Subgroups — #11874973

Order 1152.

534 conjugacy classes of subgroups.

The subgroup lattice has 237 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10589619	C₂	1	2	yes	maximal
2	C₉×#427	C₂	1	1	yes	maximal, characteristic
3	#10588787	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic
5	#10588676	C₂	1	2	yes	maximal
6	#10588558	C₂	1	2	yes	maximal, complemented
7	#10588554	C₂	1	2	yes	maximal, complemented
8	#10588538	C₂	1	2	yes	maximal, complemented
9	#10588535	C₂	1	2	yes	maximal, complemented
10	#79170	—	3	1	no	maximal
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₉×#117	C₂²	1	2	yes	—
13	#63488	C₂²	1	2	yes	—
14	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
15	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	#63452	C₂²	1	4	yes	—
17	#63450	C₂²	1	4	yes	—
18	#63440	—	2	4	no	—
19	C₄×Dic₁₈	—	2	4	no	—
20	C₉×#107	C₂²	1	2	yes	—
21	C₉×#106	C₂²	1	2	yes	—
22	#63378	C₂²	1	2	yes	—
23	#63377	C₂²	1	2	yes	—
24	C₈×Dic₉	C₂²	1	1	yes	characteristic
25	C₈×Dic₉	C₂²	1	1	yes	characteristic
26	C₈×Dic₉	C₂²	1	1	yes	characteristic
27	C₈×Dic₉	C₂²	1	1	yes	characteristic
28	#63367	C₂²	1	4	yes	—
29	#63366	C₂²	1	4	yes	—
30	#5517	—	3	2	no	—
31	C₃×#427	D₃	1	1	yes	characteristic
32	#5106	—	3	1	no	—
33	C₂×C₈×Dic₃	—	3	1	no	—
34	#4998	—	3	2	no	—
35	#4878	—	3	2	no	—
36	#4874	—	3	2	no	—
37	#4858	—	3	2	no	—
38	#4855	—	3	2	no	—
39	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂²×Dic₉	D₄	1	2	yes	—
41	C₂×Dic₁₈	—	2	4	no	—
42	C₂×C₈×C₉	Q₈	1	4	yes	complemented
43	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
45	C₉×C₄⋊C₄	C₂³	1	4	yes	—
46	C₉×C₂²⋊C₄	—	2	2	no	—
47	C₄²×C₉	—	2	2	no	—
48	#3676	—	2	2	no	—
49	C₄⋊Dic₉	—	2	4	no	—
50	C₄⋊Dic₉	C₂³	1	4	yes	—
51	Dic₉⋊C₄	—	2	4	no	—
52	Dic₉⋊C₄	—	2	4	no	—
53	C₄×Dic₉	—	2	2	no	—
54	C₄×Dic₉	D₄	1	2	yes	—
55	C₄×Dic₉	C₂³	1	1	yes	characteristic
56	C₄×Dic₉	C₂³	1	1	yes	characteristic
57	C₂×C₉⋊C₈	Q₈	1	4	yes	complemented
58	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
59	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
60	#3160	—	9	1	no	Sylow
61	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
62	C₃×#117	C₂×D₃	1	2	yes	—
63	#926	—	3	2	no	—
64	C₂×C₄×Dic₃	—	3	1	no	—
65	C₂²×C₃⋊C₈	—	3	1	no	—
66	#890	—	3	4	no	—
67	#888	—	3	4	no	—
68	#878	—	6	4	no	—
69	C₄×Dic₆	—	6	4	no	—
70	C₃×#107	C₂×D₃	1	2	yes	—
71	C₃×#106	C₂×D₃	1	2	yes	—
72	#818	—	3	2	no	—
73	#817	—	3	2	no	—
74	C₈×Dic₃	—	3	1	no	—
75	C₈×Dic₃	—	3	1	no	—
76	C₈×Dic₃	—	3	1	no	—
77	C₈×Dic₃	—	3	1	no	—
78	#808	—	3	4	no	—
79	#807	—	3	4	no	—
