Subgroups — #36707622

Order 22784.

674 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152016	C₂	1	2	yes	maximal, complemented
2	#26028965	C₂	1	1	yes	maximal, characteristic
3	#26022076	C₂	1	8	yes	maximal, complemented
4	#26021867	C₂	1	2	yes	maximal, complemented
5	C₂×#19151374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15668018	C₂²	1	2	yes	—
8	#19152148	C₄	1	1	yes	characteristic
9	#19152016	—	2	8	no	—
10	C₂×#15667951	C₂²	1	4	yes	—
11	C₂²×#13083549	C₂²	1	1	yes	characteristic
12	C₂²×#13083549	C₂²	1	1	yes	characteristic
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	#19151377	C₂²	1	8	yes	—
19	#19151374	C₂²	1	4	yes	—
20	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₈₉	C₂²	1	4	yes	—
23	C₂×#13083574	C₂×C₄	1	2	yes	—
24	#15668018	—	2	4	no	—
25	#15668018	—	2	4	no	—
26	C₂×#13083558	C₂×C₄	1	2	yes	—
27	C₂×#13083557	C₂×C₄	1	4	yes	—
28	C₂×C₈×D₈₉	C₂³	1	4	yes	—
29	#15668012	C₈	1	1	yes	characteristic
30	#15668009	C₈	1	1	yes	characteristic
31	#15668000	C₈	1	2	yes	—
32	#15667986	C₈	1	4	yes	—
33	C₄×D₃₅₆	C₂×C₄	1	4	yes	—
34	#15667951	—	2	8	no	—
35	C₂×#13083549	—	2	2	no	—
36	C₂×#13083549	D₄	1	4	yes	complemented
37	C₂×#13083549	C₂³	1	2	yes	—
38	C₂×#13083549	C₂³	1	2	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	M₃₂×C₈₉	—	2	4	no	—
46	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
47	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₈₉	C₂³	1	2	yes	—
49	#13083574	—	2	4	no	—
50	#13083574	—	2	4	no	—
51	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
52	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
53	C₂×D₃₅₆	C₂×C₈	1	4	yes	—
54	C₂×C₂²⋊D₈₉	C₂×C₈	1	4	yes	—
55	C₂×C₄×D₈₉	C₂²×C₄	1	4	yes	—
56	#13083558	—	2	4	no	—
57	#13083557	—	2	8	no	—
58	C₂×#10623494	C₂²×C₄	1	4	yes	—
59	C₈×D₈₉	—	2	4	no	—
60	C₈×D₈₉	—	2	4	no	—
61	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
62	#13083551	C₂×C₈	1	8	yes	—
63	#13083549	—	2	2	no	—
64	#13083549	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₈₉	—	2	2	no	—
68	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₈₉	#41	1	2	yes	—
72	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
73	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
76	C₁₆×C₈₉	—	2	4	no	—
77	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
78	C₂²×D₈₉	C₂²×C₈	1	4	yes	—
79	C₂×C₄×C₈₉	—	4	2	no	—
80	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₈₉	#131	1	2	yes	—
85	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
86	C₂²⋊D₈₉	—	2	8	no	—
87	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
88	D₃₅₆	—	2	4	no	—
89	D₃₅₆	—	4	4	no	—
90	C₄×D₈₉	—	2	4	no	—
91	C₄×D₈₉	—	2	4	no	—
92	Dic₁₇₈	—	2	4	no	—
93	C₈×C₈₉	—	2	2	no	—
94	C₈×C₈₉	C₄×D₄	1	2	yes	—
95	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
97	#10623494	—	2	4	no	—
98	C₂²×C₈₉	—	2	2	no	—
99	C₂²×C₈₉	C₈×D₄	1	2	yes	—
100	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
101	C₂×D₈₉	—	2	8	no	—
102	C₂×D₈₉	—	2	4	no	—
103	C₄×C₈₉	—	2	2	no	—
104	C₄×C₈₉	—	2	2	no	—
105	C₄×C₈₉	C₈×D₄	1	2	yes	—
106	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
107	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
108	Dic₈₉	—	2	4	no	—
109	#33435	—	89	1	no	maximal, Sylow
110	C₂×C₈₉	—	2	2	no	—
111	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
112	C₂×C₈₉	#3407	1	2	yes	—
113	D₈₉	—	4	4	no	—
114	C₂×#504	—	89	2	no	—
115	#2931	—	89	1	no	—
116	#2168	—	89	8	no	—
117	#2150	D₈₉	1	1	yes	characteristic, complemented
118	#2113	—	89	2	no	—
119	C₂×#346	—	89	1	no	—
120	C₈₉	#33435	1	1	yes	characteristic, complemented, Sylow
121	C₂×#131	—	89	2	no	—
122	#532	—	89	1	no	—
123	#504	—	178	8	no	—
