Subgroups — C₂×#26022513 #36709827

Order 22784.

854 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	C₂²×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#15667957	C₂	1	2	yes	maximal, complemented
3	C₂²×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151982	C₂	1	2	yes	maximal, complemented
5	#26022513	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₃₅₆	C₂²	1	2	yes	—
8	C₂²×#13083563	—	2	2	no	—
9	C₂²×D₇₁₂	C₂²	1	2	yes	—
10	C₂×#15668000	—	2	2	no	—
11	C₂×Dic₇₁₂	—	2	4	no	—
12	C₂×Dic₇₁₂	C₂²	1	4	yes	complemented
13	C₂×#15667957	—	2	8	no	—
14	C₂×#15667957	C₂²	1	8	yes	complemented
15	C₂×D₁₄₂₄	—	2	4	no	—
16	C₂×D₁₄₂₄	C₂²	1	4	yes	—
17	#19151982	—	2	4	no	—
18	#19151982	C₂²	1	8	yes	complemented
19	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₈₉	C₂²	1	4	yes	complemented
23	C₂²×Dic₁₇₈	—	2	2	no	—
24	C₂²×D₃₅₆	—	2	2	no	—
25	C₂×Dic₃₅₆	—	2	4	no	—
26	C₂×Dic₃₅₆	—	2	4	no	—
27	C₂×Dic₃₅₆	—	2	2	no	—
28	C₂×Dic₃₅₆	C₂³	1	4	yes	—
29	C₂×#13083563	—	2	8	no	—
30	C₂×#13083563	—	4	8	no	—
31	C₂×D₇₁₂	—	2	4	no	—
32	C₂×D₇₁₂	—	2	4	no	—
33	C₂×D₇₁₂	—	2	2	no	—
34	C₂×D₇₁₂	C₂³	1	4	yes	—
35	#15668000	—	2	8	no	—
36	#15668000	—	4	4	no	—
37	C₂×#13083551	—	4	4	no	—
38	Dic₇₁₂	—	2	8	no	—
39	#15667957	—	2	16	no	—
40	D₁₄₂₄	—	2	8	no	—
41	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
45	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
46	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
47	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
48	C₄×C₈×C₈₉	D₄	1	2	yes	—
49	C₄×C₈×C₈₉	C₂³	1	2	yes	—
50	C₂³×D₈₉	—	4	2	no	—
51	C₂×Dic₁₇₈	—	2	4	no	—
52	C₂×Dic₁₇₈	—	4	4	no	—
53	C₂×Dic₁₇₈	—	4	4	no	—
54	C₂×Dic₁₇₈	—	4	2	no	—
55	C₂×D₃₅₆	—	2	4	no	—
56	C₂×D₃₅₆	—	4	4	no	—
57	C₂×D₃₅₆	—	4	4	no	—
58	C₂×D₃₅₆	—	4	2	no	—
59	C₂²×Dic₈₉	—	4	2	no	—
60	Dic₃₅₆	—	2	4	no	—
61	Dic₃₅₆	—	2	4	no	—
62	Dic₃₅₆	—	4	4	no	—
63	#13083563	—	4	16	no	—
64	D₇₁₂	—	2	4	no	—
65	D₇₁₂	—	2	4	no	—
66	D₇₁₂	—	4	4	no	—
67	#13083551	—	4	16	no	—
68	C₂²×C₄×C₈₉	D₈	1	2	yes	—
69	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₈₉	D₈	1	4	yes	—
71	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
76	C₄²×C₈₉	D₈	1	2	yes	—
77	C₄²×C₈₉	C₂×D₄	1	2	yes	—
78	C₁₆×C₈₉	C₂×D₄	1	4	yes	—
79	C₁₆×C₈₉	C₂×D₄	1	4	yes	—
80	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
81	C₂²×D₈₉	—	4	4	no	—
82	C₂²×D₈₉	—	8	4	no	—
83	C₂²×D₈₉	—	8	8	no	—
84	C₂×C₄×C₈₉	—	2	4	no	—
85	C₂×C₄×C₈₉	#127	1	2	yes	—
86	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
87	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
90	C₂×Dic₈₉	—	4	4	no	—
91	C₂×Dic₈₉	—	8	4	no	—
92	D₃₅₆	—	4	4	no	—
93	D₃₅₆	—	4	4	no	—
94	D₃₅₆	—	8	4	no	—
