Subgroups — #36465726

Order 22784.

310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030766	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151351	C₂	1	1	yes	maximal, characteristic
3	#26018309	C₂	1	2	yes	maximal, complemented
4	#26018306	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic
6	#19152155	—	2	1	no	—
7	#19152003	C₂²	1	1	yes	characteristic
8	#19152003	C₂²	1	1	yes	characteristic
9	#19152000	—	2	1	no	—
10	#19152000	C₄	1	2	yes	—
11	#19151996	—	2	1	no	—
12	#19151636	—	2	1	no	—
13	#19151636	C₄	1	2	yes	—
14	C₂×C₄⋊#10623494	—	2	1	no	—
15	#19151351	—	2	1	no	—
16	#19151351	C₂²	1	2	yes	—
17	C₂×M₃₂×C₈₉	—	2	1	no	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆×C₈₉	—	2	1	no	—
20	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
21	C₂²×#10623494	—	4	1	no	—
22	#15668012	—	2	1	no	—
23	#15668012	—	2	1	no	—
24	#15668004	—	2	2	no	—
25	#15667991	—	4	1	no	—
26	#15667991	—	4	1	no	—
27	C₄×Dic₁₇₈	—	4	1	no	—
28	C₄×Dic₁₇₈	—	4	1	no	—
29	#15667971	—	2	2	no	—
30	#15667946	—	2	1	no	—
31	#15667946	—	2	1	no	—
32	#15667946	C₂×C₄	1	2	yes	—
33	#15667946	C₂×C₄	1	2	yes	—
34	#15667944	—	2	2	no	—
35	#15667944	—	2	2	no	—
36	C₄⋊#10623494	—	2	2	no	—
37	C₄⋊#10623494	—	4	1	no	—
38	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
41	M₃₂×C₈₉	—	4	2	no	—
42	C₂×C₁₆×C₈₉	—	2	2	no	—
43	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	C₂×Dic₁₇₈	—	4	1	no	—
48	C₂×Dic₁₇₈	—	4	1	no	—
49	C₂×#10623494	—	4	2	no	—
50	C₂×#10623494	—	8	2	no	—
51	C₄⋊Dic₈₉	—	2	2	no	—
52	C₄⋊Dic₈₉	—	2	2	no	—
53	C₄⋊Dic₈₉	—	4	1	no	—
54	C₄⋊Dic₈₉	—	4	1	no	—
55	#13083553	—	4	2	no	—
56	#13083553	—	4	2	no	—
57	C₂²⋊Dic₈₉	—	4	2	no	—
58	C₄×Dic₈₉	—	4	2	no	—
59	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
62	C₂×C₈×C₈₉	—	2	2	no	—
63	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
66	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
67	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₈₉	D₈	1	1	yes	characteristic
71	C₄²×C₈₉	SD₁₆	1	1	yes	characteristic
72	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₈₉	—	4	2	no	—
74	C₂³×C₈₉	#102	1	1	yes	characteristic
75	C₂×C₄×C₈₉	—	2	1	no	—
76	C₂×C₄×C₈₉	—	2	1	no	—
77	C₂×C₄×C₈₉	—	4	1	no	—
78	C₂×C₄×C₈₉	#101	1	2	yes	—
79	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
80	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
82	C₂×C₄×C₈₉	#103	1	2	yes	—
83	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×Dic₈₉	—	4	2	no	—
87	C₂×Dic₈₉	—	4	2	no	—
88	Dic₁₇₈	—	8	2	no	—
89	Dic₁₇₈	—	8	1	no	—
90	Dic₁₇₈	—	8	1	no	—
91	C₈×C₈₉	—	2	1	no	—
92	C₈×C₈₉	—	2	1	no	—
93	C₈×C₈₉	#103	1	2	yes	—
94	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
95	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
96	#10623494	—	8	2	no	—
97	C₂²×C₈₉	—	2	1	no	—
98	C₂²×C₈₉	—	2	1	no	—
99	C₂²×C₈₉	#332	1	2	yes	—
100	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
101	C₄×C₈₉	—	2	1	no	—
102	C₄×C₈₉	—	2	1	no	—
103	C₄×C₈₉	—	2	1	no	—
104	C₄×C₈₉	#332	1	2	yes	—
105	C₄×C₈₉	C₂×#101	1	1	yes	characteristic
106	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
107	C₄×C₈₉	#509	1	1	yes	characteristic
108	C₄×C₈₉	#510	1	1	yes	characteristic
109	Dic₈₉	—	8	2	no	—
110	Dic₈₉	—	8	2	no	—
111	#11942	—	89	1	no	maximal, Sylow
112	C₂×C₈₉	—	2	2	no	—
113	C₂×C₈₉	C₂×#332	1	1	yes	characteristic
114	C₂×C₈₉	#2139	1	1	yes	characteristic
115	C₂×C₈₉	#2148	1	1	yes	characteristic
116	#3160	—	89	1	no	—
117	#2150	D₈₉	1	1	yes	characteristic
118	C₂×#335	—	89	1	no	—
119	#1365	—	89	2	no	—
120	#1364	—	89	2	no	—
121	C₈₉	#11942	1	1	yes	characteristic, complemented, Sylow
122	#533	—	178	1	no	—
123	C₂×M₃₂	—	2	1	no	—
124	#500	—	89	1	no	—
