Subgroups — C₂×#26028615 #36760408

Order 22784.

2558 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15668004	C₂	1	2	yes	maximal
2	#26028615	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19151496	C₂	1	8	yes	maximal
4	C₂×#19151468	C₂	1	2	yes	maximal
5	C₂×#19151464	C₂	1	2	yes	maximal
6	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#15668004	—	2	16	no	—
8	C₂×#15668004	C₂²	1	16	yes	—
9	C₂²×C₄⋊Dic₈₉	C₂²	1	8	yes	—
10	C₂²×C₄⋊Dic₈₉	C₂²	1	4	yes	—
11	C₂²×#13083553	C₂²	1	8	yes	—
12	#19151496	C₂²	1	64	yes	—
13	#19151468	C₂²	1	16	yes	—
14	#19151464	C₂²	1	16	yes	—
15	C₂×#15667919	C₂²	1	8	yes	—
16	C₂²×C₄²×C₈₉	C₂²	1	4	yes	—
17	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
20	C₄³×C₈₉	C₂²	1	8	yes	—
21	C₂³×Dic₈₉	C₂³	1	8	yes	—
22	#15668004	—	2	64	no	—
23	C₂×C₄⋊Dic₈₉	—	2	16	no	—
24	C₂×C₄⋊Dic₈₉	—	2	32	no	—
25	C₂×C₄⋊Dic₈₉	C₂³	1	16	yes	—
26	C₂×C₄⋊Dic₈₉	C₂³	1	32	yes	—
27	C₂×#13083553	—	2	32	no	—
28	C₂×#13083553	C₂³	1	32	yes	—
29	#15667919	C₂³	1	32	yes	—
30	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
31	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
32	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
34	C₂×C₄²×C₈₉	—	2	16	no	—
35	C₂×C₄²×C₈₉	D₄	1	8	yes	—
36	C₂×C₄²×C₈₉	Q₈	1	8	yes	—
37	C₂×C₄²×C₈₉	Q₈	1	8	yes	—
38	C₂×C₄²×C₈₉	C₂³	1	16	yes	—
39	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
40	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
41	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
42	C₂²×Dic₈₉	—	2	16	no	—
43	C₂²×Dic₈₉	—	2	32	no	—
44	C₂²×Dic₈₉	C₂⁴	1	16	yes	—
45	C₄⋊Dic₈₉	—	2	32	no	—
46	C₄⋊Dic₈₉	—	2	64	no	—
47	#13083553	—	2	64	no	—
48	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	—	2	8	no	—
50	C₂²×C₄×C₈₉	—	2	8	no	—
51	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
52	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
53	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
54	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
55	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
56	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₈₉	#41	1	16	yes	—
58	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
59	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
60	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
61	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
62	C₄²×C₈₉	—	2	32	no	—
63	C₄²×C₈₉	C₂×D₄	1	16	yes	—
64	C₄²×C₈₉	C₂×Q₈	1	16	yes	—
65	C₄²×C₈₉	C₂×Q₈	1	16	yes	—
66	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
67	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
68	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
69	C₂³×C₈₉	C₂×#41	1	8	yes	—
70	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
71	C₂×C₄×C₈₉	—	2	16	no	—
