Subgroups — #36525925

Order 22784.

630 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029203	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#26019160	C₂	1	2	yes	maximal, complemented
4	#26019122	C₂	1	4	yes	maximal, complemented
5	#26019070	C₂	1	2	yes	maximal, complemented
6	#26018680	C₂	1	2	yes	maximal
7	#26018541	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₅₆	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₇₈	C₂²	1	2	yes	—
11	#19151727	—	2	4	no	—
12	#19151725	C₂²	1	4	yes	—
13	#19151722	C₂²	1	4	yes	—
14	C₄×Dic₃₅₆	C₄	1	1	yes	characteristic
15	C₄×Dic₃₅₆	C₄	1	2	yes	—
16	C₄×Dic₃₅₆	C₄	1	1	yes	characteristic
17	C₄×Dic₃₅₆	C₂²	1	2	yes	—
18	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
19	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
20	#19151651	C₂²	1	1	yes	characteristic
21	C₂×#15667946	C₄	1	1	yes	characteristic
22	#19151621	—	2	2	no	—
23	C₂×#15667937	C₄	1	2	yes	—
24	C₄×#13083558	C₂²	1	1	yes	characteristic
25	M₁₆×Dic₈₉	—	2	2	no	—
26	#19151328	C₂²	1	4	yes	—
27	#19151328	C₂²	1	4	yes	—
28	#19151320	C₂²	1	2	yes	—
29	#19151319	C₂²	1	2	yes	—
30	#423×C₈₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	#334×C₈₉	C₂²	1	2	yes	—
33	#334×C₈₉	C₂²	1	2	yes	—
34	C₂²×Dic₁₇₈	C₂×C₄	1	2	yes	—
35	C₂×Dic₃₅₆	—	2	4	no	—
36	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
37	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₃₅₆	C₂×C₄	1	1	yes	characteristic
39	C₂×#13083558	D₄	1	2	yes	—
40	C₄×Dic₁₇₈	—	2	2	no	—
41	C₄×Dic₁₇₈	—	2	2	no	—
42	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₇₈	C₂³	1	2	yes	—
45	C₄×Dic₁₇₈	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
47	C₂×#13083553	—	2	2	no	—
48	C₂×C₄×Dic₈₉	—	2	2	no	—
49	#15667946	C₂×C₄	1	2	yes	—
50	#15667946	C₂×C₄	1	1	yes	characteristic
51	#15667946	C₂×C₄	1	1	yes	characteristic
52	C₄⋊#10623494	—	2	1	no	—
53	#15667941	—	2	4	no	—
54	#15667941	—	2	4	no	—
55	#15667937	C₂×C₄	1	2	yes	—
56	#15667937	C₂×C₄	1	2	yes	—
57	#15667937	C₂×C₄	1	2	yes	—
58	#15667937	C₂×C₄	1	2	yes	—
59	C₂²⋊#10623494	—	2	1	no	—
60	#15667923	D₄	1	2	yes	—
61	#15667922	—	2	4	no	—
62	C₄×#10623494	C₂³	1	2	yes	—
63	C₄×#10623494	C₂³	1	2	yes	—
64	C₈×Dic₈₉	—	2	4	no	—
65	C₂×M₁₆×C₈₉	—	2	2	no	—
66	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₈₉	—	2	2	no	—
70	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
71	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₇₈	—	2	4	no	—
76	C₂×Dic₁₇₈	—	2	2	no	—
77	C₂×Dic₁₇₈	—	4	2	no	—
78	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
80	C₂²×Dic₈₉	—	2	2	no	—
81	C₂²×Dic₈₉	—	2	2	no	—
82	Dic₃₅₆	—	2	4	no	—
83	Dic₃₅₆	—	2	2	no	—
84	Dic₃₅₆	M₁₆	1	4	yes	complemented
85	#13083558	—	2	4	no	—
86	C₂×#10623494	—	2	2	no	—
87	C₂×#10623494	C₂×D₄	1	4	yes	—
88	C₄⋊Dic₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
90	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
91	#13083553	—	2	2	no	—
92	#13083553	—	2	2	no	—
93	#13083553	—	2	2	no	—
94	#13083553	—	2	2	no	—
95	C₄×Dic₈₉	—	2	2	no	—
96	C₄×Dic₈₉	—	2	2	no	—
97	C₄×Dic₈₉	—	2	2	no	—
98	C₄×Dic₈₉	—	2	2	no	—
99	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
