Subgroups — C₂²×#19151499 #36712779

Order 22784.

9534 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₃₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂³×#13083551	C₂	1	4	yes	maximal, complemented
3	C₂³×C₄×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151499	C₂	1	24	yes	maximal, direct product factor
5	C₂³×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁵×D₈₉	C₂²	1	2	yes	—
7	C₂³×D₃₅₆	—	2	8	no	—
8	C₂³×D₃₅₆	—	2	8	no	—
9	C₂³×D₃₅₆	C₄	1	16	yes	complemented
10	C₂³×D₃₅₆	C₂²	1	12	yes	—
11	C₂⁴×Dic₈₉	C₂²	1	2	yes	—
12	C₂²×#13083551	—	2	32	no	—
13	C₂²×#13083551	C₂²	1	48	yes	complemented
14	C₂²×C₄×Dic₈₉	—	2	8	no	—
15	C₂²×C₄×Dic₈₉	C₂²	1	12	yes	complemented
16	#19151499	C₂²	1	64	yes	direct product factor
17	C₂⁴×C₄×C₈₉	C₂²	1	2	yes	—
18	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₈₉	—	2	8	no	—
20	C₂²×C₄⋊C₄×C₈₉	C₂²	1	12	yes	complemented
21	C₂⁴×D₈₉	—	2	8	no	—
22	C₂⁴×D₈₉	—	2	8	no	—
23	C₂⁴×D₈₉	—	2	16	no	—
24	C₂⁴×D₈₉	—	2	16	no	—
25	C₂⁴×D₈₉	C₂×C₄	1	16	yes	—
26	C₂⁴×D₈₉	C₂³	1	12	yes	—
27	C₂²×D₃₅₆	—	2	128	no	—
28	C₂²×D₃₅₆	—	2	48	no	—
29	C₂²×D₃₅₆	—	2	48	no	—
30	C₂²×D₃₅₆	C₂×C₄	1	96	yes	complemented
31	C₂²×D₃₅₆	C₂³	1	16	yes	—
32	C₂³×Dic₈₉	—	2	8	no	—
33	C₂³×Dic₈₉	—	2	8	no	—
34	C₂³×Dic₈₉	D₄	1	16	yes	complemented
35	C₂³×Dic₈₉	C₂³	1	12	yes	—
36	C₂×#13083551	—	2	192	no	—
37	C₂×#13083551	C₂³	1	64	yes	complemented
38	C₂×C₄×Dic₈₉	—	2	48	no	—
39	C₂×C₄×Dic₈₉	C₂³	1	16	yes	complemented
40	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₈₉	—	2	8	no	—
42	C₂³×C₄×C₈₉	—	2	16	no	—
43	C₂³×C₄×C₈₉	C₂×C₄	1	8	yes	—
44	C₂³×C₄×C₈₉	D₄	1	8	yes	—
45	C₂³×C₄×C₈₉	D₄	1	8	yes	—
46	C₂³×C₄×C₈₉	C₂³	1	12	yes	—
47	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
48	C₂×C₄⋊C₄×C₈₉	—	2	48	no	—
49	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	complemented
50	C₂³×D₈₉	—	2	64	no	—
51	C₂³×D₈₉	—	2	128	no	—
52	C₂³×D₈₉	—	2	24	no	—
53	C₂³×D₈₉	—	2	24	no	—
54	C₂³×D₈₉	—	2	48	no	—
55	C₂³×D₈₉	—	2	48	no	—
56	C₂³×D₈₉	—	4	64	no	—
57	C₂³×D₈₉	—	4	32	no	—
58	C₂³×D₈₉	—	4	32	no	—
59	C₂³×D₈₉	C₂²×C₄	1	48	yes	—
60	C₂³×D₈₉	C₂⁴	1	8	yes	—
61	C₂×D₃₅₆	—	2	384	no	—
62	C₂×D₃₅₆	—	2	32	no	—
63	C₂×D₃₅₆	—	2	32	no	—