72	C₂×C₄×C₈₉	—	2	16	no	—
73	C₂×C₄×C₈₉	—	2	8	no	—
74	C₂×C₄×C₈₉	—	2	8	no	—
75	C₂×C₄×C₈₉	#125	1	32	yes	complemented
76	C₂×C₄×C₈₉	C₄⋊Q₈	1	16	yes	—
77	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
78	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
79	C₂×C₄×C₈₉	C₂²×Q₈	1	8	yes	—
80	C₂×C₄×C₈₉	C₂²×Q₈	1	8	yes	—
81	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
82	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
83	C₂×C₄×C₈₉	C₂×#41	1	16	yes	—
84	C₂×Dic₈₉	—	2	16	no	—
85	C₂×Dic₈₉	—	2	32	no	—
86	C₂×Dic₈₉	—	4	32	no	—
87	C₂²×C₈₉	C₂×#125	1	8	yes	—
88	C₂²×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
89	C₂²×C₈₉	#532	1	8	yes	—
90	C₂²×C₈₉	#533	1	8	yes	—
91	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
92	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
93	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
94	C₂²×C₈₉	C₂²×#41	1	4	yes	—
95	C₄×C₈₉	—	2	8	no	—
96	C₄×C₈₉	—	2	8	no	—
97	C₄×C₈₉	C₂×#125	1	16	yes	complemented
98	C₄×C₈₉	C₂×C₄⋊Q₈	1	8	yes	—
99	Dic₈₉	—	4	16	no	—
100	C₂×#2859	—	89	1	no	maximal, Sylow
101	C₂×C₈₉	#2859	1	8	yes	complemented
102	C₂×C₈₉	C₂²×#125	1	2	yes	—
103	C₂×C₈₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₈₉	C₂×#532	1	2	yes	—
105	C₂×C₈₉	C₂×#533	1	2	yes	—
106	C₂²×#125	—	89	2	no	—
107	#2859	—	89	16	no	—
108	C₂×#389	—	89	8	no	—
109	C₂×#384	—	89	2	no	—
110	C₂×#382	—	89	2	no	—
111	C₂×C₄³	D₈₉	1	1	yes	characteristic
112	C₈₉	C₂×#2859	1	1	yes	characteristic, complemented, Sylow
113	C₂×#125	—	89	16	no	—
114	C₂×#125	—	178	16	no	—
115	C₂²×C₄⋊C₄	—	89	8	no	—
116	C₂²×C₄⋊C₄	—	89	4	no	—
117	C₂²×C₄⋊C₄	—	89	8	no	—
118	C₂²×C₄²	C₂×D₈₉	1	4	yes	—
119	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
120	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
121	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
122	#389	—	89	64	no	—
123	#384	—	89	16	no	—
124	#382	—	89	16	no	—
125	C₂×#95	—	89	8	no	—
126	C₄³	C₂×D₈₉	1	8	yes	—
127	C₂³×C₄	—	89	8	no	—
128	C₂³×C₄	C₂²×D₈₉	1	2	yes	—
129	C₂³×C₄	C₂²×D₈₉	1	2	yes	—
130	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
131	C₂³×C₄	C₂²×D₈₉	1	2	yes	—
132	#125	—	178	64	no	—
133	C₂×C₄⋊C₄	—	89	16	no	—
134	C₂×C₄⋊C₄	—	89	32	no	—
135	C₂×C₄⋊C₄	—	89	32	no	—
136	C₂×C₄⋊C₄	—	178	32	no	—
137	C₂×C₄⋊C₄	—	178	16	no	—
138	C₂×C₄⋊C₄	—	178	32	no	—
139	C₂×C₄²	—	2	16	no	—
140	C₂×C₄²	Dic₁₇₈	1	8	yes	—
141	C₂×C₄²	Dic₁₇₈	1	8	yes	—
142	C₂×C₄²	D₃₅₆	1	8	yes	—
143	C₂×C₄²	C₂²×D₈₉	1	16	yes	—
144	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
145	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
146	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
147	#95	—	89	32	no	—
148	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	—	2	8	no	—
151	C₂²×C₄	—	89	16	no	—
