Subgroups — #36448455

Order 22784.

420 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028816	C₂	1	1	yes	maximal, characteristic
2	#26020236	C₂	1	1	yes	maximal, characteristic
3	#26019023	C₂	1	2	yes	maximal, complemented
4	#26018148	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1458	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	—	2	1	no	—
7	#19151625	—	2	4	no	—
8	#19151622	—	2	1	no	—
9	#19151622	C₂²	1	2	yes	—
10	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊#10623494	—	2	1	no	—
13	#19151353	—	2	1	no	—
14	#19151349	—	2	4	no	—
15	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
16	#431×C₈₉	C₂²	1	2	yes	—
17	#324×C₈₉	C₄	1	2	yes	—
18	#323×C₈₉	C₄	1	2	yes	—
19	C₂×#13083558	—	2	2	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	4	2	no	—
22	C₂×#13083558	—	4	2	no	—
23	C₂²×#10623494	—	2	2	no	—
24	C₂²×#10623494	—	2	1	no	—
25	C₂²×#10623494	—	2	1	no	—
26	C₄⋊#10623494	—	2	4	no	—
27	C₄⋊#10623494	—	2	2	no	—
28	C₄⋊#10623494	—	2	2	no	—
29	C₄⋊#10623494	—	2	2	no	—
30	C₄⋊#10623494	D₄	1	4	yes	—
31	C₂²⋊#10623494	—	4	2	no	—
32	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
33	#117×C₈₉	—	2	2	no	—
34	#117×C₈₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₈₉	Q₈	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₈₉	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₈₉	—	2	2	no	—
42	C₂²⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₈₉	—	2	2	no	—
44	#13083558	—	4	4	no	—
45	#13083558	—	4	2	no	—
46	#13083558	—	4	2	no	—
47	C₂×#10623494	—	2	4	no	—
48	C₂×#10623494	—	4	4	no	—
49	C₂×#10623494	—	4	1	no	—
50	C₂×#10623494	—	4	1	no	—
51	C₂×#10623494	—	4	1	no	—
52	C₂×#10623494	—	4	1	no	—
53	C₂×#10623494	—	4	1	no	—
54	C₂×#10623494	—	4	1	no	—
55	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	—	2	1	no	—
57	C₂²×C₄×C₈₉	—	2	1	no	—
58	C₂²×C₄×C₈₉	—	2	1	no	—
59	C₂²×C₄×C₈₉	—	2	1	no	—
60	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₈₉	—	2	4	no	—
69	C₄⋊C₄×C₈₉	—	2	2	no	—
70	C₄⋊C₄×C₈₉	—	2	2	no	—
71	C₂²⋊C₄×C₈₉	—	2	2	no	—
72	C₂²⋊C₄×C₈₉	—	2	2	no	—
73	C₄²×C₈₉	—	2	1	no	—
74	C₄²×C₈₉	—	2	1	no	—
75	C₄²×C₈₉	C₂×D₄	1	2	yes	—
76	C₄²×C₈₉	#41	1	2	yes	—
77	C₂³×C₈₉	—	2	1	no	—
78	C₂³×C₈₉	—	2	1	no	—
79	C₂³×C₈₉	—	4	1	no	—
80	C₂³×C₈₉	—	4	1	no	—
81	C₂³×C₈₉	#106	1	2	yes	—
82	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₈₉	—	2	1	no	—
84	C₂×C₄×C₈₉	—	2	1	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	—	2	2	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	4	1	no	—
92	C₂×C₄×C₈₉	—	4	1	no	—
93	C₂×C₄×C₈₉	—	4	1	no	—
94	C₂×C₄×C₈₉	—	4	1	no	—
95	C₂×C₄×C₈₉	#106	1	2	yes	—
96	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
104	C₈×C₈₉	—	4	4	no	—
105	#10623494	—	4	4	no	—
106	#10623494	—	8	1	no	—
107	#10623494	—	8	1	no	—
108	C₂²×C₈₉	—	2	1	no	—
109	C₂²×C₈₉	—	2	1	no	—
110	C₂²×C₈₉	—	2	2	no	—
111	C₂²×C₈₉	—	4	2	no	—
112	C₂²×C₈₉	—	4	1	no	—
113	C₂²×C₈₉	—	4	1	no	—
114	C₂²×C₈₉	—	8	1	no	—
115	C₂²×C₈₉	#385	1	1	yes	characteristic
116	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
117	C₂²×C₈₉	#428	1	1	yes	characteristic
118	C₄×C₈₉	—	2	2	no	—
119	C₄×C₈₉	—	2	2	no	—
120	C₄×C₈₉	—	2	2	no	—
121	C₄×C₈₉	—	4	1	no	—
122	C₄×C₈₉	—	4	1	no	—
123	C₄×C₈₉	#448	1	1	yes	characteristic
124	C₄×C₈₉	#450	1	1	yes	characteristic
125	C₄×C₈₉	#481	1	1	yes	characteristic
126	C₄×C₈₉	#483	1	1	yes	characteristic
127	#8862	—	89	1	no	maximal, Sylow
128	C₂×C₈₉	—	2	1	no	—
129	C₂×C₈₉	—	2	1	no	—
130	C₂×C₈₉	—	4	2	no	—
131	C₂×C₈₉	#1853	1	1	yes	characteristic
132	C₂×C₈₉	#1858	1	1	yes	characteristic
133	C₂×C₈₉	#1868	1	1	yes	characteristic
134	#2905	—	89	1	no	—
135	#1798	—	89	1	no	—
136	#1570	—	89	2	no	—
137	#1458	D₈₉	1	1	yes	characteristic
138	#1301	—	89	2	no	—
