Subgroups — #36454408

Order 22784.

528 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668012	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	#26018092	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₇₈	—	2	1	no	—
7	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
8	C₂×C₄×#10623494	—	2	1	no	—
9	#19151504	—	2	1	no	—
10	#19151496	—	2	1	no	—
11	#19151496	C₄	1	2	yes	—
12	#19151468	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₈₉	—	2	1	no	—
14	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
15	#19151353	—	2	1	no	—
16	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
18	C₄³×C₈₉	C₂²	1	1	yes	characteristic
19	#336×C₈₉	—	2	1	no	—
20	C₂²×Dic₁₇₈	—	2	1	no	—
21	C₂²×#10623494	—	2	2	no	—
22	#15668012	—	2	2	no	—
23	#15668012	C₈	1	4	yes	complemented
24	C₄×Dic₁₇₈	—	4	4	no	—
25	C₂×C₄⋊Dic₈₉	—	2	1	no	—
26	C₂×C₄⋊Dic₈₉	—	2	1	no	—
27	C₂×C₄⋊Dic₈₉	—	2	1	no	—
28	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
30	C₂×#13083553	—	2	2	no	—
31	C₂×C₄×Dic₈₉	—	2	2	no	—
32	C₄⋊#10623494	—	2	2	no	—
33	C₄⋊#10623494	Q₈	1	4	yes	complemented
34	C₄×#10623494	—	4	2	no	—
35	#15667919	—	2	2	no	—
36	#15667919	—	2	2	no	—
37	C₂²×C₈×C₈₉	—	2	2	no	—
38	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₈₉	—	2	4	no	—
46	C₄×C₈×C₈₉	—	2	4	no	—
47	C₂×Dic₁₇₈	—	2	4	no	—
48	C₂×Dic₁₇₈	—	4	2	no	—
49	C₂×Dic₁₇₈	—	4	2	no	—
50	C₂²×Dic₈₉	—	2	2	no	—
51	C₂²×Dic₈₉	—	2	2	no	—
52	C₂×#10623494	—	2	4	no	—
53	C₂×#10623494	—	4	2	no	—
54	C₂×#10623494	—	4	2	no	—
55	C₄⋊Dic₈₉	—	2	4	no	—
56	C₄⋊Dic₈₉	—	2	2	no	—
57	C₄⋊Dic₈₉	—	2	4	no	—
58	C₄⋊Dic₈₉	—	4	2	no	—
59	C₄⋊Dic₈₉	—	4	2	no	—
60	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
61	#13083553	—	4	4	no	—
62	C₄×Dic₈₉	—	4	4	no	—
63	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₈₉	—	2	4	no	—
71	C₂×C₈×C₈₉	—	4	2	no	—
72	C₂×C₈×C₈₉	—	4	2	no	—
73	C₄²×C₈₉	—	2	2	no	—
74	C₄²×C₈₉	—	2	2	no	—
75	C₄²×C₈₉	—	2	2	no	—
76	C₄²×C₈₉	—	4	1	no	—
77	C₄²×C₈₉	C₂×C₈	1	2	yes	—
78	C₄²×C₈₉	M₁₆	1	2	yes	—
79	C₄²×C₈₉	SD₁₆	1	2	yes	—
80	C₄²×C₈₉	Q₁₆	1	2	yes	—
81	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
82	C₄²×C₈₉	#41	1	2	yes	—
83	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₈₉	—	2	1	no	—
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
95	C₂×C₄×C₈₉	#103	1	2	yes	—
96	C₂×C₄×C₈₉	#103	1	2	yes	—
97	C₂×C₄×C₈₉	#104	1	2	yes	—
98	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#131	1	2	yes	—
105	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
108	C₂×Dic₈₉	—	2	4	no	—
109	C₂×Dic₈₉	—	4	2	no	—
110	C₂×Dic₈₉	—	4	2	no	—
111	C₂×Dic₈₉	—	4	2	no	—
