Subgroups — #36445080

Order 22784.

530 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	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667923	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#10623494	C₂	1	2	yes	maximal
4	#26018093	C₂	1	2	yes	maximal
5	C₈₉×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×#15667923	—	2	2	no	—
8	C₂×C₄×#10623494	—	2	1	no	—
9	C₂×C₄×#10623494	C₂²	1	2	yes	—
10	#19151353	—	2	1	no	—
11	Q₈⋊#10623494	C₄	1	8	yes	complemented
12	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	#391×C₈₉	C₄	1	2	yes	—
14	#389×C₈₉	C₄	1	2	yes	—
15	#387×C₈₉	—	2	2	no	—
16	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
17	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
18	#376×C₈₉	—	2	1	no	—
19	C₄³×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×#10623494	—	2	2	no	—
21	C₂²×#10623494	—	2	2	no	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	C₂×C₄	1	4	yes	—
24	#15667923	—	2	8	no	—
25	C₄×#10623494	—	2	2	no	—
26	C₄×#10623494	—	4	2	no	—
27	C₄×#10623494	C₂×C₄	1	4	yes	—
28	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₄×Q₈×C₈₉	—	2	2	no	—
30	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₈₉	—	2	1	no	—
36	C₂×C₄²×C₈₉	—	2	1	no	—
37	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	D₄	1	2	yes	—
41	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
42	#95×C₈₉	—	2	2	no	—
43	#95×C₈₉	—	2	2	no	—
44	#95×C₈₉	—	2	2	no	—
45	#95×C₈₉	—	2	1	no	—
46	C₂×#10623494	—	2	8	no	—
47	C₂×#10623494	—	2	4	no	—
48	C₂×#10623494	—	2	4	no	—
49	C₂×#10623494	—	4	2	no	—
50	C₂×#10623494	—	4	2	no	—
51	C₂×Q₈×C₈₉	—	2	1	no	—
52	C₂×Q₈×C₈₉	—	2	4	no	—
53	C₂×Q₈×C₈₉	C₄²	1	2	yes	—
54	C₂²×C₄×C₈₉	—	2	1	no	—
55	C₂²×C₄×C₈₉	—	2	1	no	—
56	C₂²×C₄×C₈₉	—	2	2	no	—
57	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₈₉	—	2	1	no	—
63	C₄⋊C₄×C₈₉	—	2	4	no	—
64	C₄⋊C₄×C₈₉	—	2	4	no	—
65	C₄⋊C₄×C₈₉	—	4	4	no	—
66	C₄⋊C₄×C₈₉	C₄²	1	2	yes	—
67	C₄²×C₈₉	—	2	4	no	—
68	C₄²×C₈₉	—	2	4	no	—
69	C₄²×C₈₉	—	2	4	no	—
70	C₄²×C₈₉	—	2	1	no	—
71	C₄²×C₈₉	—	2	1	no	—
72	C₄²×C₈₉	—	2	2	no	—
73	C₄²×C₈₉	—	4	1	no	—
74	C₄²×C₈₉	—	4	1	no	—
75	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
76	C₄²×C₈₉	#41	1	2	yes	—
77	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₈₉	—	2	4	no	—
79	Q₈×C₈₉	—	4	2	no	—
80	C₂×C₄×C₈₉	—	2	2	no	—
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	—	2	2	no	—
83	C₂×C₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	—	2	2	no	—
85	C₂×C₄×C₈₉	—	2	2	no	—
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	—	4	2	no	—
89	C₂×C₄×C₈₉	—	4	2	no	—
90	C₂×C₄×C₈₉	—	4	2	no	—
91	C₂×C₄×C₈₉	—	4	2	no	—
92	C₂×C₄×C₈₉	—	4	2	no	—
93	C₂×C₄×C₈₉	#104	1	4	yes	—
94	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₈₉	#131	1	4	yes	—
99	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#137	1	2	yes	—
102	#10623494	—	2	8	no	—
103	#10623494	—	4	4	no	—
104	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²×C₈₉	#408	1	2	yes	—
106	C₂²×C₈₉	#417	1	1	yes	characteristic
107	C₂²×C₈₉	#419	1	1	yes	characteristic
108	C₂²×C₈₉	C₂×#104	1	2	yes	—
109	C₄×C₈₉	—	2	4	no	—
110	C₄×C₈₉	—	2	2	no	—
111	C₄×C₈₉	—	4	2	no	—
112	C₄×C₈₉	—	4	2	no	—
113	C₄×C₈₉	—	4	2	no	—
114	C₄×C₈₉	—	8	2	no	—
115	C₄×C₈₉	#405	1	2	yes	—
116	C₄×C₈₉	#433	1	2	yes	—
117	C₄×C₈₉	#440	1	2	yes	—
118	C₄×C₈₉	#441	1	2	yes	—
119	#8180	—	89	1	no	maximal, Sylow
120	C₂×C₈₉	#1483	1	2	yes	—
