Subgroups — #36707617

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×#19152015	C₂	1	2	yes	maximal, complemented
2	#26022084	C₂	1	4	yes	maximal, complemented
3	D₄×#13083549	C₂	1	4	yes	maximal, complemented
4	#26022008	C₂	1	1	yes	maximal, characteristic, complemented
5	#26021866	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2931	C₂	1	1	yes	maximal, characteristic
8	#19152015	—	2	8	no	—
9	C₂×#15667952	C₂²	1	2	yes	—
10	C₂×#15667952	C₂²	1	4	yes	—
11	C₂²×#13083549	C₂²	1	4	yes	—
12	C₂²×#13083549	C₂²	1	1	yes	characteristic
13	C₂²×#13083549	C₂²	1	1	yes	characteristic
14	C₄⋊#13083549	C₂²	1	4	yes	—
15	C₂²⋊#13083549	C₂²	1	8	yes	—
16	C₄×#13083549	C₂²	1	2	yes	—
17	C₄×#13083549	C₂²	1	2	yes	—
18	C₂×#131×C₈₉	C₂²	1	2	yes	—
19	#532×C₈₉	C₄	1	1	yes	characteristic
20	#436×C₈₉	C₄	1	4	yes	—
21	C₈×D₄×C₈₉	C₂²	1	4	yes	—
22	#423×C₈₉	C₄	1	1	yes	characteristic
23	#408×C₈₉	C₄	1	2	yes	—
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	#15667952	—	2	8	no	—
26	#15667952	—	2	4	no	—
27	C₂×#13083549	—	2	8	no	—
28	C₂×#13083549	—	2	2	no	—
29	C₂×#13083549	D₄	1	4	yes	complemented
30	C₂×#13083549	C₂³	1	2	yes	—
31	C₂×#13083549	C₂³	1	4	yes	—
32	C₂×#13083549	C₂³	1	1	yes	characteristic
33	C₂×#13083549	C₂³	1	1	yes	characteristic
34	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
35	#131×C₈₉	—	2	4	no	—
36	#131×C₈₉	—	2	4	no	—
37	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
41	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₈₉	C₈	1	1	yes	characteristic
43	#127×C₈₉	C₈	1	1	yes	characteristic
44	#124×C₈₉	C₈	1	2	yes	—
45	#121×C₈₉	C₈	1	4	yes	—
46	C₄×D₄×C₈₉	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
49	C₂²⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₈₉	C₂³	1	2	yes	—
52	#13083549	—	2	4	no	—
53	#13083549	—	2	2	no	—
54	#13083549	C₂×D₄	1	2	yes	—
55	#13083549	C₂×D₄	1	2	yes	—
56	#41×C₈₉	—	2	4	no	—
57	#41×C₈₉	—	2	4	no	—
58	C₂×Q₈×C₈₉	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₈₉	C₂×C₈	1	4	yes	—
60	C₂×D₄×C₈₉	C₂×C₈	1	4	yes	—
61	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	M₁₆×C₈₉	—	2	4	no	—
66	M₁₆×C₈₉	—	2	8	no	—
67	C₂×C₈×C₈₉	—	2	2	no	—
68	C₂×C₈×C₈₉	—	2	4	no	—
69	C₂×C₈×C₈₉	—	2	4	no	—
70	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₈₉	#41	1	2	yes	—
75	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
77	C₂²⋊C₄×C₈₉	C₂×C₈	1	8	yes	—
78	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
81	C₂³×C₈₉	C₂²×C₈	1	4	yes	—
82	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
83	Q₈×C₈₉	—	2	4	no	—
84	D₄×C₈₉	—	2	8	no	—
85	D₄×C₈₉	—	2	4	no	—
86	D₄×C₈₉	—	2	8	no	—
87	C₂×C₄×C₈₉	—	2	4	no	—
88	C₂×C₄×C₈₉	—	2	4	no	—
89	C₂×C₄×C₈₉	—	2	4	no	—
90	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₈₉	#131	1	2	yes	—
96	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
97	C₈×C₈₉	—	2	4	no	—
98	C₈×C₈₉	—	2	2	no	—
99	C₈×C₈₉	C₄×D₄	1	2	yes	—
100	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
102	C₂²×C₈₉	—	2	2	no	—
103	C₂²×C₈₉	—	2	8	no	—
104	C₂²×C₈₉	—	2	4	no	—
105	C₂²×C₈₉	C₈×D₄	1	2	yes	—
106	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
107	C₄×C₈₉	—	2	2	no	—
108	C₄×C₈₉	—	2	4	no	—
109	C₄×C₈₉	C₈×D₄	1	2	yes	—
110	C₄×C₈₉	#504	1	4	yes	complemented
111	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
112	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
113	#33434	—	89	1	no	maximal, Sylow
114	C₂×C₈₉	—	2	2	no	—
115	C₂×C₈₉	—	4	4	no	—
116	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
117	C₂×C₈₉	C₂×#504	1	2	yes	—
118	C₂×#504	—	89	2	no	—
119	#2931	D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
125	C₈₉	#33434	1	1	yes	characteristic, complemented, Sylow
126	C₂×#131	C₂×D₈₉	1	2	yes	—
127	#532	Dic₈₉	1	1	yes	characteristic
128	#504	—	178	8	no	—
129	C₂×M₃₂	—	89	2	no	—
130	C₂×M₃₂	—	89	4	no	—
131	C₂²×C₁₆	—	89	1	no	—
132	C₂²×C₁₆	—	89	1	no	—