64	C₂×D₃₅₆	C₂²×C₄	1	64	yes	complemented
65	C₂²×Dic₈₉	—	2	64	no	—
66	C₂²×Dic₈₉	—	2	24	no	—
67	C₂²×Dic₈₉	—	2	24	no	—
68	C₂²×Dic₈₉	C₂×D₄	1	48	yes	complemented
69	C₂²×Dic₈₉	C₂⁴	1	8	yes	—
70	#13083551	—	2	128	no	—
71	C₄×Dic₈₉	—	2	32	no	—
72	C₂⁴×C₈₉	C₂²×C₄	1	4	yes	—
73	C₂⁴×C₈₉	C₂×D₄	1	4	yes	—
74	C₂⁴×C₈₉	C₂×D₄	1	4	yes	—
75	C₂⁴×C₈₉	C₂×D₄	1	8	yes	—
76	C₂⁴×C₈₉	#41	1	8	yes	—
77	C₂⁴×C₈₉	C₂⁴	1	3	yes	—
78	C₂²×C₄×C₈₉	—	2	24	no	—
79	C₂²×C₄×C₈₉	—	2	48	no	—
80	C₂²×C₄×C₈₉	—	4	32	no	—
81	C₂²×C₄×C₈₉	C₂²⋊C₄	1	64	yes	complemented
82	C₂²×C₄×C₈₉	C₂²×C₄	1	24	yes	—
83	C₂²×C₄×C₈₉	C₂×D₄	1	24	yes	—
84	C₂²×C₄×C₈₉	C₂×D₄	1	24	yes	—
85	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
86	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
87	C₄⋊C₄×C₈₉	—	2	32	no	—
88	C₂³×C₈₉	C₂×C₂²⋊C₄	1	16	yes	—
89	C₂³×C₈₉	C₄×D₄	1	32	yes	—
90	C₂³×C₈₉	#121	1	32	yes	—
91	C₂³×C₈₉	#124	1	16	yes	—
92	C₂³×C₈₉	#127	1	16	yes	—
93	C₂³×C₈₉	C₂³×C₄	1	6	yes	—
94	C₂³×C₈₉	C₂²×D₄	1	12	yes	—
95	C₂³×C₈₉	C₂²×D₄	1	6	yes	—
96	C₂³×C₈₉	C₂²×D₄	1	6	yes	—
97	C₂³×C₈₉	C₂×#41	1	12	yes	—
98	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
99	C₂²×D₈₉	—	2	96	no	—
100	C₂²×D₈₉	—	2	192	no	—
101	C₂²×D₈₉	—	2	16	no	—
102	C₂²×D₈₉	—	2	8	no	—
103	C₂²×D₈₉	—	2	8	no	—
104	C₂²×D₈₉	—	2	16	no	—
105	C₂²×D₈₉	—	4	96	no	—
106	C₂²×D₈₉	—	4	48	no	—
107	C₂²×D₈₉	—	4	48	no	—
108	C₂²×D₈₉	—	4	256	no	—
109	C₂²×D₈₉	C₂³×C₄	1	16	yes	—
110	C₂×C₄×C₈₉	—	2	8	no	—
111	C₂×C₄×C₈₉	—	2	16	no	—
112	C₂×C₄×C₈₉	—	4	48	no	—
113	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	96	yes	complemented
114	C₂×C₄×C₈₉	C₂³×C₄	1	8	yes	—
115	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
116	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
117	C₂×Dic₈₉	—	2	96	no	—
118	C₂×Dic₈₉	—	2	8	no	—
119	C₂×Dic₈₉	—	2	8	no	—
120	C₂×Dic₈₉	C₂²×D₄	1	16	yes	complemented
121	D₃₅₆	—	2	128	no	—
122	C₂²×C₈₉	#390	1	64	yes	complemented
123	C₂²×C₈₉	C₂²×C₂²⋊C₄	1	12	yes	—
124	C₂²×C₈₉	C₂×C₄×D₄	1	24	yes	—
125	C₂²×C₈₉	C₂×#121	1	24	yes	—
126	C₂²×C₈₉	C₂×#124	1	12	yes	—
127	C₂²×C₈₉	C₂×#127	1	12	yes	—
128	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
129	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
