Subgroups — C₂×#26028619 #36760413

Order 22784.

2838 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×#15668006	C₂	1	2	yes	maximal, complemented
2	#26028619	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19151500	C₂	1	3	yes	maximal
4	C₂×#19151496	C₂	1	3	yes	maximal, complemented
5	C₂×#19151489	C₂	1	6	yes	maximal, complemented
6	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₃₅₆	C₂²	1	1	yes	characteristic
8	C₂×#15668006	—	2	16	no	—
9	C₂×#15668006	C₂²	1	16	yes	complemented
10	C₂²×C₄⋊Dic₈₉	C₂²	1	3	yes	—
11	C₂²×#13083553	C₂²	1	6	yes	—
12	C₂²×#13083551	C₂²	1	6	yes	—
13	C₂²×#13083551	C₂²	1	6	yes	—
14	#19151500	C₂²	1	24	yes	—
15	#19151496	C₂²	1	24	yes	complemented
16	#19151489	C₂²	1	48	yes	complemented
17	C₂×#15667919	C₂²	1	6	yes	—
18	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
19	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
20	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	C₄³×C₈₉	C₂²	1	8	yes	complemented
22	C₂⁴×D₈₉	C₂³	1	2	yes	—
23	C₂²×D₃₅₆	—	2	12	no	—
24	C₂²×D₃₅₆	C₂³	1	4	yes	—
25	C₂³×Dic₈₉	C₂³	1	6	yes	—
26	#15668006	—	2	64	no	—
27	C₂×C₄⋊Dic₈₉	—	2	12	no	—
28	C₂×C₄⋊Dic₈₉	C₂³	1	12	yes	—
29	C₂×#13083553	—	2	24	no	—
30	C₂×#13083553	C₂³	1	24	yes	—
31	C₂×#13083551	—	2	24	no	—
32	C₂×#13083551	—	2	24	no	—
33	C₂×#13083551	C₂³	1	24	yes	—
34	C₂×#13083551	C₂³	1	24	yes	—
35	#15667919	C₂³	1	24	yes	—
36	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
37	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
38	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	—	2	12	no	—
40	C₂×C₄²×C₈₉	—	2	12	no	—
41	C₂×C₄²×C₈₉	D₄	1	8	yes	complemented
42	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
43	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
44	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
45	C₂³×D₈₉	—	2	4	no	—
46	C₂³×D₈₉	—	2	12	no	—
47	C₂³×D₈₉	—	2	12	no	—
48	C₂³×D₈₉	C₂⁴	1	4	yes	—
49	C₂×D₃₅₆	—	2	24	no	—
50	C₂×D₃₅₆	—	4	16	no	—
51	C₂²×Dic₈₉	—	2	12	no	—
52	C₂²×Dic₈₉	—	2	12	no	—
53	C₂²×Dic₈₉	—	2	12	no	—
54	C₂²×Dic₈₉	C₂⁴	1	12	yes	—
55	C₄⋊Dic₈₉	—	2	24	no	—
56	#13083553	—	2	48	no	—
57	#13083551	—	2	48	no	—
58	#13083551	—	2	48	no	—
59	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	—	2	6	no	—
61	C₂²×C₄×C₈₉	—	2	6	no	—
62	C₂²×C₄×C₈₉	—	2	6	no	—
63	C₂²×C₄×C₈₉	—	2	6	no	—
64	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
65	C₂²×C₄×C₈₉	#41	1	12	yes	—
66	C₂²×C₄×C₈₉	#41	1	12	yes	—
67	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
68	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
69	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
70	C₄²×C₈₉	—	2	24	no	—
71	C₄²×C₈₉	—	2	24	no	—
72	C₄²×C₈₉	C₂×D₄	1	16	yes	complemented
73	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
74	C₂³×C₈₉	C₂×#41	1	6	yes	—
75	C₂³×C₈₉	C₂×#41	1	6	yes	—
76	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
77	C₂²×D₈₉	—	2	4	no	—
78	C₂²×D₈₉	—	2	12	no	—
79	C₂²×D₈₉	—	2	12	no	—
80	C₂²×D₈₉	—	4	8	no	—
