Subgroups — #36464233

Order 22784.

362 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#26022006	C₂	1	1	yes	maximal, characteristic
3	C₄×#15667951	C₂	1	1	yes	maximal, characteristic
4	#26018232	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#13083549	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018230	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₈×D₈₉	C₄	1	1	yes	characteristic
9	C₂×#15667951	—	2	1	no	—
10	C₂×#15667951	—	2	1	no	—
11	Q₈×#10623494	C₄	1	1	yes	characteristic
12	C₈×Dic₁₇₈	—	2	1	no	—
13	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
14	#19151722	—	2	1	no	—
15	#19151722	C₄	1	1	yes	characteristic
16	C₄⋊C₈×D₈₉	—	2	1	no	—
17	C₄⋊C₈×D₈₉	C₄	1	1	yes	characteristic
18	C₄×C₈×D₈₉	—	2	1	no	—
19	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
20	C₄⋊#13083549	C₂²	1	1	yes	characteristic
21	#19151416	—	2	1	no	—
22	#19151373	—	2	1	no	—
23	C₄×#13083549	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×Q₈×D₈₉	C₈	1	1	yes	characteristic
28	C₂×C₈×D₈₉	—	2	1	no	—
29	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
30	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
31	Q₈×Dic₈₉	C₈	1	1	yes	characteristic
32	C₄×Dic₁₇₈	—	2	1	no	—
33	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
34	#15667977	—	2	1	no	—
35	#15667977	C₈	1	1	yes	characteristic
36	C₄⋊C₄×D₈₉	—	2	1	no	—
37	C₄⋊C₄×D₈₉	C₈	1	1	yes	characteristic
38	C₄²×D₈₉	—	2	1	no	—
39	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
40	#15667951	—	2	4	no	—
41	#15667951	—	4	2	no	—
42	C₂×#13083549	—	2	1	no	—
43	C₂×#13083549	—	2	1	no	—
44	C₄⋊#10623494	—	2	1	no	—
45	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
46	#15667941	—	2	1	no	—
47	#15667941	—	2	1	no	—
48	#15667941	—	2	1	no	—
49	C₄×#10623494	—	2	1	no	—
50	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
51	C₈×Dic₈₉	—	2	1	no	—
52	C₈×Dic₈₉	—	2	1	no	—
53	C₈×Dic₈₉	D₄	1	1	yes	characteristic
54	C₈×Dic₈₉	D₄	1	1	yes	characteristic
55	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₈₉	—	2	1	no	—
57	C₂×C₁₆×C₈₉	—	2	1	no	—
58	C₄⋊C₈×C₈₉	—	2	1	no	—
59	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₈₉	—	2	1	no	—
61	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₇₈	—	2	1	no	—
63	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
64	Q₈×D₈₉	—	2	2	no	—
65	Q₈×D₈₉	—	2	2	no	—
66	C₂×C₄×D₈₉	—	2	1	no	—
67	C₂×C₄×D₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₈₉	M₁₆	1	1	yes	characteristic
71	C₂×#10623494	—	2	1	no	—
72	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
74	C₈×D₈₉	—	2	2	no	—
75	C₈×D₈₉	—	4	2	no	—
76	C₈×D₈₉	SD₁₆	1	2	yes	—
77	C₈×D₈₉	Q₁₆	1	2	yes	—
78	C₄⋊Dic₈₉	—	2	1	no	—
79	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	C₄×Dic₈₉	—	2	1	no	—
84	C₄×Dic₈₉	—	2	1	no	—
85	C₄×Dic₈₉	—	2	1	no	—
86	C₄×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
88	C₄×Dic₈₉	M₁₆	1	1	yes	characteristic
89	#13083549	—	2	2	no	—
90	#13083549	—	4	1	no	—
91	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
92	C₂×C₈×C₈₉	—	2	1	no	—
93	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₄⋊C₄×C₈₉	—	2	1	no	—
96	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₈₉	—	2	1	no	—
98	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₈₉	—	2	2	no	—
