Subgroups — #36469700

Order 22784.

486 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151711	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151412	C₂	1	1	yes	maximal, characteristic
3	#26021969	C₂	1	1	yes	maximal, characteristic
4	#26020136	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018358	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018351	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#347×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₃₅₆	—	2	1	no	—
10	#19151711	—	2	2	no	—
11	#19151711	C₄	1	4	yes	complemented
12	C₂×#15667948	C₂²	1	1	yes	characteristic
13	#19151636	C₂²	1	1	yes	characteristic
14	C₂×#15667944	C₂²	1	1	yes	characteristic
15	C₂×#15667934	—	2	1	no	—
16	#19151553	—	2	1	no	—
17	#19151412	C₄	1	4	yes	complemented
18	#19151400	C₄	1	4	yes	complemented
19	#19151358	—	2	1	no	—
20	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
23	#347×C₈₉	—	2	2	no	—
24	C₂²×D₃₅₆	—	2	1	no	—
25	C₂²×C₄×D₈₉	—	4	1	no	—
26	C₂×D₇₁₂	—	2	1	no	—
27	C₂×D₇₁₂	—	2	1	no	—
28	C₂×D₇₁₂	—	2	1	no	—
29	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
30	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
31	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
32	C₂×#13083558	—	2	1	no	—
33	C₄×D₃₅₆	—	2	4	no	—
34	C₄×D₃₅₆	—	4	2	no	—
35	#15667971	—	2	1	no	—
36	C₂×C₄⋊Dic₈₉	—	2	1	no	—
37	C₂×#13083551	—	4	1	no	—
38	#15667948	—	2	1	no	—
39	#15667948	C₂×C₄	1	2	yes	complemented
40	#15667946	C₂×C₄	1	2	yes	complemented
41	#15667944	—	2	1	no	—
42	#15667944	—	2	1	no	—
43	#15667944	C₂×C₄	1	2	yes	—
44	#15667934	—	2	4	no	—
45	#15667922	—	4	1	no	—
46	C₈×Dic₈₉	—	4	1	no	—
47	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
50	M₃₂×C₈₉	—	2	1	no	—
51	M₃₂×C₈₉	—	2	1	no	—
52	M₃₂×C₈₉	—	2	1	no	—
53	M₃₂×C₈₉	C₂×C₄	1	2	yes	complemented
54	C₂×C₁₆×C₈₉	—	2	1	no	—
55	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	complemented
56	#97×C₈₉	C₂×C₄	1	2	yes	—
57	#97×C₈₉	D₄	1	2	yes	—
58	C₂³×D₈₉	—	4	1	no	—
59	C₂×D₃₅₆	—	2	2	no	—
60	C₂×D₃₅₆	—	2	2	no	—
61	C₂×D₃₅₆	—	2	2	no	—
62	C₂×D₃₅₆	—	4	1	no	—
63	C₂×D₃₅₆	—	4	1	no	—
64	C₂×D₃₅₆	—	4	1	no	—
65	C₂²×Dic₈₉	—	4	1	no	—
66	C₂×C₄×D₈₉	—	4	4	no	—
67	C₂×C₄×D₈₉	—	8	4	no	—
68	D₇₁₂	—	2	2	no	—
69	D₇₁₂	—	2	2	no	—
70	D₇₁₂	—	2	2	no	—
71	D₇₁₂	C₄²	1	4	yes	—
72	#13083558	—	2	2	no	—
73	#13083558	—	4	1	no	—
74	C₂×#10623494	—	4	1	no	—
75	C₄⋊Dic₈₉	—	2	2	no	—
76	C₄⋊Dic₈₉	—	2	2	no	—
77	C₄⋊Dic₈₉	—	4	1	no	—
78	C₂²⋊Dic₈₉	—	4	1	no	—
79	#13083551	—	4	4	no	—
80	C₄×Dic₈₉	—	4	1	no	—
81	C₂²×C₄×C₈₉	—	2	1	no	—
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	—	2	1	no	—
84	C₂×C₈×C₈₉	—	2	1	no	—
85	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
91	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
92	C₄²×C₈₉	—	2	1	no	—
93	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
94	C₁₆×C₈₉	—	2	2	no	—
95	C₁₆×C₈₉	—	2	2	no	—
96	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂²×D₈₉	—	4	2	no	—
98	C₂²×D₈₉	—	4	2	no	—
99	C₂²×D₈₉	—	4	2	no	—
100	C₂²×D₈₉	—	8	2	no	—
101	C₂²×D₈₉	—	8	2	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	—	4	2	no	—
