Subgroups — #36526207

Order 22784.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030838	C₂	1	1	yes	maximal, characteristic
2	#26029199	C₂	1	2	yes	maximal
3	#26019150	C₂	1	2	yes	maximal, complemented
4	#26019104	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₃₅₆	C₂	1	2	yes	maximal, complemented
6	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
7	#26018656	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19152155	C₂²	1	2	yes	—
10	#19152003	C₄	1	1	yes	characteristic
11	#19151978	C₄	1	1	yes	characteristic
12	#19151949	C₄	1	2	yes	—
13	#19151839	C₄	1	2	yes	—
14	#19151839	C₄	1	2	yes	—
15	#19151728	C₂²	1	4	yes	—
16	#19151726	—	2	4	no	—
17	C₈×Dic₁₇₈	C₂²	1	4	yes	—
18	C₄×Dic₃₅₆	C₂²	1	2	yes	—
19	C₄×Dic₃₅₆	C₂²	1	2	yes	—
20	#19151626	—	2	2	no	—
21	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
22	#19151556	—	2	2	no	—
23	#19151351	C₂²	1	2	yes	—
24	#19151351	C₂²	1	1	yes	characteristic
25	#19151351	C₂²	1	1	yes	characteristic
26	#19151330	C₂²	1	4	yes	—
27	#19151330	C₂²	1	4	yes	—
28	#431×C₈₉	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	#322×C₈₉	C₂²	1	2	yes	—
32	C₈²×C₈₉	C₂²	1	2	yes	—
33	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
34	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
35	C₂²×#10623494	—	2	2	no	—
36	#15668012	C₂×C₄	1	2	yes	—
37	#15668012	C₂×C₄	1	2	yes	—
38	#15668004	—	2	2	no	—
39	#15667991	—	2	2	no	—
40	#15667991	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₇₈	—	2	2	no	—
42	C₄×Dic₁₇₈	C₂³	1	4	yes	—
43	#15667971	—	2	2	no	—
44	#15667946	C₂×C₄	1	2	yes	—
45	#15667946	C₂×C₄	1	2	yes	—
46	C₄⋊#10623494	—	2	2	no	—
47	C₄⋊#10623494	C₂³	1	2	yes	—
48	C₄⋊#10623494	C₂³	1	2	yes	—
49	#15667941	—	2	4	no	—
50	#15667941	—	2	4	no	—
51	#15667937	C₂×C₄	1	4	yes	—
52	#15667937	C₂×C₄	1	4	yes	—
53	#15667922	—	2	4	no	—
54	C₈×Dic₈₉	—	2	4	no	—
55	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₈₉	—	2	1	no	—
61	C₂²⋊C₈×C₈₉	—	2	1	no	—
62	#97×C₈₉	D₄	1	2	yes	—
63	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₈₉	C₂³	1	2	yes	—
66	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₈₉	C₂³	1	2	yes	—
69	C₂×Dic₁₇₈	—	2	2	no	—
70	C₂×Dic₁₇₈	C₂²×C₄	1	4	yes	—
71	Dic₃₅₆	—	2	4	no	—
72	Dic₃₅₆	—	4	2	no	—
73	C₂×#10623494	—	2	2	no	—
74	C₂×#10623494	—	2	2	no	—
75	C₂×#10623494	—	4	4	no	—
76	C₄⋊Dic₈₉	—	2	2	no	—
77	C₄⋊Dic₈₉	—	2	2	no	—
78	C₄⋊Dic₈₉	—	2	2	no	—
79	C₄⋊Dic₈₉	C₂²×C₄	1	4	yes	—
80	#13083553	—	2	4	no	—
81	#13083553	—	2	4	no	—
82	C₂²⋊Dic₈₉	—	2	4	no	—
83	C₄×Dic₈₉	—	2	4	no	—
84	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
87	M₁₆×C₈₉	—	2	2	no	—
88	M₁₆×C₈₉	Q₁₆	1	4	yes	complemented
89	C₂×C₈×C₈₉	—	2	2	no	—
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	4	1	no	—
92	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
98	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₈₉	#41	1	1	yes	characteristic
101	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
102	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	4	1	no	—
106	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₈₉	#135	1	2	yes	—
111	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
114	C₂×Dic₈₉	—	2	4	no	—
115	C₂×Dic₈₉	—	2	4	no	—
116	Dic₁₇₈	—	2	4	no	—
117	Dic₁₇₈	—	2	4	no	—
118	Dic₁₇₈	—	8	2	no	—
119	C₈×C₈₉	—	2	1	no	—
120	C₈×C₈₉	—	2	1	no	—
121	C₈×C₈₉	—	2	1	no	—
122	C₈×C₈₉	—	2	2	no	—
123	C₈×C₈₉	—	4	2	no	—
124	C₈×C₈₉	#131	1	2	yes	—
125	C₈×C₈₉	C₂×Q₁₆	1	4	yes	—
126	#10623494	—	4	2	no	—
127	#10623494	—	4	2	no	—
128	C₂²×C₈₉	—	2	1	no	—
129	C₂²×C₈₉	—	2	1	no	—
130	C₂²×C₈₉	C₄×Q₁₆	1	2	yes	—
131	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
132	C₄×C₈₉	—	2	1	no	—
133	C₄×C₈₉	—	2	1	no	—
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
136	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
137	C₄×C₈₉	#568	1	1	yes	characteristic
138	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
139	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
140	Dic₈₉	—	4	4	no	—
141	Dic₈₉	—	4	4	no	—
142	#18068	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	—	2	2	no	—
144	C₂×C₈₉	#2935	1	1	yes	characteristic
145	C₂×C₈₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
146	C₂×C₈₉	#2957	1	1	yes	characteristic
147	#3170	—	89	1	no	—
148	#2962	—	89	2	no	—
149	#1591	—	89	2	no	—
150	#1584	—	89	4	no	—
151	C₈×Q₁₆	—	89	2	no	—
152	C₂×#335	—	89	1	no	—
153	#1481	—	89	2	no	—
