Subgroups — C₂×#26022045 #36707482

Order 22784.

642 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667951	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₈₉	C₂	1	2	yes	maximal
3	#26022045	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151428	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151427	C₂	1	2	yes	maximal, complemented
6	C₂×#368×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×M₁₆×D₈₉	—	2	2	no	—
8	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
9	C₂×#15667951	C₄	1	4	yes	—
10	C₂×#15667951	C₄	1	4	yes	—
11	C₂×#15667951	C₂²	1	4	yes	—
12	C₂²×#13083549	C₂²	1	1	yes	characteristic
13	C₂×#15667948	C₂²	1	2	yes	—
14	C₂×#15667947	—	2	2	no	—
15	#108×D₈₉	—	2	4	no	—
16	#108×D₈₉	C₂²	1	8	yes	—
17	#19151428	C₂²	1	4	yes	complemented
18	#19151427	C₂²	1	8	yes	complemented
19	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#108×C₈₉	C₂²	1	2	yes	—
21	#368×C₈₉	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083557	—	4	2	no	—
25	M₁₆×D₈₉	—	2	8	no	—
26	M₁₆×D₈₉	—	4	4	no	—
27	C₂²×#10623494	D₄	1	1	yes	characteristic
28	C₂×C₈×D₈₉	—	4	2	no	—
29	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₈₉	D₄	1	2	yes	—
32	C₂×C₈×D₈₉	Q₈	1	2	yes	—
33	C₂×C₈×D₈₉	Q₈	1	2	yes	—
34	C₂×C₈×D₈₉	C₂³	1	2	yes	—
35	#15667951	C₂×C₄	1	16	yes	—
36	C₂×#13083549	C₂×C₄	1	2	yes	—
37	C₂×#13083549	C₂×C₄	1	2	yes	—
38	C₂×#13083549	C₂³	1	2	yes	—
39	#15667948	—	2	2	no	—
40	#15667948	C₂³	1	4	yes	—
41	#15667947	—	2	8	no	—
42	C₂×M₁₆×C₈₉	—	2	2	no	—
43	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
45	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
46	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
47	#108×C₈₉	—	2	2	no	—
48	#108×C₈₉	C₂³	1	4	yes	—
49	C₂³×D₈₉	Q₁₆	1	1	yes	characteristic
50	C₂²×Dic₈₉	Q₁₆	1	1	yes	characteristic
51	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
53	C₂×C₄×D₈₉	D₈	1	2	yes	—
54	C₂×C₄×D₈₉	D₈	1	2	yes	—
55	C₂×C₄×D₈₉	Q₁₆	1	2	yes	—
56	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
57	#13083558	—	2	4	no	—
58	#13083558	—	4	2	no	—
59	#13083557	—	4	8	no	—
60	C₂×#10623494	—	4	2	no	—
61	C₂×#10623494	C₄⋊C₄	1	2	yes	—
62	C₂×#10623494	C₄⋊C₄	1	2	yes	—
63	C₂×#10623494	C₂×D₄	1	2	yes	—
64	C₈×D₈₉	—	4	8	no	—
65	C₈×D₈₉	C₄⋊C₄	1	4	yes	—
66	C₈×D₈₉	C₄⋊C₄	1	4	yes	—
67	C₈×D₈₉	C₂²×C₄	1	4	yes	—
68	C₈×D₈₉	C₂×Q₈	1	4	yes	—
69	#13083549	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
71	M₁₆×C₈₉	—	2	4	no	—
72	M₁₆×C₈₉	—	4	2	no	—
73	C₂×C₈×C₈₉	—	4	2	no	—
74	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
80	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
81	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
82	C₂²×D₈₉	—	4	2	no	—
83	C₂²×D₈₉	#107	1	2	yes	—
84	C₂²×D₈₉	#107	1	2	yes	—
85	C₂²×D₈₉	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₈₉	#107	1	2	yes	—
93	C₂×Dic₈₉	#107	1	2	yes	—
94	C₂×Dic₈₉	C₂×Q₁₆	1	2	yes	—
95	C₄×D₈₉	#107	1	4	yes	—
96	C₄×D₈₉	#107	1	4	yes	—
97	C₄×D₈₉	C₂×C₄⋊C₄	1	4	yes	—
98	C₄×D₈₉	C₂×D₈	1	4	yes	—
99	C₈×C₈₉	—	4	4	no	—
100	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
104	#10623494	—	4	4	no	—
105	#10623494	C₂×C₄⋊C₄	1	4	yes	—
106	C₂²×C₈₉	—	2	2	no	—
107	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
108	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
109	C₂²×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
110	C₂×D₈₉	—	2	4	no	—
111	C₂×D₈₉	—	4	2	no	—
112	C₂×D₈₉	—	4	2	no	—
113	C₂×D₈₉	C₂×#107	1	4	yes	—
114	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
115	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
116	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
118	Dic₈₉	C₂×#107	1	4	yes	—
119	C₂×#2161	—	89	1	no	maximal, Sylow
120	C₂×C₈₉	—	2	1	no	—
121	C₂×C₈₉	—	2	1	no	—
122	C₂×C₈₉	#2161	1	2	yes	complemented
123	C₂×C₈₉	C₂²×#107	1	1	yes	characteristic
124	D₈₉	—	2	4	no	—
125	C₂²×M₃₂	—	89	1	no	—
126	C₂²×#108	—	89	2	no	—
127	#2161	—	89	8	no	—
128	C₂×#368	—	89	1	no	—
129	C₂×#368	—	89	2	no	—
130	C₂×#368	D₈₉	1	1	yes	characteristic, complemented
131	C₈₉	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
132	C₂²×M₁₆	—	178	2	no	—
133	C₂³×C₈	—	89	1	no	—
134	C₂×M₃₂	—	89	4	no	—
135	C₂×M₃₂	—	89	4	no	—
136	C₂×M₃₂	—	89	4	no	—
137	C₂²×C₁₆	—	89	1	no	—
138	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
139	C₂×#108	—	89	2	no	—
140	C₂×#108	—	89	8	no	—