81	C₂²×D₈₉	—	4	24	no	—
82	C₂²×D₈₉	—	4	24	no	—
83	C₂×C₄×C₈₉	—	2	24	no	—
84	C₂×C₄×C₈₉	—	2	24	no	—
85	C₂×C₄×C₈₉	—	2	6	no	—
86	C₂×C₄×C₈₉	—	2	6	no	—
87	C₂×C₄×C₈₉	—	2	6	no	—
88	C₂×C₄×C₈₉	—	2	6	no	—
89	C₂×C₄×C₈₉	#126	1	16	yes	complemented
90	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
91	C₂×C₄×C₈₉	C₂×#41	1	12	yes	—
92	C₂×C₄×C₈₉	C₂×#41	1	12	yes	—
93	C₂×Dic₈₉	—	2	12	no	—
94	C₂×Dic₈₉	—	2	12	no	—
95	C₂×Dic₈₉	—	2	12	no	—
96	C₂×Dic₈₉	—	4	24	no	—
97	D₃₅₆	—	4	16	no	—
98	C₂²×C₈₉	C₂×#126	1	4	yes	—
99	C₂²×C₈₉	#529	1	12	yes	—
100	C₂²×C₈₉	#532	1	12	yes	—
101	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
102	C₂²×C₈₉	C₂²×#41	1	3	yes	—
103	C₂²×C₈₉	C₂²×#41	1	3	yes	—
104	C₂×D₈₉	—	4	4	no	—
105	C₂×D₈₉	—	4	12	no	—
106	C₂×D₈₉	—	4	12	no	—
107	C₂×D₈₉	—	8	16	no	—
108	C₄×C₈₉	—	2	12	no	—
109	C₄×C₈₉	—	2	12	no	—
110	C₄×C₈₉	C₂×#126	1	8	yes	complemented
111	Dic₈₉	—	4	12	no	—
112	C₂×#2860	—	89	1	no	maximal, Sylow
113	C₂×C₈₉	#2860	1	8	yes	complemented
114	C₂×C₈₉	C₂²×#126	1	1	yes	characteristic
115	C₂×C₈₉	C₂×#529	1	3	yes	—
116	C₂×C₈₉	C₂×#532	1	3	yes	—
117	D₈₉	—	8	4	no	—
118	C₂²×#126	—	89	2	no	—
119	#2860	—	89	16	no	—
120	C₂×#390	—	89	3	no	—
121	C₂×#389	—	89	3	no	—
122	C₂×#388	—	89	6	no	—
123	C₂×C₄³	D₈₉	1	1	yes	characteristic, complemented
124	C₈₉	C₂×#2860	1	1	yes	characteristic, complemented, Sylow
125	C₂³×D₄	—	89	1	no	—
126	C₂×#126	—	89	16	no	—
127	C₂×#126	—	178	16	no	—
128	C₂²×C₄⋊C₄	—	89	6	no	—
129	C₂²×C₄⋊C₄	—	89	3	no	—
130	C₂²×C₂²⋊C₄	—	89	6	no	—
131	C₂²×C₂²⋊C₄	—	89	6	no	—
132	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
133	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
134	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
135	#390	—	89	24	no	—
136	#389	—	89	24	no	—
137	#388	—	89	48	no	—
138	C₂×#95	—	89	6	no	—
139	C₄³	C₂×D₈₉	1	8	yes	complemented
140	C₂⁵	—	89	2	no	—
141	C₂²×D₄	—	89	4	no	—
142	C₂²×D₄	—	178	12	no	—
143	C₂³×C₄	—	89	6	no	—
144	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
145	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
146	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
147	#126	—	178	64	no	—
148	C₂×C₄⋊C₄	—	89	24	no	—
149	C₂×C₄⋊C₄	—	89	12	no	—
150	C₂×C₄⋊C₄	—	178	24	no	—
151	C₂×C₄⋊C₄	—	178	12	no	—
152	C₂×C₂²⋊C₄	—	89	24	no	—
153	C₂×C₂²⋊C₄	—	89	24	no	—
154	C₂×C₂²⋊C₄	—	178	24	no	—
155	C₂×C₂²⋊C₄	—	178	24	no	—
156	C₂×C₄²	—	2	12	no	—
157	C₂×C₄²	—	2	12	no	—
158	C₂×C₄²	D₃₅₆	1	8	yes	complemented
159	C₂×C₄²	C₂²×D₈₉	1	12	yes	—
160	C₂×C₄²	C₂²×D₈₉	1	12	yes	—
161	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
162	#95	—	89	24	no	—
163	C₂⁴	—	89	4	no	—
164	C₂⁴	—	178	4	no	—
165	C₂⁴	—	178	12	no	—
166	C₂⁴	—	178	12	no	—
167	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
168	C₂×D₄	—	356	16	no	—
169	C₂×D₄	—	178	24	no	—
170	C₂²×C₄	—	2	6	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	2	6	no	—
173	C₂²×C₄	—	2	6	no	—
174	C₂²×C₄	—	89	12	no	—
175	C₂²×C₄	—	178	12	no	—
176	C₂²×C₄	—	178	12	no	—
