Subgroups — C₂×#26022057 #36707501

Order 22784.

610 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151552	C₂	1	1	yes	maximal, characteristic
2	#26022057	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151416	C₂	1	2	yes	maximal, complemented
4	C₂×#19151377	C₂	1	2	yes	maximal
5	C₂×#19151374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#13083557	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
9	C₂×#15667969	C₄	1	1	yes	characteristic
10	C₂²×#13083549	C₂²	1	2	yes	—
11	#19151552	C₄	1	4	yes	—
12	#19151552	C₂²	1	4	yes	—
13	C₂×#15667923	C₄	1	1	yes	characteristic
14	C₂×#15667922	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
16	#19151416	C₂²	1	8	yes	complemented
17	#19151377	C₂²	1	8	yes	—
18	#19151374	C₂²	1	4	yes	complemented
19	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₄×C₁₆×C₈₉	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×#13083557	—	2	4	no	—
24	C₂×#13083557	C₂×C₄	1	4	yes	—
25	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₈₉	—	2	4	no	—
28	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×D₈₉	C₂³	1	2	yes	—
30	#15667969	C₈	1	4	yes	—
31	#15667969	C₂×C₄	1	4	yes	—
32	C₂×#13083553	C₈	1	2	yes	—
33	C₂×#13083551	C₈	1	2	yes	—
34	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×#13083549	—	2	4	no	—
36	C₂×#13083549	C₂³	1	4	yes	—
37	#15667923	C₂×C₄	1	4	yes	—
38	#15667922	C₂×C₄	1	4	yes	—
39	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
40	C₈×Dic₈₉	C₂³	1	2	yes	—
41	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₈₉	—	2	4	no	—
45	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
46	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₈₉	C₂³	1	2	yes	—
48	C₂³×D₈₉	C₂×C₈	1	1	yes	characteristic
49	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
50	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
51	C₂×C₄×D₈₉	—	2	2	no	—
52	C₂×C₄×D₈₉	—	2	2	no	—
53	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
54	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
55	#13083557	—	2	8	no	—
56	C₂×#10623494	—	2	2	no	—
57	C₂×#10623494	—	2	2	no	—
58	C₂×#10623494	C₂²×C₄	1	2	yes	—
59	C₂×#10623494	C₂²×C₄	1	2	yes	—
60	C₈×D₈₉	—	2	4	no	—
61	C₈×D₈₉	—	2	4	no	—
62	#13083553	C₂×C₈	1	8	yes	—
63	#13083551	C₂×C₈	1	8	yes	—
64	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
65	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
66	#13083549	—	2	4	no	—
67	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
70	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₈₉	#41	1	4	yes	—
73	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
74	C₄²×C₈₉	C₂×C₈	1	2	yes	—
75	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
76	C₁₆×C₈₉	—	2	4	no	—
77	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
78	C₂²×D₈₉	—	2	2	no	—
79	C₂²×D₈₉	—	2	4	no	—
80	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₈₉	—	2	4	no	—
82	C₂×C₄×C₈₉	#117	1	2	yes	—
83	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
87	C₂×Dic₈₉	—	2	2	no	—
88	C₂×Dic₈₉	—	2	4	no	—
89	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
90	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
91	C₄×D₈₉	—	2	2	no	—
92	C₄×D₈₉	—	2	2	no	—
93	C₄×D₈₉	—	2	4	no	—
94	C₈×C₈₉	#117	1	2	yes	—
95	C₈×C₈₉	C₂×M₁₆	1	4	yes	—
96	C₈×C₈₉	C₂×#41	1	2	yes	—
97	#10623494	—	2	2	no	—
98	#10623494	—	2	2	no	—
99	C₂²×C₈₉	#431	1	2	yes	—
100	C₂²×C₈₉	#504	1	4	yes	—
101	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
102	C₂×D₈₉	—	2	4	no	—
103	C₂×D₈₉	—	2	2	no	—
104	C₂×D₈₉	—	4	4	no	—
105	C₄×C₈₉	—	2	4	no	—
106	C₄×C₈₉	#431	1	2	yes	—
107	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
108	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
109	Dic₈₉	—	2	4	no	—
110	Dic₈₉	—	2	2	no	—
111	C₂×#2164	—	89	1	no	maximal, Sylow
112	C₂×C₈₉	#2164	1	4	yes	complemented
113	C₂×C₈₉	C₂×#431	1	1	yes	characteristic
114	C₂×C₈₉	C₂×#504	1	2	yes	—
115	D₈₉	—	4	2	no	—
116	C₂×#405	—	89	1	no	—
117	#2164	—	89	8	no	—
118	C₂×C₄⋊C₁₆	—	89	2	no	—
119	C₂×C₂²⋊C₁₆	—	89	2	no	—
120	C₂×#346	—	89	1	no	—