141	C₂×#108	—	178	4	no	—
142	C₂×#108	—	178	2	no	—
143	C₂×#108	C₂×D₈₉	1	2	yes	—
144	#368	—	89	4	no	—
145	#368	—	89	8	no	—
146	#368	C₂×D₈₉	1	4	yes	complemented
147	C₂³×C₄	—	89	1	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	356	2	no	—
150	C₂×M₁₆	—	356	4	no	—
151	C₂×M₁₆	—	178	8	no	—
152	C₂×M₁₆	—	178	2	no	—
153	C₂²×C₈	—	89	1	no	—
154	C₂²×C₈	—	89	2	no	—
155	C₂²×C₈	—	89	2	no	—
156	C₂²×C₈	—	89	2	no	—
157	C₂²×C₈	—	89	2	no	—
158	C₂²×C₈	—	89	2	no	—
159	C₂²×C₈	—	89	2	no	—
160	C₂²×C₈	—	356	2	no	—
161	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
162	M₃₂	—	89	16	no	—
163	C₂×C₁₆	—	89	2	no	—
164	C₂×C₁₆	—	89	2	no	—
165	C₂×C₁₆	—	89	2	no	—
166	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
167	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
168	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
169	#108	—	2	2	no	—
170	#108	—	89	4	no	—
171	#108	—	178	8	no	—
172	#108	—	178	2	no	—
173	#108	C₂²×D₈₉	1	4	yes	—
174	C₂⁴	—	89	1	no	—
175	C₂²×C₄	—	89	1	no	—
176	C₂²×C₄	—	89	2	no	—
177	C₂²×C₄	—	89	2	no	—
178	C₂²×C₄	—	89	2	no	—
179	C₂²×C₄	—	89	2	no	—
180	C₂²×C₄	—	89	2	no	—
181	C₂²×C₄	—	89	2	no	—
182	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
183	M₁₆	—	2	4	no	—
184	M₁₆	—	4	2	no	—
185	M₁₆	—	356	8	no	—
186	M₁₆	—	356	2	no	—
187	M₁₆	—	178	4	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	89	2	no	—
190	C₂×C₈	—	89	2	no	—
191	C₂×C₈	—	89	2	no	—
192	C₂×C₈	—	89	4	no	—
193	C₂×C₈	—	89	4	no	—
194	C₂×C₈	—	89	4	no	—
195	C₂×C₈	—	89	4	no	—
196	C₂×C₈	—	356	8	no	—
197	C₂×C₈	—	356	2	no	—
198	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
200	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
201	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
202	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
203	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
204	C₁₆	—	89	4	no	—
205	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
206	C₂³	—	89	2	no	—
207	C₂³	—	89	2	no	—
208	C₂³	—	89	2	no	—
209	C₂³	—	356	2	no	—
210	C₂³	Q₁₆×D₈₉	1	1	yes	characteristic
211	C₂×C₄	—	89	2	no	—
212	C₂×C₄	—	89	2	no	—
213	C₂×C₄	—	89	2	no	—
214	C₂×C₄	—	89	4	no	—
215	C₂×C₄	—	89	4	no	—
216	C₂×C₄	—	89	4	no	—
217	C₂×C₄	—	89	4	no	—
218	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
219	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
220	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
221	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
222	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
223	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
224	C₈	—	4	4	no	—
225	C₈	—	89	4	no	—
226	C₈	—	356	4	no	—
227	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
228	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
229	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
230	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
231	C₂²	—	2	2	no	—
232	C₂²	—	89	4	no	—
233	C₂²	—	356	2	no	—
234	C₂²	—	356	2	no	—
235	C₂²	—	178	4	no	—
236	C₂²	#107×D₈₉	1	1	yes	characteristic
237	C₂²	#107×D₈₉	1	1	yes	characteristic
238	C₂²	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
239	C₄	—	89	4	no	—
240	C₄	#107×D₈₉	1	1	yes	characteristic
241	C₄	#107×D₈₉	1	1	yes	characteristic
242	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
243	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	—	178	4	no	—
247	C₂	#26022045	1	2	yes	direct product factor
248	C₂	C₂×#107×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#368×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×D₈₉	Q₁₆
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#368	D₈₉
hypocenter	C₈₉	C₂×#2161
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
3. #26022045	C₂	direct (N × H)
4. C₂×#19151428	C₂	semidirect
5. C₂×#19151427	C₂	semidirect
6. C₂×#368×C₈₉	C₂	semidirect
17. #19151428	C₂²	semidirect
18. #19151427	C₂²	semidirect
21. #368×C₈₉	C₂²	semidirect
122. C₂×C₈₉	#2161	semidirect
130. C₂×#368	D₈₉	semidirect
131. C₈₉	C₂×#2161	semidirect
146. #368	C₂×D₈₉	semidirect
247. C₂	#26022045	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×#15667951	2
C₂×#108×D₈₉	2
#26022045	2
C₂×#19151428	2
C₂×#19151427	2
C₂×#368×C₈₉	2
C₂×#2161	89

Subgroups — C2×#26022045 #36707482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022045 #36707482