130	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
131	C₂²×C₈₉	C₂³×D₄	1	2	yes	—
132	C₂²×C₈₉	C₂²×#41	1	2	yes	—
133	C₂×D₈₉	—	2	16	no	—
134	C₂×D₈₉	—	2	32	no	—
135	C₂×D₈₉	—	4	16	no	—
136	C₂×D₈₉	—	4	8	no	—
137	C₂×D₈₉	—	4	8	no	—
138	C₂×D₈₉	—	4	192	no	—
139	C₄×C₈₉	—	4	8	no	—
140	C₄×C₈₉	C₂²×C₂²⋊C₄	1	16	yes	complemented
141	Dic₈₉	—	2	16	no	—
142	C₂²×#390	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	C₂×#390	1	24	yes	complemented
144	C₂×C₈₉	C₂³×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₈₉	C₂²×C₄×D₄	1	2	yes	—
146	C₂×C₈₉	C₂²×#121	1	2	yes	—
147	C₂×C₈₉	C₂²×#124	1	1	yes	characteristic
148	C₂×C₈₉	C₂²×#127	1	1	yes	characteristic
149	D₈₉	—	4	16	no	—
150	C₂⁴×D₄	—	89	1	no	—
151	C₂³×C₄⋊C₄	D₈₉	1	1	yes	characteristic, complemented
152	C₂³×C₂²⋊C₄	—	89	4	no	—
153	C₂³×C₄²	—	89	1	no	—
154	C₂×#390	—	89	24	no	—
155	C₈₉	C₂²×#390	1	1	yes	characteristic, complemented, Sylow
156	C₂⁶	—	89	2	no	—
157	C₂³×D₄	—	89	16	no	—
158	C₂³×D₄	—	89	12	no	—
159	C₂³×D₄	—	178	8	no	—
160	C₂³×D₄	—	178	8	no	—
161	C₂⁴×C₄	—	89	2	no	—
162	C₂⁴×C₄	C₂×D₈₉	1	2	yes	—
163	C₂⁴×C₄	C₂×D₈₉	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	—	2	8	no	—
165	C₂²×C₄⋊C₄	C₂×D₈₉	1	12	yes	complemented
166	C₂²×C₂²⋊C₄	—	89	48	no	—
167	C₂²×C₂²⋊C₄	—	178	32	no	—
168	C₂²×C₄²	—	89	12	no	—
169	C₂²×C₄²	—	178	8	no	—
170	#390	—	89	64	no	—
171	C₂⁵	—	89	16	no	—
172	C₂⁵	—	89	12	no	—
173	C₂⁵	—	178	16	no	—
174	C₂⁵	—	178	8	no	—
175	C₂⁵	—	178	8	no	—
176	C₂⁵	—	178	16	no	—
177	C₂⁵	C₂²×D₈₉	1	1	yes	characteristic
178	C₂²×D₄	—	89	96	no	—
179	C₂²×D₄	—	89	16	no	—
180	C₂²×D₄	—	178	128	no	—
181	C₂²×D₄	—	178	48	no	—
182	C₂²×D₄	—	178	48	no	—
183	C₂³×C₄	—	2	16	no	—
184	C₂³×C₄	—	2	8	no	—
185	C₂³×C₄	—	89	16	no	—
186	C₂³×C₄	—	89	12	no	—
187	C₂³×C₄	—	178	8	no	—
188	C₂³×C₄	—	178	8	no	—
189	C₂³×C₄	C₄×D₈₉	1	8	yes	—
190	C₂³×C₄	D₃₅₆	1	8	yes	—
191	C₂³×C₄	C₂²⋊D₈₉	1	8	yes	—
192	C₂³×C₄	C₂²×D₈₉	1	12	yes	—
193	C₂³×C₄	C₂²×D₈₉	1	6	yes	—
194	C₂×C₄⋊C₄	—	2	48	no	—
195	C₂×C₄⋊C₄	C₂²×D₈₉	1	16	yes	complemented
196	C₂×C₂²⋊C₄	—	89	64	no	—
197	C₂×C₂²⋊C₄	—	178	192	no	—
198	C₂×C₄²	—	89	16	no	—
199	C₂×C₄²	—	178	48	no	—
200	C₂⁴	—	89	48	no	—
201	C₂⁴	—	89	8	no	—
202	C₂⁴	—	356	64	no	—
