Subgroups — #36454435

Order 22784.

566 conjugacy classes of subgroups.

The subgroup lattice has 302 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	#26020404	C₂	1	1	yes	maximal, characteristic
3	#26018092	C₂	1	4	yes	maximal
4	C₄²×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	—	2	1	no	—
8	C₂×C₄⋊#10623494	—	2	1	no	—
9	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
10	#19151504	—	2	1	no	—
11	#19151496	—	2	1	no	—
12	#19151496	C₄	1	2	yes	—
13	#19151468	C₄	1	1	yes	characteristic
14	C₄×C₄⋊Dic₈₉	—	2	1	no	—
15	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	—	2	1	no	—
17	C₂×C₄×C₈×C₈₉	—	2	1	no	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
19	C₄³×C₈₉	C₂²	1	1	yes	characteristic
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	2	no	—
25	C₄×Dic₁₇₈	—	4	2	no	—
26	C₂×C₄⋊Dic₈₉	—	2	1	no	—
27	C₂×C₄⋊Dic₈₉	—	2	1	no	—
28	C₂×C₄⋊Dic₈₉	—	2	1	no	—
29	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
31	C₂×#13083553	—	2	2	no	—
32	C₂×C₄×Dic₈₉	—	2	2	no	—
33	C₄⋊#10623494	—	2	4	no	—
34	C₄⋊#10623494	—	4	2	no	—
35	C₄⋊#10623494	—	4	2	no	—
36	#15667919	—	2	2	no	—
37	#15667919	—	2	2	no	—
38	C₂²×C₈×C₈₉	—	2	2	no	—
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	C₂×C₄	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	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₈₉	—	2	4	no	—
47	C₄×C₈×C₈₉	—	2	4	no	—
48	C₄×C₈×C₈₉	—	2	2	no	—
49	C₄×C₈×C₈₉	Q₈	1	4	yes	complemented
50	C₂×Dic₁₇₈	—	2	4	no	—
51	C₂×Dic₁₇₈	—	4	2	no	—
52	C₂×Dic₁₇₈	—	4	2	no	—
53	C₂²×Dic₈₉	—	2	2	no	—
54	C₂²×Dic₈₉	—	2	2	no	—
55	C₂×#10623494	—	4	2	no	—
56	C₂×#10623494	—	4	2	no	—
57	C₂×#10623494	—	4	2	no	—
58	C₄⋊Dic₈₉	—	2	4	no	—
59	C₄⋊Dic₈₉	—	2	2	no	—
60	C₄⋊Dic₈₉	—	2	4	no	—
61	C₄⋊Dic₈₉	—	4	2	no	—
62	C₄⋊Dic₈₉	—	4	2	no	—
63	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
64	#13083553	—	4	4	no	—
65	C₄×Dic₈₉	—	4	2	no	—
66	C₄×Dic₈₉	—	4	2	no	—
67	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	—	2	4	no	—
75	C₂×C₈×C₈₉	—	2	4	no	—
76	C₂×C₈×C₈₉	—	2	4	no	—
77	C₄²×C₈₉	—	2	2	no	—
78	C₄²×C₈₉	—	2	2	no	—
79	C₄²×C₈₉	—	2	2	no	—
80	C₄²×C₈₉	C₂×C₈	1	2	yes	—
81	C₄²×C₈₉	M₁₆	1	2	yes	—
82	C₄²×C₈₉	D₈	1	2	yes	—
83	C₄²×C₈₉	SD₁₆	1	2	yes	—
84	C₄²×C₈₉	SD₁₆	1	2	yes	—
85	C₄²×C₈₉	Q₁₆	1	2	yes	—
86	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
87	C₄²×C₈₉	#41	1	2	yes	—
88	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₈₉	#102	1	2	yes	—
98	C₂×C₄×C₈₉	#102	1	2	yes	—
99	C₂×C₄×C₈₉	#103	1	2	yes	—
100	C₂×C₄×C₈₉	#103	1	2	yes	—
101	C₂×C₄×C₈₉	#104	1	2	yes	—
102	C₂×C₄×C₈₉	#104	1	2	yes	—
103	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₄⋊Q₈	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×C₄×C₈₉	#131	1	2	yes	—
110	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×Dic₈₉	—	2	4	no	—
115	C₂×Dic₈₉	—	4	2	no	—
116	C₂×Dic₈₉	—	4	2	no	—
117	C₂×Dic₈₉	—	4	2	no	—
118	C₂×Dic₈₉	—	4	2	no	—
