Subgroups — #11785339

Order 1152.

504 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₉×#380	C₂	1	1	yes	maximal, characteristic
3	#10588892	C₂	1	1	yes	maximal, characteristic
4	C₂×#63367	C₂	1	2	yes	maximal, complemented
5	#10588206	C₂	1	2	yes	maximal, complemented
6	#69511	—	3	1	no	maximal
7	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
10	C₂×#3676	—	2	1	no	—
11	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
12	C₂²×C₉⋊C₈	—	2	4	no	—
13	C₂²×C₉⋊C₈	C₄	1	4	yes	complemented
14	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₉×#95	—	2	1	no	—
17	C₉⋊#95	—	2	1	no	—
18	#63367	—	2	4	no	—
19	C₂³×C₃⋊C₈	—	3	1	no	—
20	C₃×#380	D₃	1	1	yes	characteristic
21	#5212	—	3	1	no	—
22	C₂×#808	—	3	2	no	—
23	#4529	—	3	2	no	—
24	C₂⁴×C₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	—	2	2	no	—
27	C₂²×C₄×C₉	—	2	2	no	—
28	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	Q₈	1	2	yes	—
31	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₉	—	2	2	no	—
33	C₉×C₄⋊C₄	—	2	2	no	—
34	C₉×C₄⋊C₄	—	2	2	no	—
35	C₉×C₂²⋊C₄	—	4	2	no	—
36	#3676	—	4	2	no	—
37	C₄⋊Dic₉	—	2	2	no	—
38	C₄⋊Dic₉	—	2	2	no	—
39	C₂×C₉⋊C₈	—	2	2	no	—
40	C₂×C₉⋊C₈	—	2	8	no	—
41	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₉⋊C₈	D₄	1	4	yes	—
44	#1807	—	9	1	no	Sylow
45	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
47	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
48	C₂×#217	—	6	1	no	—
49	C₂×C₄⋊Dic₃	—	3	2	no	—
50	C₂²×C₃⋊C₈	—	3	1	no	—
51	C₂²×C₃⋊C₈	—	3	4	no	—
52	C₂²×C₃⋊C₈	—	3	1	no	—
53	C₂²×C₃⋊C₈	—	6	4	no	—
54	C₃×#95	—	2	1	no	—
55	C₃⋊#95	—	6	1	no	—
56	#808	—	6	4	no	—
57	C₂³×C₉	—	2	2	no	—
58	C₂³×C₉	—	2	2	no	—
59	C₂³×C₉	C₄⋊C₄	1	2	yes	—
60	C₂³×C₉	SD₁₆	1	2	yes	—
61	C₂³×C₉	SD₁₆	1	2	yes	—
62	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₉	—	2	2	no	—
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	—	2	2	no	—
66	C₂×C₄×C₉	—	2	4	no	—
67	C₂×C₄×C₉	—	4	2	no	—
68	C₂×C₄×C₉	—	4	2	no	—
69	C₂×C₄×C₉	—	4	2	no	—
70	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₉	#41	1	2	yes	—
75	C₂×Dic₉	—	4	2	no	—
76	C₂×Dic₉	—	4	2	no	—
77	C₂×Dic₉	—	4	2	no	—
78	C₉⋊C₈	—	2	2	no	—
79	C₉⋊C₈	C₂²⋊C₄	1	4	yes	complemented
80	C₂³×C₈	—	9	1	no	—
81	C₂×#106	—	9	2	no	—
82	#380	—	9	1	no	—
83	#380	D₉	1	1	yes	characteristic
84	#340	—	9	2	no	—
85	C₂⁴×C₃	#67	1	1	yes	characteristic
86	C₂²×C₃×C₄	—	2	2	no	—
87	C₂²×C₃×C₄	—	2	2	no	—
88	C₂²×C₃×C₄	—	2	2	no	—
89	C₂²×C₃×C₄	Dic₆	1	2	yes	—
90	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
91	C₂²×C₃×C₄	#67	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
93	C₂²×Dic₃	—	6	2	no	—
94	C₃×C₄⋊C₄	—	2	2	no	—
95	C₃×C₄⋊C₄	—	2	2	no	—
96	C₃×C₂²⋊C₄	—	4	2	no	—
97	#217	—	12	2	no	—
98	C₄⋊Dic₃	—	6	2	no	—
99	C₄⋊Dic₃	—	6	2	no	—
100	C₂×C₃⋊C₈	—	3	2	no	—
101	C₂×C₃⋊C₈	—	3	4	no	—
102	C₂×C₃⋊C₈	—	3	2	no	—
103	C₂×C₃⋊C₈	—	6	8	no	—
104	C₂×C₃⋊C₈	—	6	2	no	—
105	C₂²×C₉	—	2	2	no	—
106	C₂²×C₉	—	2	2	no	—
107	C₂²×C₉	—	2	4	no	—
108	C₂²×C₉	—	2	4	no	—
109	C₂²×C₉	#106	1	4	yes	complemented
110	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₉	#121	1	2	yes	—
112	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
113	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
114	C₂²×C₉	#135	1	2	yes	—
115	C₄×C₉	—	2	2	no	—
116	C₄×C₉	—	8	2	no	—
117	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₄×C₉	#117	1	1	yes	characteristic
119	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
120	Dic₉	—	8	2	no	—
121	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
122	C₂²×C₈	—	9	1	no	—
123	C₂²×C₈	—	9	4	no	—
124	C₂²×C₈	—	9	1	no	—
125	C₂²×C₈	—	18	4	no	—
126	C₂×C₄⋊C₄	—	9	2	no	—
127	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
128	C₂×C₂²⋊C₄	—	2	1	no	—
129	C₂×C₂²⋊C₄	—	18	1	no	—
130	#106	—	18	4	no	—
131	#95	—	2	1	no	—
132	#95	—	18	1	no	—
133	C₂³×C₃	—	2	2	no	—
