Subgroups — #36472442

Order 22784.

290 conjugacy classes of subgroups.

The subgroup lattice has 232 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	1	yes	maximal, characteristic
2	#26022012	C₂	1	1	yes	maximal, characteristic
3	#26022011	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151373	C₂	1	1	yes	maximal, characteristic
5	#26020132	C₂	1	1	yes	maximal, characteristic
6	#26018340	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1379	C₂	1	1	yes	maximal, characteristic
8	C₂×#15667952	C₂²	1	1	yes	characteristic
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	C₂×#15667947	C₄	1	1	yes	characteristic
11	#19151634	C₄	1	1	yes	characteristic
12	#19151553	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
14	#19151418	—	2	2	no	—
15	C₄⋊#13083549	—	2	2	no	—
16	#19151416	—	2	2	no	—
17	#19151373	—	2	2	no	—
18	#19151359	C₄	1	1	yes	characteristic
19	#19151357	C₄	1	1	yes	characteristic
20	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
22	#405×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
24	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
25	#15667971	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
27	#15667952	—	2	1	no	—
28	#15667952	—	2	1	no	—
29	C₂×#13083549	—	2	1	no	—
30	C₂×#13083549	—	2	1	no	—
31	C₂×#13083549	—	2	1	no	—
32	C₂×#13083549	D₄	1	2	yes	—
33	#15667947	—	2	2	no	—
34	#15667945	C₈	1	2	yes	—
35	#15667943	C₈	1	2	yes	—
36	#15667922	—	2	1	no	—
37	C₈×Dic₈₉	—	2	1	no	—
38	C₈×Dic₈₉	D₄	1	2	yes	—
39	C₈×Dic₈₉	Q₈	1	2	yes	—
40	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
43	M₃₂×C₈₉	—	2	1	no	—
44	M₃₂×C₈₉	—	2	1	no	—
45	C₂×C₁₆×C₈₉	—	2	1	no	—
46	C₂×C₁₆×C₈₉	—	2	1	no	—
47	C₂×C₁₆×C₈₉	—	2	1	no	—
48	C₂×C₁₆×C₈₉	Q₈	1	2	yes	—
49	#97×C₈₉	—	2	1	no	—
50	C₄×C₈×C₈₉	—	2	1	no	—
51	C₂²×Dic₈₉	—	2	1	no	—
52	#13083558	—	2	1	no	—
53	#13083558	—	2	1	no	—
54	C₂×#10623494	—	2	1	no	—
55	C₂×#10623494	C₄⋊C₄	1	2	yes	—
56	C₂×#10623494	C₂×C₈	1	1	yes	characteristic
57	C₂×#10623494	C₂×C₈	1	1	yes	characteristic
58	C₂×#10623494	M₁₆	1	2	yes	—
59	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
60	C₂²⋊Dic₈₉	—	2	1	no	—
61	C₄×Dic₈₉	—	2	1	no	—
62	C₄×Dic₈₉	—	2	1	no	—
63	C₄×Dic₈₉	C₄⋊C₄	1	2	yes	—
64	#13083549	—	2	2	no	—
65	#13083549	—	4	1	no	—
66	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	M₁₆×C₈₉	—	2	1	no	—
68	M₁₆×C₈₉	—	2	1	no	—
69	C₂×C₈×C₈₉	—	2	1	no	—
70	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
75	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
76	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
77	C₂²⋊C₄×C₈₉	—	2	1	no	—
78	C₄²×C₈₉	—	2	1	no	—
79	C₁₆×C₈₉	—	2	2	no	—
80	C₁₆×C₈₉	—	4	1	no	—
81	C₂³×C₈₉	#131	1	1	yes	characteristic
82	C₂×C₄×C₈₉	—	2	2	no	—
83	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
89	C₂×Dic₈₉	—	2	2	no	—
90	C₂×Dic₈₉	—	2	2	no	—
91	C₂×Dic₈₉	—	4	2	no	—
92	C₈×C₈₉	—	4	1	no	—
93	C₈×C₈₉	#121	1	1	yes	characteristic
94	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
95	C₈×C₈₉	#125	1	1	yes	characteristic
96	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
97	#10623494	—	4	1	no	—
98	#10623494	C₄⋊C₈	1	2	yes	—
99	#10623494	C₄⋊C₈	1	2	yes	—
100	C₂²×C₈₉	—	2	2	no	—
101	C₂²×C₈₉	#424	1	1	yes	characteristic
102	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
103	C₂²×C₈₉	#446	1	1	yes	characteristic
104	C₄×C₈₉	—	4	2	no	—
105	C₄×C₈₉	#389	1	1	yes	characteristic
106	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₄×C₈₉	#436	1	1	yes	characteristic
108	C₄×C₈₉	C₈×Q₈	1	1	yes	characteristic
109	Dic₈₉	—	4	2	no	—
110	Dic₈₉	—	4	2	no	—
111	#13162	—	89	1	no	maximal, Sylow
112	C₂×C₈₉	—	2	1	no	—
113	C₂×C₈₉	—	2	1	no	—
114	C₂×C₈₉	#1939	1	1	yes	characteristic
115	C₂×C₈₉	#2171	1	2	yes	—
116	C₂×#364	—	89	1	no	—
117	#2151	—	89	1	no	—
118	#2151	—	89	1	no	—
119	C₂×#346	—	89	1	no	—
120	#1772	—	89	1	no	—
121	#1379	—	89	1	no	—
