Subgroups — C₂×#26022410 #36709044

Order 22784.

606 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152000	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
3	#26022410	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151426	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151424	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151400	C₂	1	1	yes	maximal, characteristic
7	C₂×#19151400	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
10	C₂×#15668004	—	2	1	no	—
11	C₂×C₄×D₃₅₆	—	2	1	no	—
12	#19152000	—	2	2	no	—
13	#19152000	C₂²	1	4	yes	complemented
14	C₄×D₇₁₂	—	2	2	no	—
15	C₄×D₇₁₂	C₂²	1	4	yes	—
16	C₂×#15667946	C₂²	1	1	yes	characteristic
17	C₂×#15667946	C₂²	1	1	yes	characteristic
18	C₂×#15667946	C₂²	1	1	yes	characteristic
19	C₂×#15667944	—	2	1	no	—
20	C₂×#15667934	—	2	1	no	—
21	#19151426	C₂²	1	4	yes	complemented
22	#19151424	C₂²	1	4	yes	complemented
23	#19151400	C₂²	1	4	yes	—
24	#19151400	C₂²	1	4	yes	—
25	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₁₆×C₈₉	C₂²	1	4	yes	complemented
29	C₂²×D₃₅₆	—	2	1	no	—
30	C₂²×C₄×D₈₉	—	4	1	no	—
31	C₂×D₇₁₂	—	2	1	no	—
32	C₂×D₇₁₂	—	2	2	no	—
33	C₂×D₇₁₂	Q₈	1	4	yes	—
34	C₂×D₇₁₂	C₂³	1	2	yes	—
35	#15668004	—	2	4	no	—
36	#15668004	—	4	2	no	—
37	C₄×D₃₅₆	—	2	4	no	—
38	C₄×D₃₅₆	—	4	2	no	—
39	C₂×C₄⋊Dic₈₉	—	2	1	no	—
40	C₂×C₄⋊Dic₈₉	—	2	1	no	—
41	C₂×C₄⋊Dic₈₉	—	2	1	no	—
42	C₂×#13083553	—	4	1	no	—
43	C₂×#13083553	—	4	1	no	—
44	C₂×#13083551	—	4	1	no	—
45	#15667946	—	2	1	no	—
46	#15667946	—	2	1	no	—
47	#15667946	—	2	1	no	—
48	#15667946	C₂³	1	2	yes	—
49	#15667946	C₂³	1	2	yes	—
50	#15667946	C₂³	1	2	yes	—
51	#15667944	—	2	4	no	—
52	#15667934	—	2	4	no	—
53	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
56	C₂×C₁₆×C₈₉	—	2	2	no	—
57	C₂×C₁₆×C₈₉	—	2	2	no	—
58	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
59	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
60	C₄×C₈×C₈₉	D₄	1	2	yes	—
61	C₄×C₈×C₈₉	C₂³	1	2	yes	—
62	C₂³×D₈₉	—	4	1	no	—
63	C₂×D₃₅₆	—	2	4	no	—
64	C₂×D₃₅₆	—	2	2	no	—
65	C₂×D₃₅₆	—	4	1	no	—
66	C₂×D₃₅₆	—	4	2	no	—
67	C₂²×Dic₈₉	—	4	1	no	—
68	C₂²×Dic₈₉	—	4	1	no	—
69	C₂²×Dic₈₉	—	4	1	no	—
70	C₂×C₄×D₈₉	—	4	4	no	—
71	C₂×C₄×D₈₉	—	8	4	no	—
72	D₇₁₂	—	2	4	no	—
73	D₇₁₂	—	2	2	no	—
74	D₇₁₂	C₂×Q₈	1	4	yes	—
75	C₄⋊Dic₈₉	—	2	2	no	—
76	C₄⋊Dic₈₉	—	2	2	no	—
77	C₄⋊Dic₈₉	—	2	2	no	—
78	C₄⋊Dic₈₉	—	4	1	no	—
79	C₄⋊Dic₈₉	—	4	1	no	—
80	C₄⋊Dic₈₉	—	4	1	no	—
81	#13083553	—	4	4	no	—
82	#13083553	—	4	4	no	—
83	#13083551	—	4	4	no	—
84	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	—	2	2	no	—
88	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₈₉	#41	1	2	yes	—
92	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
93	C₄²×C₈₉	D₈	1	2	yes	—
94	C₄²×C₈₉	C₂×D₄	1	2	yes	—
95	C₁₆×C₈₉	—	2	2	no	—
96	C₁₆×C₈₉	—	2	2	no	—
97	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
98	C₂²×D₈₉	—	4	2	no	—
99	C₂²×D₈₉	—	4	4	no	—
100	C₂²×D₈₉	—	8	2	no	—
101	C₂²×D₈₉	—	8	2	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#137	1	2	yes	—
109	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
110	C₂×Dic₈₉	—	4	2	no	—