203	C₂⁴	—	356	32	no	—
204	C₂⁴	—	356	32	no	—
205	C₂⁴	—	178	128	no	—
206	C₂⁴	—	178	48	no	—
207	C₂⁴	—	178	24	no	—
208	C₂⁴	—	178	24	no	—
209	C₂⁴	—	178	48	no	—
210	C₂⁴	—	178	64	no	—
211	C₂⁴	C₂×C₄×D₈₉	1	4	yes	—
212	C₂⁴	D₄×D₈₉	1	8	yes	—
213	C₂⁴	C₂×C₂²⋊D₈₉	1	4	yes	—
214	C₂⁴	C₂×D₃₅₆	1	4	yes	—
215	C₂⁴	#13083575	1	8	yes	—
216	C₂⁴	C₂³×D₈₉	1	3	yes	—
217	C₂×D₄	—	89	64	no	—
218	C₂×D₄	—	178	32	no	—
219	C₂×D₄	—	178	32	no	—
220	C₂×D₄	—	178	384	no	—
221	C₂²×C₄	—	2	48	no	—
222	C₂²×C₄	—	2	24	no	—
223	C₂²×C₄	—	4	32	no	—
224	C₂²×C₄	—	89	48	no	—
225	C₂²×C₄	—	89	8	no	—
226	C₂²×C₄	—	178	64	no	—
227	C₂²×C₄	—	178	24	no	—
228	C₂²×C₄	—	178	24	no	—
229	C₂²×C₄	#13083551	1	64	yes	complemented
230	C₂²×C₄	C₂×C₄×D₈₉	1	24	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₈₉	1	24	yes	—
232	C₂²×C₄	C₂×D₃₅₆	1	24	yes	—
233	C₂²×C₄	C₂³×D₈₉	1	8	yes	—
234	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
235	C₄⋊C₄	—	2	32	no	—
236	C₂²⋊C₄	—	178	128	no	—
237	C₄²	—	178	32	no	—
238	C₂³	—	89	16	no	—
239	C₂³	—	356	48	no	—
240	C₂³	—	356	256	no	—
241	C₂³	—	356	96	no	—
242	C₂³	—	356	48	no	—
243	C₂³	—	178	16	no	—
244	C₂³	—	178	8	no	—
245	C₂³	—	178	8	no	—
246	C₂³	—	178	16	no	—
247	C₂³	—	178	96	no	—
248	C₂³	—	178	192	no	—
249	C₂³	C₂×#13083551	1	16	yes	—
250	C₂³	#15667974	1	32	yes	—
251	C₂³	#15667987	1	32	yes	—
252	C₂³	#15668001	1	16	yes	—
253	C₂³	#15668008	1	16	yes	—
254	C₂³	C₂²×C₄×D₈₉	1	6	yes	—
255	C₂³	C₂×D₄×D₈₉	1	12	yes	—
256	C₂³	C₂²×C₂²⋊D₈₉	1	6	yes	—
257	C₂³	C₂²×D₃₅₆	1	6	yes	—
258	C₂³	C₂×#13083575	1	12	yes	—
259	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
260	D₄	—	178	128	no	—
261	C₂×C₄	—	2	16	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	4	48	no	—
264	C₂×C₄	—	89	16	no	—
265	C₂×C₄	—	178	8	no	—
266	C₂×C₄	—	178	8	no	—
267	C₂×C₄	—	178	96	no	—
268	C₂×C₄	C₂×#13083551	1	96	yes	complemented
269	C₂×C₄	C₂²×C₄×D₈₉	1	8	yes	—
270	C₂×C₄	C₂²×C₂²⋊D₈₉	1	8	yes	—
271	C₂×C₄	C₂²×D₃₅₆	1	8	yes	—
272	C₂²	—	356	192	no	—
273	C₂²	—	356	16	no	—
274	C₂²	—	356	8	no	—
275	C₂²	—	356	8	no	—
276	C₂²	—	178	16	no	—
277	C₂²	—	178	32	no	—
278	C₂²	#19151499	1	64	yes	direct product factor
279	C₂²	C₂²×#13083551	1	12	yes	—
280	C₂²	C₂×#15667974	1	24	yes	—
