Subgroups — C₂×#26022038 #36707460

Order 22784.

718 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667952	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151636	C₂	1	1	yes	maximal, characteristic
3	#26022038	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151425	C₂	1	2	yes	maximal, complemented
5	C₂×#19151423	C₂	1	2	yes	maximal, complemented
6	C₂²×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#15667971	—	2	1	no	—
8	C₂×#15667952	C₄	1	8	yes	complemented
9	C₂×#15667952	C₂²	1	4	yes	—
10	C₂²×#13083549	C₂²	1	2	yes	—
11	#19151636	—	2	2	no	—
12	#19151636	C₂²	1	4	yes	—
13	C₂×#15667946	C₂²	1	1	yes	characteristic
14	C₂×#15667946	C₂²	1	1	yes	characteristic
15	C₂×#15667944	—	2	1	no	—
16	#19151425	C₂²	1	8	yes	complemented
17	#19151423	C₂²	1	8	yes	complemented
18	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₈₉	—	2	1	no	—
20	C₂×#107×C₈₉	—	2	1	no	—
21	C₂×#107×C₈₉	—	2	2	no	—
22	C₂×#107×C₈₉	C₂²	1	2	yes	—
23	C₂×#107×C₈₉	C₂²	1	4	yes	—
24	#15667971	—	2	4	no	—
25	#15667971	—	4	2	no	—
26	C₂×C₄⋊Dic₈₉	—	2	1	no	—
27	C₂×C₄⋊Dic₈₉	—	2	1	no	—
28	C₂×C₂²⋊Dic₈₉	—	4	1	no	—
29	C₂×C₄×Dic₈₉	—	4	1	no	—
30	#15667952	C₂×C₄	1	16	yes	complemented
31	C₂×#13083549	C₂×C₄	1	8	yes	complemented
32	C₂×#13083549	C₂³	1	4	yes	—
33	#15667946	—	2	1	no	—
34	#15667946	—	2	1	no	—
35	#15667946	C₂³	1	2	yes	—
36	#15667946	C₂³	1	2	yes	—
37	#15667944	—	2	4	no	—
38	C₂³×C₄×C₈₉	—	4	1	no	—
39	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	D₄	1	2	yes	—
43	C₂²×C₈×C₈₉	Q₈	1	4	yes	—
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
47	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₈₉	—	4	2	no	—
49	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
50	#107×C₈₉	—	2	4	no	—
51	#107×C₈₉	—	2	2	no	—
52	#107×C₈₉	C₂³	1	4	yes	—
53	C₂²×Dic₈₉	—	4	1	no	—
54	C₂²×Dic₈₉	—	4	1	no	—
55	C₄⋊Dic₈₉	—	2	2	no	—
56	C₄⋊Dic₈₉	—	2	2	no	—
57	C₄⋊Dic₈₉	—	4	1	no	—
58	C₄⋊Dic₈₉	—	4	1	no	—
59	C₂²⋊Dic₈₉	—	4	4	no	—
60	C₄×Dic₈₉	—	4	4	no	—
61	#13083549	C₂²×C₄	1	8	yes	complemented
62	C₂⁴×C₈₉	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	—	4	4	no	—
64	C₂²×C₄×C₈₉	—	4	2	no	—
65	C₂²×C₄×C₈₉	—	8	4	no	—
66	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₈₉	D₈	1	2	yes	—
68	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	Q₁₆	1	4	yes	—
70	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
73	C₂×C₈×C₈₉	C₄⋊C₄	1	8	yes	—
74	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
80	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₈₉	—	2	4	no	—
82	C₄⋊C₄×C₈₉	—	4	4	no	—
83	C₄⋊C₄×C₈₉	—	4	2	no	—
84	C₂³×C₈₉	—	2	2	no	—
85	C₂³×C₈₉	—	2	2	no	—
86	C₂³×C₈₉	#107	1	4	yes	—
87	C₂³×C₈₉	C₂×D₈	1	2	yes	—
88	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₈₉	—	4	4	no	—
90	C₂×C₄×C₈₉	—	8	8	no	—
91	C₂×C₄×C₈₉	—	8	4	no	—
92	C₂×C₄×C₈₉	#107	1	8	yes	—
93	C₂×C₄×C₈₉	#107	1	4	yes	—
94	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