104	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
110	C₂×Dic₈₉	—	4	2	no	—
111	C₂×Dic₈₉	—	4	2	no	—
112	C₂×Dic₈₉	—	8	2	no	—
113	D₃₅₆	—	2	4	no	—
114	D₃₅₆	—	4	2	no	—
115	D₃₅₆	—	4	2	no	—
116	D₃₅₆	—	4	2	no	—
117	C₄×D₈₉	—	8	8	no	—
118	C₈×C₈₉	—	2	2	no	—
119	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
120	C₈×C₈₉	#117	1	1	yes	characteristic
121	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
122	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
123	#10623494	—	4	2	no	—
124	C₂²×C₈₉	—	2	2	no	—
125	C₂²×C₈₉	#418	1	1	yes	characteristic
126	C₂²×C₈₉	#440	1	1	yes	characteristic
127	C₂²×C₈₉	#442	1	1	yes	characteristic
128	C₂×D₈₉	—	4	4	no	—
129	C₂×D₈₉	—	8	2	no	—
130	C₂×D₈₉	—	8	2	no	—
131	C₂×D₈₉	—	8	4	no	—
132	C₂×D₈₉	—	8	4	no	—
133	C₄×C₈₉	—	4	2	no	—
134	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₈₉	#417	1	1	yes	characteristic
136	C₄×C₈₉	#440	1	1	yes	characteristic
137	C₄×C₈₉	#442	1	1	yes	characteristic
138	Dic₈₉	—	8	2	no	—
139	Dic₈₉	—	8	2	no	—
140	#12647	—	89	1	no	maximal, Sylow
141	C₂×C₈₉	—	2	1	no	—
142	C₂×C₈₉	—	2	1	no	—
143	C₂×C₈₉	#1762	1	1	yes	characteristic
144	C₂×C₈₉	AGL(1,ℤ/16)	1	2	yes	—
145	D₈₉	—	8	4	no	—
146	C₂×#442	—	89	1	no	—
147	C₂×#361	—	89	1	no	—
148	#2140	—	89	1	no	—
149	C₂×#347	D₈₉	1	1	yes	characteristic, complemented
150	#1773	—	89	1	no	—
151	#1386	—	89	1	no	—
152	#1383	—	89	1	no	—
153	C₈₉	#12647	1	1	yes	characteristic, complemented, Sylow
154	C₂²×D₈	—	89	1	no	—
155	C₂×C₄×D₄	—	178	1	no	—
156	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
157	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
158	#442	—	89	4	no	—
159	#442	—	178	2	no	—
160	C₂×#108	—	89	1	no	—
161	#427	—	89	1	no	—
162	C₂×#106	—	89	1	no	—
163	C₂×#102	—	178	1	no	—
164	#405	—	178	1	no	—
165	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
166	#361	—	89	4	no	—
167	#357	—	89	4	no	—
168	#347	—	2	2	no	—
169	#339	—	178	1	no	—
170	C₂²×D₄	—	178	1	no	—
171	C₂³×C₄	—	356	1	no	—
172	C₂×D₈	—	89	2	no	—
173	C₂×D₈	—	89	2	no	—
174	C₂×D₈	—	89	2	no	—
175	C₂×D₈	—	178	1	no	—
176	C₂×D₈	—	178	1	no	—
177	C₂×D₈	—	178	1	no	—
178	C₂×M₁₆	—	178	1	no	—
179	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
181	C₄×D₄	—	356	2	no	—
182	C₄×D₄	—	178	4	no	—
183	#117	—	178	1	no	—
184	C₂×C₄⋊C₄	—	178	1	no	—
185	C₂×C₂²⋊C₄	—	356	1	no	—
186	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	2	1	no	—
190	M₃₂	C₄×D₈₉	1	2	yes	complemented
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	C₄×D₈₉	1	2	yes	complemented
193	#108	—	89	2	no	—
194	#108	—	178	1	no	—
195	#107	—	89	2	no	—
196	#106	—	89	2	no	—
197	#106	—	178	1	no	—
198	#106	—	178	1	no	—
199	#102	—	178	4	no	—
200	#97	—	356	1	no	—
201	#97	C₄×D₈₉	1	2	yes	—
202	#97	C₂²⋊D₈₉	1	2	yes	—
203	C₄×C₈	—	356	1	no	—
204	C₂⁴	—	356	1	no	—
205	C₂×D₄	—	356	1	no	—
206	C₂×D₄	—	356	1	no	—
207	C₂×D₄	—	356	1	no	—
208	C₂×D₄	—	178	2	no	—
209	C₂×D₄	—	178	2	no	—
210	C₂×D₄	—	178	2	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	356	4	no	—
213	C₂²×C₄	—	356	1	no	—
214	C₂²×C₄	—	712	4	no	—
215	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
