Subgroups — #36707623

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×#19152015	C₂	1	2	yes	maximal, complemented
2	#26022078	C₂	1	8	yes	maximal, complemented
3	#26022008	C₂	1	1	yes	maximal, characteristic, complemented
4	#26021866	C₂	1	2	yes	maximal, complemented
5	C₂×#19151374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2931	C₂	1	1	yes	maximal, characteristic
7	#19152015	—	2	8	no	—
8	C₂×#15667952	C₂²	1	2	yes	—
9	C₂×#15667952	C₂²	1	4	yes	—
10	C₂²×#13083549	C₂²	1	4	yes	—
11	C₂²×#13083549	C₂²	1	1	yes	characteristic
12	C₂²×#13083549	C₂²	1	1	yes	characteristic
13	C₄⋊#13083549	C₂²	1	4	yes	—
14	C₂²⋊#13083549	C₂²	1	8	yes	—
15	#19151374	C₂²	1	4	yes	—
16	C₂×#131×C₈₉	C₂²	1	2	yes	—
17	#532×C₈₉	C₄	1	1	yes	characteristic
18	#436×C₈₉	C₄	1	4	yes	—
19	C₈×D₄×C₈₉	C₂²	1	4	yes	—
20	#423×C₈₉	C₄	1	1	yes	characteristic
21	#408×C₈₉	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	#15667952	—	2	8	no	—
24	#15667952	—	2	4	no	—
25	C₂×#13083549	—	2	8	no	—
26	C₂×#13083549	—	2	2	no	—
27	C₂×#13083549	D₄	1	4	yes	complemented
28	C₂×#13083549	C₂³	1	2	yes	—
29	C₂×#13083549	C₂³	1	4	yes	—
30	C₂×#13083549	C₂³	1	2	yes	—
31	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
32	#131×C₈₉	—	2	4	no	—
33	#131×C₈₉	—	2	4	no	—
34	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
38	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₈₉	C₈	1	1	yes	characteristic
40	#127×C₈₉	C₈	1	1	yes	characteristic
41	#124×C₈₉	C₈	1	2	yes	—
42	#121×C₈₉	C₈	1	4	yes	—
43	C₄×D₄×C₈₉	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
46	C₂²⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
47	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₈₉	C₂³	1	2	yes	—
49	#13083549	—	2	4	no	—
50	#13083549	—	2	2	no	—
51	#13083549	C₂×D₄	1	4	yes	—
52	#41×C₈₉	—	2	4	no	—
53	#41×C₈₉	—	2	4	no	—
54	C₂×Q₈×C₈₉	C₂×C₈	1	2	yes	—
55	C₂×D₄×C₈₉	C₂×C₈	1	4	yes	—
56	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
57	C₂×D₄×C₈₉	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₈₉	—	2	4	no	—
62	M₁₆×C₈₉	—	2	8	no	—
63	C₂×C₈×C₈₉	—	2	2	no	—
64	C₂×C₈×C₈₉	—	2	4	no	—
65	C₂×C₈×C₈₉	—	2	4	no	—
66	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₈₉	#41	1	2	yes	—
71	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
73	C₂²⋊C₄×C₈₉	C₂×C₈	1	8	yes	—
74	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
77	C₂³×C₈₉	C₂²×C₈	1	4	yes	—
78	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
79	Q₈×C₈₉	—	2	4	no	—
80	D₄×C₈₉	—	2	8	no	—
81	D₄×C₈₉	—	2	4	no	—
82	D₄×C₈₉	—	4	4	no	—
83	C₂×C₄×C₈₉	—	2	4	no	—
84	C₂×C₄×C₈₉	—	2	4	no	—
85	C₂×C₄×C₈₉	—	4	2	no	—
86	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
89	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₈₉	#131	1	2	yes	—
92	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
93	C₈×C₈₉	—	2	4	no	—
94	C₈×C₈₉	—	2	2	no	—
95	C₈×C₈₉	C₄×D₄	1	2	yes	—
96	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
97	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
98	C₂²×C₈₉	—	2	2	no	—
99	C₂²×C₈₉	—	2	8	no	—
100	C₂²×C₈₉	—	2	4	no	—
101	C₂²×C₈₉	C₈×D₄	1	2	yes	—
102	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
103	C₄×C₈₉	—	2	2	no	—
104	C₄×C₈₉	—	2	2	no	—
105	C₄×C₈₉	—	2	4	no	—
106	C₄×C₈₉	C₈×D₄	1	2	yes	—
107	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
108	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
109	#33435	—	89	1	no	maximal, Sylow
110	C₂×C₈₉	—	2	2	no	—
111	C₂×C₈₉	—	4	4	no	—
112	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
113	C₂×C₈₉	#3407	1	2	yes	—
114	C₂×#504	—	89	2	no	—
115	#2931	D₈₉	1	1	yes	characteristic
116	#2168	—	89	8	no	—
117	#2150	—	89	1	no	—
118	#2113	—	89	2	no	—
119	C₂×#346	—	89	1	no	—
120	C₈₉	#33435	1	1	yes	characteristic, complemented, Sylow
