Subgroups — #36707512

Order 22784.

590 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151553	C₂	1	1	yes	maximal, characteristic
2	#26022056	C₂	1	8	yes	maximal, complemented
3	#26022007	C₂	1	2	yes	maximal, complemented
4	#26021863	C₂	1	1	yes	maximal, characteristic
5	M₃₂×Dic₈₉	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2112	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13083558	C₄	1	1	yes	characteristic
8	C₂×#15667971	C₄	1	1	yes	characteristic
9	C₂×#15667952	C₂²	1	2	yes	—
10	#19151553	C₄	1	4	yes	—
11	#19151553	C₂²	1	4	yes	—
12	C₂×#15667922	C₄	1	1	yes	characteristic
13	C₂×#15667922	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
16	#19151416	C₂²	1	8	yes	complemented
17	C₂²⋊#13083549	C₂²	1	4	yes	—
18	#19151373	C₂²	1	4	yes	complemented
19	C₁₆×Dic₈₉	C₂²	1	4	yes	complemented
20	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
22	C₂²⋊C₁₆×C₈₉	C₂²	1	4	yes	—
23	C₂×#13083558	—	2	4	no	—
24	C₂×#13083558	C₂×C₄	1	4	yes	—
25	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
27	#15667971	C₈	1	4	yes	—
28	#15667971	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊Dic₈₉	C₈	1	2	yes	—
30	C₂×C₂²⋊Dic₈₉	C₈	1	2	yes	—
31	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
33	#15667952	—	2	4	no	—
34	C₂×#13083549	C₂³	1	4	yes	—
35	#15667922	C₂×C₄	1	4	yes	—
36	#15667922	C₂×C₄	1	4	yes	—
37	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
38	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
39	C₈×Dic₈₉	C₂³	1	2	yes	—
40	C₈×Dic₈₉	C₂³	1	2	yes	—
41	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	—	2	4	no	—
43	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
46	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	M₃₂×C₈₉	—	2	4	no	—
48	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
49	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
50	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
51	#13083558	—	2	8	no	—
52	C₂×#10623494	—	2	2	no	—
53	C₂×#10623494	M₁₆	1	4	yes	—
54	C₂×#10623494	C₂²×C₄	1	2	yes	—
55	C₂×#10623494	C₂²×C₄	1	2	yes	—
56	C₄⋊Dic₈₉	C₂×C₈	1	8	yes	—
57	C₂²⋊Dic₈₉	C₂×C₈	1	8	yes	—
58	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
59	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
60	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
61	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
62	#13083549	—	2	4	no	—
63	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	—	2	2	no	—
65	C₂²×C₄×C₈₉	—	2	2	no	—
66	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₈₉	—	2	2	no	—
71	C₂×C₈×C₈₉	—	2	4	no	—
72	C₂×C₈×C₈₉	—	2	4	no	—
73	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	#41	1	4	yes	—
76	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
77	C₁₆×C₈₉	—	2	4	no	—
78	C₂³×C₈₉	—	2	2	no	—
79	C₂³×C₈₉	—	4	2	no	—
80	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
81	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₈₉	—	2	2	no	—
83	C₂×C₄×C₈₉	—	2	2	no	—
84	C₂×C₄×C₈₉	—	2	2	no	—
85	C₂×C₄×C₈₉	—	2	4	no	—
86	C₂×C₄×C₈₉	#117	1	2	yes	—
87	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
90	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
91	C₂×Dic₈₉	—	4	2	no	—
92	C₂×Dic₈₉	—	4	2	no	—
93	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
94	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
95	C₈×C₈₉	—	2	2	no	—
96	C₈×C₈₉	#117	1	2	yes	—
97	C₈×C₈₉	C₂×#41	1	2	yes	—
98	#10623494	—	2	2	no	—
99	#10623494	C₂×M₁₆	1	4	yes	—
100	C₂²×C₈₉	—	2	2	no	—
101	C₂²×C₈₉	—	2	4	no	—
102	C₂²×C₈₉	—	2	2	no	—
103	C₂²×C₈₉	—	4	4	no	—
104	C₂²×C₈₉	#431	1	2	yes	—
105	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
106	C₄×C₈₉	—	2	2	no	—
107	C₄×C₈₉	#431	1	2	yes	—
108	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
109	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
110	Dic₈₉	—	2	4	no	—
111	Dic₈₉	—	2	4	no	—
112	#33419	—	89	1	no	maximal, Sylow
113	C₂×C₈₉	—	2	2	no	—
114	C₂×C₈₉	—	4	2	no	—
