Subgroups — C₂×#26022477 #36709713

Order 22784.

726 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151975	C₂	1	1	yes	maximal, characteristic
2	#26022477	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151398	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#500×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₁₂	—	2	1	no	—
7	C₂²×D₇₁₂	C₂²	1	2	yes	—
8	C₂×#15668009	—	2	1	no	—
9	C₂²×#13083549	C₂²	1	2	yes	—
10	#19151975	—	2	2	no	—
11	#19151975	C₂²	1	4	yes	—
12	#19151398	C₂²	1	16	yes	complemented
13	C₄×#13083549	C₂²	1	4	yes	complemented
14	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
15	#500×C₈₉	—	2	2	no	—
16	#500×C₈₉	C₂²	1	4	yes	complemented
17	C₂×#107×C₈₉	—	2	1	no	—
18	C₂×#107×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×D₃₅₆	—	2	2	no	—
21	C₂²×D₃₅₆	—	4	1	no	—
22	C₂²×D₃₅₆	—	4	1	no	—
23	C₂×D₇₁₂	—	2	2	no	—
24	C₂×D₇₁₂	—	2	4	no	—
25	C₂×D₇₁₂	—	2	8	no	—
26	C₂×D₇₁₂	—	4	4	no	—
27	C₂×D₇₁₂	C₂³	1	4	yes	—
28	#15668009	—	2	4	no	—
29	#15668009	—	4	2	no	—
30	C₂×#13083549	—	2	4	no	—
31	C₂×#13083549	C₂³	1	4	yes	—
32	C₂²×Q₈×C₈₉	—	4	1	no	—
33	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₈₉	—	2	4	no	—
36	C₄⋊Q₈×C₈₉	—	4	2	no	—
37	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
39	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
40	#107×C₈₉	—	2	4	no	—
41	#107×C₈₉	—	2	2	no	—
42	#107×C₈₉	C₂³	1	4	yes	—
43	C₄×C₈×C₈₉	D₄	1	2	yes	—
44	C₄×C₈×C₈₉	C₂³	1	2	yes	—
45	C₂³×D₈₉	—	4	2	no	—
46	C₂×D₃₅₆	—	2	4	no	—
47	C₂×D₃₅₆	—	4	4	no	—
48	C₂×D₃₅₆	—	4	4	no	—
49	C₂×D₃₅₆	—	4	4	no	—
50	C₂×D₃₅₆	—	4	4	no	—
51	C₂×D₃₅₆	—	4	2	no	—
52	C₂×D₃₅₆	—	8	4	no	—
53	C₂×D₃₅₆	—	8	4	no	—
54	D₇₁₂	—	2	8	no	—
55	D₇₁₂	—	4	4	no	—
56	D₇₁₂	—	4	8	no	—
57	#13083549	—	2	4	no	—
58	C₂×Q₈×C₈₉	—	4	4	no	—
59	C₂×Q₈×C₈₉	—	8	4	no	—
60	C₂²×C₄×C₈₉	—	4	2	no	—
61	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
65	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₈₉	#41	1	4	yes	—
68	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₈₉	—	2	4	no	—
70	C₄⋊C₄×C₈₉	—	4	2	no	—
71	C₄⋊C₄×C₈₉	—	4	4	no	—
72	C₄²×C₈₉	D₈	1	2	yes	—
73	C₄²×C₈₉	C₂×D₄	1	2	yes	—
74	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
75	C₂²×D₈₉	—	4	4	no	—
76	C₂²×D₈₉	—	8	4	no	—
77	C₂²×D₈₉	—	8	4	no	—
78	C₂²×D₈₉	—	8	4	no	—
79	Q₈×C₈₉	—	8	8	no	—
80	C₂×C₄×C₈₉	—	4	4	no	—
81	C₂×C₄×C₈₉	—	8	4	no	—
82	C₂×C₄×C₈₉	D₁₆	1	4	yes	—
83	C₂×C₄×C₈₉	SD₃₂	1	4	yes	—
84	C₂×C₄×C₈₉	#124	1	2	yes	—
85	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
90	D₃₅₆	—	4	4	no	—
91	D₃₅₆	—	4	4	no	—
92	D₃₅₆	—	8	4	no	—
93	D₃₅₆	—	8	8	no	—
94	D₃₅₆	—	8	8	no	—
95	C₈×C₈₉	#124	1	2	yes	—
96	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
97	C₈×C₈₉	C₂×#41	1	2	yes	—
98	C₂²×C₈₉	#486	1	2	yes	—
99	C₂²×C₈₉	C₂×D₁₆	1	2	yes	—
100	C₂²×C₈₉	C₂×SD₃₂	1	2	yes	—
101	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
102	C₂×D₈₉	—	8	4	no	—
103	C₂×D₈₉	—	8	4	no	—
104	C₂×D₈₉	—	8	4	no	—
105	C₂×D₈₉	—	16	8	no	—
106	C₄×C₈₉	—	8	4	no	—
107	C₄×C₈₉	#486	1	2	yes	—
108	C₄×C₈₉	C₂×D₁₆	1	4	yes	—
109	C₄×C₈₉	C₂×SD₃₂	1	4	yes	—
110	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
111	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
112	C₂×#2240	—	89	1	no	maximal, Sylow
113	C₂×C₈₉	#2240	1	4	yes	complemented
114	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
115	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
116	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
117	D₈₉	—	16	4	no	—
118	C₂×#500	D₈₉	1	1	yes	characteristic, complemented
119	C₂×#493	—	89	1	no	—
120	#2240	—	89	8	no	—
121	C₂×#357	—	89	4	no	—
122	C₂×C₄×C₁₆	—	89	1	no	—
123	C₈₉	C₂×#2240	1	1	yes	characteristic, complemented, Sylow
124	C₂²×D₈	—	89	2	no	—
125	C₂²×D₈	—	178	1	no	—
126	C₂×C₄⋊Q₈	—	2	1	no	—
127	C₂×#127	—	178	1	no	—
128	C₂²×C₁₆	—	89	2	no	—
129	#500	—	2	2	no	—
130	#500	C₂×D₈₉	1	4	yes	complemented