95	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
99	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×Dic₈₉	—	4	2	no	—
102	C₂×Dic₈₉	—	4	2	no	—
103	C₂×Dic₈₉	—	8	2	no	—
104	C₂×Dic₈₉	—	8	2	no	—
105	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₈×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
107	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
109	C₂²×C₈₉	—	2	2	no	—
110	C₂²×C₈₉	—	2	2	no	—
111	C₂²×C₈₉	—	2	8	no	—
112	C₂²×C₈₉	C₂×#107	1	4	yes	—
113	C₂²×C₈₉	C₂×#107	1	2	yes	—
114	C₂²×C₈₉	#509	1	2	yes	—
115	C₂²×C₈₉	#510	1	2	yes	—
116	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
117	C₄×C₈₉	—	8	4	no	—
118	C₄×C₈₉	C₂×#107	1	4	yes	—
119	C₄×C₈₉	C₂×#107	1	2	yes	—
120	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
122	Dic₈₉	—	8	2	no	—
123	Dic₈₉	—	8	2	no	—
124	C₂×#2159	—	89	1	no	maximal, Sylow
125	C₂×C₈₉	—	2	4	no	—
126	C₂×C₈₉	#2159	1	4	yes	complemented
127	C₂×C₈₉	C₂²×#107	1	1	yes	characteristic
128	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
129	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
130	C₂²×M₃₂	—	89	1	no	—
131	C₂×#427	—	89	1	no	—
132	C₂²×#107	D₈₉	1	1	yes	characteristic
133	#2159	—	89	8	no	—
134	C₂×#367	—	89	2	no	—
135	C₂×#366	—	89	2	no	—
136	C₈₉	C₂×#2159	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
138	C₂×#117	—	178	1	no	—
139	C₂²×C₄⋊C₄	—	2	1	no	—
140	C₂×M₃₂	—	89	8	no	—
141	C₂×M₃₂	—	89	4	no	—
142	C₂²×C₁₆	—	89	2	no	—
143	#427	—	89	4	no	—
144	#427	—	178	2	no	—
145	C₂×#107	—	2	1	no	—
146	C₂×#107	—	2	2	no	—
147	C₂×#107	—	89	1	no	—
148	C₂×#107	—	89	1	no	—
149	C₂×#107	C₂×D₈₉	1	2	yes	—
150	C₂×#107	C₂×D₈₉	1	4	yes	—
151	C₂×#106	—	178	1	no	—
152	#367	—	89	8	no	—
153	#366	—	89	8	no	—
154	C₂³×C₄	—	4	1	no	—
155	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
156	C₂²×C₈	Dic₁₇₈	1	4	yes	—
157	C₂²×C₈	C₄×D₈₉	1	4	yes	—
158	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
160	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
162	#117	—	356	2	no	—
163	#117	—	178	4	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	4	2	no	—
167	C₂×C₄⋊C₄	—	4	1	no	—
168	C₂×C₄⋊C₄	—	178	1	no	—
169	C₂×C₄⋊C₄	—	178	1	no	—
170	C₂×C₂²⋊C₄	—	356	1	no	—
171	C₂×C₄²	—	356	1	no	—
172	M₃₂	—	89	16	no	—
173	C₂×C₁₆	—	89	8	no	—
174	C₂×C₁₆	—	89	4	no	—
175	#107	—	2	4	no	—
176	#107	—	2	2	no	—
177	#107	—	89	2	no	—
178	#107	—	89	2	no	—
179	#107	—	178	1	no	—
180	#107	—	178	1	no	—
181	#107	C₂²×D₈₉	1	4	yes	—
182	#106	—	178	4	no	—
183	C₂⁴	#13083559	1	1	yes	characteristic
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	8	4	no	—
187	C₂²×C₄	—	356	1	no	—
188	C₂²×C₄	—	356	1	no	—
189	C₂²×C₄	#13083553	1	4	yes	—
190	C₂²×C₄	#13083559	1	2	yes	—
191	C₂²×C₄	#13083559	1	1	yes	characteristic
192	C₂²×C₄	#13083564	1	4	yes	—
193	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
195	C₂×C₈	#13083553	1	4	yes	—
196	C₂×C₈	#13083553	1	8	yes	—