95	Dic₁₇₈	—	4	4	no	—
96	Dic₁₇₈	—	4	4	no	—
97	Dic₁₇₈	—	8	4	no	—
98	C₈×C₈₉	#127	1	2	yes	—
99	C₈×C₈₉	C₂×D₈	1	4	yes	—
100	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
102	C₂²×C₈₉	#493	1	2	yes	—
103	C₂²×C₈₉	#508	1	4	yes	—
104	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
105	C₂×D₈₉	—	8	4	no	—
106	C₂×D₈₉	—	8	8	no	—
107	C₂×D₈₉	—	16	8	no	—
108	C₄×C₈₉	—	2	4	no	—
109	C₄×C₈₉	#493	1	2	yes	—
110	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
111	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
112	Dic₈₉	—	8	4	no	—
113	C₂×#2249	—	89	1	no	maximal, Sylow
114	C₂×C₈₉	#2249	1	4	yes	complemented
115	C₂×C₈₉	C₂×#493	1	1	yes	characteristic
116	C₂×C₈₉	C₂×#508	1	2	yes	—
117	D₈₉	—	16	4	no	—
118	C₂²×Q₃₂	—	89	1	no	—
119	C₂²×SD₃₂	—	89	2	no	—
120	C₂²×D₁₆	—	89	1	no	—
121	C₂×#495	—	89	2	no	—
122	#2249	—	89	8	no	—
123	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
124	C₈₉	C₂×#2249	1	1	yes	characteristic, complemented, Sylow
125	C₂²×Q₁₆	—	89	2	no	—
126	C₂²×SD₁₆	—	178	2	no	—
127	C₂²×D₈	—	89	2	no	—
128	C₂×#124	—	178	2	no	—
129	C₂×Q₃₂	—	89	4	no	—
130	C₂×Q₃₂	—	178	4	no	—
131	C₂×SD₃₂	—	89	8	no	—
132	C₂×SD₃₂	—	178	8	no	—
133	C₂×D₁₆	—	89	4	no	—
134	C₂×D₁₆	—	178	4	no	—
135	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
136	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
137	#495	—	89	8	no	—
138	#495	—	178	4	no	—
139	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
140	C₄×C₁₆	C₂×D₈₉	1	4	yes	complemented
141	C₂²×Q₈	—	178	2	no	—
142	C₂²×D₄	—	178	2	no	—
143	C₂×Q₁₆	—	89	4	no	—
144	C₂×Q₁₆	—	178	2	no	—
145	C₂×Q₁₆	—	178	4	no	—
146	C₂×Q₁₆	—	178	4	no	—
147	C₂×SD₁₆	—	356	8	no	—
148	C₂×SD₁₆	—	178	8	no	—
149	C₂×D₈	—	89	4	no	—
150	C₂×D₈	—	178	2	no	—
151	C₂×D₈	—	178	4	no	—
152	C₂×D₈	—	178	4	no	—
153	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
154	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
155	#124	—	356	4	no	—
156	#124	—	178	8	no	—
157	C₂×C₂²⋊C₄	—	356	4	no	—
158	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
159	Q₃₂	—	178	8	no	—
160	SD₃₂	—	178	16	no	—
161	D₁₆	—	178	8	no	—
162	C₂×C₁₆	D₃₅₆	1	4	yes	—
163	C₂×C₁₆	D₃₅₆	1	4	yes	—
164	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
165	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
166	C₄×C₈	D₃₅₆	1	2	yes	—
167	C₄×C₈	C₂²×D₈₉	1	2	yes	—
168	C₂⁴	—	356	2	no	—
169	C₂×Q₈	—	356	2	no	—
170	C₂×Q₈	—	356	4	no	—
171	C₂×Q₈	—	356	4	no	—
172	C₂×Q₈	—	178	4	no	—
173	C₂×D₄	—	356	4	no	—
174	C₂×D₄	—	356	4	no	—
175	C₂×D₄	—	356	2	no	—
176	C₂×D₄	—	178	4	no	—
177	C₂²×C₄	—	356	2	no	—
178	C₂²×C₄	D₇₁₂	1	2	yes	—
179	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
