Subgroups — #36469823

Order 22784.

310 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151427	C₂	1	1	yes	maximal, characteristic
2	#26022034	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151374	C₂	1	1	yes	maximal, characteristic
4	#26020131	C₂	1	1	yes	maximal, characteristic
5	#26018349	C₂	1	1	yes	maximal, characteristic
6	#26018347	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1772	C₂	1	1	yes	maximal, characteristic
8	C₂²×#13083549	C₂²	1	1	yes	characteristic
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	C₂×#15667948	C₂²	1	1	yes	characteristic
11	#19151636	C₂²	1	1	yes	characteristic
12	#19151553	—	2	1	no	—
13	#19151427	C₄	1	4	yes	complemented
14	#19151425	C₄	1	2	yes	complemented
15	#19151423	C₄	1	2	yes	complemented
16	#19151374	C₄	1	4	yes	complemented
17	#19151358	—	2	1	no	—
18	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
19	#427×C₈₉	C₂²	1	1	yes	characteristic
20	#405×C₈₉	—	2	1	no	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	#339×C₈₉	—	2	1	no	—
23	C₂×#13083558	—	2	1	no	—
24	#15667971	—	2	1	no	—
25	C₂×#13083549	—	2	1	no	—
26	C₂×#13083549	—	2	1	no	—
27	C₂×#13083549	C₂×C₄	1	2	yes	—
28	C₂×#13083549	C₂×C₄	1	2	yes	—
29	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
30	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
31	C₂×#13083549	C₂×C₄	1	2	yes	—
32	#15667948	—	2	1	no	—
33	#15667948	C₂×C₄	1	2	yes	—
34	#15667946	C₂×C₄	1	2	yes	—
35	#15667944	—	2	1	no	—
36	#15667922	—	4	1	no	—
37	C₈×Dic₈₉	—	4	1	no	—
38	C₂×M₁₆×C₈₉	—	2	1	no	—
39	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
41	#117×C₈₉	—	2	1	no	—
42	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
43	#108×C₈₉	—	2	1	no	—
44	#108×C₈₉	C₂×C₄	1	2	yes	—
45	#107×C₈₉	C₂×C₄	1	2	yes	—
46	#106×C₈₉	—	2	1	no	—
47	#97×C₈₉	—	4	1	no	—
48	C₄×C₈×C₈₉	—	4	1	no	—
49	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₈₉	Q₈	1	2	yes	—
51	#13083558	—	2	2	no	—
52	#13083558	—	4	1	no	—
53	C₂×#10623494	—	4	1	no	—
54	C₄⋊Dic₈₉	—	2	2	no	—
55	C₂²⋊Dic₈₉	—	4	1	no	—
56	C₄×Dic₈₉	—	4	1	no	—
57	#13083549	—	2	2	no	—
58	#13083549	C₄²	1	2	yes	—
59	#13083549	C₄²	1	2	yes	—
60	C₂²×C₄×C₈₉	—	2	1	no	—
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	M₁₆×C₈₉	—	2	2	no	—
63	M₁₆×C₈₉	—	4	1	no	—
64	C₂×C₈×C₈₉	—	4	1	no	—
65	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
66	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
67	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
68	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
69	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
74	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
75	C₄⋊C₄×C₈₉	—	2	2	no	—
76	C₂²⋊C₄×C₈₉	—	4	1	no	—
77	C₄²×C₈₉	—	2	1	no	—
78	C₄²×C₈₉	—	4	1	no	—
79	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
80	C₂³×C₈₉	#135	1	1	yes	characteristic
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	—	4	2	no	—
83	C₂×C₄×C₈₉	—	4	2	no	—
84	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
90	C₂×Dic₈₉	—	4	2	no	—
91	C₈×C₈₉	—	4	2	no	—
92	C₈×C₈₉	#107	1	2	yes	—
93	C₈×C₈₉	#107	1	2	yes	—
94	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
95	C₈×C₈₉	#117	1	1	yes	characteristic
96	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
97	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
98	#10623494	—	4	2	no	—
99	C₂²×C₈₉	—	2	2	no	—
100	C₂²×C₈₉	#427	1	1	yes	characteristic
101	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
102	C₂²×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
103	C₄×C₈₉	—	4	2	no	—
104	C₄×C₈₉	—	8	2	no	—
105	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
106	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
107	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
108	C₄×C₈₉	C₄×Q₁₆	1	1	yes	characteristic
109	Dic₈₉	—	8	2	no	—
110	#12671	—	89	1	no	maximal, Sylow
111	C₂×C₈₉	—	2	1	no	—
112	C₂×C₈₉	—	2	1	no	—
113	C₂×C₈₉	C₄×#107	1	1	yes	characteristic
114	C₂×C₈₉	#2179	1	2	yes	—
115	C₂×#368	—	89	1	no	—
116	#2158	—	89	1	no	—
