Subgroups — C₂×#26020416 #36554835

Order 22784.

1182 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×#15667946	C₂	1	2	yes	maximal
2	C₂²×#15667944	C₂	1	2	yes	maximal
3	C₂×#19151464	C₂	1	2	yes	maximal
4	#26020416	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₈₉	C₂²	1	4	yes	—
7	C₂×#15667946	—	2	8	no	—
8	C₂×#15667946	C₂²	1	8	yes	—
9	C₂×#15667944	—	2	8	no	—
10	C₂×#15667944	C₂²	1	8	yes	—
11	#19151464	—	2	4	no	—
12	#19151464	C₂²	1	8	yes	—
13	C₂×#15667919	—	2	4	no	—
14	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₈₉	C₄	1	8	yes	complemented
18	C₂×C₄×C₈×C₈₉	C₂²	1	4	yes	—
19	C₂³×Dic₈₉	—	2	4	no	—
20	C₂×C₄⋊Dic₈₉	—	2	8	no	—
21	C₂×C₄⋊Dic₈₉	—	2	8	no	—
22	C₂×C₄⋊Dic₈₉	—	2	4	no	—
23	C₂×C₄⋊Dic₈₉	C₂³	1	8	yes	—
24	#15667946	—	2	16	no	—
25	#15667944	—	2	16	no	—
26	#15667919	—	2	16	no	—
27	C₂³×C₄×C₈₉	D₄	1	2	yes	—
28	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₈₉	D₄	1	4	yes	—
32	C₂²×C₈×C₈₉	D₄	1	4	yes	—
33	C₂²×C₈×C₈₉	Q₈	1	4	yes	—
34	C₂²×C₈×C₈₉	Q₈	1	4	yes	—
35	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
36	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
37	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₈₉	D₄	1	2	yes	—
39	C₂×C₄²×C₈₉	Q₈	1	4	yes	—
40	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
41	C₄×C₈×C₈₉	C₂×C₄	1	16	yes	complemented
42	C₂²×Dic₈₉	—	2	8	no	—
43	C₂²×Dic₈₉	—	4	8	no	—
44	C₂²×Dic₈₉	—	4	16	no	—
45	C₄⋊Dic₈₉	—	2	8	no	—
46	C₄⋊Dic₈₉	—	2	8	no	—
47	C₄⋊Dic₈₉	—	4	8	no	—
48	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	—	2	8	no	—
50	C₂²×C₄×C₈₉	C₄⋊C₄	1	8	yes	—
51	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
53	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₈₉	C₄⋊C₄	1	16	yes	—
61	C₂×C₈×C₈₉	C₄⋊C₄	1	16	yes	—
62	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
66	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
67	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
68	C₄²×C₈₉	C₄⋊C₄	1	8	yes	—
69	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
70	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
71	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₈₉	#127	1	2	yes	—
73	C₂³×C₈₉	C₄⋊Q₈	1	2	yes	—
74	C₂³×C₈₉	#135	1	8	yes	—
75	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₈₉	—	2	8	no	—
77	C₂×C₄×C₈₉	—	2	16	no	—
78	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	8	yes	—
79	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
80	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₈₉	#127	1	2	yes	—
83	C₂×C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
84	C₂×C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
85	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
92	C₂×Dic₈₉	—	4	8	no	—
93	C₂×Dic₈₉	—	4	16	no	—
94	C₂×Dic₈₉	—	8	16	no	—
95	C₈×C₈₉	C₂×C₄⋊C₄	1	8	yes	—
96	C₈×C₈₉	C₂×C₄⋊C₄	1	8	yes	—
97	C₂²×C₈₉	#384	1	4	yes	—
98	C₂²×C₈₉	#427	1	8	yes	—
99	C₂²×C₈₉	#495	1	8	yes	—
100	C₂²×C₈₉	#499	1	8	yes	—
101	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
103	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₈₉	C₂×#135	1	4	yes	—
105	C₄×C₈₉	—	2	8	no	—
106	C₄×C₈₉	#384	1	4	yes	—
107	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
110	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
111	Dic₈₉	—	8	8	no	—
112	C₂×#1846	—	89	1	no	maximal, Sylow
113	C₂×C₈₉	#1846	1	8	yes	complemented
114	C₂×C₈₉	C₂×#384	1	1	yes	characteristic
115	C₂×C₈₉	C₂×#427	1	2	yes	—
116	C₂×C₈₉	C₂×#495	1	2	yes	—
117	C₂×C₈₉	C₂×#499	1	2	yes	—
118	C₂²×#107	—	89	2	no	—
119	C₂²×#106	—	89	2	no	—
120	C₂²×C₄×C₈	D₈₉	1	1	yes	characteristic
121	C₂×#382	—	89	2	no	—
122	#1846	—	89	8	no	—
123	C₈₉	C₂×#1846	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
125	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
126	C₂²×C₄⋊C₄	—	89	4	no	—
127	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
128	C₂×#107	—	89	8	no	—
129	C₂×#107	—	178	8	no	—
130	C₂×#106	—	89	8	no	—
131	C₂×#106	—	178	8	no	—
132	C₂×C₄×C₈	Dic₈₉	1	8	yes	complemented
133	C₂×C₄×C₈	C₂×D₈₉	1	4	yes	—
134	#382	—	89	8	no	—
135	#382	—	178	4	no	—
136	C₂×#95	—	178	4	no	—
137	C₂³×C₄	—	178	4	no	—
138	C₂³×C₄	D₃₅₆	1	2	yes	—
139	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
140	C₂²×C₈	Dic₁₇₈	1	4	yes	—
141	C₂²×C₈	Dic₁₇₈	1	4	yes	—