112	C₂×Dic₈₉	—	4	2	no	—
113	C₂×Dic₈₉	—	4	2	no	—
114	Dic₁₇₈	—	4	8	no	—
115	C₈×C₈₉	—	4	4	no	—
116	#10623494	—	4	4	no	—
117	C₂²×C₈₉	#391	1	1	yes	characteristic
118	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₈₉	#408	1	1	yes	characteristic
120	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
121	C₂²×C₈₉	#418	1	1	yes	characteristic
122	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
123	C₂²×C₈₉	#421	1	1	yes	characteristic
124	C₄×C₈₉	—	2	1	no	—
125	C₄×C₈₉	—	2	1	no	—
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	2	2	no	—
128	C₄×C₈₉	—	4	1	no	—
129	C₄×C₈₉	Q₈⋊C₈	1	4	yes	—
130	C₄×C₈₉	C₈×Q₈	1	2	yes	—
131	C₄×C₈₉	#446	1	2	yes	—
132	C₄×C₈₉	#476	1	2	yes	—
133	C₄×C₈₉	#477	1	2	yes	—
134	Dic₈₉	—	4	4	no	—
135	Dic₈₉	—	8	2	no	—
136	#9864	—	89	1	no	maximal, Sylow
137	C₂×C₈₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
138	C₂×C₈₉	#1478	1	1	yes	characteristic
139	C₂×C₈₉	#1481	1	1	yes	characteristic
140	C₂×C₈₉	#1484	1	1	yes	characteristic
141	C₂×C₈₉	#1984	1	1	yes	characteristic
142	C₂×C₈₉	#1986	1	1	yes	characteristic
143	C₂×C₈₉	#1987	1	1	yes	characteristic
144	C₄×C₄⋊Q₈	—	89	1	no	—
145	C₄×C₄⋊C₈	—	89	1	no	—
146	C₄×C₄⋊C₈	D₈₉	1	1	yes	characteristic
147	#1279	—	89	4	no	—
148	C₈₉	#9864	1	1	yes	characteristic, complemented, Sylow
149	C₂×C₄⋊Q₈	—	89	1	no	—
150	C₂×C₄×Q₈	—	178	1	no	—
151	C₂×C₄⋊C₈	—	2	1	no	—
152	C₂×C₄⋊C₈	—	89	2	no	—
153	C₂×C₄×C₈	—	178	1	no	—
154	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
155	#391	—	178	1	no	—
156	#389	—	89	2	no	—
157	#389	—	178	1	no	—
158	#384	—	89	1	no	—
159	C₄×C₄⋊C₄	—	89	2	no	—
160	C₄×C₄⋊C₄	—	178	1	no	—
161	C₄³	C₂×D₈₉	1	1	yes	characteristic
162	#336	—	2	1	no	—
163	#336	—	178	1	no	—
164	C₂²×Q₈	—	178	1	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	178	2	no	—
167	C₄⋊Q₈	—	89	4	no	—
168	C₄⋊Q₈	—	178	2	no	—
169	C₄×Q₈	—	356	4	no	—
170	C₂×C₄⋊C₄	—	89	2	no	—
171	C₂×C₄⋊C₄	—	89	2	no	—
172	C₂×C₄⋊C₄	—	178	1	no	—
173	C₂×C₄⋊C₄	—	178	2	no	—
174	C₂×C₄⋊C₄	—	178	1	no	—
175	C₂×C₄⋊C₄	—	178	1	no	—
176	C₂×C₄²	—	178	2	no	—
177	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
179	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
180	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
184	C₄⋊C₈	—	2	4	no	—
185	C₄⋊C₈	—	89	4	no	—
186	C₄⋊C₈	—	178	2	no	—
187	C₄×C₈	—	2	4	no	—
188	C₄×C₈	—	356	2	no	—
189	#95	—	178	2	no	—
190	#95	—	178	2	no	—
191	C₂×Q₈	—	356	2	no	—
192	C₂×Q₈	—	356	2	no	—
193	C₂×Q₈	—	178	4	no	—
194	C₂²×C₄	—	178	2	no	—
195	C₂²×C₄	—	178	2	no	—
196	C₂²×C₄	#13083551	1	1	yes	characteristic
197	C₂²×C₄	#13083551	1	1	yes	characteristic