133	C₂²×C₁₆	—	89	4	no	—
134	#436	Dic₈₉	1	4	yes	—
135	C₈×D₄	C₂×D₈₉	1	4	yes	—
136	#423	Dic₈₉	1	1	yes	characteristic
137	#408	Dic₈₉	1	2	yes	—
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	—	89	2	no	—
142	C₄×C₁₆	—	89	2	no	—
143	C₂×#41	C₂×Dic₈₉	1	2	yes	—
144	#131	—	2	4	no	—
145	#131	—	2	4	no	—
146	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
147	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
148	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
149	C₂²×C₈	C₂²×D₈₉	1	4	yes	—
150	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
151	C₄⋊Q₈	#10623494	1	1	yes	characteristic
152	#127	#10623494	1	1	yes	characteristic
153	#124	#10623494	1	2	yes	—
154	#121	#10623494	1	4	yes	—
155	C₄×D₄	C₂×Dic₈₉	1	4	yes	—
156	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
157	M₃₂	—	178	8	no	—
158	M₃₂	—	178	4	no	—
159	C₂×C₁₆	—	89	4	no	—
160	C₂×C₁₆	—	89	1	no	—
161	C₂×C₁₆	—	89	1	no	—
162	C₂×C₁₆	—	89	2	no	—
163	C₂×C₁₆	—	89	4	no	—
164	C₂×C₁₆	—	178	8	no	—
165	C₂×C₁₆	—	178	2	no	—
166	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
167	C₂²⋊C₈	C₂×Dic₈₉	1	8	yes	—
168	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
169	C₄×C₈	C₂²×D₈₉	1	2	yes	—
170	#41	—	2	4	no	—
171	#41	—	2	4	no	—
172	C₂×Q₈	C₂×#10623494	1	2	yes	—
173	C₂×D₄	C₂×#10623494	1	4	yes	—
174	C₂×D₄	C₂×#10623494	1	4	yes	—
175	C₂×D₄	C₂×#10623494	1	2	yes	—
176	C₂²×C₄	C₂×#10623494	1	2	yes	—
177	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
178	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
179	M₁₆	—	2	4	no	—
180	M₁₆	—	2	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
185	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
186	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
187	C₂×C₈	D₄×D₈₉	1	2	yes	—
188	C₂×C₈	#13083573	1	2	yes	—
189	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
190	C₄⋊C₄	C₂×#10623494	1	4	yes	—
191	C₂²⋊C₄	C₂×#10623494	1	8	yes	—
192	C₄²	C₂×#10623494	1	1	yes	characteristic
193	C₄²	C₂×#10623494	1	1	yes	characteristic
194	C₄²	C₂²×Dic₈₉	1	2	yes	—
195	C₁₆	—	89	2	no	—
196	C₁₆	—	89	2	no	—
197	C₁₆	—	178	4	no	—
198	C₁₆	—	178	2	no	—
199	C₂³	C₂²×#10623494	1	4	yes	—
200	C₂³	C₂²×#10623494	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	D₄	—	2	8	no	—
203	D₄	—	2	4	no	—
204	D₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
209	C₂×C₄	C₂²×#10623494	1	2	yes	—
210	C₂×C₄	C₂²×#10623494	1	4	yes	—
211	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
212	C₂×C₄	C₂²×#10623494	1	2	yes	—
213	C₂×C₄	#15668019	1	2	yes	—
214	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	2	2	no	—
217	C₈	D₄×Dic₈₉	1	2	yes	—
218	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
219	C₈	C₂×#13083573	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	8	no	—
223	C₂²	D₄×#10623494	1	2	yes	—
224	C₂²	C₂³×#10623494	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	2	2	no	—
227	C₄	D₄×#10623494	1	2	yes	—
228	C₄	#19152015	1	4	yes	complemented
229	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
230	C₄	C₂×#15668019	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	4	4	no	—
233	C₂	C₂×D₄×#10623494	1	1	yes	characteristic
234	C₂	C₂×#19152015	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2931	C₂
prime radical	#532×C₈₉	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2931	D₈₉
hypocenter	C₈₉	#33434
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. C₂×#19152015	C₂	semidirect
2. #26022084	C₂	semidirect
3. D₄×#13083549	C₂	semidirect
4. #26022008	C₂	semidirect
5. #26021866	C₂	semidirect
6. C₂×C₄×#13083549	C₂	semidirect
29. C₂×#13083549	D₄	semidirect
110. C₄×C₈₉	#504	semidirect
125. C₈₉	#33434	semidirect
228. C₄	#19152015	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152015	2
#26022084	2
D₄×#13083549	2
#26022008	2
#26021866	2
C₂×C₄×#13083549	2
C₈₉×#2931	2
#33434	89