yes	characteristic
103	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
104	C₈×C₈₉	#123	1	1	yes	characteristic
105	#10623494	—	2	4	no	—
106	C₂²×C₈₉	—	2	1	no	—
107	C₂²×C₈₉	—	2	1	no	—
108	C₂²×C₈₉	—	2	1	no	—
109	C₂²×C₈₉	—	2	1	no	—
110	C₂²×C₈₉	—	2	1	no	—
111	C₂²×C₈₉	—	2	1	no	—
112	C₂²×C₈₉	—	2	1	no	—
113	C₂²×C₈₉	—	4	4	no	—
114	C₂²×C₈₉	#364	1	2	yes	complemented
115	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂²×C₈₉	#435	1	2	yes	—
117	C₄×C₈₉	—	2	1	no	—
118	C₄×C₈₉	—	2	1	no	—
119	C₄×C₈₉	#385	1	1	yes	characteristic
120	C₄×C₈₉	#424	1	1	yes	characteristic
121	C₄×C₈₉	C₈×D₄	1	2	yes	—
122	Dic₈₉	—	4	4	no	—
123	#12914	—	89	1	no	maximal, Sylow
124	C₂×C₈₉	—	2	2	no	—
125	C₂×C₈₉	—	2	2	no	—
126	C₂×C₈₉	—	4	1	no	—
127	C₂×C₈₉	#1850	1	1	yes	characteristic
128	C₂×C₈₉	C₂×#364	1	1	yes	characteristic
129	C₂×C₈₉	#2153	1	1	yes	characteristic
130	C₂²×M₃₂	D₈₉	1	1	yes	characteristic
131	C₂×#364	—	89	2	no	—
132	#2112	—	89	1	no	—
133	C₈×C₂²⋊C₄	—	89	1	no	—
134	#1376	—	89	2	no	—
135	C₈₉	#12914	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
137	C₂×M₃₂	—	2	1	no	—
138	C₂×M₃₂	—	2	4	no	—
139	C₂×M₃₂	—	89	1	no	—
140	C₂×M₃₂	—	89	1	no	—
141	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
142	C₂²×C₁₆	—	2	1	no	—
143	C₂×C₂²⋊C₈	—	89	1	no	—
144	C₂×C₄×C₈	—	89	2	no	—
145	C₄×C₂²⋊C₄	—	89	1	no	—
146	#364	—	178	4	no	—
147	C₂²⋊C₁₆	—	178	1	no	—
148	C₂²⋊C₁₆	—	178	1	no	—
149	#336	—	89	2	no	—
150	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	89	2	no	—
154	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
155	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
156	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
158	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	—	178	1	no	—
161	C₂×C₄²	—	89	2	no	—
162	M₃₂	—	2	4	no	—
163	M₃₂	—	2	2	no	—
164	M₃₂	—	178	2	no	—
165	M₃₂	—	178	2	no	—
166	M₃₂	C₂²⋊D₈₉	1	4	yes	—
167	C₂×C₁₆	—	2	4	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	178	1	no	—
170	C₂×C₁₆	—	178	1	no	—
171	C₂²⋊C₈	—	89	2	no	—
172	C₂²⋊C₈	—	89	2	no	—
173	C₄×C₈	—	178	2	no	—
174	C₄×C₈	—	178	2	no	—
175	#95	—	178	1	no	—
176	C₂⁴	#13083557	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	178	2	no	—
180	C₂²×C₄	#13083553	1	1	yes	characteristic
181	C₂²×C₄	#13083553	1	1	yes	characteristic
182	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
183	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
184	C₂²×C₄	#13083557	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	89	4	no	—
194	C₂×C₈	—	178	2	no	—
195	C₂×C₈	—	178	2	no	—
196	C₂×C₈	#13083553	1	1	yes	characteristic
197	C₂×C₈	#13083553	1	1	yes	characteristic
198	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
199	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
200	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
201	C₂×C₈	#13083574	1	2	yes	—
202	C₂²⋊C₄	—	178	4	no	—
203	C₄²	—	178	2	no	—
204	C₄²	—	178	2	no	—
205	C₁₆	—	2	4	no	—
206	C₁₆	—	356	1	no	—
207	C₁₆	—	356	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	1	no	—
211	C₂³	#15667941	1	1	yes	characteristic
212	C₂³	#15667941	1	1	yes	characteristic
213	C₂³	C₂×#13083557	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	356	2	no	—
222	C₂×C₄	—	356	2	no	—
223	C₂×C₄	—	178	4	no	—
224	C₂×C₄	#15667941	1	1	yes	characteristic
225	C₂×C₄	#15667941	1	1	yes	characteristic
226	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
227	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
228	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
229	C₂×C₄	#15668018	1	2	yes	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	178	4	no	—
233	C₈	#15667981	1	1	yes	characteristic
234	C₈	#15667991	1	2	yes	—
235	C₈	#15667995	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	#19151418	1	2	yes	complemented
245	C₂²	C₂×#15667941	1	1	yes	characteristic
246	C₂²	#19151669	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	356	4	no	—
250	C₄	#19151471	1	1	yes	characteristic
251	C₄	#19151624	1	1	yes	characteristic
252	C₄	C₈×C₂²⋊D₈₉	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	4	1	no	—
256	C₂	#26020430	1	1	yes	characteristic
257	C₂	C₂×#19151418	1	1	yes	characteristic
258	C₂	#26022017	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂²×M₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₈
prime radical	C₂⁴×C₈₉	M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂²×M₃₂	D₈₉
hypocenter	C₈₉	#12914
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#19151418	C₂	semidirect
4. #26018333	C₂	semidirect
114. C₂²×C₈₉	#364	semidirect
135. C₈₉	#12914	semidirect
244. C₂²	#19151418	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151418	2
#26021863	2
C₈×C₂²⋊Dic₈₉	2
#26018333	2
C₂²×M₃₂×C₈₉	2
#12914	89