Subgroups — #36453289

Order 22784.

606 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	C₂×C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
2	#26021045	C₂	1	2	yes	maximal, complemented
3	C₂×#19151325	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#335×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₁₂	C₄	1	1	yes	characteristic
7	C₂×C₄×D₃₅₆	C₂²	1	2	yes	—
8	C₄×D₇₁₂	—	2	2	no	—
9	C₄×D₇₁₂	—	2	2	no	—
10	C₂×#15667946	C₄	1	1	yes	characteristic
11	C₂×#15667934	C₄	1	2	yes	—
12	C₂×#15667924	—	2	4	no	—
13	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
14	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
15	#19151325	—	2	4	no	—
16	C₈×#10623494	—	2	2	no	—
17	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#335×C₈₉	—	2	2	no	—
20	C₂²×D₃₅₆	C₂×C₄	1	2	yes	—
21	C₂²×C₄×D₈₉	—	2	2	no	—
22	C₂×D₇₁₂	—	2	2	no	—
23	C₂×D₇₁₂	—	2	2	no	—
24	C₂×D₇₁₂	C₈	1	4	yes	complemented
25	C₂²×#10623494	—	2	1	no	—
26	C₂²×#10623494	D₄	1	2	yes	—
27	C₄×D₃₅₆	—	2	2	no	—
28	C₄×D₃₅₆	—	2	2	no	—
29	C₄×D₃₅₆	—	2	2	no	—
30	C₄×D₃₅₆	—	2	2	no	—
31	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
32	C₂×#13083551	—	2	2	no	—
33	#15667946	—	2	2	no	—
34	#15667934	—	2	4	no	—
35	#15667924	—	4	8	no	—
36	C₄×#10623494	—	2	1	no	—
37	C₄×#10623494	—	2	1	no	—
38	C₄×#10623494	—	2	1	no	—
39	C₄×#10623494	D₄	1	2	yes	—
40	C₂²×C₈×C₈₉	—	2	2	no	—
41	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₈₉	—	2	2	no	—
45	C₄⋊C₈×C₈₉	—	2	2	no	—
46	C₄×C₈×C₈₉	D₄	1	2	yes	—
47	C₄×C₈×C₈₉	D₄	1	2	yes	—
48	C₂³×D₈₉	—	2	2	no	—
49	C₂×D₃₅₆	—	2	2	no	—
50	C₂×D₃₅₆	—	2	2	no	—
51	C₂×D₃₅₆	—	2	2	no	—
52	C₂×D₃₅₆	—	2	2	no	—
53	C₂×D₃₅₆	—	2	2	no	—
54	C₂×D₃₅₆	C₂×C₈	1	4	yes	—
55	C₂²×Dic₈₉	—	2	2	no	—
56	C₂×C₄×D₈₉	—	4	2	no	—
57	C₂×C₄×D₈₉	—	4	2	no	—
58	C₂×C₄×D₈₉	—	4	4	no	—
59	C₂×C₄×D₈₉	—	4	4	no	—
60	D₇₁₂	—	2	8	no	—
61	C₂×#10623494	—	2	4	no	—
62	C₂×#10623494	—	2	2	no	—
63	C₂×#10623494	—	2	2	no	—
64	C₂×#10623494	—	2	2	no	—
65	C₂×#10623494	D₈	1	4	yes	complemented
66	C₄⋊Dic₈₉	—	2	2	no	—
67	C₄⋊Dic₈₉	—	2	2	no	—
68	#13083551	—	4	2	no	—
69	#13083551	—	4	2	no	—
70	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
73	C₂×C₈×C₈₉	—	2	1	no	—
74	C₂×C₈×C₈₉	—	2	1	no	—
75	C₂×C₈×C₈₉	—	4	2	no	—
76	C₂×C₈×C₈₉	—	4	4	no	—
77	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
81	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₈₉	#41	1	1	yes	characteristic
85	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
86	C₂²×D₈₉	—	2	4	no	—
87	C₂²×D₈₉	—	4	2	no	—
88	C₂²×D₈₉	—	4	2	no	—
89	C₂²×D₈₉	—	4	4	no	—
90	C₂²×D₈₉	—	4	4	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
96	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₈₉	#135	1	2	yes	—
104	C₂×Dic₈₉	—	4	2	no	—
105	C₂×Dic₈₉	—	4	4	no	—
106	D₃₅₆	—	2	4	no	—
107	D₃₅₆	—	2	4	no	—
108	D₃₅₆	—	4	2	no	—
109	D₃₅₆	—	4	2	no	—
110	C₄×D₈₉	—	8	4	no	—
111	C₄×D₈₉	—	8	2	no	—
112	C₄×D₈₉	—	8	2	no	—
113	C₈×C₈₉	—	2	2	no	—
114	C₈×C₈₉	—	8	2	no	—
115	C₈×C₈₉	C₂²⋊C₈	1	4	yes	complemented
116	#10623494	—	2	4	no	—
