Subgroups — #36526458

Order 22784.

564 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029354	C₂	1	2	yes	maximal
2	#26019161	C₂	1	2	yes	maximal, complemented
3	#26019103	C₂	1	4	yes	maximal, complemented
4	#26019071	C₂	1	2	yes	maximal, complemented
5	#26019000	C₂	1	1	yes	maximal, characteristic
6	#26018673	C₂	1	2	yes	maximal
7	#26018541	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#3170	C₂	1	1	yes	maximal, characteristic
9	#19151729	—	2	2	no	—
10	#19151729	C₂²	1	4	yes	—
11	Q₈×#10623494	—	2	2	no	—
12	Q₈×#10623494	C₂²	1	4	yes	—
13	#19151659	—	2	4	no	—
14	#19151651	C₂²	1	2	yes	—
15	#19151651	C₂²	1	1	yes	characteristic
16	C₄×#13083558	C₂²	1	1	yes	characteristic
17	#19151351	C₂²	1	2	yes	—
18	#19151349	C₂²	1	1	yes	characteristic
19	#19151349	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	#19151319	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	—	2	1	no	—
34	C₄⋊#10623494	—	2	4	no	—
35	C₄⋊#10623494	—	2	4	no	—
36	C₄⋊#10623494	D₄	1	2	yes	—
37	C₄⋊#10623494	C₂³	1	2	yes	—
38	C₄⋊#10623494	C₂³	1	2	yes	—
39	C₂²⋊#10623494	—	2	2	no	—
40	C₂²⋊#10623494	D₄	1	2	yes	—
41	#15667923	—	2	4	no	—
42	#15667923	—	2	1	no	—
43	C₄×#10623494	—	2	4	no	—
44	C₄×#10623494	—	2	2	no	—
45	C₄×#10623494	C₂³	1	2	yes	—
46	C₄×#10623494	C₂³	1	2	yes	—
47	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
48	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
52	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
53	#125×C₈₉	—	2	2	no	—
54	C₂²⋊Q₈×C₈₉	—	2	2	no	—
55	C₂²⋊Q₈×C₈₉	C₂×C₄	1	4	yes	—
56	C₄×Q₈×C₈₉	—	2	2	no	—
57	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
58	#117×C₈₉	—	2	2	no	—
59	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	#107×C₈₉	C₂×C₄	1	2	yes	—
61	#107×C₈₉	C₂×C₄	1	2	yes	—
62	#103×C₈₉	C₂×C₄	1	4	yes	—
63	#103×C₈₉	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	#13083558	—	4	2	no	—
69	C₂×#10623494	—	2	2	no	—
70	C₂×#10623494	—	2	2	no	—
71	C₂×#10623494	—	2	2	no	—
72	C₂×#10623494	C₂×D₄	1	4	yes	—
73	C₂×Q₈×C₈₉	—	2	2	no	—
74	C₂×Q₈×C₈₉	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
78	Q₁₆×C₈₉	—	2	4	no	—
79	Q₁₆×C₈₉	—	4	2	no	—
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	—	4	1	no	—
82	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₄⋊C₄×C₈₉	—	2	2	no	—
87	C₄⋊C₄×C₈₉	—	2	2	no	—
88	C₄⋊C₄×C₈₉	—	2	2	no	—
89	C₄⋊C₄×C₈₉	—	2	4	no	—
90	C₄⋊C₄×C₈₉	—	2	4	no	—
91	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
92	C₂²⋊C₄×C₈₉	—	2	4	no	—
93	C₄²×C₈₉	—	2	4	no	—
94	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₈₉	#41	1	1	yes	characteristic
97	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
98	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
99	Q₈×C₈₉	—	2	4	no	—
100	Q₈×C₈₉	—	2	4	no	—
101	Q₈×C₈₉	—	4	4	no	—
102	C₂×C₄×C₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	2	4	no	—
107	C₂×C₄×C₈₉	—	2	4	no	—
108	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
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	C₈×C₈₉	—	2	1	no	—
115	C₈×C₈₉	—	2	1	no	—
116	C₈×C₈₉	—	2	1	no	—
117	C₈×C₈₉	#131	1	2	yes	—
118	#10623494	—	2	4	no	—
119	#10623494	—	4	2	no	—
120	#10623494	—	4	2	no	—
121	#10623494	—	4	2	no	—
122	C₂²×C₈₉	—	2	1	no	—
123	C₂²×C₈₉	—	2	1	no	—
124	C₂²×C₈₉	—	2	1	no	—
125	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₈₉	—	2	1	no	—
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	—	4	4	no	—
130	C₄×C₈₉	—	4	4	no	—
131	C₄×C₈₉	#444	1	2	yes	—
132	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
133	C₄×C₈₉	#568	1	1	yes	characteristic
134	C₄×C₈₉	#578	1	2	yes	—
135	#18090	—	89	1	no	maximal, Sylow
136	C₂×C₈₉	—	4	1	no	—
137	C₂×C₈₉	#2935	1	1	yes	characteristic
138	C₂×C₈₉	#2950	1	1	yes	characteristic
139	C₂×C₈₉	C₂×#444	1	1	yes	characteristic
140	#3170	D₈₉	1	1	yes	characteristic
141	#2986	—	89	2	no	—
142	#1593	—	89	2	no	—
143	#1584	—	89	4	no	—
144	#1579	—	89	2	no	—
145	#1564	—	89	1	no	—
146	#1485	—	89	2	no	—