198	C₂²×C₄	#13083551	1	1	yes	characteristic
199	C₂²×C₄	#13083551	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	356	2	no	—
207	C₂×C₈	—	356	2	no	—
208	C₂×C₈	—	178	4	no	—
209	C₄⋊C₄	—	89	4	no	—
210	C₄⋊C₄	—	356	4	no	—
211	C₄⋊C₄	—	356	2	no	—
212	C₄⋊C₄	—	356	2	no	—
213	C₄⋊C₄	—	178	4	no	—
214	C₄⋊C₄	—	178	2	no	—
215	C₄⋊C₄	—	178	4	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	356	4	no	—
221	C₄²	C₈×D₈₉	1	2	yes	—
222	C₄²	#13083557	1	2	yes	—
223	C₄²	#13083563	1	2	yes	—
224	C₄²	Dic₃₅₆	1	2	yes	—
225	C₄²	Q₈×D₈₉	1	2	yes	—
226	C₄²	#13083573	1	2	yes	—
227	C₂³	C₂×#13083551	1	1	yes	characteristic
228	Q₈	—	356	8	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	356	2	no	—
239	C₂×C₄	—	356	2	no	—
240	C₂×C₄	—	356	2	no	—
241	C₂×C₄	—	356	2	no	—
242	C₂×C₄	—	356	2	no	—
243	C₂×C₄	—	178	4	no	—
244	C₂×C₄	#15667924	1	2	yes	—
245	C₂×C₄	#15667924	1	2	yes	—
246	C₂×C₄	#15667937	1	2	yes	—
247	C₂×C₄	#15667937	1	2	yes	—
248	C₂×C₄	#15667940	1	2	yes	—
249	C₂×C₄	#15667977	1	2	yes	—
250	C₂×C₄	#15667992	1	2	yes	—
251	C₂×C₄	#15667999	1	1	yes	characteristic
252	C₂×C₄	#15668010	1	1	yes	characteristic
253	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
254	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
255	C₂×C₄	#15668020	1	2	yes	—
256	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
257	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
258	C₂×C₄	#15668040	1	2	yes	—
259	C₈	—	4	4	no	—
260	C₈	—	356	4	no	—
261	C₂²	#19151503	1	1	yes	characteristic
262	C₂²	C₂×#15667924	1	1	yes	characteristic
263	C₂²	#19151566	1	1	yes	characteristic
264	C₂²	C₂×#15667937	1	1	yes	characteristic
265	C₂²	#19151604	1	1	yes	characteristic
266	C₂²	C₂×#15667940	1	1	yes	characteristic
267	C₂²	#19151616	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	356	4	no	—
274	C₄	—	712	2	no	—
275	C₄	#19151330	1	4	yes	—
276	C₄	#19151722	1	2	yes	—
277	C₄	#19151725	1	2	yes	—
278	C₄	#19151889	1	2	yes	—
279	C₄	#19151894	1	2	yes	—
280	C₂	C₂×#19151330	1	1	yes	characteristic
281	C₂	#26018644	1	1	yes	characteristic
282	C₂	#26018656	1	1	yes	characteristic
283	C₂	#26018670	1	1	yes	characteristic
284	C₂	#26021187	1	1	yes	characteristic
285	C₂	#26021195	1	1	yes	characteristic
286	C₂	#26021199	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#9864
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
23. #15668012	C₈	semidirect
33. C₄⋊#10623494	Q₈	semidirect
148. C₈₉	#9864	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668012	2
C₄×C₄⋊#10623494	2
#26018092	2
C₄×C₄⋊C₈×C₈₉	2
#9864	89