102	M₁₆×C₈₉	—	4	4	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	—	2	4	no	—
105	C₂×C₈×C₈₉	—	4	1	no	—
106	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₈₉	#41	1	1	yes	characteristic
113	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
114	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	4	1	no	—
118	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#137	1	4	yes	—
122	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₈₉	—	2	2	no	—
126	C₂×Dic₈₉	—	2	2	no	—
127	C₂×Dic₈₉	—	2	2	no	—
128	C₂×Dic₈₉	—	2	2	no	—
129	C₂×Dic₈₉	—	4	4	no	—
130	C₂×Dic₈₉	—	8	2	no	—
131	Dic₁₇₈	—	2	2	no	—
132	Dic₁₇₈	—	2	2	no	—
133	Dic₁₇₈	—	2	2	no	—
134	Dic₁₇₈	—	4	4	no	—
135	Dic₁₇₈	C₂×M₁₆	1	4	yes	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	—	2	1	no	—
138	C₈×C₈₉	—	2	1	no	—
139	C₈×C₈₉	—	4	4	no	—
140	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
141	#10623494	—	2	4	no	—
142	#10623494	—	4	2	no	—
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	#441	1	2	yes	—
146	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	#441	1	2	yes	—
151	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
152	C₄×C₈₉	#568	1	1	yes	characteristic
153	C₄×C₈₉	C₂×#137	1	2	yes	—
154	Dic₈₉	—	2	4	no	—
155	Dic₈₉	—	4	2	no	—
156	Dic₈₉	—	4	2	no	—
157	Dic₈₉	—	4	2	no	—
158	#18042	—	89	1	no	maximal, Sylow
159	C₂×C₈₉	—	2	2	no	—
160	C₂×C₈₉	D₄×M₁₆	1	1	yes	characteristic
161	C₂×C₈₉	C₂×#441	1	1	yes	characteristic
162	C₂×C₈₉	#2957	1	1	yes	characteristic
163	Q₈×M₁₆	—	89	2	no	—
164	C₂×C₄×Q₁₆	—	89	1	no	—
165	#1593	—	89	2	no	—
166	#1587	—	89	4	no	—
167	#1579	—	89	2	no	—
168	#1566	D₈₉	1	1	yes	characteristic
169	#1487	—	89	2	no	—
170	#1451	—	89	1	no	—
171	C₈₉	#18042	1	1	yes	characteristic, complemented, Sylow
172	C₂²×Q₁₆	—	89	1	no	—
173	C₂×C₄×Q₈	—	89	2	no	—
174	#446	—	89	4	no	—
175	#446	—	178	4	no	—
176	C₈×Q₈	—	89	4	no	—
177	C₄×Q₁₆	—	89	1	no	—
178	C₄×Q₁₆	—	89	2	no	—
179	C₄×Q₁₆	—	89	1	no	—
180	C₄×Q₁₆	—	89	1	no	—
181	C₄×Q₁₆	—	89	2	no	—
182	C₄×Q₁₆	—	89	1	no	—
183	#431	—	89	1	no	—
184	C₂×#107	—	89	1	no	—
185	#423	—	178	2	no	—
186	#423	C₂×D₈₉	1	2	yes	—
187	C₂×#103	—	89	2	no	—
188	C₄×M₁₆	—	89	1	no	—
189	C₄×M₁₆	—	178	2	no	—
190	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
191	#334	C₂×D₈₉	1	2	yes	—
192	#334	C₂×D₈₉	1	2	yes	—
193	Q₈⋊C₈	—	89	4	no	—
194	Q₈⋊C₈	—	89	4	no	—
195	#322	—	89	2	no	—
196	#322	—	89	2	no	—
197	C₂²×Q₈	—	89	2	no	—
198	C₂×Q₁₆	—	89	1	no	—
199	C₂×Q₁₆	—	89	2	no	—
200	C₂×Q₁₆	—	89	1	no	—
201	C₂×Q₁₆	—	178	4	no	—
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	—	89	2	no	—
204	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
205	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
206	C₄×Q₈	—	89	2	no	—
207	C₄×Q₈	—	89	2	no	—
208	C₄×Q₈	—	89	2	no	—
209	C₄×Q₈	—	89	2	no	—
210	C₄×Q₈	—	178	2	no	—
211	C₄×Q₈	—	178	2	no	—
212	C₂×C₄⋊C₄	—	89	2	no	—
213	C₂×C₄⋊C₄	—	178	2	no	—
214	C₂×C₄²	—	178	2	no	—
215	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
216	#107	—	89	2	no	—
217	#107	—	89	1	no	—
218	#107	—	89	1	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	178	4	no	—
221	C₄⋊C₈	—	178	4	no	—