177	C₂²×C₄	—	178	12	no	—
178	C₂²×C₄	C₂×D₃₅₆	1	12	yes	—
179	C₂²×C₄	#13083574	1	12	yes	—
180	C₂²×C₄	#13083574	1	12	yes	—
181	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
182	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
183	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
184	C₄⋊C₄	—	178	48	no	—
185	C₄⋊C₄	—	178	24	no	—
186	C₂²⋊C₄	—	178	48	no	—
187	C₂²⋊C₄	—	178	48	no	—
188	C₄²	—	2	24	no	—
189	C₄²	—	2	24	no	—
190	C₄²	C₂×D₃₅₆	1	16	yes	complemented
191	C₂³	—	356	24	no	—
192	C₂³	—	356	24	no	—
193	C₂³	—	356	8	no	—
194	C₂³	—	178	4	no	—
195	C₂³	—	178	12	no	—
196	C₂³	—	178	12	no	—
197	C₂³	C₂²×D₃₅₆	1	2	yes	—
198	C₂³	C₂×#13083574	1	6	yes	—
199	C₂³	C₂×#13083574	1	6	yes	—
200	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
201	D₄	—	356	16	no	—
202	C₂×C₄	—	2	24	no	—
203	C₂×C₄	—	2	24	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	2	6	no	—
206	C₂×C₄	—	2	6	no	—
207	C₂×C₄	—	2	6	no	—
208	C₂×C₄	—	356	24	no	—
209	C₂×C₄	—	178	12	no	—
210	C₂×C₄	—	178	12	no	—
211	C₂×C₄	—	178	12	no	—
212	C₂×C₄	#15668006	1	16	yes	complemented
213	C₂×C₄	C₂²×D₃₅₆	1	12	yes	—
214	C₂×C₄	C₂×#13083574	1	12	yes	—
215	C₂×C₄	C₂×#13083574	1	12	yes	—
216	C₂²	—	356	4	no	—
217	C₂²	—	356	12	no	—
218	C₂²	—	356	12	no	—
219	C₂²	—	712	16	no	—
220	C₂²	C₂×#15668006	1	4	yes	—
221	C₂²	#19152132	1	12	yes	—
222	C₂²	#19152148	1	12	yes	—
223	C₂²	C₂³×D₃₅₆	1	1	yes	characteristic
224	C₂²	C₂²×#13083574	1	3	yes	—
225	C₂²	C₂²×#13083574	1	3	yes	—
226	C₄	—	2	12	no	—
227	C₄	—	2	12	no	—
228	C₄	—	356	12	no	—
229	C₄	C₂×#15668006	1	8	yes	complemented
230	C₂	—	712	4	no	—
231	C₂	#26028619	1	8	yes	direct product factor
232	C₂	C₂²×#15668006	1	1	yes	characteristic
233	C₂	C₂×#19152132	1	3	yes	—
234	C₂	C₂×#19152148	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₈₉
commutator, cosocle	C₂³×C₈₉	C₂⁵
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×C₄³×C₈₉	C₂
socle	C₂⁴×C₈₉	C₂⁴
prime radical	C₂³×D₃₅₆	C₂²
hypercenter	C₂×C₄³	D₈₉
hypocenter	C₈₉	C₂×#2860
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
1. C₂²×#15668006	C₂	semidirect
2. #26028619	C₂	direct (N × H)
4. C₂×#19151496	C₂	semidirect
5. C₂×#19151489	C₂	semidirect
6. C₂×C₄³×C₈₉	C₂	semidirect
9. C₂×#15668006	C₂²	semidirect
15. #19151496	C₂²	semidirect
16. #19151489	C₂²	semidirect
21. C₄³×C₈₉	C₂²	semidirect
41. C₂×C₄²×C₈₉	D₄	semidirect
72. C₄²×C₈₉	C₂×D₄	semidirect
89. C₂×C₄×C₈₉	#126	semidirect
110. C₄×C₈₉	C₂×#126	semidirect
113. C₂×C₈₉	#2860	semidirect
123. C₂×C₄³	D₈₉	semidirect
124. C₈₉	C₂×#2860	semidirect
139. C₄³	C₂×D₈₉	semidirect
158. C₂×C₄²	D₃₅₆	semidirect
190. C₄²	C₂×D₃₅₆	semidirect
212. C₂×C₄	#15668006	semidirect
229. C₄	C₂×#15668006	semidirect
231. C₂	#26028619	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×#15668006	2
#26028619	2
C₂×#19151500	2
C₂×#19151496	2
C₂×#19151489	2
C₂×C₄³×C₈₉	2
C₂×#2860	89

Subgroups — C2×#26028619 #36760413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26028619 #36760413