121	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
122	C₈₉	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
123	C₂²×M₁₆	—	89	1	no	—
124	C₂³×C₈	—	89	1	no	—
125	C₂×#117	—	89	1	no	—
126	C₂²×C₁₆	—	89	2	no	—
127	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
128	#405	—	89	4	no	—
129	#405	—	89	4	no	—
130	C₂×#97	—	89	1	no	—
131	C₂×#97	—	89	1	no	—
132	C₂×C₄×C₈	—	89	1	no	—
133	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
134	C₄⋊C₁₆	—	89	8	no	—
135	C₂²⋊C₁₆	—	89	8	no	—
136	#346	—	89	4	no	—
137	C₄×C₁₆	C₂×D₈₉	1	4	yes	complemented
138	C₂³×C₄	—	89	1	no	—
139	C₂×M₁₆	—	89	4	no	—
140	C₂×M₁₆	—	178	4	no	—
141	C₂²×C₈	—	89	1	no	—
142	C₂²×C₈	—	89	1	no	—
143	C₂²×C₈	—	89	2	no	—
144	C₂²×C₈	—	89	2	no	—
145	C₂²×C₈	—	178	4	no	—
146	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
148	#117	—	89	4	no	—
149	#117	—	89	4	no	—
150	C₂×C₄⋊C₄	—	89	2	no	—
151	C₂×C₂²⋊C₄	—	89	2	no	—
152	C₂×C₄²	—	89	1	no	—
153	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
154	C₂×C₁₆	—	2	4	no	—
155	C₂×C₁₆	—	89	4	no	—
156	C₂×C₁₆	—	178	4	no	—
157	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
158	#97	—	89	4	no	—
159	#97	—	89	4	no	—
160	C₄×C₈	—	89	2	no	—
161	C₄×C₈	—	89	2	no	—
162	C₄×C₈	C₄×D₈₉	1	2	yes	—
163	C₄×C₈	C₂²×D₈₉	1	2	yes	—
164	C₂⁴	—	89	1	no	—
165	C₂²×C₄	—	89	2	no	—
166	C₂²×C₄	—	89	1	no	—
167	C₂²×C₄	—	89	2	no	—
168	C₂²×C₄	—	89	2	no	—
169	C₂²×C₄	—	178	2	no	—
170	C₂²×C₄	—	178	2	no	—
171	C₂²×C₄	C₈×D₈₉	1	2	yes	—
172	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
173	M₁₆	—	178	8	no	—
174	C₂×C₈	—	89	2	no	—
175	C₂×C₈	—	89	2	no	—
176	C₂×C₈	—	178	2	no	—
177	C₂×C₈	—	178	2	no	—
178	C₂×C₈	—	178	4	no	—
179	C₂×C₈	—	178	4	no	—
180	C₂×C₈	#13083557	1	4	yes	—
181	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
182	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
183	C₂×C₈	#13083574	1	4	yes	—
184	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
185	C₄⋊C₄	—	89	8	no	—
186	C₂²⋊C₄	—	89	8	no	—
187	C₄²	—	89	2	no	—
188	C₄²	—	89	2	no	—
189	C₄²	C₈×D₈₉	1	2	yes	—
190	C₄²	C₂×C₄×D₈₉	1	2	yes	—
191	C₁₆	—	2	4	no	—
192	C₁₆	—	178	4	no	—
193	C₂³	—	89	2	no	—
194	C₂³	—	178	2	no	—
195	C₂³	—	178	4	no	—
196	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	89	4	no	—
199	C₂×C₄	—	89	2	no	—
200	C₂×C₄	—	178	2	no	—
201	C₂×C₄	—	178	4	no	—
202	C₂×C₄	—	178	4	no	—
203	C₂×C₄	—	178	2	no	—
204	C₂×C₄	—	178	2	no	—
205	C₂×C₄	#15667969	1	2	yes	—
206	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
207	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
208	C₂×C₄	C₂×#13083557	1	2	yes	—
209	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
210	C₈	—	178	2	no	—
211	C₈	—	178	2	no	—
212	C₈	#15667969	1	2	yes	—
213	C₈	C₂×#13083557	1	4	yes	—
214	C₈	C₂×#13083574	1	2	yes	—
215	C₂²	—	356	4	no	—
216	C₂²	—	178	2	no	—
217	C₂²	—	178	4	no	—
218	C₂²	#19151649	1	2	yes	—
219	C₂²	#19152014	1	4	yes	—
220	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	178	2	no	—
223	C₄	—	178	4	no	—
224	C₄	#19151649	1	2	yes	—
225	C₄	C₂×#15667969	1	1	yes	characteristic
226	C₄	C₂²×#13083557	1	1	yes	characteristic
227	C₂	—	356	2	no	—
228	C₂	#26022057	1	4	yes	direct product factor
229	C₂	C₂×#19151649	1	1	yes	characteristic
230	C₂	C₂×#19152014	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂²×C₈
prime radical	C₂³×D₈₉	C₂×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	C₂×#2164
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #26022057	C₂	direct (N × H)
3. C₂×#19151416	C₂	semidirect
5. C₂×#19151374	C₂	semidirect
6. C₂×C₄×C₁₆×C₈₉	C₂	semidirect
16. #19151416	C₂²	semidirect
18. #19151374	C₂²	semidirect
21. C₄×C₁₆×C₈₉	C₂²	semidirect
112. C₂×C₈₉	#2164	semidirect
121. C₂×C₄×C₁₆	D₈₉	semidirect
122. C₈₉	C₂×#2164	semidirect
137. C₄×C₁₆	C₂×D₈₉	semidirect
228. C₂	#26022057	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×#19151552	2
#26022057	2
C₂×#19151416	2
C₂×#19151377	2
C₂×#19151374	2
C₂×C₄×C₁₆×C₈₉	2
C₂×#2164	89

Subgroups — C2×#26022057 #36707501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022057 #36707501