Subgroups — #36469484

Order 22784.

482 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668018	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151383	C₂	1	2	yes	maximal
3	C₂×#19151374	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018342	C₂	1	2	yes	maximal, complemented
5	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15668018	C₄	1	1	yes	characteristic
7	C₂×#15668018	C₂²	1	1	yes	characteristic
8	C₄×#13083574	C₄	1	1	yes	characteristic
9	C₂²×#13083549	C₂²	1	2	yes	—
10	#19151553	C₄	1	1	yes	characteristic
11	#19151553	C₂²	1	1	yes	characteristic
12	#19151552	—	2	1	no	—
13	#19151552	—	2	1	no	—
14	C₄×#13083558	C₄	1	1	yes	characteristic
15	C₄×#13083557	—	2	1	no	—
16	C₄×C₈×D₈₉	—	2	1	no	—
17	#19151383	C₄	1	8	yes	complemented
18	#19151374	—	2	1	no	—
19	#19151374	—	2	1	no	—
20	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	#346×C₈₉	—	2	1	no	—
23	C₄×C₁₆×C₈₉	—	2	1	no	—
24	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
25	#15668018	—	2	2	no	—
26	#15668018	—	2	2	no	—
27	#15668018	C₈	1	4	yes	—
28	#15668018	C₂×C₄	1	4	yes	—
29	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
30	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
31	C₂×#13083557	—	2	1	no	—
32	C₂×#13083557	—	2	1	no	—
33	C₂×C₈×D₈₉	—	2	1	no	—
34	C₂×C₈×D₈₉	—	2	1	no	—
35	C₄×Dic₁₇₈	C₈	1	2	yes	—
36	C₄×D₃₅₆	C₈	1	2	yes	—
37	C₄×C₂²⋊D₈₉	—	2	2	no	—
38	#15667971	C₂×C₄	1	1	yes	characteristic
39	#15667969	—	2	1	no	—
40	C₄²×D₈₉	—	2	1	no	—
41	C₂×#13083549	—	2	2	no	—
42	C₂×#13083549	—	2	2	no	—
43	C₂×#13083549	C₂×C₄	1	4	yes	complemented
44	#15667923	—	2	1	no	—
45	#15667922	—	2	1	no	—
46	#15667922	—	2	1	no	—
47	C₄×#10623494	—	2	1	no	—
48	C₈×Dic₈₉	—	2	1	no	—
49	C₈×Dic₈₉	—	2	1	no	—
50	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₈₉	—	2	2	no	—
54	M₃₂×C₈₉	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₆×C₈₉	—	2	2	no	—
56	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
57	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
59	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
60	#13083574	—	2	1	no	—
61	#13083574	—	2	1	no	—
62	#13083574	C₄²	1	2	yes	—
63	#13083574	C₂×C₈	1	2	yes	—
64	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
65	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	2	no	—
69	#13083558	—	2	1	no	—
70	#13083558	—	2	1	no	—
71	#13083558	C₄²	1	2	yes	—
72	#13083558	C₂×C₈	1	2	yes	—
73	#13083557	—	2	4	no	—
74	#13083557	—	2	4	no	—
75	C₂×#10623494	—	2	1	no	—
76	C₂×#10623494	—	2	1	no	—
77	C₈×D₈₉	—	2	4	no	—
78	C₈×D₈₉	—	2	4	no	—
79	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
80	#13083553	—	2	2	no	—
81	C₂²⋊Dic₈₉	—	2	1	no	—
82	#13083551	—	2	2	no	—
83	C₄×Dic₈₉	—	2	2	no	—
84	C₄×Dic₈₉	—	2	1	no	—
85	#13083549	—	2	4	no	—
86	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
87	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
91	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
92	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	#41	1	2	yes	—
96	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₈₉	C₂×C₈	1	2	yes	—