147	#1451	—	89	1	no	—
148	C₈₉	#18090	1	1	yes	characteristic, complemented, Sylow
149	#533	C₂×D₈₉	1	2	yes	—
150	#500	Dic₈₉	1	1	yes	characteristic
151	#494	Dic₈₉	1	1	yes	characteristic
152	#487	Dic₈₉	1	2	yes	—
153	#467	Dic₈₉	1	2	yes	—
154	#467	Dic₈₉	1	2	yes	—
155	#446	—	89	4	no	—
156	#446	—	178	2	no	—
157	C₈×Q₈	—	89	4	no	—
158	C₈×Q₈	—	178	2	no	—
159	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
160	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
161	#433	—	178	4	no	—
162	#431	—	89	1	no	—
163	#431	—	89	2	no	—
164	C₄×M₁₆	—	89	1	no	—
165	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
166	#335	—	89	2	no	—
167	#334	—	89	1	no	—
168	#334	—	89	1	no	—
169	Q₈⋊C₈	—	89	4	no	—
170	Q₈⋊C₈	—	89	4	no	—
171	#322	—	89	2	no	—
172	#322	—	89	2	no	—
173	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
174	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
175	C₂×M₁₆	—	178	1	no	—
176	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
178	C₄⋊Q₈	C₂×Dic₈₉	1	2	yes	—
179	C₄⋊Q₈	C₂×Dic₈₉	1	2	yes	—
180	#125	—	2	2	no	—
181	C₂²⋊Q₈	—	2	2	no	—
182	C₂²⋊Q₈	C₂×Dic₈₉	1	4	yes	—
183	C₄×Q₈	—	2	2	no	—
184	C₄×Q₈	C₂²×D₈₉	1	4	yes	—
185	#117	—	2	2	no	—
186	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
187	#107	C₂×Dic₈₉	1	2	yes	—
188	#107	C₂×Dic₈₉	1	2	yes	—
189	C₄⋊C₈	—	89	2	no	—
190	C₄⋊C₈	—	89	2	no	—
191	C₄⋊C₈	—	89	2	no	—
192	C₄⋊C₈	—	178	4	no	—
193	C₄⋊C₈	—	178	4	no	—
194	#103	C₂×Dic₈₉	1	4	yes	—
195	#103	C₂×Dic₈₉	1	4	yes	—
196	C₂²⋊C₈	—	89	2	no	—
197	C₂²⋊C₈	—	178	2	no	—
198	#97	—	178	4	no	—
199	#97	—	178	1	no	—
200	C₄×C₈	—	89	2	no	—
201	C₄×C₈	—	89	2	no	—
202	C₄×C₈	—	178	4	no	—
203	C₄×C₈	—	178	2	no	—
204	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
205	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	C₂²×Dic₈₉	1	4	yes	—
210	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
211	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
212	C₂²×C₄	#13083573	1	1	yes	characteristic
213	Q₁₆	—	2	4	no	—
214	Q₁₆	—	4	2	no	—
215	M₁₆	—	356	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	89	4	no	—
219	C₂×C₈	—	178	2	no	—
220	C₂×C₈	—	178	2	no	—
221	C₂×C₈	—	178	2	no	—
222	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
223	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
224	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
225	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	C₂²×Dic₈₉	1	4	yes	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
235	C₄²	D₄×D₈₉	1	1	yes	characteristic
236	C₄²	#13083573	1	1	yes	characteristic
237	C₄²	C₂³×D₈₉	1	1	yes	characteristic
238	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
239	Q₈	—	2	4	no	—
240	Q₈	—	2	4	no	—
241	Q₈	—	4	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
249	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
250	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
251	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
252	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
253	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	356	2	no	—
258	C₈	—	356	2	no	—
259	C₈	—	356	2	no	—
260	C₈	—	178	4	no	—
261	C₈	#15668019	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	4	no	—
271	C₄	#19151720	1	2	yes	—
272	C₄	C₂×#15668019	1	1	yes	characteristic
273	C₄	#19152365	1	1	yes	characteristic
274	C₄	#19152411	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	#26028996	1	1	yes	characteristic
277	C₂	#26029120	1	1	yes	characteristic
278	C₂	C₂×#19151720	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	#18090
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. #26019161	C₂	semidirect
3. #26019103	C₂	semidirect
4. #26019071	C₂	semidirect
148. C₈₉	#18090	semidirect

Maximal subgroups

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

subgroup	index
#26029354	2
#26019161	2
#26019103	2
#26019071	2
#26019000	2
#26018673	2
#26018541	2
C₈₉×#3170	2
#18090	89