152	C₂²×C₄	—	178	16	no	—
153	C₂²×C₄	—	178	32	no	—
154	C₂²×C₄	C₂×D₃₅₆	1	8	yes	—
155	C₂²×C₄	C₂×D₃₅₆	1	4	yes	—
156	C₂²×C₄	C₂×Dic₁₇₈	1	8	yes	—
157	C₂²×C₄	C₂×Dic₁₇₈	1	8	yes	—
158	C₂²×C₄	C₂×Dic₁₇₈	1	4	yes	—
159	C₂²×C₄	C₂×Dic₁₇₈	1	4	yes	—
160	C₂²×C₄	#13083574	1	16	yes	—
161	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
162	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
163	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
164	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
165	C₄⋊C₄	—	178	64	no	—
166	C₄⋊C₄	—	178	32	no	—
167	C₄⋊C₄	—	178	64	no	—
168	C₄²	—	2	32	no	—
169	C₄²	C₂×D₃₅₆	1	16	yes	—
170	C₄²	C₂×Dic₁₇₈	1	16	yes	—
171	C₄²	C₂×Dic₁₇₈	1	16	yes	—
172	C₂³	C₂²×D₃₅₆	1	2	yes	—
173	C₂³	C₂²×Dic₁₇₈	1	2	yes	—
174	C₂³	C₂²×Dic₁₇₈	1	2	yes	—
175	C₂³	C₂×#13083574	1	8	yes	—
176	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
177	C₂×C₄	—	2	16	no	—
178	C₂×C₄	—	2	16	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	356	32	no	—
182	C₂×C₄	—	178	16	no	—
183	C₂×C₄	—	178	32	no	—
184	C₂×C₄	#15668004	1	32	yes	complemented
185	C₂×C₄	#15668012	1	16	yes	—
186	C₂×C₄	C₂²×D₃₅₆	1	8	yes	—
187	C₂×C₄	C₂²×D₃₅₆	1	4	yes	—
188	C₂×C₄	C₂²×Dic₁₇₈	1	8	yes	—
189	C₂×C₄	C₂²×Dic₁₇₈	1	4	yes	—
190	C₂×C₄	C₂²×Dic₁₇₈	1	4	yes	—
191	C₂×C₄	C₂²×Dic₁₇₈	1	8	yes	—
192	C₂×C₄	C₂×#13083574	1	16	yes	—
193	C₂²	C₂×#15668004	1	8	yes	—
194	C₂²	C₂×#15668012	1	4	yes	—
195	C₂²	#19152148	1	8	yes	—
196	C₂²	#19152155	1	8	yes	—
197	C₂²	C₂³×D₃₅₆	1	1	yes	characteristic
198	C₂²	C₂³×Dic₁₇₈	1	1	yes	characteristic
199	C₂²	C₂³×Dic₁₇₈	1	1	yes	characteristic
200	C₂²	C₂²×#13083574	1	4	yes	—
201	C₄	—	2	8	no	—
202	C₄	—	2	8	no	—
203	C₄	—	356	16	no	—
204	C₄	C₂×#15668004	1	16	yes	complemented
205	C₄	C₂×#15668012	1	8	yes	—
206	C₂	#26028615	1	8	yes	direct product factor
207	C₂	C₂²×#15668004	1	2	yes	—
208	C₂	C₂²×#15668012	1	1	yes	characteristic
209	C₂	C₂×#19152148	1	2	yes	—
210	C₂	C₂×#19152155	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₈₉
commutator, cosocle	C₂³×C₈₉	C₂⁵
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂×C₄³×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂⁴
hypercenter	C₂×C₄³	D₈₉
hypocenter	C₈₉	C₂×#2859
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
2. #26028615	C₂	direct (N × H)
75. C₂×C₄×C₈₉	#125	semidirect
97. C₄×C₈₉	C₂×#125	semidirect
101. C₂×C₈₉	#2859	semidirect
112. C₈₉	C₂×#2859	semidirect
184. C₂×C₄	#15668004	semidirect
204. C₄	C₂×#15668004	semidirect
206. C₂	#26028615	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×#15668004	2
#26028615	2
C₂×#19151496	2
C₂×#19151468	2
C₂×#19151464	2
C₂×C₄³×C₈₉	2
C₂×#2859	89

Subgroups — C2×#26028615 #36760408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26028615 #36760408