Subgroups — #36707133

Order 22784.

1114 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26034963	C₂	1	1	yes	maximal, characteristic
2	#26033355	C₂	1	1	yes	maximal, characteristic, complemented
3	#26021879	C₂	1	6	yes	maximal, complemented
4	C₂×#19151381	C₂	1	6	yes	maximal, complemented
5	C₂×#504×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152434	C₄	1	1	yes	characteristic
7	#19152365	C₄	1	1	yes	characteristic
8	#19152364	C₄	1	6	yes	—
9	#19152363	—	2	4	no	—
10	C₂×#15668018	C₂²	1	6	yes	—
11	#131×D₈₉	—	2	4	no	—
12	#19152015	—	2	4	no	—
13	C₂×#15667952	C₂²	1	6	yes	complemented
14	#19151381	C₂²	1	4	yes	complemented
15	#19151381	C₂²	1	12	yes	complemented
16	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
17	#504×C₈₉	—	2	4	no	—
18	C₂×M₃₂×C₈₉	C₂²	1	3	yes	complemented
19	C₂²×C₁₆×C₈₉	C₂²	1	3	yes	complemented
20	#15668060	—	2	1	no	—
21	#15668059	C₈	1	4	yes	—
22	#15668058	C₈	1	4	yes	—
23	#15668057	—	2	3	no	—
24	C₂×#13083574	C₂×C₄	1	6	yes	—
25	#41×D₈₉	—	2	4	no	—
26	#15668020	—	2	12	no	—
27	#15668019	—	2	4	no	—
28	#15668018	—	2	6	no	—
29	#15668018	—	2	6	no	—
30	#15668018	C₂×C₄	1	12	yes	—
31	#15668018	C₂³	1	4	yes	—
32	C₂×#13083558	C₂×C₄	1	6	yes	—
33	C₂×#13083557	—	2	6	no	—
34	M₁₆×D₈₉	—	2	12	no	—
35	C₂×C₈×D₈₉	—	2	6	no	—
36	#15667952	—	2	6	no	—
37	#15667952	C₂³	1	4	yes	—
38	C₂×#13083549	—	2	6	no	—
39	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
40	#131×C₈₉	D₄	1	4	yes	—
41	#131×C₈₉	D₄	1	4	yes	—
42	C₂×M₁₆×C₈₉	C₂×C₄	1	3	yes	—
43	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
44	M₃₂×C₈₉	—	2	6	no	—
45	C₂×C₁₆×C₈₉	—	2	6	no	—
46	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₈₉	C₂³	1	3	yes	—
48	#13083575	—	2	4	no	—
49	#13083574	—	2	6	no	—
50	#13083574	—	2	6	no	—
51	#13083574	C₂×C₈	1	12	yes	—
52	#13083574	C₂²×C₄	1	4	yes	—
53	#13083573	—	2	12	no	—
54	C₂×Dic₁₇₈	C₂×C₈	1	6	yes	—
55	Q₈×D₈₉	—	2	4	no	—
56	C₂×D₃₅₆	C₂×C₈	1	6	yes	—
57	C₂×C₂²⋊D₈₉	—	2	6	no	—
58	D₄×D₈₉	—	2	12	no	—
59	C₂×C₄×D₈₉	—	2	6	no	—
60	#13083558	—	2	6	no	—
61	#13083558	C₂²×C₄	1	4	yes	—
62	#13083557	—	2	12	no	—
63	#13083557	—	2	4	no	—
64	C₂×#10623494	—	2	6	no	—
65	C₈×D₈₉	—	2	12	no	—
66	C₈×D₈₉	—	2	4	no	—
67	#13083549	—	2	4	no	—
68	#41×C₈₉	C₂²⋊C₄	1	4	yes	—
69	#41×C₈₉	C₂×C₈	1	4	yes	—
70	C₂×Q₈×C₈₉	M₁₆	1	1	yes	characteristic
71	C₂×D₄×C₈₉	M₁₆	1	3	yes	—
72	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
73	M₁₆×C₈₉	C₂²⋊C₄	1	12	yes	—
74	C₂×C₈×C₈₉	C₂²×C₄	1	3	yes	—
75	C₂×C₈×C₈₉	C₂×D₄	1	6	yes	—
76	C₂×C₈×C₈₉	C₂×D₄	1	6	yes	—
77	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₁₆×C₈₉	—	2	1	no	—
