Subgroups — #36709942

Order 22784.

474 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030788	C₂	1	2	yes	maximal
2	#26022537	C₂	1	4	yes	maximal, complemented
3	#26022533	C₂	1	2	yes	maximal, complemented
4	#26022530	C₂	1	2	yes	maximal, complemented
5	#26022038	C₂	1	2	yes	maximal
6	#26022036	C₂	1	1	yes	maximal, characteristic
7	#26021848	C₂	1	1	yes	maximal, characteristic
8	C₂×#367×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#19152229	—	2	2	no	—
10	C₂×#15667952	C₂²	1	1	yes	characteristic
11	#19152006	—	2	4	no	—
12	#19152002	C₂²	1	4	yes	—
13	#19151999	C₂²	1	4	yes	—
14	#19151638	—	2	2	no	—
15	#19151636	C₂²	1	2	yes	—
16	#19151553	C₂²	1	1	yes	characteristic
17	#19151426	C₂²	1	1	yes	characteristic
18	#19151426	C₂²	1	1	yes	characteristic
19	#19151425	C₂²	1	4	yes	—
20	#19151424	C₂²	1	2	yes	—
21	#19151423	C₂²	1	4	yes	—
22	#19151373	C₂²	1	2	yes	—
23	C₁₆×Dic₈₉	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#107×C₈₉	C₂²	1	2	yes	—
26	#367×C₈₉	C₂²	1	2	yes	—
27	#367×C₈₉	C₂²	1	2	yes	—
28	C₂×#13083558	D₄	1	1	yes	characteristic
29	#15668011	—	2	4	no	—
30	#15668011	—	4	2	no	—
31	#15668003	—	2	4	no	—
32	#15668003	—	4	2	no	—
33	#15667991	—	4	4	no	—
34	#15667971	—	2	2	no	—
35	#15667971	D₄	1	1	yes	characteristic
36	#15667952	Q₈	1	4	yes	complemented
37	C₂×#13083549	C₂³	1	2	yes	—
38	#15667946	C₂³	1	2	yes	—
39	#15667946	C₂³	1	2	yes	—
40	#15667945	—	2	4	no	—
41	#15667944	—	2	2	no	—
42	#15667943	—	2	4	no	—
43	#15667922	D₄	1	2	yes	—
44	C₈×Dic₈₉	C₂³	1	2	yes	—
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
47	C₂×C₁₆×C₈₉	Q₈	1	4	yes	complemented
48	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
50	#107×C₈₉	—	2	2	no	—
51	#107×C₈₉	C₂³	1	4	yes	—
52	C₂×Dic₁₇₈	—	4	4	no	—
53	#13083558	Q₁₆	1	4	yes	—
54	C₂×#10623494	C₂×D₄	1	2	yes	—
55	C₄⋊Dic₈₉	—	2	2	no	—
56	C₄⋊Dic₈₉	—	2	2	no	—
57	C₄⋊Dic₈₉	—	4	4	no	—
58	C₄⋊Dic₈₉	D₈	1	2	yes	—
59	#13083553	—	4	4	no	—
60	#13083553	—	4	4	no	—
61	C₂²⋊Dic₈₉	—	4	2	no	—
62	C₂²⋊Dic₈₉	D₈	1	2	yes	—
63	C₄×Dic₈₉	—	4	2	no	—
64	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
65	#13083549	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₈₉	—	4	2	no	—
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₈₉	—	2	4	no	—
73	C₄⋊C₄×C₈₉	—	4	2	no	—
74	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
75	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
76	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₈₉	—	4	4	no	—
78	C₂×C₄×C₈₉	—	16	2	no	—
79	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₈₉	—	4	2	no	—
84	C₂×Dic₈₉	—	4	2	no	—
85	C₂×Dic₈₉	C₂×D₈	1	4	yes	—
86	Dic₁₇₈	—	16	4	no	—
87	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
88	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
89	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
90	#10623494	C₂×Q₁₆	1	4	yes	—
91	C₂²×C₈₉	—	2	2	no	—
92	C₂²×C₈₉	#500	1	2	yes	—
93	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₈₉	—	8	4	no	—
95	C₄×C₈₉	#500	1	2	yes	—
96	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
97	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
98	Dic₈₉	—	2	4	no	—
99	Dic₈₉	—	8	2	no	—
100	Dic₈₉	—	8	2	no	—
101	#33730	—	89	1	no	maximal, Sylow
102	C₂×C₈₉	—	2	2	no	—
103	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
104	C₂×C₈₉	#3416	1	2	yes	—
105	#3164	—	89	2	no	—
106	#2255	—	89	4	no	—
107	#2254	—	89	2	no	—