119	C₂×Dic₈₉	—	4	2	no	—
120	Dic₁₇₈	—	4	8	no	—
121	C₈×C₈₉	—	2	8	no	—
122	#10623494	—	8	2	no	—
123	C₂²×C₈₉	#391	1	1	yes	characteristic
124	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₂²×C₈₉	#408	1	1	yes	characteristic
126	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
127	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
128	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
129	C₂²×C₈₉	C₂×#104	1	1	yes	characteristic
130	C₄×C₈₉	—	2	2	no	—
131	C₄×C₈₉	—	2	2	no	—
132	C₄×C₈₉	—	2	2	no	—
133	C₄×C₈₉	D₄⋊C₈	1	4	yes	—
134	C₄×C₈₉	Q₈⋊C₈	1	4	yes	—
135	C₄×C₈₉	C₈×Q₈	1	2	yes	—
136	C₄×C₈₉	#446	1	2	yes	—
137	C₄×C₈₉	#486	1	1	yes	characteristic
138	C₄×C₈₉	#487	1	1	yes	characteristic
139	C₄×C₈₉	#498	1	1	yes	characteristic
140	C₄×C₈₉	#500	1	1	yes	characteristic
141	Dic₈₉	—	4	4	no	—
142	Dic₈₉	—	8	2	no	—
143	#9870	—	89	1	no	maximal, Sylow
144	C₂×C₈₉	C₂×D₄⋊C₈	1	1	yes	characteristic
145	C₂×C₈₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
146	C₂×C₈₉	#1492	1	1	yes	characteristic
147	C₂×C₈₉	#1493	1	1	yes	characteristic
148	C₂×C₈₉	#1984	1	1	yes	characteristic
149	C₂×C₈₉	#1990	1	1	yes	characteristic
150	C₂×C₈₉	#1994	1	1	yes	characteristic
151	C₄×C₄⋊Q₈	—	89	1	no	—
152	#1842	—	89	1	no	—
153	C₄²×C₈	D₈₉	1	1	yes	characteristic
154	#1279	—	89	4	no	—
155	C₈₉	#9870	1	1	yes	characteristic, complemented, Sylow
156	C₂×C₄⋊Q₈	—	89	1	no	—
157	C₂×C₄×Q₈	—	178	1	no	—
158	C₂×C₄⋊C₈	—	89	2	no	—
159	C₂×C₄⋊C₈	—	178	1	no	—
160	C₂×C₄⋊C₈	—	178	1	no	—
161	C₂×C₄×C₈	—	2	1	no	—
162	C₂×C₄×C₈	—	2	1	no	—
163	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
164	#391	—	178	1	no	—
165	#389	—	89	2	no	—
166	#389	—	178	1	no	—
167	#384	—	89	1	no	—
168	C₄×C₄⋊C₄	—	89	2	no	—
169	C₄×C₄⋊C₄	—	178	1	no	—
170	C₄³	C₂×D₈₉	1	1	yes	characteristic
171	C₂²×Q₈	—	178	1	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	178	2	no	—
174	C₄⋊Q₈	—	89	4	no	—
175	C₄⋊Q₈	—	178	2	no	—
176	C₄×Q₈	—	356	2	no	—
177	C₄×Q₈	—	356	2	no	—
178	C₂×C₄⋊C₄	—	89	2	no	—
179	C₂×C₄⋊C₄	—	89	2	no	—
180	C₂×C₄⋊C₄	—	178	1	no	—
181	C₂×C₄⋊C₄	—	178	2	no	—
182	C₂×C₄⋊C₄	—	178	1	no	—
183	C₂×C₄⋊C₄	—	178	1	no	—
184	C₂×C₄²	—	178	2	no	—
185	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
187	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
188	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
192	C₄⋊C₈	—	356	2	no	—
193	C₄⋊C₈	—	356	2	no	—
194	C₄⋊C₈	—	178	4	no	—
195	C₄×C₈	—	2	4	no	—
196	C₄×C₈	—	2	4	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	Dic₁₇₈	1	4	yes	complemented
199	#95	—	178	2	no	—
200	#95	—	178	2	no	—
201	C₂×Q₈	—	356	2	no	—
202	C₂×Q₈	—	356	2	no	—
203	C₂×Q₈	—	178	4	no	—
204	C₂²×C₄	—	178	2	no	—
205	C₂²×C₄	—	178	2	no	—
206	C₂²×C₄	#13083551	1	1	yes	characteristic
207	C₂²×C₄	#13083551	1	1	yes	characteristic
208	C₂²×C₄	#13083551	1	1	yes	characteristic
209	C₂²×C₄	#13083551	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	356	2	no	—