117	#10623494	—	2	4	no	—
118	C₂²×C₈₉	#390	1	1	yes	characteristic
119	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
120	C₂²×C₈₉	#436	1	1	yes	characteristic
121	C₂²×C₈₉	C₄×D₈	1	2	yes	—
122	C₂²×C₈₉	#443	1	2	yes	—
123	C₂×D₈₉	—	4	8	no	—
124	C₂×D₈₉	—	4	4	no	—
125	C₂×D₈₉	—	8	4	no	—
126	C₂×D₈₉	—	8	2	no	—
127	C₂×D₈₉	—	8	2	no	—
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	—	2	1	no	—
130	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₄×C₈₉	#408	1	1	yes	characteristic
132	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
133	C₄×C₈₉	#436	1	1	yes	characteristic
134	C₄×C₈₉	#466	1	2	yes	—
135	C₄×C₈₉	#493	1	1	yes	characteristic
136	C₄×C₈₉	#495	1	1	yes	characteristic
137	Dic₈₉	—	8	2	no	—
138	#9657	—	89	1	no	maximal, Sylow
139	C₂×C₈₉	C₈×D₈	1	2	yes	—
140	C₂×C₈₉	#1589	1	2	yes	—
141	C₂×C₈₉	#1954	1	1	yes	characteristic
142	C₂×C₈₉	#1970	1	1	yes	characteristic
143	C₂×C₈₉	#1974	1	1	yes	characteristic
144	D₈₉	—	8	4	no	—
145	C₂×C₄×D₈	—	89	1	no	—
146	#1954	—	89	2	no	—
147	C₂×#335	D₈₉	1	1	yes	characteristic, complemented
148	C₂×D₄⋊C₈	—	89	2	no	—
149	C₂×C₈²	—	89	1	no	—
150	C₈₉	#9657	1	1	yes	characteristic, complemented, Sylow
151	C₂²×D₈	—	89	1	no	—
152	C₂×C₄×D₄	—	89	2	no	—
153	C₄×D₈	—	178	2	no	—
154	C₄×D₈	—	178	2	no	—
155	C₂×#107	—	89	1	no	—
156	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
157	C₂×#102	—	89	2	no	—
158	C₂×C₂²⋊C₈	—	178	4	no	—
159	C₂×C₄×C₈	—	89	1	no	—
160	C₂×C₄×C₈	—	89	1	no	—
161	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
162	#335	—	2	2	no	—
163	D₄⋊C₈	—	178	4	no	—
164	C₈²	—	178	2	no	—
165	C₂²×D₄	—	89	2	no	—
166	C₂³×C₄	—	178	2	no	—
167	C₂×D₈	—	89	4	no	—
168	C₂×D₈	—	178	2	no	—
169	C₂×D₈	—	178	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	89	2	no	—
172	C₂²×C₈	—	178	1	no	—
173	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
174	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
175	C₄×D₄	—	178	2	no	—
176	C₄×D₄	—	178	2	no	—
177	C₄×D₄	—	178	2	no	—
178	C₄×D₄	—	178	2	no	—
179	C₂×C₄⋊C₄	—	89	2	no	—
180	C₂×C₂²⋊C₄	—	178	2	no	—
181	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
182	#107	—	178	2	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	—	2	2	no	—
185	#102	—	178	4	no	—
186	C₂²⋊C₈	—	356	8	no	—
187	C₄×C₈	—	89	2	no	—
188	C₄×C₈	—	178	1	no	—
189	C₄×C₈	—	178	1	no	—
190	C₄×C₈	—	178	1	no	—
191	C₄×C₈	D₃₅₆	1	2	yes	—
192	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
193	C₂⁴	—	178	2	no	—
194	C₂×D₄	—	89	4	no	—
195	C₂×D₄	—	178	2	no	—
196	C₂×D₄	—	178	2	no	—
197	C₂×D₄	—	178	2	no	—
198	C₂×D₄	—	178	2	no	—
199	C₂×D₄	—	178	2	no	—
200	C₂²×C₄	—	356	2	no	—
201	C₂²×C₄	—	356	2	no	—
202	C₂²×C₄	—	356	4	no	—
203	C₂²×C₄	—	356	4	no	—
204	C₂²×C₄	—	178	2	no	—
205	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
207	C₂²×C₄	#13083575	1	1	yes	characteristic
208	D₈	—	178	8	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	4	no	—
213	C₂×C₈	—	89	4	no	—
214	C₂×C₈	—	178	4	no	—
215	C₂×C₈	—	178	2	no	—
216	C₂×C₈	—	178	2	no	—
