Subgroups — #36526205

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	#26029201	C₂	1	2	yes	maximal
2	#26019149	C₂	1	2	yes	maximal, complemented
3	#26019103	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#10623494	C₂	1	2	yes	maximal, complemented
5	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
6	#26018655	C₂	1	2	yes	maximal
7	C₈×#13083558	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#3170	C₂	1	1	yes	maximal, characteristic
9	#19151729	—	2	4	no	—
10	#19151729	C₂²	1	4	yes	—
11	Q₈×#10623494	C₂²	1	4	yes	—
12	#19151651	C₂²	1	1	yes	characteristic
13	#19151625	—	2	2	no	—
14	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
15	#19151554	—	2	2	no	—
16	C₄×#13083558	C₂²	1	1	yes	characteristic
17	#19151351	C₂²	1	2	yes	—
18	#19151351	C₂²	1	1	yes	characteristic
19	#19151351	C₂²	1	1	yes	characteristic
20	Q₈⋊#10623494	C₂²	1	4	yes	—
21	Q₈⋊#10623494	C₂²	1	4	yes	—
22	#19151320	C₂²	1	2	yes	—
23	C₈×#10623494	C₂²	1	2	yes	—
24	#533×C₈₉	C₂²	1	2	yes	—
25	#500×C₈₉	C₄	1	1	yes	characteristic
26	#494×C₈₉	C₄	1	1	yes	characteristic
27	#487×C₈₉	C₄	1	2	yes	—
28	#467×C₈₉	C₄	1	2	yes	—
29	#467×C₈₉	C₄	1	2	yes	—
30	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
31	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#13083558	D₄	1	1	yes	characteristic
34	C₂×#13083558	D₄	1	1	yes	characteristic
35	C₂²×#10623494	—	2	2	no	—
36	C₄⋊#10623494	—	2	4	no	—
37	C₄⋊#10623494	—	2	4	no	—
38	C₄⋊#10623494	—	2	2	no	—
39	C₄⋊#10623494	—	2	1	no	—
40	C₄⋊#10623494	C₂³	1	2	yes	—
41	C₄⋊#10623494	C₂³	1	2	yes	—
42	C₂²⋊#10623494	—	2	1	no	—
43	#15667923	—	2	4	no	—
44	#15667923	D₄	1	2	yes	—
45	C₄×#10623494	—	2	4	no	—
46	C₄×#10623494	C₂³	1	2	yes	—
47	C₄×#10623494	C₂³	1	2	yes	—
48	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
49	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
53	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
54	#125×C₈₉	—	2	2	no	—
55	C₂²⋊Q₈×C₈₉	—	2	2	no	—
56	C₂²⋊Q₈×C₈₉	C₂×C₄	1	4	yes	—
57	C₄×Q₈×C₈₉	—	2	2	no	—
58	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
59	#117×C₈₉	—	2	2	no	—
60	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
61	#107×C₈₉	C₂×C₄	1	2	yes	—
62	#107×C₈₉	C₂×C₄	1	2	yes	—
63	#103×C₈₉	C₂×C₄	1	4	yes	—
64	#103×C₈₉	C₂×C₄	1	4	yes	—
65	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
69	#13083558	—	2	2	no	—
70	#13083558	Q₁₆	1	4	yes	complemented
71	C₂×#10623494	—	2	2	no	—
72	C₂×#10623494	—	2	2	no	—
73	C₂×#10623494	—	2	2	no	—
74	C₂×#10623494	—	4	4	no	—
75	C₂×#10623494	C₂×D₄	1	2	yes	—
76	C₂×#10623494	C₂×D₄	1	2	yes	—
77	C₂×Q₈×C₈₉	—	2	2	no	—
78	C₂×Q₈×C₈₉	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
82	Q₁₆×C₈₉	—	2	4	no	—
83	Q₁₆×C₈₉	—	4	2	no	—
84	C₂×C₈×C₈₉	—	2	2	no	—
85	C₂×C₈×C₈₉	—	4	1	no	—
86	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₄⋊C₄×C₈₉	—	2	2	no	—
91	C₄⋊C₄×C₈₉	—	2	2	no	—
92	C₄⋊C₄×C₈₉	—	2	2	no	—
93	C₄⋊C₄×C₈₉	—	2	4	no	—
94	C₄⋊C₄×C₈₉	—	2	4	no	—
95	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
96	C₂²⋊C₄×C₈₉	—	2	4	no	—
97	C₄²×C₈₉	—	2	4	no	—
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	Q₈×C₈₉	—	2	4	no	—
104	Q₈×C₈₉	—	2	4	no	—
105	Q₈×C₈₉	—	8	2	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	4	no	—
109	C₂×C₄×C₈₉	—	2	4	no	—
110	C₂×C₄×C₈₉	—	4	1	no	—
111	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
115	C₂×C₄×C₈₉	#135	1	2	yes	—
116	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
119	C₈×C₈₉	—	2	1	no	—
120	C₈×C₈₉	—	2	1	no	—
121	C₈×C₈₉	—	2	1	no	—
122	C₈×C₈₉	#131	1	2	yes	—
123	#10623494	—	2	2	no	—
124	#10623494	—	4	2	no	—
125	#10623494	—	4	2	no	—
126	#10623494	—	4	2	no	—
127	#10623494	C₂×Q₁₆	1	4	yes	—
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	—	4	4	no	—
136	C₄×C₈₉	—	4	4	no	—
137	C₄×C₈₉	C₄×Q₁₆	1	2	yes	—
138	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
139	C₄×C₈₉	#568	1	1	yes	characteristic
140	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
141	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
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	D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