139	C₈₉	#8862	1	1	yes	characteristic, complemented, Sylow
140	C₂²×M₁₆	—	178	1	no	—
141	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
142	#431	C₂×D₈₉	1	2	yes	—
143	#424	—	178	4	no	—
144	#423	—	89	2	no	—
145	#423	—	178	1	no	—
146	C₂×C₄⋊C₈	—	89	1	no	—
147	C₂×C₄⋊C₈	—	89	1	no	—
148	C₂×C₂²⋊C₈	—	178	1	no	—
149	#336	—	178	1	no	—
150	#334	—	178	4	no	—
151	#324	Dic₈₉	1	2	yes	—
152	#323	Dic₈₉	1	2	yes	—
153	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
154	C₂×M₁₆	—	356	2	no	—
155	C₂×M₁₆	—	356	2	no	—
156	C₂×M₁₆	—	178	2	no	—
157	C₂×M₁₆	—	178	2	no	—
158	C₂²×C₈	—	178	1	no	—
159	C₂²×C₈	—	178	1	no	—
160	C₂²×C₈	—	178	2	no	—
161	#117	—	2	2	no	—
162	#117	—	2	2	no	—
163	C₂×C₄⋊C₄	Dic₁₇₈	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	Dic₁₇₈	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
167	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
168	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	89	4	no	—
171	C₄⋊C₈	—	178	4	no	—
172	C₄⋊C₈	—	178	2	no	—
173	C₄⋊C₈	—	178	2	no	—
174	C₄⋊C₈	—	178	2	no	—
175	C₂²⋊C₈	—	356	2	no	—
176	C₂²⋊C₈	C₂×Dic₈₉	1	4	yes	—
177	C₄×C₈	—	2	2	no	—
178	C₂⁴	C₄⋊Dic₈₉	1	1	yes	characteristic
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
184	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
185	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
186	C₂²×C₄	#13083563	1	2	yes	—
187	C₂²×C₄	#13083563	1	2	yes	—
188	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
190	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
191	M₁₆	—	356	4	no	—
192	M₁₆	—	356	2	no	—
193	M₁₆	—	356	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	356	4	no	—
196	C₂×C₈	—	356	1	no	—
197	C₂×C₈	—	356	1	no	—
198	C₂×C₈	—	356	1	no	—
199	C₂×C₈	—	356	1	no	—
200	C₂×C₈	—	356	1	no	—
201	C₂×C₈	—	356	1	no	—
202	C₂×C₈	—	178	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	D₄×D₈₉	1	2	yes	—
210	C₄²	#13083573	1	2	yes	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	#15667944	1	2	yes	—
216	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	#15667944	1	2	yes	—
230	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
231	C₂×C₄	#15667982	1	1	yes	characteristic
232	C₂×C₄	#15667989	1	2	yes	—
233	C₂×C₄	#15667996	1	1	yes	characteristic
234	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
235	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
236	C₂×C₄	#15668040	1	1	yes	characteristic
237	C₂×C₄	#15668045	1	1	yes	characteristic
238	C₈	—	4	4	no	—
239	C₈	—	356	4	no	—
240	C₈	—	712	1	no	—
241	C₈	—	712	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	8	1	no	—
249	C₂²	#19151473	1	1	yes	characteristic
250	C₂²	C₂×#15667944	1	1	yes	characteristic
251	C₂²	#19151640	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	1	no	—
257	C₄	#19151739	1	1	yes	characteristic
258	C₄	#19151751	1	1	yes	characteristic
259	C₄	#19151917	1	1	yes	characteristic
260	C₄	#19151929	1	1	yes	characteristic
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	4	2	no	—
264	C₂	#26020447	1	1	yes	characteristic
265	C₂	#26020474	1	1	yes	characteristic
266	C₂	#26020540	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667982
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1458	C₂
socle	C₂²×C₈₉	#385
prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1458	D₈₉
hypocenter	C₈₉	#8862
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1458

Complements

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

N	H ≅ G/N	type
3. #26019023	C₂	semidirect
4. #26018148	C₂	semidirect
139. C₈₉	#8862	semidirect

Maximal subgroups

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

subgroup	index
#26028816	2
#26020236	2
#26019023	2
#26018148	2
C₈₉×#1458	2
#8862	89