Subgroups — #36447284

Order 22784.

478 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020232	C₂	1	2	yes	maximal
2	C₂×#19151351	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151349	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018087	C₂	1	2	yes	maximal, complemented
5	C₂×#431×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	C₂×C₂²⋊#10623494	—	2	2	no	—
9	#19151353	—	2	2	no	—
10	#19151351	—	2	2	no	—
11	#19151349	—	2	2	no	—
12	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
13	#431×C₈₉	—	2	2	no	—
14	#431×C₈₉	—	2	2	no	—
15	C₂×C₄⋊C₈×C₈₉	C₄	1	2	yes	—
16	C₂×C₂²⋊C₈×C₈₉	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×#10623494	—	2	2	no	—
20	C₂²×#10623494	—	2	2	no	—
21	C₄⋊#10623494	—	2	2	no	—
22	C₄⋊#10623494	—	2	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	C₄⋊#10623494	—	2	2	no	—
25	C₂²⋊#10623494	—	4	4	no	—
26	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₈₉	—	2	1	no	—
32	C₂×C₄²×C₈₉	—	2	1	no	—
33	C₂×C₄²×C₈₉	—	2	1	no	—
34	C₂×C₄²×C₈₉	—	2	1	no	—
35	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₈₉	—	2	4	no	—
40	C₂²⋊C₈×C₈₉	—	2	4	no	—
41	C₄×C₈×C₈₉	—	2	1	no	—
42	C₄×C₈×C₈₉	—	2	1	no	—
43	C₄×C₈×C₈₉	D₄	1	2	yes	—
44	C₄×C₈×C₈₉	D₄	1	2	yes	—
45	C₂×#10623494	—	4	2	no	—
46	C₂×#10623494	—	4	2	no	—
47	C₂×#10623494	—	4	2	no	—
48	C₂×#10623494	—	4	2	no	—
49	C₂×#10623494	—	4	2	no	—
50	C₂×#10623494	—	4	2	no	—
51	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	—	2	2	no	—
53	C₂²×C₄×C₈₉	—	2	2	no	—
54	C₂²×C₄×C₈₉	—	2	1	no	—
55	C₂²×C₄×C₈₉	—	2	1	no	—
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	—	2	4	no	—
61	C₂²×C₄×C₈₉	—	2	2	no	—
62	C₂²×C₄×C₈₉	—	2	2	no	—
63	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₈₉	—	2	4	no	—
70	C₂×C₈×C₈₉	—	2	2	no	—
71	C₂×C₈×C₈₉	—	2	4	no	—
72	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
73	C₄²×C₈₉	—	2	1	no	—
74	C₄²×C₈₉	—	2	1	no	—
75	C₄²×C₈₉	—	2	1	no	—
76	C₄²×C₈₉	—	2	1	no	—
77	C₄²×C₈₉	—	2	1	no	—
78	C₄²×C₈₉	—	2	1	no	—
79	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₈₉	#41	1	1	yes	characteristic
82	C₄²×C₈₉	#41	1	1	yes	characteristic
83	C₂³×C₈₉	—	2	1	no	—
84	C₂³×C₈₉	—	2	2	no	—
85	C₂³×C₈₉	—	2	2	no	—
86	C₂³×C₈₉	—	2	1	no	—
87	C₂³×C₈₉	#108	1	2	yes	—
88	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₈₉	—	2	4	no	—
90	C₂×C₄×C₈₉	—	2	4	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	2	2	no	—
99	C₂×C₄×C₈₉	—	4	2	no	—
100	C₂×C₄×C₈₉	—	4	2	no	—
101	C₂×C₄×C₈₉	—	4	2	no	—
102	C₂×C₄×C₈₉	—	4	1	no	—
103	C₂×C₄×C₈₉	—	4	1	no	—
104	C₂×C₄×C₈₉	#108	1	2	yes	—
105	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#131	1	4	yes	—
112	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₈₉	#137	1	2	yes	—
114	C₈×C₈₉	—	2	4	no	—
115	C₈×C₈₉	—	4	2	no	—
116	#10623494	—	8	2	no	—
117	#10623494	—	8	2	no	—
118	C₂²×C₈₉	—	2	2	no	—
119	C₂²×C₈₉	—	2	4	no	—
120	C₂²×C₈₉	—	4	1	no	—
121	C₂²×C₈₉	—	4	1	no	—
122	C₂²×C₈₉	—	4	2	no	—
123	C₂²×C₈₉	#380	1	1	yes	characteristic
124	C₂²×C₈₉	#424	1	2	yes	—
125	C₂²×C₈₉	#428	1	2	yes	—
126	C₂²×C₈₉	C₂×#108	1	1	yes	characteristic
127	C₂²×C₈₉	C₂×#108	1	1	yes	characteristic
128	C₄×C₈₉	—	2	2	no	—
129	C₄×C₈₉	—	2	1	no	—
130	C₄×C₈₉	—	2	1	no	—
131	C₄×C₈₉	—	4	1	no	—
132	C₄×C₈₉	—	4	1	no	—
133	C₄×C₈₉	#408	1	2	yes	—
134	C₄×C₈₉	#433	1	2	yes	—
135	C₄×C₈₉	#468	1	2	yes	—
136	C₄×C₈₉	#469	1	1	yes	characteristic
137	C₄×C₈₉	#470	1	1	yes	characteristic
138	#8600	—	89	1	no	maximal, Sylow
139	C₂×C₈₉	—	4	2	no	—
140	C₂×C₈₉	#1573	1	2	yes	—
141	C₂×C₈₉	#1574	1	2	yes	—
142	C₂×C₈₉	#1798	1	1	yes	characteristic
143	C₂×C₈₉	#1809	1	1	yes	characteristic
144	C₂×C₈₉	#1810	1	1	yes	characteristic
145	C₂×#431	D₈₉	1	1	yes	characteristic
146	#1797	—	89	2	no	—
147	C₂×#335	—	89	1	no	—
