Subgroups — C₂×#26028539 #36760265

Order 22784.

7502 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667991	C₂	1	3	yes	maximal, complemented
2	#26028539	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19151471	C₂	1	6	yes	maximal, complemented
4	C₂×#19151458	C₂	1	3	yes	maximal, complemented
5	C₂×#19151454	C₂	1	2	yes	maximal
6	C₂⁵×C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₇₈	C₂²	1	1	yes	characteristic, complemented
8	C₂×#15667991	—	2	24	no	—
9	C₂×#15667991	C₂²	1	24	yes	complemented
10	C₂²×C₄⋊Dic₈₉	C₂²	1	3	yes	complemented
11	C₂²×#13083553	C₂²	1	6	yes	complemented
12	C₂²×C₂²⋊Dic₈₉	C₂²	1	6	yes	—
13	C₂²×C₂²⋊Dic₈₉	C₂²	1	6	yes	—
14	#19151471	C₂²	1	48	yes	complemented
15	#19151458	C₂²	1	24	yes	complemented
16	#19151454	C₂²	1	16	yes	—
17	C₂×#15667919	C₂²	1	6	yes	complemented
18	C₂⁶×C₈₉	C₂²	1	1	yes	characteristic
19	C₂⁴×C₄×C₈₉	—	2	24	no	—
20	C₂⁴×C₄×C₈₉	C₂²	1	3	yes	—
21	C₂⁴×C₄×C₈₉	C₂²	1	8	yes	—
22	C₂⁴×C₄×C₈₉	C₂²	1	3	yes	—
23	C₂²×Dic₁₇₈	—	2	12	no	—
24	C₂²×Dic₁₇₈	C₂³	1	4	yes	complemented
25	C₂³×Dic₈₉	C₂³	1	2	yes	—
26	C₂³×Dic₈₉	C₂³	1	6	yes	—
27	#15667991	—	2	96	no	—
28	C₂×C₄⋊Dic₈₉	—	2	12	no	—
29	C₂×C₄⋊Dic₈₉	C₂³	1	12	yes	complemented
30	C₂×#13083553	—	2	24	no	—
31	C₂×#13083553	C₂³	1	24	yes	complemented
32	C₂×C₂²⋊Dic₈₉	—	2	24	no	—
33	C₂×C₂²⋊Dic₈₉	—	2	24	no	—
34	C₂×C₂²⋊Dic₈₉	C₂³	1	24	yes	—
35	C₂×C₂²⋊Dic₈₉	C₂³	1	24	yes	—
36	#15667919	C₂³	1	24	yes	complemented
37	C₂⁵×C₈₉	—	2	12	no	—
38	C₂⁵×C₈₉	—	2	12	no	—
39	C₂⁵×C₈₉	Q₈	1	8	yes	complemented
40	C₂⁵×C₈₉	C₂³	1	4	yes	—
41	C₂⁵×C₈₉	C₂³	1	3	yes	—
42	C₂³×C₄×C₈₉	—	2	128	no	—
43	C₂³×C₄×C₈₉	—	2	12	no	—
44	C₂³×C₄×C₈₉	—	2	96	no	—
45	C₂³×C₄×C₈₉	—	2	24	no	—
46	C₂³×C₄×C₈₉	—	2	24	no	—
47	C₂³×C₄×C₈₉	D₄	1	24	yes	—
48	C₂³×C₄×C₈₉	C₂³	1	12	yes	—
49	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
50	C₂³×C₄×C₈₉	C₂³	1	12	yes	—
51	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
52	C₂×Dic₁₇₈	—	2	24	no	—
53	C₂×Dic₁₇₈	—	4	16	no	—
54	C₂²×Dic₈₉	—	2	12	no	—
55	C₂²×Dic₈₉	—	2	12	no	—
56	C₂²×Dic₈₉	—	2	24	no	—
57	C₂²×Dic₈₉	C₂⁴	1	4	yes	—
58	C₂²×Dic₈₉	C₂⁴	1	12	yes	—
59	C₄⋊Dic₈₉	—	2	24	no	—
60	#13083553	—	2	48	no	—
61	C₂²⋊Dic₈₉	—	2	48	no	—
62	C₂²⋊Dic₈₉	—	2	48	no	—
63	C₂⁴×C₈₉	—	2	32	no	—
64	C₂⁴×C₈₉	—	2	96	no	—
65	C₂⁴×C₈₉	—	2	96	no	—
66	C₂⁴×C₈₉	—	2	24	no	—
67	C₂⁴×C₈₉	—	2	12	no	—
68	C₂⁴×C₈₉	—	2	24	no	—
69	C₂⁴×C₈₉	—	2	12	no	—
70	C₂⁴×C₈₉	C₂×D₄	1	12	yes	—
71	C₂⁴×C₈₉	C₂×Q₈	1	12	yes	—
72	C₂⁴×C₈₉	C₂×Q₈	1	16	yes	complemented
73	C₂⁴×C₈₉	#41	1	12	yes	—
74	C₂⁴×C₈₉	C₂⁴	1	6	yes	—