79	C₁₆×C₈₉	—	2	3	no	—
80	C₂³×C₈₉	C₂×M₁₆	1	3	yes	—
81	C₂²×D₈₉	—	2	6	no	—
82	Q₈×C₈₉	C₂²⋊C₈	1	4	yes	—
83	D₄×C₈₉	C₂²⋊C₈	1	12	yes	—
84	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
85	C₂×C₄×C₈₉	C₂²×C₈	1	6	yes	—
86	C₂×C₄×C₈₉	C₂×M₁₆	1	3	yes	—
87	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
88	C₂²⋊D₈₉	—	2	4	no	—
89	C₂²⋊D₈₉	—	2	12	no	—
90	C₂×Dic₈₉	—	2	6	no	—
91	D₃₅₆	—	2	6	no	—
92	D₃₅₆	C₂²×C₈	1	4	yes	—
93	C₄×D₈₉	—	2	4	no	—
94	C₄×D₈₉	—	2	12	no	—
95	Dic₁₇₈	—	2	6	no	—
96	Dic₁₇₈	C₂²×C₈	1	4	yes	—
97	C₈×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
98	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
100	#10623494	—	2	4	no	—
101	C₂²×C₈₉	—	4	3	no	—
102	C₂²×C₈₉	C₂×C₂²⋊C₈	1	6	yes	—
103	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
104	C₂×D₈₉	—	2	4	no	—
105	C₂×D₈₉	—	8	6	no	—
106	C₄×C₈₉	C₂×C₂²⋊C₈	1	6	yes	—
107	C₄×C₈₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
109	Dic₈₉	—	2	4	no	—
110	#33340	—	89	1	no	maximal, Sylow
111	C₂×C₈₉	—	2	6	no	—
112	C₂×C₈₉	—	2	1	no	—
113	C₂×C₈₉	C₂²×C₂²⋊C₈	1	1	yes	characteristic
114	D₈₉	—	4	4	no	—
115	#3573	—	89	1	no	—
116	#3407	—	89	1	no	—
117	C₂×#504	D₈₉	1	1	yes	characteristic, complemented
118	#2117	—	89	6	no	—
119	C₂×#350	—	89	6	no	—
120	C₈₉	#33340	1	1	yes	characteristic, complemented, Sylow
121	#585	—	89	1	no	—
122	#568	—	89	1	no	—
123	#568	—	89	6	no	—
124	#568	—	178	4	no	—
125	C₂×#131	—	89	6	no	—
126	C₂×#131	—	178	4	no	—
127	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
128	#504	—	2	4	no	—
129	#504	—	178	4	no	—
130	C₂×M₃₂	—	89	6	no	—
131	C₂×M₃₂	C₂×D₈₉	1	3	yes	complemented
132	C₂²×C₁₆	C₂×D₈₉	1	3	yes	complemented
133	#350	—	89	4	no	—
134	#350	—	89	12	no	—
135	2₋¹⁺⁴	—	89	4	no	—
136	2₋¹⁺⁴	—	178	1	no	—
137	H(5,𝔽₂)	—	89	4	no	—
138	H(5,𝔽₂)	—	178	3	no	—
139	C₂×#41	—	89	6	no	—
140	C₂×#41	—	178	4	no	—
141	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
142	#131	—	89	12	no	—
143	#131	—	89	4	no	—
144	#131	—	178	4	no	—
145	#131	—	178	12	no	—
146	#131	—	178	6	no	—
147	#131	—	178	6	no	—
148	#131	D₃₅₆	1	4	yes	—
149	#131	C₂²⋊D₈₉	1	4	yes	—
150	C₂×M₁₆	—	89	6	no	—
151	C₂×M₁₆	—	178	12	no	—
152	C₂×M₁₆	—	178	6	no	—
153	C₂×M₁₆	C₄×D₈₉	1	3	yes	—
154	C₂²×C₈	—	178	6	no	—
155	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
156	M₃₂	—	2	6	no	—
157	M₃₂	—	89	4	no	—
158	M₃₂	—	178	6	no	—
159	C₂×C₁₆	—	2	6	no	—
160	C₂×C₁₆	—	178	6	no	—
161	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
162	C₂×C₁₆	C₂²×D₈₉	1	3	yes	—
163	#41	—	89	12	no	—
164	#41	—	89	4	no	—
165	#41	—	178	4	no	—
166	#41	—	178	12	no	—