111	C₂×Dic₈₉	—	4	2	no	—
112	C₂×Dic₈₉	—	4	2	no	—
113	C₂×Dic₈₉	—	8	2	no	—
114	C₂×Dic₈₉	—	8	2	no	—
115	C₂×Dic₈₉	—	8	2	no	—
116	D₃₅₆	—	2	4	no	—
117	D₃₅₆	—	4	2	no	—
118	D₃₅₆	—	4	4	no	—
119	C₄×D₈₉	—	8	8	no	—
120	C₈×C₈₉	—	2	2	no	—
121	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
122	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
123	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
124	C₂²×C₈₉	#474	1	2	yes	—
125	C₂²×C₈₉	#508	1	2	yes	—
126	C₂²×C₈₉	#509	1	2	yes	—
127	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
128	C₂×D₈₉	—	4	4	no	—
129	C₂×D₈₉	—	8	2	no	—
130	C₂×D₈₉	—	8	2	no	—
131	C₂×D₈₉	—	8	8	no	—
132	C₄×C₈₉	—	2	2	no	—
133	C₄×C₈₉	—	2	2	no	—
134	C₄×C₈₉	#474	1	2	yes	—
135	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
136	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
137	Dic₈₉	—	8	2	no	—
138	Dic₈₉	—	8	2	no	—
139	Dic₈₉	—	8	2	no	—
140	C₂×#2228	—	89	1	no	maximal, Sylow
141	C₂×C₈₉	#2228	1	4	yes	complemented
142	C₂×C₈₉	C₂×#474	1	1	yes	characteristic
143	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
144	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
145	D₈₉	—	8	4	no	—
146	C₂×#499	—	89	1	no	—
147	C₂×C₄×D₈	—	89	1	no	—
148	#2228	—	89	8	no	—
149	C₂×#367	—	89	1	no	—
150	C₂×#366	—	89	1	no	—
151	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic, complemented
152	C₂×#357	—	89	1	no	—
153	C₂×#357	—	89	1	no	—
154	C₈₉	C₂×#2228	1	1	yes	characteristic, complemented, Sylow
155	C₂²×D₈	—	89	1	no	—
156	C₂×#125	—	178	1	no	—
157	C₂×C₄×D₄	—	178	1	no	—
158	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
159	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
160	#499	—	89	4	no	—
161	#499	—	178	2	no	—
162	C₄×D₈	—	89	4	no	—
163	C₄×D₈	—	178	2	no	—
164	C₂×#107	—	89	1	no	—
165	C₂×#107	—	89	1	no	—
166	C₂×#107	—	89	1	no	—
167	C₂×#106	—	178	1	no	—
168	C₂×#102	—	178	1	no	—
169	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
170	#367	—	89	4	no	—
171	#366	—	89	4	no	—
172	C₄⋊C₁₆	C₂×D₈₉	1	4	yes	complemented
173	#357	—	89	4	no	—
174	#357	—	89	4	no	—
175	C₂²×D₄	—	178	1	no	—
176	C₂³×C₄	—	356	1	no	—
177	C₂×D₈	—	89	4	no	—
178	C₂×D₈	—	89	2	no	—
179	C₂×D₈	—	178	1	no	—
180	C₂×D₈	—	178	2	no	—
181	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
183	#125	—	356	2	no	—
184	#125	—	178	4	no	—
185	C₄×D₄	—	356	2	no	—
186	C₄×D₄	—	178	4	no	—
187	C₂×C₄⋊C₄	—	356	1	no	—
188	C₂×C₄⋊C₄	—	356	1	no	—
189	C₂×C₄⋊C₄	—	178	1	no	—
190	C₂×C₄⋊C₄	—	178	1	no	—
191	C₂×C₄⋊C₄	—	178	1	no	—
192	C₂×C₂²⋊C₄	—	356	1	no	—
193	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
197	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
198	#107	—	89	2	no	—
199	#107	—	89	2	no	—
200	#107	—	89	2	no	—
201	#107	—	178	1	no	—
202	#107	—	178	1	no	—
203	#107	—	178	1	no	—
204	#106	—	178	4	no	—
205	#102	—	178	4	no	—
206	C₄×C₈	D₃₅₆	1	2	yes	—
207	C₄×C₈	C₂²×D₈₉	1	2	yes	—
208	C₂⁴	—	356	1	no	—
209	C₂×D₄	—	356	1	no	—
210	C₂×D₄	—	356	2	no	—
211	C₂×D₄	—	178	4	no	—
212	C₂×D₄	—	178	2	no	—
213	C₂²×C₄	—	356	1	no	—
214	C₂²×C₄	—	356	1	no	—
215	C₂²×C₄	—	356	1	no	—