131	#493	—	89	4	no	—
132	#493	—	178	2	no	—
133	C₂×#107	—	2	1	no	—
134	C₂×#107	C₂×D₈₉	1	2	yes	—
135	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
136	#357	—	89	16	no	—
137	C₄×C₁₆	—	89	4	no	—
138	C₂²×Q₈	—	4	1	no	—
139	C₂²×D₄	—	356	1	no	—
140	C₂²×D₄	—	356	1	no	—
141	C₂²×D₄	—	178	2	no	—
142	C₂×D₈	—	89	4	no	—
143	C₂×D₈	—	356	4	no	—
144	C₂×D₈	—	178	2	no	—
145	C₂×D₈	—	178	4	no	—
146	C₂×D₈	—	178	8	no	—
147	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	4	no	—
150	C₄⋊Q₈	—	4	2	no	—
151	#127	—	356	2	no	—
152	#127	—	178	4	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	4	1	no	—
155	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
156	C₂×C₁₆	—	89	4	no	—
157	C₂×C₁₆	—	178	4	no	—
158	#107	—	2	4	no	—
159	#107	—	2	2	no	—
160	#107	C₂²×D₈₉	1	4	yes	—
161	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
162	C₄×C₈	C₂²×D₈₉	1	2	yes	—
163	C₂⁴	—	356	2	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	8	4	no	—
166	C₂×D₄	—	356	4	no	—
167	C₂×D₄	—	356	2	no	—
168	C₂×D₄	—	356	4	no	—
169	C₂×D₄	—	356	4	no	—
170	C₂×D₄	—	356	4	no	—
171	C₂×D₄	—	178	4	no	—
172	C₂×D₄	—	712	4	no	—
173	C₂×D₄	—	712	4	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	#13083559	1	1	yes	characteristic
176	C₂²×C₄	#13083559	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
178	D₈	—	356	4	no	—
179	D₈	—	356	8	no	—
180	D₈	—	178	8	no	—
181	C₂×C₈	#13083561	1	4	yes	—
182	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
183	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
184	C₂×C₈	#13083575	1	4	yes	—
185	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄²	#13083559	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
191	C₁₆	—	178	4	no	—
192	C₂³	—	356	4	no	—
193	C₂³	—	712	4	no	—
194	C₂³	—	712	4	no	—
195	C₂³	—	712	4	no	—
196	C₂³	C₂×#13083559	1	1	yes	characteristic
197	Q₈	—	8	8	no	—
198	D₄	—	356	4	no	—
199	D₄	—	356	4	no	—
200	D₄	—	712	4	no	—
201	D₄	—	712	8	no	—
202	D₄	—	712	8	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	#15667953	1	4	yes	—
206	C₂×C₄	#15667955	1	4	yes	—
207	C₂×C₄	#15668001	1	2	yes	—
208	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
209	C₂×C₄	C₂×#13083559	1	2	yes	—
210	C₂×C₄	C₂×#13083559	1	2	yes	—
211	C₂×C₄	C₂×#13083561	1	2	yes	—
212	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
213	C₈	#15668001	1	2	yes	—
214	C₈	C₂×#13083561	1	4	yes	—
215	C₈	C₂×#13083575	1	2	yes	—
216	C₂²	—	1424	8	no	—
217	C₂²	—	712	4	no	—
218	C₂²	—	712	4	no	—
219	C₂²	—	712	4	no	—
220	C₂²	#19151946	1	2	yes	—
221	C₂²	C₂×#15667953	1	2	yes	—
222	C₂²	C₂×#15667955	1	2	yes	—
223	C₂²	C₂²×#13083559	1	1	yes	characteristic
224	C₄	—	8	4	no	—
225	C₄	#19151946	1	2	yes	—
226	C₄	C₂×#15667953	1	4	yes	—
227	C₄	C₂×#15667955	1	4	yes	—
228	C₄	C₂×#15668001	1	1	yes	characteristic
229	C₄	C₂²×#13083561	1	1	yes	characteristic
230	C₂	—	1424	4	no	—
231	C₂	#26022477	1	4	yes	direct product factor
232	C₂	C₂×#19151946	1	1	yes	characteristic
233	C₂	C₂²×#15667953	1	1	yes	characteristic
234	C₂	C₂²×#15667955	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#500×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂×#19151975	C₂
hypercenter	C₂×#500	D₈₉
hypocenter	C₈₉	C₂×#2240
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #26022477	C₂	direct (N × H)
3. C₂×#19151398	C₂	semidirect
4. C₂×C₄×#13083549	C₂	semidirect
5. C₂×#500×C₈₉	C₂	semidirect
12. #19151398	C₂²	semidirect
13. C₄×#13083549	C₂²	semidirect
16. #500×C₈₉	C₂²	semidirect
113. C₂×C₈₉	#2240	semidirect
118. C₂×#500	D₈₉	semidirect
123. C₈₉	C₂×#2240	semidirect
130. #500	C₂×D₈₉	semidirect
231. C₂	#26022477	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×#19151975	2
#26022477	2
C₂×#19151398	2
C₂×C₄×#13083549	2
C₂×#500×C₈₉	2
C₂×#2240	89

Subgroups — C2×#26022477 #36709713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022477 #36709713