155	C₈₉	#18068	1	1	yes	characteristic, complemented, Sylow
156	#533	C₂×D₈₉	1	2	yes	—
157	#500	Dic₈₉	1	1	yes	characteristic
158	#494	Dic₈₉	1	1	yes	characteristic
159	#487	Dic₈₉	1	2	yes	—
160	#467	Dic₈₉	1	2	yes	—
161	#467	Dic₈₉	1	2	yes	—
162	#446	—	89	4	no	—
163	#446	—	178	4	no	—
164	C₈×Q₈	—	89	4	no	—
165	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
166	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
167	#431	—	89	1	no	—
168	#424	—	178	2	no	—
169	C₂×C₄⋊C₈	—	89	2	no	—
170	#405	—	178	2	no	—
171	C₄×M₁₆	—	89	1	no	—
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	—	89	2	no	—
179	C₈²	—	89	2	no	—
180	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
181	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
182	C₂×M₁₆	—	89	1	no	—
183	C₂×M₁₆	—	89	1	no	—
184	C₂²×C₈	—	178	2	no	—
185	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
187	C₄⋊Q₈	C₂×Dic₈₉	1	2	yes	—
188	C₄⋊Q₈	C₂×Dic₈₉	1	2	yes	—
189	#125	—	2	2	no	—
190	C₂²⋊Q₈	—	2	2	no	—
191	C₂²⋊Q₈	C₂×Dic₈₉	1	4	yes	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	C₂²×D₈₉	1	4	yes	—
194	#117	—	2	2	no	—
195	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
196	#107	C₂×Dic₈₉	1	2	yes	—
197	#107	C₂×Dic₈₉	1	2	yes	—
198	C₄⋊C₈	—	89	2	no	—
199	C₄⋊C₈	—	89	2	no	—
200	C₄⋊C₈	—	178	4	no	—
201	C₄⋊C₈	—	178	4	no	—
202	C₄⋊C₈	—	178	2	no	—
203	C₄⋊C₈	—	178	1	no	—
204	#103	C₂×Dic₈₉	1	4	yes	—
205	#103	C₂×Dic₈₉	1	4	yes	—
206	C₂²⋊C₈	—	178	1	no	—
207	#97	—	89	2	no	—
208	#97	—	178	4	no	—
209	C₄×C₈	—	89	2	no	—
210	C₄×C₈	—	89	2	no	—
211	C₄×C₈	—	178	4	no	—
212	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
213	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	C₂²×Dic₈₉	1	4	yes	—
218	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
220	C₂²×C₄	#13083573	1	1	yes	characteristic
221	Q₁₆	—	2	4	no	—
222	Q₁₆	—	4	2	no	—
223	M₁₆	—	89	4	no	—
224	M₁₆	—	178	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	89	2	no	—
228	C₂×C₈	—	89	2	no	—
229	C₂×C₈	—	356	4	no	—
230	C₂×C₈	—	178	2	no	—
231	C₂×C₈	—	178	2	no	—
232	C₂×C₈	—	178	2	no	—
233	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
235	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
236	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	C₂²×Dic₈₉	1	4	yes	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
246	C₄²	D₄×D₈₉	1	1	yes	characteristic
247	C₄²	#13083573	1	1	yes	characteristic
248	C₄²	C₂³×D₈₉	1	1	yes	characteristic
249	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	2	4	no	—
252	Q₈	—	8	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
259	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
260	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
261	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
262	C₂×C₄	#15668035	1	2	yes	—
263	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
265	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	89	4	no	—
270	C₈	—	356	2	no	—
271	C₈	—	356	2	no	—
272	C₈	—	356	2	no	—
273	C₈	—	178	2	no	—
274	C₈	#15668019	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	Q₁₆×Dic₈₉	1	2	yes	—
278	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	4	no	—
284	C₄	Q₁₆×Dic₈₉	1	2	yes	—
285	C₄	C₂×#15668019	1	1	yes	characteristic
286	C₄	#19152365	1	1	yes	characteristic
287	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
288	C₄	C₂×#15668035	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#26028996	1	1	yes	characteristic
291	C₂	C₂×Q₁₆×Dic₈₉	1	1	yes	characteristic
292	C₂	#26029169	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×#3170	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#3170	D₈₉
hypocenter	C₈₉	#18068
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #26019149	C₂	semidirect
3. #26019103	C₂	semidirect
4. Q₁₆×#10623494	C₂	semidirect
70. #13083558	Q₁₆	semidirect
155. C₈₉	#18068	semidirect

Maximal subgroups

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

subgroup	index
#26029201	2
#26019149	2
#26019103	2
Q₁₆×#10623494	2
C₂×#19151351	2
#26018655	2
C₈×#13083558	2
C₈₉×#3170	2
#18068	89