180	Q₁₆	—	356	4	no	—
181	Q₁₆	—	178	4	no	—
182	Q₁₆	—	178	4	no	—
183	SD₁₆	—	356	16	no	—
184	D₈	—	356	4	no	—
185	D₈	—	178	4	no	—
186	D₈	—	178	4	no	—
187	C₂×C₈	D₇₁₂	1	4	yes	—
188	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
189	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
190	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
191	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
192	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
193	C₂²⋊C₄	—	356	16	no	—
194	C₄²	D₇₁₂	1	2	yes	—
195	C₄²	C₂×D₃₅₆	1	2	yes	—
196	C₁₆	C₂×D₃₅₆	1	4	yes	—
197	C₁₆	C₂×D₃₅₆	1	4	yes	—
198	C₂³	—	356	4	no	—
199	C₂³	—	712	4	no	—
200	C₂³	—	712	8	no	—
201	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
202	Q₈	—	356	4	no	—
203	Q₈	—	356	4	no	—
204	Q₈	—	712	4	no	—
205	D₄	—	356	4	no	—
206	D₄	—	356	4	no	—
207	D₄	—	712	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	356	4	no	—
210	C₂×C₄	—	712	4	no	—
211	C₂×C₄	#15668009	1	2	yes	—
212	C₂×C₄	C₂×D₇₁₂	1	4	yes	—
213	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
214	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
215	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
216	C₈	#15668009	1	2	yes	—
217	C₈	C₂×D₇₁₂	1	4	yes	—
218	C₈	C₂²×D₃₅₆	1	1	yes	characteristic
219	C₈	C₂²×D₃₅₆	1	1	yes	characteristic
220	C₂²	—	1424	8	no	—
221	C₂²	—	712	4	no	—
222	C₂²	—	712	8	no	—
223	C₂²	#19151975	1	2	yes	—
224	C₂²	#19152030	1	4	yes	—
225	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	712	4	no	—
228	C₄	#19151975	1	2	yes	—
229	C₄	C₂×#15668009	1	1	yes	characteristic
230	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
231	C₂	—	1424	4	no	—
232	C₂	#26022513	1	4	yes	direct product factor
233	C₂	C₂×#19151975	1	1	yes	characteristic
234	C₂	C₂×#19152030	1	2	yes	—

	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×D₁₄₂₄	C₂
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	C₂×#2249
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. C₂²×Dic₇₁₂	C₂	semidirect
2. C₂²×#15667957	C₂	semidirect
4. C₂×#19151982	C₂	semidirect
5. #26022513	C₂	direct (N × H)
6. C₂×C₄×C₁₆×C₈₉	C₂	semidirect
12. C₂×Dic₇₁₂	C₂²	semidirect
14. C₂×#15667957	C₂²	semidirect
18. #19151982	C₂²	semidirect
22. C₄×C₁₆×C₈₉	C₂²	semidirect
114. C₂×C₈₉	#2249	semidirect
123. C₂×C₄×C₁₆	D₈₉	semidirect
124. C₈₉	C₂×#2249	semidirect
140. C₄×C₁₆	C₂×D₈₉	semidirect
232. C₂	#26022513	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×Dic₇₁₂	2
C₂²×#15667957	2
C₂²×D₁₄₂₄	2
C₂×#19151982	2
#26022513	2
C₂×C₄×C₁₆×C₈₉	2
C₂×#2249	89

Subgroups — C2×#26022513 #36709827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022513 #36709827