117	C₂×#346	—	89	1	no	—
118	#1772	—	89	1	no	—
119	#1772	D₈₉	1	1	yes	characteristic
120	#1382	—	89	1	no	—
121	#1382	—	89	1	no	—
122	C₈₉	#12671	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₁₆	—	89	1	no	—
124	C₂²×C₁₆	—	89	1	no	—
125	C₂×#108	—	89	1	no	—
126	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
127	#427	—	89	1	no	—
128	#427	C₂×D₈₉	1	1	yes	characteristic
129	#405	—	2	1	no	—
130	#405	—	178	1	no	—
131	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
132	#368	—	89	4	no	—
133	#367	—	89	2	no	—
134	#366	—	89	2	no	—
135	#346	—	89	4	no	—
136	#339	—	2	1	no	—
137	#339	—	178	1	no	—
138	C₂×M₁₆	—	2	1	no	—
139	C₂×M₁₆	—	178	1	no	—
140	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
142	#117	—	2	1	no	—
143	#117	—	178	1	no	—
144	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
145	C₂×C₁₆	—	89	2	no	—
146	C₂×C₁₆	—	89	1	no	—
147	C₂×C₁₆	—	89	1	no	—
148	C₂×C₁₆	—	89	2	no	—
149	C₂×C₁₆	—	89	2	no	—
150	C₂×C₁₆	—	178	1	no	—
151	C₂×C₁₆	—	178	1	no	—
152	#108	—	2	1	no	—
153	#108	—	89	2	no	—
154	#108	—	178	1	no	—
155	#108	C₄×D₈₉	1	2	yes	—
156	#107	—	89	2	no	—
157	#107	C₄×D₈₉	1	2	yes	—
158	#106	—	2	1	no	—
159	#106	—	178	1	no	—
160	#97	—	4	1	no	—
161	#97	—	356	1	no	—
162	C₄×C₈	—	4	1	no	—
163	C₄×C₈	—	356	1	no	—
164	C₄×C₈	Dic₁₇₈	1	2	yes	—
165	C₄×C₈	C₄×D₈₉	1	2	yes	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
168	M₁₆	—	2	2	no	—
169	M₁₆	—	4	1	no	—
170	M₁₆	—	356	1	no	—
171	M₁₆	—	178	2	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	356	1	no	—
174	C₂×C₈	#13083553	1	2	yes	—
175	C₂×C₈	#13083559	1	1	yes	characteristic
176	C₂×C₈	#13083559	1	1	yes	characteristic
177	C₂×C₈	#13083564	1	2	yes	—
178	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
179	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
180	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
181	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
182	C₂×C₈	#13083574	1	1	yes	characteristic
183	C₂×C₈	#13083574	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	178	2	no	—
186	C₂²⋊C₄	—	4	1	no	—
187	C₂²⋊C₄	—	356	1	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	4	1	no	—
190	C₄²	—	356	1	no	—
191	C₄²	#13083553	1	2	yes	—
192	C₁₆	—	89	2	no	—
193	C₁₆	—	89	2	no	—
194	C₁₆	—	178	2	no	—
195	C₂³	#15668032	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	356	2	no	—
200	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
201	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
202	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
203	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
204	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
205	C₂×C₄	#15668032	1	1	yes	characteristic
206	C₈	—	4	2	no	—
207	C₈	—	356	2	no	—
208	C₈	#15667945	1	2	yes	—
209	C₈	#15667945	1	2	yes	—
210	C₈	C₄²×D₈₉	1	1	yes	characteristic
211	C₈	#15667969	1	1	yes	characteristic
212	C₈	C₄×Dic₁₇₈	1	1	yes	characteristic
213	C₈	C₄×Dic₁₇₈	1	1	yes	characteristic
214	C₂²	—	2	2	no	—
215	C₂²	#19151634	1	1	yes	characteristic
216	C₂²	C₄×#13083559	1	1	yes	characteristic
217	C₂²	C₄×#13083564	1	1	yes	characteristic
218	C₄	—	4	2	no	—
219	C₄	—	8	2	no	—
220	C₄	—	712	2	no	—
221	C₄	C₄×#13083553	1	1	yes	characteristic
222	C₄	C₂×#15667945	1	1	yes	characteristic
223	C₄	C₄×#13083559	1	1	yes	characteristic
224	C₄	C₄×#13083564	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	C₄×#15667945	1	1	yes	characteristic
228	C₂	#26022136	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#13083559
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1772	C₂
socle	C₂²×C₈₉	C₄×D₈
prime radical	C₂³×C₈₉	#135
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1772	D₈₉
hypocenter	C₈₉	#12671
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
13. #19151427	C₄	semidirect
14. #19151425	C₄	semidirect
15. #19151423	C₄	semidirect
16. #19151374	C₄	semidirect
122. C₈₉	#12671	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151427	2
#26022034	2
C₂×#19151374	2
#26020131	2
#26018349	2
#26018347	2
C₈₉×#1772	2
#12671	89