216	C₂²×C₄	—	356	4	no	—
217	C₂²×C₄	—	712	4	no	—
218	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
219	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
221	D₈	—	89	4	no	—
222	D₈	—	178	4	no	—
223	D₈	—	178	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
226	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
227	C₂×C₈	Q₈×D₈₉	1	2	yes	—
228	C₂×C₈	#13083573	1	2	yes	—
229	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
230	C₄⋊C₄	—	356	4	no	—
231	C₄⋊C₄	—	356	4	no	—
232	C₄⋊C₄	—	356	1	no	—
233	C₄⋊C₄	—	356	1	no	—
234	C₄⋊C₄	—	356	1	no	—
235	C₄⋊C₄	—	178	2	no	—
236	C₄⋊C₄	—	178	2	no	—
237	C₄⋊C₄	—	178	2	no	—
238	C₂²⋊C₄	—	356	4	no	—
239	C₄²	D₇₁₂	1	2	yes	—
240	C₄²	C₂×D₃₅₆	1	2	yes	—
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	2	no	—
243	C₂³	—	356	4	no	—
244	C₂³	—	356	2	no	—
245	C₂³	—	712	2	no	—
246	C₂³	—	712	2	no	—
247	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
248	D₄	—	356	4	no	—
249	D₄	—	356	2	no	—
250	D₄	—	178	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	356	2	no	—
254	C₂×C₄	—	356	2	no	—
255	C₂×C₄	—	356	2	no	—
256	C₂×C₄	—	712	8	no	—
257	C₂×C₄	—	712	2	no	—
258	C₂×C₄	—	712	2	no	—
259	C₂×C₄	—	712	2	no	—
260	C₂×C₄	#15667992	1	2	yes	—
261	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
262	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
263	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
264	C₂×C₄	#15668045	1	2	yes	—
265	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	#15667992	1	2	yes	—
268	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
269	C₈	C₂×#13083573	1	1	yes	characteristic
270	C₂²	—	356	4	no	—
271	C₂²	—	712	8	no	—
272	C₂²	—	712	2	no	—
273	C₂²	—	712	2	no	—
274	C₂²	#19151879	1	2	yes	—
275	C₂²	#19152030	1	2	yes	—
276	C₂²	#19152036	1	2	yes	—
277	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	712	2	no	—
281	C₄	—	712	2	no	—
282	C₄	—	712	2	no	—
283	C₄	#19151879	1	2	yes	—
284	C₄	C₂×#15667992	1	1	yes	characteristic
285	C₄	C₂×#15668045	1	1	yes	characteristic
286	C₂	—	712	4	no	—
287	C₂	#26022410	1	4	yes	direct product factor
288	C₂	C₂×#19151879	1	1	yes	characteristic
289	C₂	C₂×#19152030	1	1	yes	characteristic
290	C₂	C₂×#19152036	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂²×D₇₁₂	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	C₂×#2228
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#19152000	C₂	semidirect
3. #26022410	C₂	direct (N × H)
4. C₂×#19151426	C₂	semidirect
5. C₂×#19151424	C₂	semidirect
8. C₂×C₄⋊C₁₆×C₈₉	C₂	semidirect
13. #19152000	C₂²	semidirect
21. #19151426	C₂²	semidirect
22. #19151424	C₂²	semidirect
28. C₄⋊C₁₆×C₈₉	C₂²	semidirect
141. C₂×C₈₉	#2228	semidirect
151. C₂×C₄⋊C₁₆	D₈₉	semidirect
154. C₈₉	C₂×#2228	semidirect
172. C₄⋊C₁₆	C₂×D₈₉	semidirect
287. C₂	#26022410	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19152000	2
C₂×C₄×D₇₁₂	2
#26022410	2
C₂×#19151426	2
C₂×#19151424	2
C₂×#19151400	2
C₂×C₄⋊C₁₆×C₈₉	2
C₂×#2228	89

Subgroups — C2×#26022410 #36709044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022410 #36709044