216	D₈	—	89	4	no	—
217	D₈	—	178	2	no	—
218	D₈	—	178	2	no	—
219	D₈	—	178	2	no	—
220	M₁₆	—	356	1	no	—
221	M₁₆	—	178	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	356	1	no	—
225	C₂×C₈	#13083551	1	2	yes	—
226	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
229	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
230	C₂×C₈	#13083574	1	1	yes	characteristic
231	C₂×C₈	#13083574	1	1	yes	characteristic
232	C₄⋊C₄	—	356	1	no	—
233	C₄⋊C₄	—	178	2	no	—
234	C₄⋊C₄	—	178	2	no	—
235	C₂²⋊C₄	—	356	4	no	—
236	C₂²⋊C₄	—	356	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	356	1	no	—
239	C₄²	#13083551	1	2	yes	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	2	2	no	—
242	C₂³	—	356	2	no	—
243	C₂³	—	356	2	no	—
244	C₂³	—	356	2	no	—
245	C₂³	—	712	2	no	—
246	C₂³	—	712	2	no	—
247	C₂³	#15668040	1	1	yes	characteristic
248	D₄	—	356	2	no	—
249	D₄	—	356	2	no	—
250	D₄	—	356	2	no	—
251	D₄	—	178	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	356	2	no	—
255	C₂×C₄	—	356	2	no	—
256	C₂×C₄	—	712	8	no	—
257	C₂×C₄	—	712	2	no	—
258	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
259	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
260	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
261	C₂×C₄	#15668040	1	1	yes	characteristic
262	C₂×C₄	#15668040	1	1	yes	characteristic
263	C₂×C₄	#15668045	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	356	2	no	—
266	C₈	C₄²×D₈₉	1	1	yes	characteristic
267	C₈	#15667969	1	1	yes	characteristic
268	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
269	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
270	C₂²	—	2	2	no	—
271	C₂²	—	356	4	no	—
272	C₂²	—	712	2	no	—
273	C₂²	—	712	2	no	—
274	C₂²	—	712	4	no	—
275	C₂²	—	712	4	no	—
276	C₂²	#19151604	1	1	yes	characteristic
277	C₂²	#19151700	1	1	yes	characteristic
278	C₂²	#19151711	1	1	yes	characteristic
279	C₄	—	4	2	no	—
280	C₄	—	712	2	no	—
281	C₄	—	712	2	no	—
282	C₄	C₄×#13083551	1	1	yes	characteristic
283	C₄	#19151600	1	1	yes	characteristic
284	C₄	#19151700	1	1	yes	characteristic
285	C₄	#19151711	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	712	4	no	—
289	C₂	#26020078	1	1	yes	characteristic
290	C₂	#26022152	1	2	yes	—

	subgroup	G/H
center	C₂²	#19151711
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#347×C₈₉	C₂
socle	C₂²×C₈₉	#442
prime radical	C₂²×D₇₁₂	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#347	D₈₉
hypocenter	C₈₉	#12647
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
4. #26020136	C₂	semidirect
5. #26018358	C₂	semidirect
6. #26018351	C₂	semidirect
7. C₂×#347×C₈₉	C₂	semidirect
11. #19151711	C₄	semidirect
17. #19151412	C₄	semidirect
18. #19151400	C₄	semidirect
39. #15667948	C₂×C₄	semidirect
40. #15667946	C₂×C₄	semidirect
53. M₃₂×C₈₉	C₂×C₄	semidirect
55. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
149. C₂×#347	D₈₉	semidirect
153. C₈₉	#12647	semidirect
190. M₃₂	C₄×D₈₉	semidirect
192. C₂×C₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151711	2
C₂×#19151412	2
#26021969	2
#26020136	2
#26018358	2
#26018351	2
C₂×#347×C₈₉	2
#12647	89