108	#2253	—	89	2	no	—
109	#2159	—	89	2	no	—
110	#2158	—	89	1	no	—
111	C₂×#367	D₈₉	1	1	yes	characteristic
112	#2108	—	89	1	no	—
113	C₈₉	#33730	1	1	yes	characteristic, complemented, Sylow
114	#544	—	178	2	no	—
115	C₂×M₃₂	—	89	1	no	—
116	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
117	#501	—	178	4	no	—
118	#500	—	89	4	no	—
119	#499	—	89	4	no	—
120	#428	—	178	2	no	—
121	#427	—	89	2	no	—
122	C₂×#107	C₂×D₈₉	1	2	yes	—
123	#405	—	89	1	no	—
124	#367	—	89	1	no	—
125	#367	—	89	1	no	—
126	#367	—	89	4	no	—
127	#367	C₂×D₈₉	1	2	yes	—
128	#367	C₂×D₈₉	1	2	yes	—
129	#366	—	89	2	no	—
130	#366	—	89	4	no	—
131	#346	—	89	2	no	—
132	C₄×C₁₆	—	89	2	no	—
133	C₂×M₁₆	—	89	1	no	—
134	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
135	C₄⋊Q₈	—	356	2	no	—
136	C₄⋊Q₈	—	178	4	no	—
137	#125	—	356	2	no	—
138	#125	—	178	4	no	—
139	C₂²⋊Q₈	—	356	4	no	—
140	#117	—	89	1	no	—
141	#117	—	178	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	M₃₂	—	89	4	no	—
144	C₂×C₁₆	—	89	2	no	—
145	C₂×C₁₆	Dic₁₇₈	1	4	yes	complemented
146	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
148	#107	—	2	2	no	—
149	#107	—	89	2	no	—
150	#107	—	89	2	no	—
151	#107	—	178	4	no	—
152	#107	C₂²×D₈₉	1	4	yes	—
153	#106	—	178	4	no	—
154	#106	—	178	2	no	—
155	#97	—	89	2	no	—
156	C₄×C₈	—	89	2	no	—
157	C₂×Q₈	—	356	4	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
160	M₁₆	—	89	4	no	—
161	C₂×C₈	—	89	2	no	—
162	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
163	C₂×C₈	Q₈×D₈₉	1	2	yes	—
164	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
165	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	89	2	no	—
169	C₄⋊C₄	—	356	4	no	—
170	C₄⋊C₄	—	356	4	no	—
171	C₄⋊C₄	—	356	4	no	—
172	C₄⋊C₄	—	178	2	no	—
173	C₄⋊C₄	—	178	2	no	—
174	C₂²⋊C₄	—	89	2	no	—
175	C₂²⋊C₄	—	356	2	no	—
176	C₄²	—	89	2	no	—
177	C₄²	—	356	2	no	—
178	C₁₆	—	89	4	no	—
179	C₁₆	C₂×Dic₁₇₈	1	2	yes	—
180	C₁₆	C₂×Dic₁₇₈	1	2	yes	—
181	C₂³	D₈×D₈₉	1	1	yes	characteristic
182	Q₈	—	1424	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	16	2	no	—
185	C₂×C₄	—	89	4	no	—
186	C₂×C₄	—	356	2	no	—
187	C₂×C₄	—	356	2	no	—
188	C₂×C₄	#15668011	1	2	yes	—
189	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
190	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
191	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
192	C₈	—	89	4	no	—
193	C₈	#15668011	1	2	yes	—
194	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
195	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	#19152002	1	2	yes	—
198	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
199	C₄	—	8	4	no	—
200	C₄	—	178	4	no	—
201	C₄	—	712	2	no	—
202	C₄	—	712	2	no	—
203	C₄	#19152002	1	2	yes	—
204	C₄	C₂×#15668011	1	1	yes	characteristic
205	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×#19152002	1	1	yes	characteristic
208	C₂	#26033394	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#367×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	C₂×#367	D₈₉
hypocenter	C₈₉	#33730
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
2. #26022537	C₂	semidirect
3. #26022533	C₂	semidirect
4. #26022530	C₂	semidirect
36. #15667952	Q₈	semidirect
47. C₂×C₁₆×C₈₉	Q₈	semidirect
113. C₈₉	#33730	semidirect
145. C₂×C₁₆	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26030788	2
#26022537	2
#26022533	2
#26022530	2
#26022038	2
#26022036	2
#26021848	2
C₂×#367×C₈₉	2
#33730	89