148	C₂×#334	—	89	1	no	—
149	#1277	—	89	2	no	—
150	C₈₉	#8600	1	1	yes	characteristic, complemented, Sylow
151	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
152	#431	—	2	2	no	—
153	#431	—	2	2	no	—
154	C₂×C₄⋊C₈	—	89	2	no	—
155	C₂×C₄⋊C₈	—	89	2	no	—
156	C₂×C₄⋊C₈	Dic₈₉	1	2	yes	—
157	C₂×C₂²⋊C₈	—	178	2	no	—
158	C₂×C₂²⋊C₈	Dic₈₉	1	2	yes	—
159	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
161	#336	—	178	2	no	—
162	#335	—	178	2	no	—
163	#334	—	178	2	no	—
164	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
165	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
166	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
167	C₂²×C₈	—	178	2	no	—
168	C₂²×C₈	—	178	2	no	—
169	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
170	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
171	C₂×C₄²	—	2	1	no	—
172	C₂×C₄²	—	2	1	no	—
173	C₂×C₄²	—	2	1	no	—
174	C₂×C₄²	—	2	1	no	—
175	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
176	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
177	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
179	C₄⋊C₈	—	2	4	no	—
180	C₄⋊C₈	—	178	2	no	—
181	C₄⋊C₈	—	178	2	no	—
182	C₄⋊C₈	—	178	2	no	—
183	C₄⋊C₈	—	178	2	no	—
184	C₂²⋊C₈	—	2	4	no	—
185	C₂²⋊C₈	—	356	4	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
189	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
190	C₂⁴	C₄⋊Dic₈₉	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
203	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
205	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
207	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	356	2	no	—
212	C₂×C₈	—	356	2	no	—
213	C₂×C₈	—	356	2	no	—
214	C₂×C₈	—	356	2	no	—
215	C₂×C₈	—	356	2	no	—
216	C₂×C₈	—	356	2	no	—
217	C₂×C₈	C₂²⋊Dic₈₉	1	4	yes	—
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
226	C₄²	#13083574	1	1	yes	characteristic
227	C₄²	#13083574	1	1	yes	characteristic
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	#15667948	1	2	yes	—
233	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	#15667948	1	2	yes	—
250	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
251	C₂×C₄	#15667971	1	1	yes	characteristic
252	C₂×C₄	#15667986	1	1	yes	characteristic
253	C₂×C₄	#15667986	1	1	yes	characteristic
254	C₂×C₄	#15667991	1	1	yes	characteristic
255	C₂×C₄	#15667991	1	1	yes	characteristic
256	C₂×C₄	#15668019	1	4	yes	—
257	C₂×C₄	#15668040	1	2	yes	—
258	C₂×C₄	#15668045	1	2	yes	—
259	C₈	—	2	4	no	—
260	C₈	—	4	2	no	—
261	C₈	—	712	2	no	—
262	C₈	—	712	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	#19151458	1	1	yes	characteristic
269	C₂²	#19151625	1	2	yes	—
270	C₂²	#19151640	1	2	yes	—
271	C₂²	C₂×#15667948	1	1	yes	characteristic
272	C₂²	C₂×#15667948	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	2	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	#19151565	1	2	yes	—
279	C₄	#19151659	1	2	yes	—
280	C₄	#19151844	1	2	yes	—
281	C₄	#19151851	1	1	yes	characteristic
282	C₄	#19151856	1	1	yes	characteristic
283	C₂	—	4	2	no	—
284	C₂	#26019037	1	2	yes	—
285	C₂	#26019041	1	2	yes	—
286	C₂	#26020236	1	1	yes	characteristic
287	C₂	#26020285	1	1	yes	characteristic
288	C₂	#26020290	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#431×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#431	D₈₉
hypocenter	C₈₉	#8600
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. C₂×#19151351	C₂	semidirect
3. C₂×#19151349	C₂	semidirect
4. #26018087	C₂	semidirect
150. C₈₉	#8600	semidirect

Maximal subgroups

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

subgroup	index
#26020232	2
C₂×#19151351	2
C₂×#19151349	2
#26018087	2
C₂×#431×C₈₉	2
#8600	89