217	C₂×C₈	—	178	2	no	—
218	C₂×C₈	#13083551	1	2	yes	—
219	C₂×C₈	#13083551	1	2	yes	—
220	C₂×C₈	C₈×D₈₉	1	2	yes	—
221	C₂×C₈	#13083557	1	2	yes	—
222	C₄⋊C₄	—	178	2	no	—
223	C₄⋊C₄	—	178	2	no	—
224	C₂²⋊C₄	—	356	2	no	—
225	C₂²⋊C₄	—	356	2	no	—
226	C₄²	D₄×D₈₉	1	1	yes	characteristic
227	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
228	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
229	C₄²	#13083575	1	1	yes	characteristic
230	C₂³	—	356	2	no	—
231	C₂³	—	356	4	no	—
232	C₂³	—	356	4	no	—
233	C₂³	—	356	2	no	—
234	C₂³	—	178	4	no	—
235	C₂³	#15667974	1	1	yes	characteristic
236	D₄	—	356	2	no	—
237	D₄	—	356	2	no	—
238	D₄	—	178	4	no	—
239	D₄	—	178	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	356	4	no	—
242	C₂×C₄	—	356	2	no	—
243	C₂×C₄	—	712	2	no	—
244	C₂×C₄	—	712	2	no	—
245	C₂×C₄	—	712	4	no	—
246	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
247	C₂×C₄	#15667974	1	1	yes	characteristic
248	C₂×C₄	#15667987	1	1	yes	characteristic
249	C₂×C₄	#15667987	1	1	yes	characteristic
250	C₂×C₄	#15668001	1	1	yes	characteristic
251	C₂×C₄	#15668008	1	1	yes	characteristic
252	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
253	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
254	C₂×C₄	#15668020	1	1	yes	characteristic
255	C₂×C₄	#15668020	1	1	yes	characteristic
256	C₂×C₄	D₈×D₈₉	1	2	yes	—
257	C₂×C₄	#15668035	1	2	yes	—
258	C₈	—	2	2	no	—
259	C₈	—	8	2	no	—
260	C₈	—	178	4	no	—
261	C₈	—	178	4	no	—
262	C₈	#15667924	1	4	yes	complemented
263	C₂²	—	356	4	no	—
264	C₂²	—	356	8	no	—
265	C₂²	—	712	2	no	—
266	C₂²	—	712	4	no	—
267	C₂²	—	712	2	no	—
268	C₂²	#19151499	1	1	yes	characteristic
269	C₂²	#19151663	1	1	yes	characteristic
270	C₂²	#19151675	1	1	yes	characteristic
271	C₂²	#19151680	1	2	yes	—
272	C₂²	#19151714	1	2	yes	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	712	2	no	—
276	C₄	C₂×#15667924	1	1	yes	characteristic
277	C₄	#19151566	1	1	yes	characteristic
278	C₄	#19151663	1	1	yes	characteristic
279	C₄	#19151675	1	1	yes	characteristic
280	C₄	#19151835	1	2	yes	—
281	C₄	#19151973	1	1	yes	characteristic
282	C₄	#19151983	1	1	yes	characteristic
283	C₂	—	712	4	no	—
284	C₂	#26019046	1	2	yes	—
285	C₂	#26019136	1	2	yes	—
286	C₂	#26021045	1	1	yes	characteristic
287	C₂	#26021116	1	1	yes	characteristic
288	C₂	#26021138	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#335×C₈₉	C₂
socle	C₂³×C₈₉	C₄×D₄
prime radical	C₂²×D₇₁₂	C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#335	D₈₉
hypocenter	C₈₉	#9657
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. #26021045	C₂	semidirect
3. C₂×#19151325	C₂	semidirect
4. C₂×C₈×#10623494	C₂	semidirect
5. C₂×#335×C₈₉	C₂	semidirect
24. C₂×D₇₁₂	C₈	semidirect
65. C₂×#10623494	D₈	semidirect
115. C₈×C₈₉	C₂²⋊C₈	semidirect
147. C₂×#335	D₈₉	semidirect
150. C₈₉	#9657	semidirect
262. C₈	#15667924	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₇₁₂	2
#26021045	2
C₂×#19151325	2
C₂×C₈×#10623494	2
C₂×#335×C₈₉	2
#9657	89