217	C₂×C₈	—	356	2	no	—
218	C₂×C₈	—	356	2	no	—
219	C₄⋊C₄	—	89	4	no	—
220	C₄⋊C₄	—	356	4	no	—
221	C₄⋊C₄	—	356	2	no	—
222	C₄⋊C₄	—	356	2	no	—
223	C₄⋊C₄	—	178	4	no	—
224	C₄⋊C₄	—	178	2	no	—
225	C₄⋊C₄	—	178	4	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	356	2	no	—
230	C₄²	—	356	2	no	—
231	C₄²	C₈×D₈₉	1	2	yes	—
232	C₄²	#13083557	1	2	yes	—
233	C₄²	D₇₁₂	1	2	yes	—
234	C₄²	#13083563	1	2	yes	—
235	C₄²	#13083563	1	2	yes	—
236	C₄²	Dic₃₅₆	1	2	yes	—
237	C₄²	C₂×Dic₁₇₈	1	2	yes	—
238	C₄²	#13083574	1	2	yes	—
239	C₂³	C₂×#13083551	1	1	yes	characteristic
240	Q₈	—	356	8	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	356	2	no	—
248	C₂×C₄	—	356	2	no	—
249	C₂×C₄	—	356	2	no	—
250	C₂×C₄	—	356	2	no	—
251	C₂×C₄	—	356	2	no	—
252	C₂×C₄	—	178	4	no	—
253	C₂×C₄	#15667924	1	2	yes	—
254	C₂×C₄	#15667924	1	2	yes	—
255	C₂×C₄	#15667934	1	2	yes	—
256	C₂×C₄	#15667934	1	2	yes	—
257	C₂×C₄	#15667937	1	2	yes	—
258	C₂×C₄	#15667937	1	2	yes	—
259	C₂×C₄	#15667940	1	2	yes	—
260	C₂×C₄	#15667940	1	2	yes	—
261	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
262	C₂×C₄	#15667991	1	2	yes	—
263	C₂×C₄	#15668000	1	1	yes	characteristic
264	C₂×C₄	#15668012	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
266	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
267	C₂×C₄	#15668018	1	2	yes	—
268	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
269	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
270	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
271	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
272	C₈	—	2	8	no	—
273	C₈	—	712	2	no	—
274	C₂²	#19151504	1	1	yes	characteristic
275	C₂²	C₂×#15667924	1	1	yes	characteristic
276	C₂²	#19151564	1	1	yes	characteristic
277	C₂²	C₂×#15667934	1	1	yes	characteristic
278	C₂²	C₂×#15667937	1	1	yes	characteristic
279	C₂²	C₂×#15667940	1	1	yes	characteristic
280	C₂²	C₂×#15667940	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	356	4	no	—
285	C₄	—	712	2	no	—
286	C₄	#19151327	1	4	yes	—
287	C₄	#19151330	1	4	yes	—
288	C₄	C₈×Dic₁₇₈	1	2	yes	—
289	C₄	#19151728	1	2	yes	—
290	C₄	#19151945	1	1	yes	characteristic
291	C₄	#19151949	1	1	yes	characteristic
292	C₄	#19151996	1	1	yes	characteristic
293	C₄	#19152003	1	1	yes	characteristic
294	C₂	C₂×#19151327	1	1	yes	characteristic
295	C₂	C₂×#19151330	1	1	yes	characteristic
296	C₂	#26018704	1	1	yes	characteristic
297	C₂	#26018708	1	1	yes	characteristic
298	C₂	#26021188	1	1	yes	characteristic
299	C₂	#26021213	1	1	yes	characteristic
300	C₂	#26021229	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	#9870
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
23. #15668012	C₈	semidirect
49. C₄×C₈×C₈₉	Q₈	semidirect
155. C₈₉	#9870	semidirect
198. C₄×C₈	Dic₁₇₈	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
#26020404	2
#26018092	2
C₄²×C₈×C₈₉	2
#9870	89