121	C₂×#131	C₂×D₈₉	1	2	yes	—
122	#532	Dic₈₉	1	1	yes	characteristic
123	#504	—	178	8	no	—
124	C₂×M₃₂	—	89	2	no	—
125	C₂×M₃₂	—	89	4	no	—
126	C₂²×C₁₆	—	89	1	no	—
127	C₂²×C₁₆	—	89	1	no	—
128	C₂²×C₁₆	—	89	4	no	—
129	#436	Dic₈₉	1	4	yes	—
130	C₈×D₄	C₂×D₈₉	1	4	yes	—
131	#423	Dic₈₉	1	1	yes	characteristic
132	#408	Dic₈₉	1	2	yes	—
133	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
134	C₄⋊C₁₆	—	89	4	no	—
135	C₂²⋊C₁₆	—	89	8	no	—
136	#346	—	89	4	no	—
137	C₂×#41	C₂×Dic₈₉	1	2	yes	—
138	#131	—	2	4	no	—
139	#131	—	2	4	no	—
140	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
141	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
142	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₈₉	1	4	yes	—
144	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
145	C₄⋊Q₈	#10623494	1	1	yes	characteristic
146	#127	#10623494	1	1	yes	characteristic
147	#124	#10623494	1	2	yes	—
148	#121	#10623494	1	4	yes	—
149	C₄×D₄	C₂×Dic₈₉	1	4	yes	—
150	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
151	M₃₂	—	178	8	no	—
152	M₃₂	—	178	4	no	—
153	C₂×C₁₆	—	89	4	no	—
154	C₂×C₁₆	—	89	2	no	—
155	C₂×C₁₆	—	89	2	no	—
156	C₂×C₁₆	—	89	4	no	—
157	C₂×C₁₆	—	178	8	no	—
158	C₂×C₁₆	—	178	2	no	—
159	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
160	C₂²⋊C₈	C₂×Dic₈₉	1	8	yes	—
161	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
162	C₄×C₈	C₂²×D₈₉	1	2	yes	—
163	#41	—	2	4	no	—
164	#41	—	2	4	no	—
165	C₂×Q₈	C₂×#10623494	1	2	yes	—
166	C₂×D₄	C₂×#10623494	1	4	yes	—
167	C₂×D₄	C₂×#10623494	1	2	yes	—
168	C₂×D₄	C₂×#10623494	1	4	yes	—
169	C₂²×C₄	C₂×#10623494	1	2	yes	—
170	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
171	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
172	M₁₆	—	2	4	no	—
173	M₁₆	—	2	8	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
178	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
179	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
180	C₂×C₈	D₄×D₈₉	1	2	yes	—
181	C₂×C₈	#13083573	1	2	yes	—
182	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
183	C₄⋊C₄	C₂×#10623494	1	4	yes	—
184	C₂²⋊C₄	C₂×#10623494	1	8	yes	—
185	C₄²	C₂×#10623494	1	1	yes	characteristic
186	C₄²	C₂×#10623494	1	1	yes	characteristic
187	C₄²	C₂²×Dic₈₉	1	2	yes	—
188	C₁₆	—	89	4	no	—
189	C₁₆	—	178	4	no	—
190	C₁₆	—	178	2	no	—
191	C₂³	C₂²×#10623494	1	4	yes	—
192	C₂³	C₂²×#10623494	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	D₄	—	2	8	no	—
195	D₄	—	2	4	no	—
196	D₄	—	4	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
201	C₂×C₄	C₂²×#10623494	1	2	yes	—
202	C₂×C₄	C₂²×#10623494	1	2	yes	—
203	C₂×C₄	C₂²×#10623494	1	4	yes	—
204	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
205	C₂×C₄	#15668019	1	2	yes	—
206	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
207	C₈	—	2	4	no	—
208	C₈	—	2	2	no	—
209	C₈	D₄×Dic₈₉	1	2	yes	—
210	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
211	C₈	C₂×#13083573	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	2	4	no	—
214	C₂²	—	2	8	no	—
215	C₂²	D₄×#10623494	1	2	yes	—
216	C₂²	C₂³×#10623494	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	D₄×#10623494	1	2	yes	—
221	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
222	C₄	C₂×#15668019	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	—	4	4	no	—
225	C₂	C₂×D₄×#10623494	1	1	yes	characteristic
226	C₂	#26033355	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	#33435
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. #26022078	C₂	semidirect
3. #26022008	C₂	semidirect
4. #26021866	C₂	semidirect
5. C₂×#19151374	C₂	semidirect
27. C₂×#13083549	D₄	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×#19152015	2
#26022078	2
#26022008	2
#26021866	2
C₂×#19151374	2
C₈₉×#2931	2
#33435	89