121	C₂×C₈₉	#1485	1	2	yes	—
122	C₂×C₈₉	#1750	1	1	yes	characteristic
123	C₂×C₈₉	C₄×#104	1	1	yes	characteristic
124	C₂×C₈₉	#1763	1	1	yes	characteristic
125	#2295	D₈₉	1	1	yes	characteristic
126	C₄×C₄⋊C₈	—	89	1	no	—
127	C₄×#97	—	89	1	no	—
128	C₂×Q₈⋊C₈	—	89	2	no	—
129	#1279	—	89	2	no	—
130	C₈₉	#8180	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
132	C₂×C₄⋊C₈	—	89	2	no	—
133	C₂×#97	—	178	2	no	—
134	C₂×C₄×C₈	—	89	2	no	—
135	C₂×C₄×C₈	—	178	1	no	—
136	#391	Dic₈₉	1	2	yes	—
137	#389	Dic₈₉	1	2	yes	—
138	#387	—	2	2	no	—
139	C₄×C₄⋊C₄	—	2	1	no	—
140	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
141	#376	—	2	1	no	—
142	C₄³	C₂×D₈₉	1	1	yes	characteristic
143	#336	—	178	1	no	—
144	Q₈⋊C₈	—	89	8	no	—
145	C₂²×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
146	C₂²×C₈	—	178	2	no	—
147	C₂²×C₈	—	178	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	C₄×D₈₉	1	4	yes	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
158	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
160	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
161	C₄⋊C₈	—	89	4	no	—
162	C₄⋊C₈	—	178	2	no	—
163	#97	—	178	8	no	—
164	C₄×C₈	—	89	4	no	—
165	C₄×C₈	—	356	2	no	—
166	C₄×C₈	—	178	2	no	—
167	#95	—	2	2	no	—
168	#95	—	2	2	no	—
169	#95	—	2	2	no	—
170	#95	—	2	1	no	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	C₄×Dic₈₉	1	2	yes	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
178	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
179	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
182	C₂×C₈	—	356	2	no	—
183	C₂×C₈	—	356	2	no	—
184	C₂×C₈	—	178	4	no	—
185	C₂×C₈	—	178	8	no	—
186	C₂×C₈	—	178	4	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	C₄×Dic₈₉	1	2	yes	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	C₂×C₄×D₈₉	1	2	yes	—
201	C₄²	#13083574	1	2	yes	—
202	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
203	Q₈	—	2	4	no	—
204	Q₈	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	#15667939	1	4	yes	—
219	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
220	C₂×C₄	#15667971	1	1	yes	characteristic
221	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
222	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
223	C₂×C₄	#15668019	1	4	yes	—
224	C₂×C₄	#15668037	1	1	yes	characteristic
225	C₂×C₄	#15668042	1	1	yes	characteristic
226	C₂×C₄	#15668043	1	2	yes	—
227	C₈	—	356	4	no	—
228	C₈	—	178	8	no	—
229	C₂²	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
230	C₂²	#19151565	1	2	yes	—
231	C₂²	#19151599	1	1	yes	characteristic
232	C₂²	#19151607	1	1	yes	characteristic
233	C₂²	C₂×#15667939	1	2	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	2	2	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	8	2	no	—
240	C₄	#19151554	1	2	yes	—
241	C₄	#19151659	1	2	yes	—
242	C₄	#19151696	1	2	yes	—
243	C₄	#19151703	1	2	yes	—
244	C₂	#26018665	1	2	yes	—
245	C₂	#26018673	1	2	yes	—
246	C₂	#26020018	1	1	yes	characteristic
247	C₂	C₄×#15667939	1	1	yes	characteristic
248	C₂	#26020082	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#2295	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#2295	D₈₉
hypocenter	C₈₉	#8180
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
11. Q₈⋊#10623494	C₄	semidirect
130. C₈₉	#8180	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10623494	2
C₄×#15667923	2
C₂×Q₈⋊#10623494	2
#26018093	2
C₈₉×#2295	2
#8180	89