80	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₉	—	2	4	no	—
82	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
84	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
85	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
86	C₂×Dic₉	—	2	4	no	—
87	C₂×Dic₉	—	2	4	no	—
88	C₂×Dic₉	C₂×D₄	1	2	yes	—
89	C₂×Dic₉	C₂×D₄	1	2	yes	—
90	Dic₁₈	—	4	8	no	—
91	C₈×C₉	C₂×Q₈	1	2	yes	—
92	C₈×C₉	C₂×Q₈	1	2	yes	—
93	C₉⋊C₈	C₂×Q₈	1	2	yes	—
94	C₉⋊C₈	C₂×Q₈	1	2	yes	—
95	#533	—	9	2	no	—
96	#500	—	9	2	no	—
97	#499	—	9	2	no	—
98	#499	—	9	2	no	—
99	#498	—	9	2	no	—
100	#427	—	9	1	no	—
101	#427	—	9	2	no	—
102	#427	D₉	1	1	yes	characteristic
103	C₂×C₄×C₈	—	9	1	no	—
104	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
105	C₂²×Dic₃	—	3	2	no	—
106	C₂×Dic₆	—	6	4	no	—
107	C₂×C₃×C₈	Dic₆	1	4	yes	—
108	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
110	C₃×C₄⋊C₄	C₂²×D₃	1	4	yes	—
111	C₃×C₂²⋊C₄	—	2	2	no	—
112	C₃×C₄²	—	2	2	no	—
113	#217	—	6	2	no	—
114	C₄⋊Dic₃	—	3	4	no	—
115	C₄⋊Dic₃	—	6	4	no	—
116	Dic₃⋊C₄	—	6	4	no	—
117	Dic₃⋊C₄	—	6	4	no	—
118	C₄×Dic₃	—	3	2	no	—
119	C₄×Dic₃	—	3	1	no	—
120	C₄×Dic₃	—	3	1	no	—
121	C₄×Dic₃	—	6	2	no	—
122	C₂×C₃⋊C₈	—	3	4	no	—
123	C₂×C₃⋊C₈	—	3	1	no	—
124	C₂×C₃⋊C₈	—	3	1	no	—
125	C₂²×C₉	—	2	2	no	—
126	C₂²×C₉	C₄⋊Q₈	1	2	yes	—
127	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₉	—	4	4	no	—
129	C₄×C₉	C₄⋊Q₈	1	2	yes	—
130	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
131	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
132	Dic₉	—	2	2	no	—
133	Dic₉	—	4	4	no	—
134	Dic₉	#135	1	4	yes	—
135	C₂²×C₈	—	9	1	no	—
136	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
137	C₄⋊Q₈	—	9	4	no	—
138	#125	—	9	4	no	—
139	C₂²⋊Q₈	—	18	4	no	—
140	C₄×Q₈	—	18	4	no	—
141	#117	—	9	2	no	—
142	#117	C₂×D₉	1	2	yes	—
143	C₂×C₄²	—	9	1	no	—
144	#107	—	9	2	no	—
145	#107	—	9	4	no	—
146	#107	C₂×D₉	1	2	yes	—
147	#106	—	9	2	no	—
148	#106	—	9	4	no	—
149	#106	C₂×D₉	1	2	yes	—
150	C₄×C₈	—	9	1	no	—
151	C₄×C₈	—	9	1	no	—
152	C₄×C₈	—	9	1	no	—
153	C₄×C₈	—	9	1	no	—
154	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
157	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
159	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
160	C₂×Dic₃	—	3	2	no	—
161	C₂×Dic₃	—	3	2	no	—
162	C₂×Dic₃	—	6	4	no	—
163	C₂×Dic₃	—	6	4	no	—
164	Dic₆	—	12	8	no	—
165	C₃×C₈	C₂×Dic₆	1	2	yes	—
166	C₃×C₈	C₂×Dic₆	1	2	yes	—
167	C₃⋊C₈	—	3	2	no	—
168	C₃⋊C₈	—	3	2	no	—
169	C₂×C₉	—	2	2	no	—
170	C₂×C₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
171	C₂×C₉	C₂×#135	1	2	yes	—
172	C₂×Q₈	—	18	4	no	—