142	C₂²×C₈	D₃₅₆	1	4	yes	—
143	C₂²×C₈	D₃₅₆	1	4	yes	—
144	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
145	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
146	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
147	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
148	C₂×C₄⋊C₄	—	89	8	no	—
149	C₂×C₄⋊C₄	—	178	4	no	—
150	C₂×C₄⋊C₄	—	178	8	no	—
151	C₂×C₄⋊C₄	—	178	8	no	—
152	C₂×C₄²	Dic₁₇₈	1	4	yes	—
153	C₂×C₄²	D₃₅₆	1	2	yes	—
154	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
155	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
156	#107	—	178	16	no	—
157	#106	—	178	16	no	—
158	C₄×C₈	C₂×Dic₈₉	1	16	yes	complemented
159	#95	—	178	16	no	—
160	C₂⁴	C₂×D₃₅₆	1	1	yes	characteristic
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	356	8	no	—
163	C₂²×C₄	—	356	16	no	—
164	C₂²×C₄	—	178	8	no	—
165	C₂²×C₄	C₄⋊Dic₈₉	1	8	yes	—
166	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
167	C₂²×C₄	C₂×D₃₅₆	1	4	yes	—
168	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
170	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
171	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
172	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
173	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
174	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
175	C₂×C₈	C₄⋊Dic₈₉	1	16	yes	—
176	C₂×C₈	C₄⋊Dic₈₉	1	16	yes	—
177	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
178	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
179	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
180	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
181	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
182	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
183	C₄⋊C₄	—	356	8	no	—
184	C₄⋊C₄	—	178	8	no	—
185	C₄⋊C₄	—	178	8	no	—
186	C₄²	C₄⋊Dic₈₉	1	8	yes	—
187	C₄²	C₂²×Dic₈₉	1	4	yes	—
188	C₄²	C₂×Dic₁₇₈	1	4	yes	—
189	C₂³	C₂×C₄⋊Dic₈₉	1	2	yes	—
190	C₂³	#15668009	1	2	yes	—
191	C₂³	#15668012	1	2	yes	—
192	C₂³	#15668034	1	8	yes	—
193	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
194	C₂×C₄	—	2	16	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	356	8	no	—
197	C₂×C₄	—	356	16	no	—
198	C₂×C₄	—	712	16	no	—
199	C₂×C₄	C₂×C₄⋊Dic₈₉	1	8	yes	—
200	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
201	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
203	C₂×C₄	#15668009	1	2	yes	—
204	C₂×C₄	#15668012	1	4	yes	—
205	C₂×C₄	#15668012	1	4	yes	—
206	C₂×C₄	#15668012	1	2	yes	—
207	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
208	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
209	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
212	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
213	C₈	C₂×C₄⋊Dic₈₉	1	8	yes	—
214	C₈	C₂×C₄⋊Dic₈₉	1	8	yes	—
215	C₂²	#19151468	1	4	yes	—
216	C₂²	#19151636	1	8	yes	—
217	C₂²	#19151982	1	8	yes	—
218	C₂²	#19152000	1	8	yes	—
219	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
220	C₂²	C₂×#15668009	1	1	yes	characteristic
221	C₂²	C₂×#15668012	1	1	yes	characteristic
222	C₂²	C₂×#15668034	1	4	yes	—
223	C₄	—	2	8	no	—
224	C₄	—	712	8	no	—
225	C₄	#19151468	1	4	yes	—
226	C₄	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
227	C₄	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
228	C₄	C₂×#15668012	1	1	yes	characteristic
229	C₄	C₂×#15668012	1	1	yes	characteristic
230	C₂	#26020416	1	8	yes	direct product factor
231	C₂	C₂×#19151468	1	1	yes	characteristic
232	C₂	C₂×#19151636	1	2	yes	—
233	C₂	C₂×#19151982	1	2	yes	—
234	C₂	C₂×#19152000	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂²×C₄×C₈×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₈₉
hypocenter	C₈₉	C₂×#1846
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #26020416	C₂	direct (N × H)
17. C₂×C₄×C₈×C₈₉	C₄	semidirect
41. C₄×C₈×C₈₉	C₂×C₄	semidirect
113. C₂×C₈₉	#1846	semidirect
123. C₈₉	C₂×#1846	semidirect
132. C₂×C₄×C₈	Dic₈₉	semidirect
158. C₄×C₈	C₂×Dic₈₉	semidirect
230. C₂	#26020416	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×#15667946	2
C₂²×#15667944	2
C₂×#19151464	2
#26020416	2
C₂²×C₄×C₈×C₈₉	2
C₂×#1846	89

Subgroups — C2×#26020416 #36554835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26020416 #36554835