Subgroups — #36460310

Order 22784.

830 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151622	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#13083558	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151367	C₂	1	4	yes	maximal
4	C₂×#382×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₂²	1	2	yes	—
6	C₂²×#13083558	C₂²	1	2	yes	—
7	#19151622	—	2	2	no	—
8	#19151622	C₄	1	4	yes	complemented
9	C₂×C₄⋊#10623494	—	2	2	no	—
10	C₄×#13083558	—	2	2	no	—
11	C₄×#13083558	C₄	1	4	yes	complemented
12	C₂×#15667923	—	2	1	no	—
13	C₂×C₄×#10623494	—	2	1	no	—
14	#19151367	—	2	8	no	—
15	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
16	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	#382×C₈₉	—	2	2	no	—
18	#382×C₈₉	—	2	1	no	—
19	#382×C₈₉	—	2	1	no	—
20	C₂×#95×C₈₉	C₄	1	4	yes	—
21	C₂×#13083558	—	2	2	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	4	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	—	2	2	no	—
26	C₂×#13083558	—	2	4	no	—
27	C₂×#13083558	C₂×C₄	1	4	yes	—
28	C₂×#13083558	C₂×C₄	1	4	yes	—
29	C₂×#13083558	D₄	1	4	yes	—
30	C₂×#13083558	Q₈	1	4	yes	—
31	C₂²×#10623494	—	2	2	no	—
32	C₂²×#10623494	—	2	2	no	—
33	C₄⋊#10623494	—	2	8	no	—
34	#15667923	—	2	4	no	—
35	C₄×#10623494	—	2	4	no	—
36	C₂³×C₄×C₈₉	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
40	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
43	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₈₉	D₄	1	2	yes	—
46	C₂×C₄²×C₈₉	D₄	1	2	yes	—
47	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
50	#95×C₈₉	—	2	4	no	—
51	#95×C₈₉	—	2	4	no	—
52	#13083558	—	2	4	no	—
53	#13083558	—	2	4	no	—
54	#13083558	—	2	4	no	—
55	#13083558	—	2	8	no	—
56	#13083558	—	4	4	no	—
57	#13083558	C₄⋊C₄	1	8	yes	—
58	C₂×#10623494	—	2	4	no	—
59	C₂×#10623494	—	2	4	no	—
60	C₂×#10623494	—	2	4	no	—
61	C₂×#10623494	—	2	4	no	—
62	C₂×#10623494	—	4	2	no	—
63	C₂×#10623494	—	4	2	no	—
64	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	—	2	4	no	—
66	C₂²×C₄×C₈₉	—	2	8	no	—
67	C₂²×C₄×C₈₉	—	4	4	no	—
68	C₂²×C₄×C₈₉	—	4	8	no	—
69	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
70	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₈₉	#41	1	2	yes	—
82	C₂²×C₄×C₈₉	#41	1	2	yes	—
83	C₄⋊C₄×C₈₉	—	4	4	no	—
84	C₄⋊C₄×C₈₉	—	4	2	no	—
85	C₄⋊C₄×C₈₉	—	4	2	no	—
86	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₈₉	C₂²⋊C₄	1	4	yes	—
89	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
90	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
91	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
92	C₂³×C₈₉	—	2	4	no	—
93	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
94	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₈₉	C₄×D₄	1	2	yes	—
97	C₂³×C₈₉	C₄×Q₈	1	2	yes	—
98	C₂×C₄×C₈₉	—	2	4	no	—
99	C₂×C₄×C₈₉	—	2	4	no	—
100	C₂×C₄×C₈₉	—	4	4	no	—
101	C₂×C₄×C₈₉	—	4	4	no	—
102	C₂×C₄×C₈₉	—	4	4	no	—
103	C₂×C₄×C₈₉	—	4	8	no	—
104	C₂×C₄×C₈₉	—	8	8	no	—
105	C₂×C₄×C₈₉	#95	1	4	yes	—
106	C₂×C₄×C₈₉	#95	1	4	yes	—
107	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#117	1	2	yes	—
