Subgroups — #36707618

Order 22784.

690 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	C₂×#19152014	C₂	1	2	yes	maximal, complemented
2	#26028965	C₂	1	1	yes	maximal, characteristic
3	#26022085	C₂	1	4	yes	maximal, complemented
4	C₁₆×D₃₅₆	C₂	1	4	yes	maximal, complemented
5	#26022008	C₂	1	1	yes	maximal, characteristic, complemented
6	#26021865	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	C₂²	1	2	yes	—
9	#19152148	C₄	1	1	yes	characteristic
10	#19152014	—	2	8	no	—
11	C₂×#15667952	C₂²	1	2	yes	—
12	C₂×#15667951	C₂²	1	4	yes	—
13	C₂×C₁₆×D₈₉	C₂²	1	4	yes	—
14	#19151677	C₄	1	4	yes	—
15	C₈×D₃₅₆	C₂²	1	4	yes	—
16	#19151622	C₄	1	1	yes	characteristic
17	#19151564	C₄	1	2	yes	—
18	C₄⋊#13083549	C₂²	1	4	yes	—
19	#19151377	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
24	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
25	C₂×#13083574	C₂×C₄	1	2	yes	—
26	#15668018	—	2	4	no	—
27	#15668018	—	2	4	no	—
28	C₂×#13083558	C₂×C₄	1	2	yes	—
29	C₂×#13083557	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₈₉	C₂³	1	4	yes	—
31	#15668012	C₈	1	1	yes	characteristic
32	#15668009	C₈	1	1	yes	characteristic
33	#15668000	C₈	1	2	yes	—
34	#15667986	C₈	1	4	yes	—
35	C₄×D₃₅₆	C₂×C₄	1	4	yes	—
36	#15667952	—	2	4	no	—
37	#15667951	—	2	8	no	—
38	C₂×#13083549	C₂³	1	4	yes	—
39	C₁₆×D₈₉	—	2	8	no	—
40	C₄⋊#10623494	C₂×C₄	1	4	yes	—
41	#15667924	C₂×C₄	1	8	yes	—
42	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆×C₈₉	—	2	2	no	—
46	C₂×C₁₆×C₈₉	D₄	1	4	yes	complemented
47	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
48	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₈₉	C₂³	1	2	yes	—
52	#13083574	—	2	4	no	—
53	#13083574	—	2	4	no	—
54	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
55	C₂×D₃₅₆	C₂×C₈	1	4	yes	—
56	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
57	C₂×C₂²⋊D₈₉	C₂×C₈	1	4	yes	—
58	C₂×C₄×D₈₉	C₂²×C₄	1	4	yes	—
59	#13083558	—	2	4	no	—
60	#13083557	—	2	8	no	—
61	C₂×#10623494	C₂²×C₄	1	4	yes	—
62	C₈×D₈₉	—	2	4	no	—
63	C₈×D₈₉	—	2	4	no	—
64	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
65	#13083551	C₂×C₈	1	8	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	—	2	2	no	—
70	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₈₉	#41	1	2	yes	—
74	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
75	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
77	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
78	C₁₆×C₈₉	—	2	2	no	—
79	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
80	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
81	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₈₉	C₂²×C₈	1	4	yes	—
83	C₂×C₄×C₈₉	—	2	4	no	—
84	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₈₉	#131	1	2	yes	—
89	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
90	C₂²⋊D₈₉	—	2	8	no	—
91	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
92	D₃₅₆	—	2	4	no	—
93	D₃₅₆	—	2	8	no	—
94	C₄×D₈₉	—	2	4	no	—
95	C₄×D₈₉	—	2	4	no	—
96	Dic₁₇₈	—	2	4	no	—
97	C₈×C₈₉	—	2	2	no	—
98	C₈×C₈₉	C₄×D₄	1	2	yes	—
99	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
101	#10623494	—	2	4	no	—
102	C₂²×C₈₉	—	2	2	no	—
103	C₂²×C₈₉	C₈×D₄	1	2	yes	—
104	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
105	C₂×D₈₉	—	2	8	no	—
106	C₂×D₈₉	—	2	4	no	—
107	C₄×C₈₉	—	2	2	no	—
108	C₄×C₈₉	C₈×D₄	1	2	yes	—
109	C₄×C₈₉	#504	1	4	yes	complemented
110	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
111	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
112	Dic₈₉	—	2	4	no	—
113	#33434	—	89	1	no	maximal, Sylow
114	C₂×C₈₉	—	2	2	no	—
115	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
116	C₂×C₈₉	C₂×#504	1	2	yes	—
117	D₈₉	—	4	4	no	—
118	C₂×#504	—	89	2	no	—
119	#2931	—	89	1	no	—
120	#2169	—	89	4	no	—
121	D₄×C₁₆	—	89	4	no	—
122	#2150	—	89	1	no	—
123	#2113	—	89	2	no	—
124	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
125	C₈₉	#33434	1	1	yes	characteristic, complemented, Sylow
126	C₂×#131	—	89	2	no	—
127	#532	—	89	1	no	—