154	C₈×M₁₆	D₈₉	1	1	yes	characteristic
155	C₈₉	#18068	1	1	yes	characteristic, complemented, Sylow
156	#533	—	89	2	no	—
157	#500	—	89	1	no	—
158	#494	—	89	1	no	—
159	#487	—	89	2	no	—
160	#467	—	89	2	no	—
161	#467	—	89	2	no	—
162	#446	—	89	4	no	—
163	#446	—	178	4	no	—
164	C₈×Q₈	—	89	4	no	—
165	C₄×Q₁₆	—	89	2	no	—
166	C₄×Q₁₆	—	89	2	no	—
167	#431	C₂×D₈₉	1	1	yes	characteristic
168	#424	—	178	2	no	—
169	C₂×C₄⋊C₈	—	89	2	no	—
170	#405	—	178	2	no	—
171	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
173	#335	—	89	1	no	—
174	#335	—	89	2	no	—
175	#335	—	89	1	no	—
176	Q₈⋊C₈	—	89	4	no	—
177	Q₈⋊C₈	—	89	4	no	—
178	#322	C₂×D₈₉	1	2	yes	—
179	C₈²	C₂×D₈₉	1	2	yes	—
180	C₂×Q₁₆	—	89	2	no	—
181	C₂×Q₁₆	—	89	2	no	—
182	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
183	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
184	C₂²×C₈	—	178	2	no	—
185	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
187	C₄⋊Q₈	—	89	2	no	—
188	C₄⋊Q₈	—	89	2	no	—
189	#125	—	178	2	no	—
190	C₂²⋊Q₈	—	89	4	no	—
191	C₂²⋊Q₈	—	178	2	no	—
192	C₄×Q₈	—	89	4	no	—
193	C₄×Q₈	—	178	2	no	—
194	#117	—	178	2	no	—
195	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
196	#107	—	89	2	no	—
197	#107	—	89	2	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	89	2	no	—
200	C₄⋊C₈	—	89	2	no	—
201	C₄⋊C₈	—	178	4	no	—
202	C₄⋊C₈	—	178	4	no	—
203	C₄⋊C₈	—	178	2	no	—
204	#103	—	89	4	no	—
205	#103	—	89	4	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	#97	—	178	4	no	—
208	#97	D₃₅₆	1	2	yes	—
209	C₄×C₈	—	178	4	no	—
210	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
211	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₈₉	1	2	yes	—
213	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₈₉	1	2	yes	—
216	C₂×Q₈	—	89	4	no	—
217	C₂×Q₈	—	178	2	no	—
218	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
220	C₂²×C₄	#13083574	1	1	yes	characteristic
221	Q₁₆	—	356	2	no	—
222	Q₁₆	—	178	4	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	Dic₃₅₆	1	4	yes	complemented
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	356	4	no	—
229	C₂×C₈	—	178	2	no	—
230	C₂×C₈	—	178	2	no	—
231	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
235	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
236	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
237	C₄⋊C₄	—	89	4	no	—
238	C₄⋊C₄	—	178	2	no	—
239	C₄⋊C₄	—	178	2	no	—
240	C₄⋊C₄	—	178	4	no	—
241	C₄⋊C₄	—	178	2	no	—
242	C₄⋊C₄	—	178	4	no	—
243	C₂²⋊C₄	—	178	4	no	—
244	C₄²	—	178	4	no	—
245	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
246	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
247	C₄²	#13083574	1	1	yes	characteristic
248	C₄²	C₂³×D₈₉	1	1	yes	characteristic
249	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
250	Q₈	—	178	4	no	—
251	Q₈	—	178	4	no	—
252	Q₈	—	712	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	178	4	no	—
257	C₂×C₄	—	178	4	no	—
258	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
259	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
260	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
261	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
262	C₂×C₄	#15668034	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
265	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	356	2	no	—
272	C₈	—	356	2	no	—
273	C₈	#15668020	1	2	yes	—
274	C₈	C₂×Dic₃₅₆	1	4	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₄×Dic₃₅₆	1	2	yes	—
278	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	356	4	no	—
283	C₄	—	356	4	no	—
284	C₄	C₄×Dic₃₅₆	1	2	yes	—
285	C₄	C₂×#15668020	1	1	yes	characteristic
286	C₄	#19152364	1	1	yes	characteristic
287	C₄	C₂²×Dic₃₅₆	1	1	yes	characteristic
288	C₄	C₂×#15668034	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#26028998	1	1	yes	characteristic
291	C₂	C₂×C₄×Dic₃₅₆	1	1	yes	characteristic
292	C₂	#26029170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₈×M₁₆	D₈₉
hypocenter	C₈₉	#18068
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #26019150	C₂	semidirect
4. #26019104	C₂	semidirect
5. C₈×Dic₃₅₆	C₂	semidirect
88. M₁₆×C₈₉	Q₁₆	semidirect
155. C₈₉	#18068	semidirect
224. M₁₆	Dic₃₅₆	semidirect

Maximal subgroups

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

subgroup	index
#26030838	2
#26029199	2
#26019150	2
#26019104	2
C₈×Dic₃₅₆	2
C₂×#19151351	2
#26018656	2
C₈×M₁₆×C₈₉	2
#18068	89