197	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
198	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
199	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
200	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
201	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
202	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
203	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	356	1	no	—
208	C₄⋊C₄	—	356	1	no	—
209	C₄⋊C₄	—	178	2	no	—
210	C₄⋊C₄	—	178	2	no	—
211	C₂²⋊C₄	—	356	4	no	—
212	C₄²	—	356	4	no	—
213	C₁₆	—	89	8	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	#15667945	1	4	yes	—
217	C₂³	C₂×#13083559	1	2	yes	—
218	C₂³	C₂×#13083559	1	1	yes	characteristic
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	8	8	no	—
221	C₂×C₄	—	8	4	no	—
222	C₂×C₄	—	356	2	no	—
223	C₂×C₄	—	356	2	no	—
224	C₂×C₄	—	712	2	no	—
225	C₂×C₄	—	712	2	no	—
226	C₂×C₄	#15667945	1	8	yes	—
227	C₂×C₄	#15667945	1	4	yes	—
228	C₂×C₄	C₂×#13083553	1	4	yes	—
229	C₂×C₄	C₂×#13083553	1	2	yes	—
230	C₂×C₄	C₂×#13083559	1	2	yes	—
231	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
232	C₂×C₄	C₂×#13083564	1	4	yes	—
233	C₂×C₄	C₂×#13083564	1	2	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
235	C₈	C₂×#13083553	1	2	yes	—
236	C₈	C₂×#13083553	1	4	yes	—
237	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
238	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	C₂×#15667945	1	4	yes	—
243	C₂²	C₂×#15667945	1	2	yes	—
244	C₂²	#19152034	1	2	yes	—
245	C₂²	#19152039	1	2	yes	—
246	C₂²	C₂²×#13083559	1	1	yes	characteristic
247	C₄	—	8	4	no	—
248	C₄	—	712	2	no	—
249	C₄	—	712	2	no	—
250	C₄	C₂×#15667945	1	4	yes	—
251	C₄	C₂×#15667945	1	2	yes	—
252	C₄	C₂²×#13083553	1	1	yes	characteristic
253	C₄	C₂²×#13083564	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#26022038	1	4	yes	direct product factor
256	C₂	C₂²×#15667945	1	1	yes	characteristic
257	C₂	C₂×#19152034	1	1	yes	characteristic
258	C₂	C₂×#19152039	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂²×#107×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂⁴×C₈₉	D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂²×#107	D₈₉
hypocenter	C₈₉	C₂×#2159
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. #26022038	C₂	direct (N × H)
4. C₂×#19151425	C₂	semidirect
5. C₂×#19151423	C₂	semidirect
8. C₂×#15667952	C₄	semidirect
16. #19151425	C₂²	semidirect
17. #19151423	C₂²	semidirect
30. #15667952	C₂×C₄	semidirect
31. C₂×#13083549	C₂×C₄	semidirect
61. #13083549	C₂²×C₄	semidirect
126. C₂×C₈₉	#2159	semidirect
136. C₈₉	C₂×#2159	semidirect
255. C₂	#26022038	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×#15667952	2
C₂×#19151636	2
#26022038	2
C₂×#19151425	2
C₂×#19151423	2
C₂²×#107×C₈₉	2
C₂×#2159	89

Subgroups — C2×#26022038 #36707460

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022038 #36707460