115	C₂×C₈₉	C₂×#431	1	1	yes	characteristic
116	C₂×C₈₉	#3407	1	2	yes	—
117	C₂×#405	—	89	1	no	—
118	#2164	—	89	8	no	—
119	#2150	—	89	2	no	—
120	#2112	—	89	1	no	—
121	#2112	D₈₉	1	1	yes	characteristic
122	C₄×M₃₂	—	89	2	no	—
123	C₈₉	#33419	1	1	yes	characteristic, complemented, Sylow
124	C₂²×M₁₆	—	89	1	no	—
125	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
126	C₂×#117	—	89	1	no	—
127	C₂×M₃₂	—	89	2	no	—
128	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
129	#405	—	89	4	no	—
130	#405	—	89	4	no	—
131	C₂×#97	—	89	1	no	—
132	C₂×#97	—	89	1	no	—
133	C₂×C₄×C₈	—	89	1	no	—
134	C₂×C₄×C₈	—	89	1	no	—
135	C₄⋊C₁₆	—	89	8	no	—
136	C₂²⋊C₁₆	—	89	4	no	—
137	C₂²⋊C₁₆	C₂×D₈₉	1	4	yes	—
138	#346	—	89	4	no	—
139	C₄×C₁₆	—	89	4	no	—
140	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
141	C₂×M₁₆	—	89	4	no	—
142	C₂×M₁₆	—	178	4	no	—
143	C₂²×C₈	—	2	4	no	—
144	C₂²×C₈	—	89	1	no	—
145	C₂²×C₈	—	89	1	no	—
146	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
147	C₂²×C₈	C₄×D₈₉	1	2	yes	—
148	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
149	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
150	#117	—	89	4	no	—
151	#117	—	89	4	no	—
152	C₂×C₄⋊C₄	—	89	2	no	—
153	C₂×C₂²⋊C₄	—	89	2	no	—
154	C₂×C₄²	—	89	1	no	—
155	C₂×C₄²	—	89	1	no	—
156	M₃₂	—	2	4	no	—
157	M₃₂	—	178	4	no	—
158	C₂×C₁₆	—	89	4	no	—
159	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
160	#97	—	89	4	no	—
161	#97	—	89	4	no	—
162	C₄×C₈	—	89	2	no	—
163	C₄×C₈	—	89	2	no	—
164	C₄×C₈	—	89	2	no	—
165	C₄×C₈	—	89	2	no	—
166	C₂⁴	C₈×D₈₉	1	1	yes	characteristic
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	89	2	no	—
170	C₂²×C₄	—	89	2	no	—
171	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
172	C₂²×C₄	C₈×D₈₉	1	2	yes	—
173	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
174	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
175	M₁₆	—	178	8	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	89	4	no	—
180	C₂×C₈	—	89	2	no	—
181	C₂×C₈	—	89	2	no	—
182	C₂×C₈	—	178	2	no	—
183	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
184	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
185	C₂×C₈	#13083573	1	4	yes	—
186	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
187	C₄⋊C₄	—	89	8	no	—
188	C₂²⋊C₄	—	89	8	no	—
189	C₄²	—	89	2	no	—
190	C₄²	—	89	2	no	—
191	C₄²	—	89	2	no	—
192	C₄²	—	89	2	no	—
193	C₁₆	—	2	4	no	—
194	C₁₆	—	178	4	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	4	2	no	—
197	C₂³	C₂×C₈×D₈₉	1	2	yes	—
198	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	89	4	no	—
204	C₂×C₄	—	89	4	no	—
205	C₂×C₄	—	356	2	no	—
206	C₂×C₄	—	356	2	no	—
207	C₂×C₄	#15667968	1	2	yes	—
208	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
209	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
210	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
211	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	89	4	no	—
214	C₈	—	178	2	no	—
215	C₈	#15667968	1	2	yes	—
216	C₈	C₂×#13083573	1	2	yes	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	4	no	—
221	C₂²	#19151648	1	2	yes	—
222	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	178	4	no	—
225	C₄	—	178	4	no	—
226	C₄	#19151648	1	2	yes	—
227	C₄	C₂×#15667968	1	1	yes	characteristic
228	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	4	2	no	—
231	C₂	C₂×#19151648	1	1	yes	characteristic
232	C₂	#26033354	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2112	C₂
prime radical	C₂⁴×C₈₉	C₂×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2112	D₈₉
hypocenter	C₈₉	#33419
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
2. #26022056	C₂	semidirect
3. #26022007	C₂	semidirect
5. M₃₂×Dic₈₉	C₂	semidirect
16. #19151416	C₂²	semidirect
18. #19151373	C₂²	semidirect
19. C₁₆×Dic₈₉	C₂²	semidirect
123. C₈₉	#33419	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151553	2
#26022056	2
#26022007	2
#26021863	2
M₃₂×Dic₈₉	2
C₈₉×#2112	2
#33419	89