122	#1379	D₈₉	1	1	yes	characteristic
123	C₈₉	#13162	1	1	yes	characteristic, complemented, Sylow
124	C₂×M₃₂	—	89	1	no	—
125	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
126	C₂²×C₁₆	—	89	1	no	—
127	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
128	C₂×#108	—	89	1	no	—
129	#427	—	89	1	no	—
130	#405	—	89	1	no	—
131	#405	C₂×D₈₉	1	1	yes	characteristic
132	C₂×C₄×C₈	—	89	1	no	—
133	#364	—	178	2	no	—
134	C₄⋊C₁₆	—	178	2	no	—
135	C₄⋊C₁₆	—	178	2	no	—
136	#346	—	178	2	no	—
137	#339	—	89	1	no	—
138	#339	—	89	1	no	—
139	C₂×M₁₆	—	89	1	no	—
140	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
141	C₂²×C₈	—	89	1	no	—
142	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
143	#117	—	89	1	no	—
144	#117	C₄×D₈₉	1	1	yes	characteristic
145	C₂×C₄²	—	89	1	no	—
146	M₃₂	—	2	1	no	—
147	M₃₂	—	2	1	no	—
148	M₃₂	—	178	1	no	—
149	M₃₂	—	178	1	no	—
150	C₂×C₁₆	—	2	1	no	—
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	—	2	1	no	—
153	C₂×C₁₆	—	89	2	no	—
154	C₂×C₁₆	—	178	1	no	—
155	C₂×C₁₆	—	178	1	no	—
156	C₂×C₁₆	—	178	1	no	—
157	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
158	#108	—	178	2	no	—
159	#107	—	89	2	no	—
160	#106	—	89	2	no	—
161	#97	—	2	1	no	—
162	#97	—	178	1	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	—	89	2	no	—
165	C₄×C₈	—	89	2	no	—
166	C₄×C₈	—	178	1	no	—
167	C₂²×C₄	—	178	1	no	—
168	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
169	M₁₆	—	2	1	no	—
170	M₁₆	—	2	1	no	—
171	M₁₆	—	178	1	no	—
172	M₁₆	—	178	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	89	1	no	—
175	C₂×C₈	—	89	1	no	—
176	C₂×C₈	—	89	2	no	—
177	C₂×C₈	—	89	2	no	—
178	C₂×C₈	—	178	1	no	—
179	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
180	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
181	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
182	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
183	C₂×C₈	#13083573	1	1	yes	characteristic
184	C₂×C₈	#13083574	1	1	yes	characteristic
185	C₄⋊C₄	—	89	2	no	—
186	C₄⋊C₄	C₈×D₈₉	1	2	yes	—
187	C₂²⋊C₄	—	2	1	no	—
188	C₂²⋊C₄	—	178	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	89	2	no	—
191	C₄²	—	178	1	no	—
192	C₄²	—	178	1	no	—
193	C₁₆	—	2	2	no	—
194	C₁₆	—	4	1	no	—
195	C₁₆	—	356	1	no	—
196	C₁₆	—	178	2	no	—
197	C₂³	#15668020	1	1	yes	characteristic
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	356	2	no	—
200	C₂×C₄	—	178	2	no	—
201	C₂×C₄	—	178	2	no	—
202	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
203	C₂×C₄	#15667972	1	1	yes	characteristic
204	C₂×C₄	C₄×Dic₁₇₈	1	1	yes	characteristic
205	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
206	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
207	C₂×C₄	#15668018	1	1	yes	characteristic
208	C₈	—	4	1	no	—
209	C₈	—	89	2	no	—
210	C₈	—	89	2	no	—
211	C₈	—	356	1	no	—
212	C₈	#15667985	1	1	yes	characteristic
213	C₈	#15667989	1	1	yes	characteristic
214	C₈	#15668002	1	1	yes	characteristic
215	C₈	#15668011	1	1	yes	characteristic
216	C₂²	—	2	2	no	—
217	C₂²	#19151623	1	1	yes	characteristic
218	C₂²	#19151661	1	1	yes	characteristic
219	C₂²	#19151726	1	1	yes	characteristic
220	C₄	—	4	2	no	—
221	C₄	—	356	2	no	—
222	C₄	—	356	2	no	—
223	C₄	#19151495	1	1	yes	characteristic
224	C₄	C₄⋊C₈×D₈₉	1	1	yes	characteristic
225	C₄	#19151673	1	1	yes	characteristic
226	C₄	C₈×Dic₁₇₈	1	1	yes	characteristic
227	C₂	—	2	1	no	—
228	C₂	—	2	1	no	—
229	C₂	#26020969	1	1	yes	characteristic
230	C₂	#26022091	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1379	C₂
socle	C₂²×C₈₉	C₈×D₄
prime radical	C₂³×C₈₉	#131
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1379	D₈₉
hypocenter	C₈₉	#13162
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
123. C₈₉	#13162	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
#26022012	2
#26022011	2
C₂×#19151373	2
#26020132	2
#26018340	2
C₈₉×#1379	2
#13162	89