75	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	—	2	96	no	—
77	C₂²×C₄×C₈₉	—	2	256	no	—
78	C₂²×C₄×C₈₉	—	2	24	no	—
79	C₂²×C₄×C₈₉	—	2	12	no	—
80	C₂²×C₄×C₈₉	—	2	96	no	—
81	C₂²×C₄×C₈₉	—	2	48	no	—
82	C₂²×C₄×C₈₉	—	2	48	no	—
83	C₂²×C₄×C₈₉	C₂×D₄	1	48	yes	—
84	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
85	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
86	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
87	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
88	C₂³×C₈₉	—	2	32	no	—
89	C₂³×C₈₉	—	2	256	no	—
90	C₂³×C₈₉	—	2	96	no	—
91	C₂³×C₈₉	—	2	24	no	—
92	C₂³×C₈₉	—	2	48	no	—
93	C₂³×C₈₉	—	2	96	no	—
94	C₂³×C₈₉	—	2	24	no	—
95	C₂³×C₈₉	—	2	48	no	—
96	C₂³×C₈₉	—	2	12	no	—
97	C₂³×C₈₉	—	2	12	no	—
98	C₂³×C₈₉	C₂²⋊Q₈	1	48	yes	complemented
99	C₂³×C₈₉	C₂²×D₄	1	12	yes	—
100	C₂³×C₈₉	C₂²×D₄	1	6	yes	—
101	C₂³×C₈₉	C₂²×Q₈	1	12	yes	—
102	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
103	C₂³×C₈₉	C₂×#41	1	12	yes	—
104	C₂³×C₈₉	C₂×#41	1	6	yes	—
105	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
106	C₂×C₄×C₈₉	—	2	96	no	—
107	C₂×C₄×C₈₉	—	2	24	no	—
108	C₂×C₄×C₈₉	—	2	12	no	—
109	C₂×C₄×C₈₉	—	2	96	no	—
110	C₂×C₄×C₈₉	#120	1	16	yes	—
111	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
112	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
113	C₂×Dic₈₉	—	2	12	no	—
114	C₂×Dic₈₉	—	2	12	no	—
115	C₂×Dic₈₉	—	2	24	no	—
116	C₂×Dic₈₉	—	4	8	no	—
117	C₂×Dic₈₉	—	4	24	no	—
118	Dic₁₇₈	—	4	16	no	—
119	C₂²×C₈₉	—	2	128	no	—
120	C₂²×C₈₉	—	2	96	no	—
121	C₂²×C₈₉	—	2	12	no	—
122	C₂²×C₈₉	—	2	24	no	—
123	C₂²×C₈₉	—	2	12	no	—
124	C₂²×C₈₉	—	2	24	no	—
125	C₂²×C₈₉	C₂×#120	1	4	yes	—
126	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	12	yes	—
127	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	24	yes	complemented
128	C₂²×C₈₉	#525	1	12	yes	—
129	C₂²×C₈₉	C₂³×D₄	1	3	yes	—
130	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
131	C₂²×C₈₉	C₂²×#41	1	3	yes	—
132	C₄×C₈₉	—	2	12	no	—
133	C₄×C₈₉	C₂×#120	1	8	yes	—
134	Dic₈₉	—	4	4	no	—
135	Dic₈₉	—	4	12	no	—
136	C₂×#2848	—	89	1	no	maximal, Sylow
137	C₂×C₈₉	—	2	24	no	—
138	C₂×C₈₉	#2848	1	8	yes	complemented
139	C₂×C₈₉	C₂²×#120	1	1	yes	characteristic
140	C₂×C₈₉	C₂²×C₂²⋊Q₈	1	3	yes	—
141	C₂×C₈₉	C₂×#525	1	3	yes	—
142	C₂⁵×C₄	D₈₉	1	1	yes	characteristic
143	C₂²×C₂²⋊Q₈	—	89	3	no	—
144	#2848	—	89	16	no	—
145	C₂×#385	—	89	6	no	—
146	C₂×#380	—	89	3	no	—
147	C₂×#379	—	89	2	no	—
148	C₈₉	C₂×#2848	1	1	yes	characteristic, complemented, Sylow
149	C₂⁶	C₂×D₈₉	1	1	yes	characteristic
150	C₂³×Q₈	—	89	1	no	—
151	C₂⁴×C₄	—	2	24	no	—