128	#504	—	178	8	no	—
129	C₂×M₃₂	—	89	2	no	—
130	C₂×M₃₂	—	89	4	no	—
131	C₂²×C₁₆	—	89	4	no	—
132	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
133	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
134	#436	—	89	4	no	—
135	C₈×D₄	—	89	4	no	—
136	#423	—	89	1	no	—
137	#408	—	89	2	no	—
138	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
139	C₄⋊C₁₆	—	89	4	no	—
140	C₂²⋊C₁₆	—	89	8	no	—
141	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
142	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
143	C₂×#41	—	89	2	no	—
144	#131	—	178	4	no	—
145	#131	—	178	4	no	—
146	C₂×M₁₆	—	89	2	no	—
147	C₂×M₁₆	—	89	4	no	—
148	C₂²×C₈	—	89	4	no	—
149	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
151	C₄⋊Q₈	—	89	1	no	—
152	#127	—	89	1	no	—
153	#124	—	89	2	no	—
154	#121	—	89	4	no	—
155	C₄×D₄	—	89	4	no	—
156	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
157	M₃₂	—	178	4	no	—
158	M₃₂	—	178	8	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	89	4	no	—
161	C₂×C₁₆	—	178	8	no	—
162	C₂×C₁₆	D₃₅₆	1	4	yes	complemented
163	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
164	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
165	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
166	C₄⋊C₈	—	89	4	no	—
167	C₂²⋊C₈	—	89	8	no	—
168	C₄×C₈	C₄×D₈₉	1	2	yes	—
169	C₄×C₈	C₂²×D₈₉	1	2	yes	—
170	#41	—	178	4	no	—
171	#41	—	178	4	no	—
172	C₂×Q₈	—	89	2	no	—
173	C₂×D₄	—	89	4	no	—
174	C₂×D₄	—	89	4	no	—
175	C₂×D₄	—	89	2	no	—
176	C₂²×C₄	—	89	4	no	—
177	C₂²×C₄	C₈×D₈₉	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
179	M₁₆	—	178	4	no	—
180	M₁₆	—	178	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	89	4	no	—
183	C₂×C₈	—	178	4	no	—
184	C₂×C₈	—	178	4	no	—
185	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
186	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
187	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
188	C₂×C₈	#13083574	1	2	yes	—
189	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
190	C₄⋊C₄	—	89	4	no	—
191	C₂²⋊C₄	—	89	8	no	—
192	C₄²	C₈×D₈₉	1	1	yes	characteristic
193	C₄²	C₈×D₈₉	1	1	yes	characteristic
194	C₄²	C₂×C₄×D₈₉	1	2	yes	—
195	C₁₆	—	2	2	no	—
196	C₁₆	—	178	4	no	—
197	C₁₆	C₂×D₃₅₆	1	2	yes	—
198	C₁₆	C₂×D₃₅₆	1	2	yes	—
199	C₂³	—	89	4	no	—
200	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
201	Q₈	—	178	4	no	—
202	D₄	—	178	8	no	—
203	D₄	—	178	4	no	—
204	D₄	—	178	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	89	4	no	—
207	C₂×C₄	—	178	4	no	—
208	C₂×C₄	—	178	4	no	—
209	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
212	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
213	C₂×C₄	#15668018	1	2	yes	—
214	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	178	4	no	—
217	C₈	C₄×D₃₅₆	1	2	yes	—
218	C₈	C₂²×D₃₅₆	1	1	yes	characteristic
219	C₈	C₂×#13083574	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	178	4	no	—
222	C₂²	—	178	8	no	—
223	C₂²	C₈×D₃₅₆	1	2	yes	—
224	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	178	4	no	—
227	C₄	C₈×D₃₅₆	1	2	yes	—
228	C₄	#19152014	1	4	yes	complemented
229	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
230	C₄	C₂×#15668018	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	356	4	no	—
233	C₂	C₂×C₈×D₃₅₆	1	1	yes	characteristic
234	C₂	C₂×#19152014	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
prime radical	#19152148	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	#33434
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#19152014	C₂	semidirect
3. #26022085	C₂	semidirect
4. C₁₆×D₃₅₆	C₂	semidirect
5. #26022008	C₂	semidirect
6. #26021865	C₂	semidirect
7. C₂×C₄×C₁₆×C₈₉	C₂	semidirect
46. C₂×C₁₆×C₈₉	D₄	semidirect
109. C₄×C₈₉	#504	semidirect
124. C₂×C₄×C₁₆	D₈₉	semidirect
125. C₈₉	#33434	semidirect
162. C₂×C₁₆	D₃₅₆	semidirect
228. C₄	#19152014	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19152014	2
#26028965	2
#26022085	2
C₁₆×D₃₅₆	2
#26022008	2
#26021865	2
C₂×C₄×C₁₆×C₈₉	2
#33434	89