124	C₂×M₃₂	—	89	4	no	—
125	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
126	C₂²×C₁₆	—	89	1	no	—
127	C₂²×C₁₆	—	89	1	no	—
128	C₂²×C₁₆	—	89	4	no	—
129	#436	—	89	4	no	—
130	C₈×D₄	—	89	4	no	—
131	#423	—	89	1	no	—
132	#408	—	89	2	no	—
133	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
134	C₄⋊C₁₆	C₂×D₈₉	1	4	yes	—
135	C₂²⋊C₁₆	—	89	8	no	—
136	#346	—	89	4	no	—
137	C₂×#41	—	89	2	no	—
138	#131	—	178	4	no	—
139	#131	—	178	4	no	—
140	C₂×M₁₆	—	89	2	no	—
141	C₂×M₁₆	—	89	4	no	—
142	C₂²×C₈	—	89	4	no	—
143	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
145	C₄⋊Q₈	—	89	1	no	—
146	#127	—	89	1	no	—
147	#124	—	89	2	no	—
148	#121	—	89	4	no	—
149	C₄×D₄	—	89	4	no	—
150	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
151	M₃₂	—	2	4	no	—
152	M₃₂	—	178	8	no	—
153	C₂×C₁₆	—	89	4	no	—
154	C₂×C₁₆	—	89	2	no	—
155	C₂×C₁₆	—	89	2	no	—
156	C₂×C₁₆	—	178	8	no	—
157	C₂×C₁₆	—	178	2	no	—
158	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
159	C₄⋊C₈	—	89	4	no	—
160	C₂²⋊C₈	—	89	8	no	—
161	C₄×C₈	C₄×D₈₉	1	2	yes	—
162	C₄×C₈	C₂²×D₈₉	1	2	yes	—
163	#41	—	178	4	no	—
164	#41	—	178	4	no	—
165	C₂×Q₈	—	89	2	no	—
166	C₂×D₄	—	89	4	no	—
167	C₂×D₄	—	89	4	no	—
168	C₂×D₄	—	89	2	no	—
169	C₂²×C₄	—	89	4	no	—
170	C₂²×C₄	C₈×D₈₉	1	2	yes	—
171	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
172	M₁₆	—	178	4	no	—
173	M₁₆	—	178	8	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	89	4	no	—
176	C₂×C₈	—	178	4	no	—
177	C₂×C₈	—	178	4	no	—
178	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
179	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
180	C₂×C₈	D₄×D₈₉	1	2	yes	—
181	C₂×C₈	#13083575	1	2	yes	—
182	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
183	C₄⋊C₄	—	89	4	no	—
184	C₂²⋊C₄	—	89	8	no	—
185	C₄²	C₈×D₈₉	1	1	yes	characteristic
186	C₄²	C₈×D₈₉	1	1	yes	characteristic
187	C₄²	C₂×C₄×D₈₉	1	2	yes	—
188	C₁₆	—	2	4	no	—
189	C₁₆	—	89	4	no	—
190	C₁₆	—	178	2	no	—
191	C₂³	—	89	4	no	—
192	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
193	Q₈	—	178	4	no	—
194	D₄	—	356	4	no	—
195	D₄	—	178	8	no	—
196	D₄	—	178	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	89	4	no	—
199	C₂×C₄	—	178	4	no	—
200	C₂×C₄	—	178	4	no	—
201	C₂×C₄	#15667974	1	2	yes	—
202	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
203	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
204	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
205	C₂×C₄	#15668020	1	2	yes	—
206	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	178	4	no	—
209	C₈	#15667974	1	2	yes	—
210	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
211	C₈	C₂×#13083575	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	178	4	no	—
214	C₂²	—	178	8	no	—
215	C₂²	#19151663	1	2	yes	—
216	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	178	4	no	—
220	C₄	#19151663	1	2	yes	—
221	C₄	C₂×#15667974	1	1	yes	characteristic
222	C₄	C₂×#15668020	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	—	356	4	no	—
225	C₂	C₂×#19151663	1	1	yes	characteristic
226	C₂	#26033354	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2150	C₂
prime radical	#19152148	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2150	D₈₉
hypocenter	C₈₉	#33435
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
1. C₂×#19152016	C₂	semidirect
3. #26022076	C₂	semidirect
4. #26021867	C₂	semidirect
5. C₂×#19151374	C₂	semidirect
6. C₈₉×#2150	C₂	semidirect
36. C₂×#13083549	D₄	semidirect
117. #2150	D₈₉	semidirect
120. C₈₉	#33435	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152016	2
#26028965	2
#26022076	2
#26021867	2
C₂×#19151374	2
C₈₉×#2150	2
#33435	89