152	C₂⁴×C₄	C₂×D₈₉	1	8	yes	—
153	C₂⁴×C₄	C₂×D₈₉	1	3	yes	—
154	C₂⁴×C₄	C₂×D₈₉	1	3	yes	—
155	C₂×C₂²⋊Q₈	—	89	24	no	—
156	C₂×C₂²⋊Q₈	—	178	24	no	—
157	C₂²×C₄⋊C₄	—	89	6	no	—
158	C₂²×C₄⋊C₄	—	89	3	no	—
159	C₂²×C₂²⋊C₄	—	89	6	no	—
160	C₂²×C₂²⋊C₄	—	89	6	no	—
161	#385	—	89	48	no	—
162	#380	—	89	24	no	—
163	#379	—	89	16	no	—
164	C₂×#95	—	89	6	no	—
165	C₂⁵	—	2	12	no	—
166	C₂⁵	—	2	12	no	—
167	C₂⁵	Dic₁₇₈	1	8	yes	complemented
168	C₂⁵	C₂²×D₈₉	1	4	yes	—
169	C₂⁵	C₂²×D₈₉	1	3	yes	—
170	C₂²×Q₈	—	89	4	no	—
171	C₂²×Q₈	—	178	12	no	—
172	C₂³×C₄	—	2	128	no	—
173	C₂³×C₄	—	2	12	no	—
174	C₂³×C₄	—	2	24	no	—
175	C₂³×C₄	—	2	96	no	—
176	C₂³×C₄	—	2	24	no	—
177	C₂³×C₄	—	89	2	no	—
178	C₂³×C₄	—	89	6	no	—
179	C₂³×C₄	C₂²⋊D₈₉	1	24	yes	—
180	C₂³×C₄	C₂²×D₈₉	1	12	yes	—
181	C₂³×C₄	C₂²×D₈₉	1	12	yes	—
182	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
183	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
184	C₂²⋊Q₈	—	178	96	no	—
185	C₂×C₄⋊C₄	—	89	12	no	—
186	C₂×C₄⋊C₄	—	89	24	no	—
187	C₂×C₄⋊C₄	—	178	24	no	—
188	C₂×C₄⋊C₄	—	178	12	no	—
189	C₂×C₂²⋊C₄	—	89	24	no	—
190	C₂×C₂²⋊C₄	—	89	24	no	—
191	C₂×C₂²⋊C₄	—	178	24	no	—
192	C₂×C₂²⋊C₄	—	178	24	no	—
193	#95	—	89	24	no	—
194	C₂⁴	—	2	32	no	—
195	C₂⁴	—	2	96	no	—
196	C₂⁴	—	2	96	no	—
197	C₂⁴	—	2	24	no	—
198	C₂⁴	—	2	12	no	—
199	C₂⁴	—	2	24	no	—
200	C₂⁴	—	2	12	no	—
201	C₂⁴	C₂×C₂²⋊D₈₉	1	12	yes	—
202	C₂⁴	C₂×Dic₁₇₈	1	12	yes	—
203	C₂⁴	C₂×Dic₁₇₈	1	16	yes	complemented
204	C₂⁴	#13083574	1	12	yes	—
205	C₂⁴	C₂³×D₈₉	1	6	yes	—
206	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
207	C₂×Q₈	—	356	16	no	—
208	C₂×Q₈	—	178	24	no	—
209	C₂²×C₄	—	2	96	no	—
210	C₂²×C₄	—	2	256	no	—
211	C₂²×C₄	—	2	24	no	—
212	C₂²×C₄	—	2	12	no	—
213	C₂²×C₄	—	2	96	no	—
214	C₂²×C₄	—	2	48	no	—
215	C₂²×C₄	—	2	48	no	—
216	C₂²×C₄	—	89	12	no	—
217	C₂²×C₄	—	89	4	no	—
218	C₂²×C₄	—	178	12	no	—
219	C₂²×C₄	—	178	12	no	—
220	C₂²×C₄	—	178	24	no	—
221	C₂²×C₄	C₂×C₂²⋊D₈₉	1	12	yes	—
222	C₂²×C₄	C₂×C₂²⋊D₈₉	1	48	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₈₉	1	12	yes	—
224	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
225	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
226	C₄⋊C₄	—	178	48	no	—
227	C₄⋊C₄	—	178	24	no	—
228	C₂²⋊C₄	—	178	48	no	—
229	C₂²⋊C₄	—	178	48	no	—
230	C₂³	—	2	32	no	—
231	C₂³	—	2	96	no	—
232	C₂³	—	2	256	no	—
233	C₂³	—	2	12	no	—
234	C₂³	—	2	12	no	—
235	C₂³	—	2	24	no	—
236	C₂³	—	2	48	no	—
237	C₂³	—	2	96	no	—
238	C₂³	—	2	24	no	—
239	C₂³	—	2	48	no	—
240	C₂³	#15667991	1	48	yes	complemented
241	C₂³	C₂²×C₂²⋊D₈₉	1	12	yes	—
242	C₂³	C₂²×C₂²⋊D₈₉	1	6	yes	—