167	#41	—	178	6	no	—
168	#41	—	178	6	no	—
169	#41	#13083551	1	4	yes	—
170	#41	C₈×D₈₉	1	4	yes	—
171	C₂×Q₈	—	89	6	no	—
172	C₂×Q₈	—	178	4	no	—
173	C₂×Q₈	#13083557	1	1	yes	characteristic
174	C₂×D₄	—	89	6	no	—
175	C₂×D₄	—	178	6	no	—
176	C₂×D₄	—	178	12	no	—
177	C₂×D₄	#13083557	1	3	yes	—
178	C₂²×C₄	—	178	6	no	—
179	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
180	M₁₆	—	89	4	no	—
181	M₁₆	—	178	6	no	—
182	M₁₆	—	178	4	no	—
183	M₁₆	—	178	12	no	—
184	M₁₆	#13083551	1	12	yes	—
185	C₂×C₈	—	178	6	no	—
186	C₂×C₈	—	178	4	no	—
187	C₂×C₈	—	178	12	no	—
188	C₂×C₈	C₂×C₄×D₈₉	1	3	yes	—
189	C₂×C₈	C₂×C₂²⋊D₈₉	1	6	yes	—
190	C₂×C₈	C₂×D₃₅₆	1	6	yes	—
191	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
192	C₁₆	—	2	1	no	—
193	C₁₆	—	2	3	no	—
194	C₁₆	—	178	4	no	—
195	C₂³	—	178	6	no	—
196	C₂³	C₂×#13083557	1	3	yes	—
197	Q₈	—	89	4	no	—
198	Q₈	—	178	6	no	—
199	Q₈	#15667924	1	4	yes	—
200	D₄	—	89	4	no	—
201	D₄	—	178	12	no	—
202	D₄	—	178	4	no	—
203	D₄	—	178	6	no	—
204	D₄	#15667924	1	12	yes	—
205	C₂×C₄	—	178	6	no	—
206	C₂×C₄	—	178	4	no	—
207	C₂×C₄	—	178	12	no	—
208	C₂×C₄	C₂×#13083551	1	6	yes	—
209	C₂×C₄	C₂×C₈×D₈₉	1	6	yes	—
210	C₂×C₄	C₂×#13083557	1	3	yes	—
211	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
212	C₈	—	178	4	no	—
213	C₈	C₂×#13083551	1	6	yes	—
214	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
215	C₈	C₂²×D₃₅₆	1	1	yes	characteristic
216	C₂²	—	4	3	no	—
217	C₂²	—	178	4	no	—
218	C₂²	—	712	6	no	—
219	C₂²	C₂×#15667924	1	6	yes	—
220	C₂²	C₂²×#13083557	1	1	yes	characteristic
221	C₄	—	178	4	no	—
222	C₄	C₂×#15667924	1	6	yes	—
223	C₄	C₂²×#13083551	1	1	yes	characteristic
224	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	6	no	—
227	C₂	—	356	4	no	—
228	C₂	C₂²×#15667924	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×D₃₅₆
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#504×C₈₉	C₂
socle	C₂×C₈₉	C₂²×C₂²⋊C₈
prime radical	#19152434	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#504	D₈₉
hypocenter	C₈₉	#33340
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #26033355	C₂	semidirect
3. #26021879	C₂	semidirect
4. C₂×#19151381	C₂	semidirect
5. C₂×#504×C₈₉	C₂	semidirect
13. C₂×#15667952	C₂²	semidirect
14. #19151381	C₂²	semidirect
15. #19151381	C₂²	semidirect
18. C₂×M₃₂×C₈₉	C₂²	semidirect
19. C₂²×C₁₆×C₈₉	C₂²	semidirect
117. C₂×#504	D₈₉	semidirect
120. C₈₉	#33340	semidirect
131. C₂×M₃₂	C₂×D₈₉	semidirect
132. C₂²×C₁₆	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26034963	2
#26033355	2
#26021879	2
C₂×#19151381	2
C₂×#504×C₈₉	2
#33340	89