113	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₈₉	#121	1	2	yes	—
116	C₂×C₄×C₈₉	#121	1	2	yes	—
117	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
120	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#125	1	2	yes	—
122	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
125	#10623494	—	2	8	no	—
126	#10623494	—	4	4	no	—
127	C₂²×C₈₉	—	2	4	no	—
128	C₂²×C₈₉	—	2	2	no	—
129	C₂²×C₈₉	—	2	2	no	—
130	C₂²×C₈₉	—	2	4	no	—
131	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
132	C₂²×C₈₉	C₄×C₄⋊C₄	1	2	yes	—
133	C₂²×C₈₉	#389	1	2	yes	—
134	C₂²×C₈₉	#390	1	1	yes	characteristic
135	C₂²×C₈₉	#391	1	1	yes	characteristic
136	C₂²×C₈₉	#413	1	4	yes	—
137	C₄×C₈₉	—	2	4	no	—
138	C₄×C₈₉	—	2	4	no	—
139	C₄×C₈₉	—	8	4	no	—
140	C₄×C₈₉	#380	1	2	yes	—
141	C₄×C₈₉	#382	1	1	yes	characteristic
142	C₄×C₈₉	#384	1	1	yes	characteristic
143	C₄×C₈₉	#389	1	2	yes	—
144	C₄×C₈₉	#390	1	1	yes	characteristic
145	C₄×C₈₉	#391	1	1	yes	characteristic
146	#10858	—	89	1	no	maximal, Sylow
147	C₂×C₈₉	—	2	2	no	—
148	C₂×C₈₉	—	2	2	no	—
149	C₂×C₈₉	C₄⋊#95	1	1	yes	characteristic
150	C₂×C₈₉	#1742	1	2	yes	—
151	C₂×C₈₉	#1967	1	2	yes	—
152	C₂×C₈₉	#1998	1	2	yes	—
153	C₂×#423	—	89	1	no	—
154	C₂×C₄×M₁₆	—	89	1	no	—
155	C₂×#382	D₈₉	1	1	yes	characteristic
156	C₂×#343	—	89	4	no	—
157	C₈₉	#10858	1	1	yes	characteristic, complemented, Sylow
158	C₂²×M₁₆	—	89	2	no	—
159	C₂²×M₁₆	—	89	2	no	—
160	C₂²×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
161	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
162	#423	—	89	4	no	—
163	#423	—	178	2	no	—
164	C₂×C₄⋊C₈	—	178	2	no	—
165	C₄×M₁₆	—	89	4	no	—
166	C₄×M₁₆	—	178	2	no	—
167	C₂×#97	—	178	1	no	—
168	C₂×C₄×C₈	—	178	1	no	—
169	#382	—	2	1	no	—
170	#382	—	2	1	no	—
171	#382	—	2	2	no	—
172	C₂×#95	Dic₈₉	1	4	yes	—
173	#343	—	178	8	no	—
174	C₂³×C₄	C₂×Dic₈₉	1	4	yes	—
175	C₂³×C₄	C₂×Dic₈₉	1	2	yes	—
176	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
177	C₂×M₁₆	—	89	4	no	—
178	C₂×M₁₆	—	89	4	no	—
179	C₂×M₁₆	—	89	4	no	—
180	C₂×M₁₆	—	89	4	no	—
181	C₂×M₁₆	—	178	4	no	—
182	C₂×M₁₆	—	178	4	no	—
183	C₂×M₁₆	—	178	2	no	—
184	C₂×M₁₆	—	178	2	no	—
185	C₂×M₁₆	—	178	2	no	—
186	C₂×M₁₆	—	178	2	no	—
187	C₂²×C₈	—	178	2	no	—
188	C₂²×C₈	—	178	2	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄²	Dic₁₇₈	1	2	yes	—
194	C₂×C₄²	C₄×D₈₉	1	2	yes	—
195	C₂×C₄²	C₄×D₈₉	1	2	yes	—
196	C₂×C₄²	D₃₅₆	1	2	yes	—
197	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
200	C₄⋊C₈	—	178	8	no	—
201	#97	—	178	4	no	—
202	C₄×C₈	—	178	4	no	—
203	#95	—	2	4	no	—
204	#95	—	2	4	no	—
205	C₂⁴	C₂²×Dic₈₉	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	4	8	no	—
210	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
211	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
212	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
213	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
214	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₈₉	1	2	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
220	C₂²×C₄	Q₈×D₈₉	1	2	yes	—