134	C₂³×C₃	—	2	2	no	—
135	C₂³×C₃	Dic₃⋊C₄	1	2	yes	—
136	C₂³×C₃	#214	1	2	yes	—
137	C₂³×C₃	#215	1	2	yes	—
138	C₂³×C₃	C₂×#67	1	1	yes	characteristic
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	4	2	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	—	4	2	no	—
146	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
147	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
149	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
150	C₂×C₃×C₄	#237	1	2	yes	—
151	C₂×Dic₃	—	12	2	no	—
152	C₂×Dic₃	—	12	2	no	—
153	C₂×Dic₃	—	12	2	no	—
154	C₃⋊C₈	—	3	4	no	—
155	C₃⋊C₈	—	6	2	no	—
156	C₂×C₉	—	2	4	no	—
157	C₂×C₉	#380	1	1	yes	characteristic
158	C₂×C₉	C₂×#106	1	1	yes	characteristic
159	C₂×C₉	#427	1	1	yes	characteristic
160	C₂×C₉	#465	1	2	yes	—
161	C₂×C₉	#465	1	2	yes	—
162	C₂⁴	#611	1	1	yes	characteristic
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	18	2	no	—
167	C₂²×C₄	Dic₁₈	1	2	yes	—
168	C₂²×C₄	C₄×D₉	1	2	yes	—
169	C₂²×C₄	#611	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
171	C₂×C₈	—	9	2	no	—
172	C₂×C₈	—	9	4	no	—
173	C₂×C₈	—	9	2	no	—
174	C₂×C₈	—	18	8	no	—
175	C₂×C₈	—	18	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	18	2	no	—
179	C₄⋊C₄	—	18	2	no	—
180	C₂²⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	36	2	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	4	no	—
185	C₂²×C₃	—	2	4	no	—
186	C₂²×C₃	#808	1	4	yes	—
187	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
188	C₂²×C₃	C₂×#214	1	1	yes	characteristic
189	C₂²×C₃	#939	1	2	yes	—
190	C₂²×C₃	C₂×#215	1	1	yes	characteristic
191	C₂²×C₃	#950	1	2	yes	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	8	2	no	—
194	C₃×C₄	#877	1	1	yes	characteristic
195	C₃×C₄	#878	1	2	yes	—
196	C₃×C₄	D₃×C₂²⋊C₄	1	1	yes	characteristic
197	Dic₃	—	24	2	no	—
198	C₉	#1807	1	1	yes	characteristic, complemented, Sylow
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	Dic₉⋊C₄	1	2	yes	—
202	C₂³	#3672	1	2	yes	—
203	C₂³	#3675	1	2	yes	—
204	C₂³	C₂×#611	1	1	yes	characteristic
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	36	2	no	—
213	C₂×C₄	—	36	2	no	—
214	C₂×C₄	—	36	2	no	—
215	C₂×C₄	Dic₉⋊C₄	1	2	yes	—
216	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
218	C₂×C₄	D₄×D₉	1	2	yes	—
219	C₂×C₄	#3699	1	2	yes	—
220	C₈	—	9	4	no	—
221	C₈	—	18	2	no	—
222	C₂×C₃	—	2	4	no	—
223	C₂×C₃	#4942	1	1	yes	characteristic
224	C₂×C₃	C₂×#808	1	1	yes	characteristic
225	C₂×C₃	#4998	1	1	yes	characteristic
226	C₂×C₃	#5041	1	2	yes	—
227	C₂×C₃	#5048	1	2	yes	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	2	no	—
232	C₂²	#63367	1	4	yes	complemented
233	C₂²	C₂×Dic₉⋊C₄	1	1	yes	characteristic
234	C₂²	C₂×#3672	1	1	yes	characteristic
235	C₂²	#63501	1	2	yes	—
236	C₂²	C₂×#3675	1	1	yes	characteristic
237	C₂²	#63513	1	2	yes	—
238	C₄	—	2	2	no	—
239	C₄	—	8	2	no	—
240	C₄	—	72	2	no	—
241	C₄	#63439	1	1	yes	characteristic
242	C₄	#63440	1	2	yes	—
243	C₄	C₂²⋊C₄×D₉	1	1	yes	characteristic
244	C₃	#69511	1	1	yes	characteristic
245	C₂	—	2	4	no	—
246	C₂	#10588622	1	1	yes	characteristic
247	C₂	C₂×#63367	1	1	yes	characteristic
248	C₂	#10588676	1	1	yes	characteristic
249	C₂	#10588717	1	2	yes	—
250	C₂	#10588731	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#380	C₂
socle	C₂³×C₃	C₂×#67
prime radical	C₂⁴×C₃	#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#380	D₉
hypocenter	C₉	#1807
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

Complements

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

N	H ≅ G/N	type
4. C₂×#63367	C₂	semidirect
5. #10588206	C₂	semidirect
13. C₂²×C₉⋊C₈	C₄	semidirect
79. C₉⋊C₈	C₂²⋊C₄	semidirect
109. C₂²×C₉	#106	semidirect
198. C₉	#1807	semidirect
232. C₂²	#63367	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₉⋊C₈	2
C₉×#380	2
#10588892	2
C₂×#63367	2
#10588206	2
#69511	3