100	C₁₆×C₈₉	—	4	1	no	—
101	C₂²×D₈₉	C₂²⋊C₈	1	1	yes	characteristic
102	Q₈×C₈₉	—	2	1	no	—
103	Q₈×C₈₉	M₃₂	1	2	yes	complemented
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
109	C₂×Dic₈₉	—	2	1	no	—
110	C₂×Dic₈₉	—	2	1	no	—
111	C₂×Dic₈₉	C₂²⋊C₈	1	1	yes	characteristic
112	C₂×Dic₈₉	C₂²⋊C₈	1	1	yes	characteristic
113	C₂×Dic₈₉	C₂²⋊C₈	1	1	yes	characteristic
114	C₄×D₈₉	—	2	4	no	—
115	C₄×D₈₉	—	2	1	no	—
116	C₄×D₈₉	—	2	1	no	—
117	C₄×D₈₉	—	4	1	no	—
118	C₄×D₈₉	—	4	2	no	—
119	C₄×D₈₉	#103	1	2	yes	—
120	C₄×D₈₉	#104	1	2	yes	—
121	Dic₁₇₈	—	2	4	no	—
122	Dic₁₇₈	—	2	1	no	—
123	Dic₁₇₈	M₃₂	1	2	yes	complemented
124	C₈×C₈₉	—	2	1	no	—
125	C₈×C₈₉	—	4	1	no	—
126	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
127	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
128	#10623494	—	2	1	no	—
129	#10623494	—	4	1	no	—
130	#10623494	#103	1	2	yes	—
131	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₂×D₈₉	—	2	1	no	—
133	C₂×D₈₉	—	2	1	no	—
134	C₂×D₈₉	Q₈⋊C₈	1	2	yes	—
135	C₄×C₈₉	—	2	2	no	—
136	C₄×C₈₉	—	2	1	no	—
137	C₄×C₈₉	—	4	1	no	—
138	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
139	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
140	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
141	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
142	Dic₈₉	—	2	1	no	—
143	Dic₈₉	—	2	2	no	—
144	Dic₈₉	—	2	1	no	—
145	Dic₈₉	—	4	1	no	—
146	Dic₈₉	Q₈⋊C₈	1	2	yes	—
147	#11607	—	89	1	no	maximal, Sylow
148	C₂×C₈₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
149	C₂×C₈₉	#2112	1	1	yes	characteristic
150	C₂×C₈₉	#2117	1	1	yes	characteristic
151	D₈₉	—	2	2	no	—
152	C₂×C₈×Q₈	—	89	1	no	—
153	#2150	—	89	1	no	—
154	C₄×M₃₂	—	89	1	no	—
155	Q₈⋊C₁₆	—	89	1	no	—
156	Q₈⋊C₁₆	—	89	1	no	—
157	Q₈⋊C₁₆	—	89	1	no	—
158	Q₈⋊C₁₆	D₈₉	1	1	yes	characteristic, complemented
159	C₈₉	#11607	1	1	yes	characteristic, complemented, Sylow
160	C₂×C₄×Q₈	—	89	1	no	—
161	C₂×M₃₂	—	178	1	no	—
162	C₂×M₃₂	—	178	1	no	—
163	C₈×Q₈	—	89	1	no	—
164	C₈×Q₈	—	89	1	no	—
165	C₈×Q₈	—	89	1	no	—
166	C₈×Q₈	—	178	1	no	—
167	C₈×Q₈	—	178	1	no	—
168	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
169	C₂×C₄⋊C₈	—	89	1	no	—
170	C₂×C₄⋊C₈	—	178	1	no	—
171	C₂×C₄×C₈	—	89	1	no	—
172	C₂×C₄×C₈	—	178	1	no	—
173	C₄⋊C₁₆	—	89	1	no	—
174	C₄⋊C₁₆	—	178	1	no	—
175	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
176	#346	—	178	1	no	—
177	C₄×C₁₆	—	89	1	no	—
178	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
179	C₂²×Q₈	—	89	1	no	—
180	C₂²×C₈	—	89	1	no	—
181	C₂²×C₈	—	89	1	no	—
182	C₂²×C₈	—	178	1	no	—
183	C₄×Q₈	—	89	1	no	—
184	C₄×Q₈	—	89	1	no	—
185	C₄×Q₈	—	89	1	no	—
186	C₄×Q₈	—	178	1	no	—
187	C₄×Q₈	—	178	1	no	—
188	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
189	C₂×C₄⋊C₄	—	89	1	no	—
190	C₂×C₄⋊C₄	—	178	1	no	—
191	C₂×C₄²	—	89	1	no	—
192	C₂×C₄²	—	178	1	no	—
193	M₃₂	—	356	2	no	—
194	M₃₂	—	178	4	no	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	178	1	no	—
198	C₂×C₁₆	—	178	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	89	1	no	—
201	C₄⋊C₈	—	178	1	no	—
202	C₄⋊C₈	—	178	1	no	—
203	C₄⋊C₈	—	178	1	no	—
204	C₄⋊C₈	—	178	1	no	—
205	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	89	1	no	—
208	C₄×C₈	—	89	1	no	—
209	C₄×C₈	—	89	1	no	—