221	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
222	C₂²×C₄	#13083573	1	2	yes	—
223	C₂²×C₄	#13083575	1	2	yes	—
224	M₁₆	—	89	8	no	—
225	M₁₆	—	356	4	no	—
226	M₁₆	—	178	8	no	—
227	M₁₆	—	178	4	no	—
228	M₁₆	—	178	4	no	—
229	M₁₆	—	178	4	no	—
230	C₂×C₈	—	356	2	no	—
231	C₂×C₈	—	356	2	no	—
232	C₂×C₈	—	178	4	no	—
233	C₂×C₈	—	178	4	no	—
234	C₂×C₈	—	178	4	no	—
235	C₂×C₈	—	178	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄²	#13083551	1	2	yes	—
240	C₄²	#13083551	1	2	yes	—
241	C₄²	#13083551	1	4	yes	—
242	C₄²	#13083553	1	4	yes	—
243	C₄²	#13083553	1	2	yes	—
244	C₄²	#13083553	1	2	yes	—
245	C₂³	—	2	4	no	—
246	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
247	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
248	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
249	C₂³	D₄×Dic₈₉	1	2	yes	—
250	C₂³	Q₈×Dic₈₉	1	2	yes	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	8	8	no	—
258	C₂×C₄	#15667919	1	4	yes	—
259	C₂×C₄	#15667919	1	4	yes	—
260	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
261	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
262	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
263	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
264	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
265	C₂×C₄	#15667970	1	2	yes	—
266	C₂×C₄	#15667974	1	2	yes	—
267	C₂×C₄	#15667977	1	2	yes	—
268	C₂×C₄	#15667983	1	2	yes	—
269	C₂×C₄	#15667987	1	2	yes	—
270	C₂×C₄	#15667988	1	2	yes	—
271	C₂×C₄	#15667992	1	2	yes	—
272	C₂×C₄	#15667999	1	1	yes	characteristic
273	C₂×C₄	#15668001	1	1	yes	characteristic
274	C₂×C₄	#15668003	1	2	yes	—
275	C₂×C₄	#15668008	1	1	yes	characteristic
276	C₂×C₄	#15668010	1	1	yes	characteristic
277	C₂×C₄	#15668011	1	2	yes	—
278	C₈	—	356	4	no	—
279	C₈	—	178	8	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	C₂×#15667919	1	1	yes	characteristic
285	C₂²	C₄⋊C₄×Dic₈₉	1	2	yes	—
286	C₂²	#19151497	1	2	yes	—
287	C₂²	#19151501	1	1	yes	characteristic
288	C₂²	#19151505	1	1	yes	characteristic
289	C₂²	#19151582	1	4	yes	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	8	4	no	—
293	C₄	#19151455	1	2	yes	—
294	C₄	#19151463	1	1	yes	characteristic
295	C₄	#19151467	1	1	yes	characteristic
296	C₄	#19151494	1	2	yes	—
297	C₄	#19151499	1	1	yes	characteristic
298	C₄	#19151503	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	2	2	no	—
301	C₂	#26018481	1	1	yes	characteristic
302	C₂	#26019991	1	2	yes	—
303	C₂	#26021102	1	2	yes	—
304	C₂	#26021246	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₈₉
commutator	C₂³×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂×#382×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂×#382	D₈₉
hypocenter	C₈₉	#10858
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#382

Complements

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

N	H ≅ G/N	type
8. #19151622	C₄	semidirect
11. C₄×#13083558	C₄	semidirect
157. C₈₉	#10858	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151622	2
C₂×C₄×#13083558	2
C₂×#19151367	2
C₂×#382×C₈₉	2
#10858	89