281	C₂²	C₂×#15667987	1	24	yes	—
282	C₂²	C₂×#15668001	1	12	yes	—
283	C₂²	C₂×#15668008	1	12	yes	—
284	C₂²	C₂³×C₄×D₈₉	1	1	yes	characteristic
285	C₂²	C₂²×D₄×D₈₉	1	2	yes	—
286	C₂²	C₂³×C₂²⋊D₈₉	1	1	yes	characteristic
287	C₂²	C₂³×D₃₅₆	1	1	yes	characteristic
288	C₂²	C₂²×#13083575	1	2	yes	—
289	C₄	—	4	8	no	—
290	C₄	—	178	16	no	—
291	C₄	C₂²×#13083551	1	16	yes	complemented
292	C₂	—	356	16	no	—
293	C₂	C₂×#19151499	1	24	yes	direct product factor
294	C₂	C₂³×#13083551	1	1	yes	characteristic
295	C₂	C₂²×#15667974	1	2	yes	—
296	C₂	C₂²×#15667987	1	2	yes	—
297	C₂	C₂²×#15668001	1	1	yes	characteristic
298	C₂	C₂²×#15668008	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂³×C₄⋊C₄×C₈₉	C₂
socle	C₂⁵×C₈₉	C₂³
prime radical	C₂⁴×D₃₅₆	C₂
cosocle	C₂³×C₈₉	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₈₉
hypocenter	C₈₉	C₂²×#390
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×#13083551	C₂	semidirect
3. C₂³×C₄×Dic₈₉	C₂	semidirect
4. C₂×#19151499	C₂	direct (N × H)
5. C₂³×C₄⋊C₄×C₈₉	C₂	semidirect
9. C₂³×D₃₅₆	C₄	semidirect
13. C₂²×#13083551	C₂²	semidirect
15. C₂²×C₄×Dic₈₉	C₂²	semidirect
16. #19151499	C₂²	direct (N × H)
20. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
30. C₂²×D₃₅₆	C₂×C₄	semidirect
34. C₂³×Dic₈₉	D₄	semidirect
37. C₂×#13083551	C₂³	semidirect
39. C₂×C₄×Dic₈₉	C₂³	semidirect
49. C₂×C₄⋊C₄×C₈₉	C₂³	semidirect
64. C₂×D₃₅₆	C₂²×C₄	semidirect
68. C₂²×Dic₈₉	C₂×D₄	semidirect
81. C₂²×C₄×C₈₉	C₂²⋊C₄	semidirect
113. C₂×C₄×C₈₉	C₂×C₂²⋊C₄	semidirect
120. C₂×Dic₈₉	C₂²×D₄	semidirect
122. C₂²×C₈₉	#390	semidirect
140. C₄×C₈₉	C₂²×C₂²⋊C₄	semidirect
143. C₂×C₈₉	C₂×#390	semidirect
151. C₂³×C₄⋊C₄	D₈₉	semidirect
155. C₈₉	C₂²×#390	semidirect
165. C₂²×C₄⋊C₄	C₂×D₈₉	semidirect
195. C₂×C₄⋊C₄	C₂²×D₈₉	semidirect
229. C₂²×C₄	#13083551	semidirect
268. C₂×C₄	C₂×#13083551	semidirect
278. C₂²	#19151499	direct (N × H)
291. C₄	C₂²×#13083551	semidirect
293. C₂	C₂×#19151499	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₃₅₆	2
C₂³×#13083551	2
C₂³×C₄×Dic₈₉	2
C₂×#19151499	2
C₂³×C₄⋊C₄×C₈₉	2
C₂²×#390	89

Subgroups — C22×#19151499 #36712779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19151499 #36712779