125	#500	—	89	1	no	—
126	#499	—	89	2	no	—
127	#499	—	178	1	no	—
128	#498	—	178	1	no	—
129	#427	—	89	2	no	—
130	#427	—	178	1	no	—
131	C₂×C₄⋊C₈	—	178	1	no	—
132	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
133	C₄⋊C₁₆	—	2	1	no	—
134	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
135	#335	—	89	2	no	—
136	#335	—	178	1	no	—
137	C₂²×C₈	—	356	1	no	—
138	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
140	C₄⋊Q₈	—	178	1	no	—
141	C₄⋊Q₈	—	178	1	no	—
142	#125	—	178	2	no	—
143	C₂²⋊Q₈	—	356	1	no	—
144	C₂²⋊Q₈	—	356	1	no	—
145	C₄×Q₈	—	356	1	no	—
146	C₄×Q₈	—	356	1	no	—
147	#117	—	178	2	no	—
148	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
149	M₃₂	—	4	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	#107	—	89	2	no	—
152	#107	—	89	2	no	—
153	#107	—	178	1	no	—
154	#107	—	178	1	no	—
155	#106	—	178	2	no	—
156	#106	—	178	2	no	—
157	C₄⋊C₈	—	356	1	no	—
158	C₄⋊C₈	—	178	2	no	—
159	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
160	C₄×C₈	D₃₅₆	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
163	C₂×Q₈	—	356	1	no	—
164	C₂×Q₈	—	356	1	no	—
165	C₂²×C₄	#13083551	1	1	yes	characteristic
166	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
167	C₂²×C₄	#13083563	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	356	2	no	—
170	C₂×C₈	—	712	2	no	—
171	C₂×C₈	#13083551	1	1	yes	characteristic
172	C₂×C₈	#13083551	1	1	yes	characteristic
173	C₂×C₈	#13083562	1	2	yes	—
174	C₂×C₈	#13083564	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
177	C₄⋊C₄	—	356	1	no	—
178	C₄⋊C₄	—	356	1	no	—
179	C₄⋊C₄	—	356	2	no	—
180	C₄⋊C₄	—	356	2	no	—
181	C₄⋊C₄	—	178	2	no	—
182	C₄⋊C₄	—	178	2	no	—
183	C₂²⋊C₄	—	356	2	no	—
184	C₄²	—	356	2	no	—
185	C₄²	#13083551	1	1	yes	characteristic
186	C₄²	D₇₁₂	1	1	yes	characteristic
187	C₄²	#13083563	1	1	yes	characteristic
188	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
189	C₁₆	—	4	2	no	—
190	C₂³	#15667934	1	1	yes	characteristic
191	Q₈	—	712	2	no	—
192	Q₈	—	712	1	no	—
193	Q₈	—	712	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	356	2	no	—
198	C₂×C₄	—	356	2	no	—
199	C₂×C₄	#15667930	1	2	yes	—
200	C₂×C₄	#15667934	1	1	yes	characteristic
201	C₂×C₄	#15667934	1	1	yes	characteristic
202	C₂×C₄	#15667934	1	1	yes	characteristic
203	C₂×C₄	#15667935	1	2	yes	—
204	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
205	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
206	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	712	2	no	—
210	C₈	#15667935	1	2	yes	—
211	C₈	C₂×#13083562	1	1	yes	characteristic
212	C₈	C₂×#13083564	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#19151343	1	2	yes	—
216	C₂²	C₂×#15667934	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	712	2	no	—
221	C₄	—	712	2	no	—
222	C₄	#19151343	1	2	yes	—
223	C₄	C₂×#15667930	1	1	yes	characteristic
224	C₄	C₂×#15667935	1	1	yes	characteristic
225	C₄	#19152036	1	1	yes	characteristic
226	C₄	#19152041	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#19151343	1	1	yes	characteristic
229	C₂	#26021965	1	1	yes	characteristic
230	C₂	#26022002	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667934
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2150	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂³×C₈₉	#102
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#2150	D₈₉
hypocenter	C₈₉	#11942
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #26018309	C₂	semidirect
4. #26018306	C₂	semidirect
121. C₈₉	#11942	semidirect

Maximal subgroups

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

subgroup	index
#26030766	2
C₂×#19151351	2
#26018309	2
#26018306	2
C₈₉×#2150	2
#11942	89