222	C₄⋊C₈	—	178	1	no	—
223	C₄⋊C₈	C₂²×D₈₉	1	4	yes	—
224	#103	—	89	2	no	—
225	#103	—	89	2	no	—
226	#103	—	89	2	no	—
227	#103	—	89	2	no	—
228	C₂²⋊C₈	—	178	1	no	—
229	#97	—	89	2	no	—
230	#97	—	178	4	no	—
231	C₄×C₈	—	89	2	no	—
232	C₄×C₈	—	89	2	no	—
233	C₄×C₈	—	178	4	no	—
234	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
235	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
238	C₂×Q₈	—	89	2	no	—
239	C₂×Q₈	—	89	2	no	—
240	C₂×Q₈	—	356	2	no	—
241	C₂×Q₈	—	178	4	no	—
242	C₂×Q₈	—	178	2	no	—
243	C₂²×C₄	—	178	2	no	—
244	C₂²×C₄	—	178	2	no	—
245	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
246	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
247	C₂²×C₄	#13083575	1	1	yes	characteristic
248	Q₁₆	—	89	4	no	—
249	Q₁₆	—	178	4	no	—
250	Q₁₆	—	178	2	no	—
251	M₁₆	—	4	4	no	—
252	M₁₆	—	178	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	89	4	no	—
257	C₂×C₈	—	178	2	no	—
258	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
262	C₄⋊C₄	—	89	2	no	—
263	C₄⋊C₄	—	89	2	no	—
264	C₄⋊C₄	—	178	2	no	—
265	C₄⋊C₄	—	178	2	no	—
266	C₄⋊C₄	—	178	2	no	—
267	C₄⋊C₄	—	178	2	no	—
268	C₄⋊C₄	—	178	2	no	—
269	C₄²	—	178	2	no	—
270	C₄²	—	178	2	no	—
271	C₄²	—	178	2	no	—
272	C₄²	—	178	2	no	—
273	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
274	C₄²	D₄×D₈₉	1	1	yes	characteristic
275	C₄²	#13083575	1	1	yes	characteristic
276	C₄²	C₂³×D₈₉	1	1	yes	characteristic
277	C₂³	#15667974	1	1	yes	characteristic
278	Q₈	—	89	4	no	—
279	Q₈	—	356	4	no	—
280	Q₈	—	178	2	no	—
281	Q₈	—	178	2	no	—
282	Q₈	—	178	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	356	4	no	—
287	C₂×C₄	—	178	2	no	—
288	C₂×C₄	—	178	2	no	—
289	C₂×C₄	—	178	2	no	—
290	C₂×C₄	—	178	2	no	—
291	C₂×C₄	—	712	2	no	—
292	C₂×C₄	#15667974	1	1	yes	characteristic
293	C₂×C₄	#15667974	1	1	yes	characteristic
294	C₂×C₄	#15667974	1	1	yes	characteristic
295	C₂×C₄	#15668041	1	4	yes	—
296	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
298	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	4	4	no	—
303	C₈	—	356	2	no	—
304	C₈	—	178	4	no	—
305	C₈	M₁₆×D₈₉	1	2	yes	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	#19151704	1	2	yes	—
309	C₂²	C₂×#15667974	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	356	2	no	—
314	C₄	—	356	2	no	—
315	C₄	—	356	2	no	—
316	C₄	—	178	4	no	—
317	C₄	#19151704	1	2	yes	—
318	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
319	C₄	#19152364	1	1	yes	characteristic
320	C₄	C₂×#15668041	1	2	yes	—
321	C₂	—	2	2	no	—
322	C₂	#26028987	1	1	yes	characteristic
323	C₂	C₂×#19151704	1	1	yes	characteristic
324	C₂	#26029168	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1566	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1566	D₈₉
hypocenter	C₈₉	#18042
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. #26019160	C₂	semidirect
4. #26019122	C₂	semidirect
5. #26019070	C₂	semidirect
84. Dic₃₅₆	M₁₆	semidirect
171. C₈₉	#18042	semidirect

Maximal subgroups

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

subgroup	index
#26029203	2
C₂×C₄×Dic₃₅₆	2
#26019160	2
#26019122	2
#26019070	2
#26018680	2
#26018541	2
C₈₉×#1566	2
#18042	89