243	C₂³	C₂²×Dic₁₇₈	1	12	yes	—
244	C₂³	C₂²×Dic₁₇₈	1	2	yes	—
245	C₂³	C₂×#13083574	1	12	yes	—
246	C₂³	C₂×#13083574	1	6	yes	—
247	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
248	Q₈	—	356	16	no	—
249	C₂×C₄	—	2	96	no	—
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	—	2	24	no	—
252	C₂×C₄	—	2	96	no	—
253	C₂×C₄	—	356	8	no	—
254	C₂×C₄	—	356	24	no	—
255	C₂×C₄	—	178	12	no	—
256	C₂×C₄	—	178	12	no	—
257	C₂×C₄	—	178	24	no	—
258	C₂×C₄	#15667981	1	16	yes	—
259	C₂×C₄	C₂²×C₂²⋊D₈₉	1	12	yes	—
260	C₂×C₄	C₂²×C₂²⋊D₈₉	1	12	yes	—
261	C₂²	—	2	128	no	—
262	C₂²	—	2	12	no	—
263	C₂²	—	2	24	no	—
264	C₂²	—	2	12	no	—
265	C₂²	—	2	24	no	—
266	C₂²	—	2	96	no	—
267	C₂²	C₂×#15667981	1	4	yes	—
268	C₂²	C₂×#15667991	1	12	yes	—
269	C₂²	C₂×#15667991	1	24	yes	complemented
270	C₂²	#19152109	1	12	yes	—
271	C₂²	C₂³×C₂²⋊D₈₉	1	3	yes	—
272	C₂²	C₂³×Dic₁₇₈	1	1	yes	characteristic
273	C₂²	C₂²×#13083574	1	3	yes	—
274	C₄	—	2	12	no	—
275	C₄	—	356	4	no	—
276	C₄	—	356	12	no	—
277	C₄	C₂×#15667981	1	8	yes	—
278	C₂	—	2	24	no	—
279	C₂	#26028539	1	8	yes	direct product factor
280	C₂	C₂²×#15667981	1	1	yes	characteristic
281	C₂	C₂²×#15667991	1	3	yes	—
282	C₂	C₂×#19152109	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₈₉
commutator, cosocle	C₂³×C₈₉	C₂⁵
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂⁵×C₄×C₈₉	C₂
socle	C₂⁴×C₈₉	C₂⁴
prime radical	C₂⁶×C₈₉	C₂²
hypercenter	C₂⁵×C₄	D₈₉
hypocenter	C₈₉	C₂×#2848
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#15667991	C₂	semidirect
2. #26028539	C₂	direct (N × H)
3. C₂×#19151471	C₂	semidirect
4. C₂×#19151458	C₂	semidirect
7. C₂³×Dic₁₇₈	C₂²	semidirect
9. C₂×#15667991	C₂²	semidirect
10. C₂²×C₄⋊Dic₈₉	C₂²	semidirect
11. C₂²×#13083553	C₂²	semidirect
14. #19151471	C₂²	semidirect
15. #19151458	C₂²	semidirect
17. C₂×#15667919	C₂²	semidirect
24. C₂²×Dic₁₇₈	C₂³	semidirect
29. C₂×C₄⋊Dic₈₉	C₂³	semidirect
31. C₂×#13083553	C₂³	semidirect
36. #15667919	C₂³	semidirect
39. C₂⁵×C₈₉	Q₈	semidirect
72. C₂⁴×C₈₉	C₂×Q₈	semidirect
98. C₂³×C₈₉	C₂²⋊Q₈	semidirect
127. C₂²×C₈₉	C₂×C₂²⋊Q₈	semidirect
138. C₂×C₈₉	#2848	semidirect
148. C₈₉	C₂×#2848	semidirect
167. C₂⁵	Dic₁₇₈	semidirect
203. C₂⁴	C₂×Dic₁₇₈	semidirect
240. C₂³	#15667991	semidirect
269. C₂²	C₂×#15667991	semidirect
279. C₂	#26028539	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15667991	2
#26028539	2
C₂×#19151471	2
C₂×#19151458	2
C₂×#19151454	2
C₂⁵×C₄×C₈₉	2
C₂×#2848	89

Subgroups — C2×#26028539 #36760265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26028539 #36760265