173	C₂²×C₄	—	9	2	no	—
174	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
175	C₂×C₈	—	9	4	no	—
176	C₂×C₈	—	9	1	no	—
177	C₂×C₈	—	9	1	no	—
178	C₂×C₈	Dic₁₈	1	4	yes	complemented
179	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
181	C₄⋊C₄	—	9	4	no	—
182	C₄⋊C₄	—	18	4	no	—
183	C₄⋊C₄	—	18	4	no	—
184	C₄⋊C₄	—	18	4	no	—
185	C₄⋊C₄	C₂²×D₉	1	4	yes	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	18	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	9	2	no	—
190	C₄²	—	9	1	no	—
191	C₄²	—	9	1	no	—
192	C₄²	—	18	2	no	—
193	C₂²×C₃	—	2	2	no	—
194	C₂²×C₃	#888	1	2	yes	—
195	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
196	C₃×C₄	—	4	4	no	—
197	C₃×C₄	#888	1	2	yes	—
198	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
199	C₃×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
200	Dic₃	—	3	4	no	—
201	Dic₃	—	6	2	no	—
202	Dic₃	—	12	4	no	—
203	C₉	#3160	1	1	yes	characteristic, complemented, Sylow
204	C₂³	D₄×D₉	1	1	yes	characteristic
205	Q₈	—	36	8	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	9	2	no	—
208	C₂×C₄	—	9	2	no	—
209	C₂×C₄	—	18	4	no	—
210	C₂×C₄	—	18	4	no	—
211	C₂×C₄	C₂×Dic₁₈	1	2	yes	—
212	C₂×C₄	D₄×D₉	1	1	yes	characteristic
213	C₂×C₄	Q₈×D₉	1	2	yes	—
214	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
215	C₈	—	9	2	no	—
216	C₈	—	9	2	no	—
217	C₈	C₂×Dic₁₈	1	2	yes	—
218	C₈	C₂×Dic₁₈	1	2	yes	—
219	C₂×C₃	—	2	2	no	—
220	C₂×C₃	C₂×#888	1	1	yes	characteristic
221	C₂×C₃	D₃×#135	1	2	yes	—
222	C₂²	—	2	2	no	—
223	C₂²	#63450	1	2	yes	—
224	C₂²	C₂×D₄×D₉	1	1	yes	characteristic
225	C₄	—	4	4	no	—
226	C₄	—	9	4	no	—
227	C₄	—	18	2	no	—
228	C₄	—	36	4	no	—
229	C₄	#63450	1	2	yes	—
230	C₄	C₂²×Dic₁₈	1	1	yes	characteristic
231	C₄	C₂×Q₈×D₉	1	1	yes	characteristic
232	C₃	#79170	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	C₂×#63450	1	1	yes	characteristic
235	C₂	D₉×#135	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₉
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₉×#427	C₂
socle	C₂²×C₃	C₂×D₃×D₄
prime radical	C₂³×C₃	D₃×D₄
hypercenter	#427	D₉
hypocenter	C₉	#3160
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
6. #10588558	C₂	semidirect
7. #10588554	C₂	semidirect
8. #10588538	C₂	semidirect
9. #10588535	C₂	semidirect
42. C₂×C₈×C₉	Q₈	semidirect
57. C₂×C₉⋊C₈	Q₈	semidirect
178. C₂×C₈	Dic₁₈	semidirect
203. C₉	#3160	semidirect

Maximal subgroups

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

subgroup	index
#10589619	2
C₉×#427	2
#10588787	2
C₂×C₈×Dic₉	2
#10588676	2
#10588558	2
#10588554	2
#10588538	2
#10588535	2
#79170	3