99	C₁₆×C₈₉	—	2	4	no	—
100	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₈₉	—	2	1	no	—
102	C₂×C₄×C₈₉	—	4	2	no	—
103	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
105	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
111	C₂²⋊D₈₉	—	2	2	no	—
112	C₂²⋊D₈₉	—	2	2	no	—
113	C₂×Dic₈₉	—	2	1	no	—
114	C₂×Dic₈₉	—	2	2	no	—
115	D₃₅₆	—	2	1	no	—
116	D₃₅₆	C₄×C₈	1	2	yes	—
117	C₄×D₈₉	—	2	2	no	—
118	C₄×D₈₉	—	2	2	no	—
119	C₄×D₈₉	—	4	4	no	—
120	Dic₁₇₈	—	2	1	no	—
121	Dic₁₇₈	C₄×C₈	1	2	yes	—
122	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
123	C₈×C₈₉	C₂²⋊C₈	1	2	yes	—
124	C₈×C₈₉	C₄×D₄	1	2	yes	—
125	C₈×C₈₉	C₄×D₄	1	2	yes	—
126	#10623494	—	2	2	no	—
127	#10623494	—	2	2	no	—
128	C₂²×C₈₉	—	2	2	no	—
129	C₂²×C₈₉	C₄×M₁₆	1	1	yes	characteristic
130	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₂²×C₈₉	#431	1	1	yes	characteristic
132	C₂×D₈₉	—	2	2	no	—
133	C₂×D₈₉	—	4	2	no	—
134	C₄×C₈₉	—	2	2	no	—
135	C₄×C₈₉	—	2	2	no	—
136	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
138	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₈₉	#431	1	1	yes	characteristic
140	Dic₈₉	—	2	2	no	—
141	Dic₈₉	—	4	2	no	—
142	#12604	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	—	2	1	no	—
144	C₂×C₈₉	—	2	1	no	—
145	C₂×C₈₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
146	C₂×C₈₉	#2117	1	2	yes	—
147	D₈₉	—	4	2	no	—
148	C₄×#131	—	89	1	no	—
149	C₂×#350	—	89	2	no	—
150	C₄×M₃₂	D₈₉	1	1	yes	characteristic, complemented
151	C₂×#346	—	89	1	no	—
152	#1379	—	89	2	no	—
153	C₈₉	#12604	1	1	yes	characteristic, complemented, Sylow
154	C₂×#131	—	89	1	no	—
155	C₂×#131	—	89	1	no	—
156	C₄×#41	—	89	1	no	—
157	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
158	C₂²×C₁₆	—	89	2	no	—
159	#405	—	89	1	no	—
160	#405	—	89	1	no	—
161	#405	—	178	1	no	—
162	#405	—	178	1	no	—
163	C₄×M₁₆	—	89	1	no	—
164	C₄×M₁₆	—	178	1	no	—
165	C₂×C₄×C₈	—	178	1	no	—
166	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
167	#350	—	89	8	no	—
168	#346	—	2	1	no	—
169	#346	—	178	1	no	—
170	#346	—	178	1	no	—
171	C₄×C₁₆	—	2	1	no	—
172	C₂×#41	—	89	1	no	—
173	#131	—	89	4	no	—
174	#131	—	89	4	no	—
175	#131	—	178	2	no	—
176	#131	—	178	2	no	—
177	C₂×M₁₆	—	89	1	no	—
178	C₂×M₁₆	—	89	1	no	—
179	C₂×M₁₆	—	178	1	no	—
180	C₂×M₁₆	—	178	1	no	—
181	C₂²×C₈	—	178	1	no	—
182	C₂²×C₈	—	178	1	no	—
183	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
185	C₄×Q₈	—	89	2	no	—
186	C₄×D₄	—	89	2	no	—
187	C₄×D₄	—	178	2	no	—
188	#117	—	89	1	no	—
189	#117	—	178	1	no	—
190	C₂×C₄²	—	178	1	no	—
191	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
192	M₃₂	—	2	2	no	—
193	M₃₂	C₄×D₈₉	1	4	yes	complemented
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	89	4	no	—
196	C₂×C₁₆	—	178	2	no	—
197	C₂×C₁₆	—	178	2	no	—
198	#97	—	178	1	no	—
199	#97	—	178	1	no	—
200	#97	—	178	1	no	—
201	C₄×C₈	—	178	1	no	—
202	C₄×C₈	—	178	1	no	—
203	C₄×C₈	—	178	1	no	—
204	C₄×C₈	D₃₅₆	1	1	yes	characteristic
205	C₄×C₈	D₃₅₆	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
208	#41	—	89	2	no	—
209	#41	—	89	2	no	—
210	#41	—	178	1	no	—