210	C₄×C₈	—	178	1	no	—
211	C₄×C₈	—	178	1	no	—
212	C₄×C₈	—	178	1	no	—
213	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
214	C₂×Q₈	—	89	1	no	—
215	C₂×Q₈	—	178	1	no	—
216	C₂×Q₈	—	178	2	no	—
217	C₂×Q₈	—	178	2	no	—
218	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
219	C₂²×C₄	—	89	1	no	—
220	C₂²×C₄	—	89	1	no	—
221	C₂²×C₄	—	89	1	no	—
222	C₂²×C₄	—	178	1	no	—
223	C₂²×C₄	—	178	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	89	1	no	—
226	C₂×C₈	—	89	2	no	—
227	C₂×C₈	—	89	2	no	—
228	C₂×C₈	—	89	1	no	—
229	C₂×C₈	—	356	2	no	—
230	C₂×C₈	—	178	2	no	—
231	C₂×C₈	—	178	1	no	—
232	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
233	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	89	1	no	—
236	C₄⋊C₄	—	178	1	no	—
237	C₄⋊C₄	—	178	1	no	—
238	C₄⋊C₄	—	178	1	no	—
239	C₄⋊C₄	—	178	1	no	—
240	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
241	C₄²	—	2	1	no	—
242	C₄²	—	89	1	no	—
243	C₄²	—	89	1	no	—
244	C₄²	—	89	1	no	—
245	C₄²	—	178	1	no	—
246	C₄²	—	178	1	no	—
247	C₄²	—	178	1	no	—
248	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
249	C₁₆	—	2	2	no	—
250	C₁₆	—	4	1	no	—
251	C₁₆	—	356	1	no	—
252	C₁₆	—	178	2	no	—
253	C₂³	—	89	1	no	—
254	Q₈	—	2	1	no	—
255	Q₈	—	89	2	no	—
256	Q₈	—	178	4	no	—
257	Q₈	—	178	1	no	—
258	Q₈	#15667951	1	2	yes	complemented
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	89	1	no	—
262	C₂×C₄	—	89	2	no	—
263	C₂×C₄	—	89	2	no	—
264	C₂×C₄	—	89	1	no	—
265	C₂×C₄	—	89	1	no	—
266	C₂×C₄	—	356	1	no	—
267	C₂×C₄	—	356	2	no	—
268	C₂×C₄	—	178	4	no	—
269	C₂×C₄	—	178	1	no	—
270	C₂×C₄	—	178	1	no	—
271	C₂×C₄	—	178	1	no	—
272	C₂×C₄	—	178	1	no	—
273	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
274	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
275	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
276	C₈	—	2	1	no	—
277	C₈	—	4	1	no	—
278	C₈	—	89	2	no	—
279	C₈	—	356	1	no	—
280	C₈	—	178	1	no	—
281	C₈	SD₁₆×D₈₉	1	1	yes	characteristic
282	C₈	Q₁₆×D₈₉	1	1	yes	characteristic
283	C₂²	—	89	2	no	—
284	C₂²	—	178	1	no	—
285	C₂²	—	178	1	no	—
286	C₂²	C₂²⋊C₈×D₈₉	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	89	2	no	—
291	C₄	—	356	1	no	—
292	C₄	—	178	1	no	—
293	C₄	—	178	2	no	—
294	C₄	—	178	1	no	—
295	C₄	#103×D₈₉	1	1	yes	characteristic
296	C₄	#104×D₈₉	1	1	yes	characteristic
297	C₄	M₃₂×D₈₉	1	1	yes	characteristic
298	C₄	C₂×#15667951	1	1	yes	characteristic
299	C₂	—	178	2	no	—
300	C₂	Q₈⋊C₈×D₈₉	1	1	yes	characteristic
301	C₂	#26021861	1	1	yes	characteristic
302	C₂	#26021878	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×Q₈⋊C₁₆	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂²×D₈₉	C₂²⋊C₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	Q₈⋊C₁₆	D₈₉
hypocenter	C₈₉	#11607
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ Q₈⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. #26018232	C₂	semidirect
5. Q₈⋊#13083549	C₂	semidirect
6. #26018230	C₂	semidirect
7. C₈₉×Q₈⋊C₁₆	C₂	semidirect
103. Q₈×C₈₉	M₃₂	semidirect
123. Dic₁₇₈	M₃₂	semidirect
158. Q₈⋊C₁₆	D₈₉	semidirect
159. C₈₉	#11607	semidirect
258. Q₈	#15667951	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₈₉	2
#26022006	2
C₄×#15667951	2
#26018232	2
Q₈⋊#13083549	2
#26018230	2
C₈₉×Q₈⋊C₁₆	2
#11607	89