211	#41	—	178	1	no	—
212	C₂×Q₈	—	89	1	no	—
213	C₂×D₄	—	89	1	no	—
214	C₂×D₄	—	178	1	no	—
215	C₂²×C₄	—	178	2	no	—
216	C₂²×C₄	—	178	1	no	—
217	C₂²×C₄	#13083557	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
219	M₁₆	—	89	2	no	—
220	M₁₆	—	89	2	no	—
221	M₁₆	—	178	1	no	—
222	M₁₆	—	178	1	no	—
223	M₁₆	—	178	4	no	—
224	M₁₆	—	178	4	no	—
225	C₂×C₈	—	178	1	no	—
226	C₂×C₈	—	178	1	no	—
227	C₂×C₈	—	178	4	no	—
228	C₂×C₈	—	178	4	no	—
229	C₂×C₈	#13083551	1	1	yes	characteristic
230	C₂×C₈	#13083551	1	1	yes	characteristic
231	C₂×C₈	C₈×D₈₉	1	2	yes	—
232	C₂×C₈	#13083557	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
234	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
235	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
236	C₂×C₈	#13083574	1	2	yes	—
237	C₄⋊C₄	—	89	2	no	—
238	C₄⋊C₄	—	178	2	no	—
239	C₂²⋊C₄	—	178	1	no	—
240	C₂²⋊C₄	—	178	2	no	—
241	C₄²	—	178	2	no	—
242	C₄²	—	178	1	no	—
243	C₄²	#13083551	1	1	yes	characteristic
244	C₄²	#13083551	1	1	yes	characteristic
245	C₄²	C₈×D₈₉	1	2	yes	—
246	C₁₆	—	2	4	no	—
247	C₁₆	—	178	4	no	—
248	C₂³	—	178	1	no	—
249	C₂³	C₂×#13083557	1	1	yes	characteristic
250	Q₈	—	89	2	no	—
251	Q₈	—	178	1	no	—
252	D₄	—	89	2	no	—
253	D₄	—	178	2	no	—
254	D₄	—	178	2	no	—
255	D₄	—	178	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	356	4	no	—
258	C₂×C₄	—	178	2	no	—
259	C₂×C₄	—	178	1	no	—
260	C₂×C₄	—	178	2	no	—
261	C₂×C₄	—	178	2	no	—
262	C₂×C₄	#15667924	1	2	yes	—
263	C₂×C₄	#15667924	1	2	yes	—
264	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
265	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
266	C₂×C₄	#15667969	1	1	yes	characteristic
267	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
268	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
269	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
270	C₈	—	178	2	no	—
271	C₈	—	178	2	no	—
272	C₈	#15667924	1	2	yes	—
273	C₈	#15667924	1	2	yes	—
274	C₈	C₄×C₂²⋊D₈₉	1	2	yes	—
275	C₈	C₄×D₃₅₆	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	356	2	no	—
278	C₂²	—	178	2	no	—
279	C₂²	C₄×#13083557	1	1	yes	characteristic
280	C₂²	C₂×#15667924	1	1	yes	characteristic
281	C₂²	#19151649	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	356	2	no	—
285	C₄	—	178	2	no	—
286	C₄	C₄×#13083551	1	1	yes	characteristic
287	C₄	C₄×C₈×D₈₉	1	1	yes	characteristic
288	C₄	C₂×#15667924	1	1	yes	characteristic
289	C₄	#19151649	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	356	2	no	—
293	C₂	C₄×#15667924	1	1	yes	characteristic
294	C₂	#26021881	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄×M₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#12604
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₂×#19151374	C₂	semidirect
4. #26018342	C₂	semidirect
5. C₄×M₃₂×C₈₉	C₂	semidirect
17. #19151383	C₄	semidirect
43. C₂×#13083549	C₂×C₄	semidirect
54. M₃₂×C₈₉	C₂×C₄	semidirect
150. C₄×M₃₂	D₈₉	semidirect
153. C₈₉	#12604	semidirect
193. M₃₂	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668018	2
C₂×#19151383	2
C₂×#19151374	2
#26018342	2
C₄×M₃₂×C₈₉	2
#12604	89