Subgroups — #11774729

Order 1152.

414 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10589579	C₂	1	1	yes	maximal, characteristic
2	C₉×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588162	C₂	1	4	yes	maximal, complemented
5	#70498	—	3	1	no	maximal
6	C₂×#3697	—	2	1	no	—
7	C₂×C₉×M₁₆	—	2	1	no	—
8	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
9	#63492	—	2	1	no	—
10	#63492	C₄	1	2	yes	—
11	C₂²×C₉⋊C₈	—	2	1	no	—
12	#63437	—	2	1	no	—
13	#63436	C₄	1	1	yes	characteristic
14	C₄×D₃₆	—	2	1	no	—
15	C₄×D₃₆	C₂²	1	2	yes	—
16	#63431	C₄	1	1	yes	characteristic
17	C₉×C₄⋊C₈	—	2	1	no	—
18	C₉×C₄⋊C₈	C₂²	1	2	yes	—
19	C₄×C₉⋊C₈	—	2	1	no	—
20	C₄×C₉⋊C₈	C₂²	1	2	yes	—
21	#5477	—	3	1	no	—
22	C₃×#423	D₃	1	1	yes	characteristic
23	C₂×C₄×C₃⋊C₈	—	3	1	no	—
24	#4483	—	3	4	no	—
25	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂×#611	—	2	2	no	—
29	#3697	—	4	2	no	—
30	#3697	—	4	2	no	—
31	C₂×D₃₆	—	2	1	no	—
32	C₂×D₃₆	—	2	1	no	—
33	C₂×D₃₆	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₉	—	2	2	no	—
35	C₂×Dic₁₈	—	2	1	no	—
36	C₉×M₁₆	—	4	2	no	—
37	C₂×C₈×C₉	—	2	2	no	—
38	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₉	D₄	1	1	yes	characteristic
40	C₄²×C₉	D₄	1	1	yes	characteristic
41	C₄²×C₉	C₂³	1	1	yes	characteristic
42	(C₂×D₉)⋊C₄	—	2	2	no	—
43	(C₂×D₉)⋊C₄	—	2	2	no	—
44	C₄⋊Dic₉	—	2	1	no	—
45	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
46	C₂×C₉⋊C₈	—	2	4	no	—
47	C₂×C₉⋊C₈	—	2	2	no	—
48	#1492	—	9	1	no	Sylow
49	C₂×#235	—	6	1	no	—
50	C₂×C₃×M₁₆	—	2	1	no	—
51	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
52	#930	—	3	2	no	—
53	#930	—	6	1	no	—
54	C₂²×C₃⋊C₈	—	6	1	no	—
55	#875	—	6	1	no	—
56	#874	—	3	1	no	—
57	C₄×D₁₂	—	3	2	no	—
58	C₄×D₁₂	—	6	1	no	—
59	#869	—	3	1	no	—
60	C₃×C₄⋊C₈	—	2	1	no	—
61	C₃×C₄⋊C₈	C₂×D₃	1	2	yes	—
62	C₄×C₃⋊C₈	—	3	2	no	—
63	C₄×C₃⋊C₈	—	6	1	no	—
64	C₂³×C₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×D₉	—	2	2	no	—
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	—	2	2	no	—
68	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₉	D₈	1	2	yes	—
72	C₂×C₄×C₉	SD₁₆	1	2	yes	—
73	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
76	#611	—	4	4	no	—
77	C₂×Dic₉	—	2	2	no	—
78	D₃₆	—	2	4	no	—
79	D₃₆	—	4	2	no	—
80	D₃₆	—	4	2	no	—
81	C₄×D₉	—	4	2	no	—
82	C₄×D₉	—	4	2	no	—
83	Dic₁₈	—	4	2	no	—
84	C₈×C₉	—	4	2	no	—
85	C₉⋊C₈	—	2	4	no	—
86	#532	—	9	1	no	—
87	#423	D₉	1	1	yes	characteristic, complemented
88	C₂×C₄×C₈	—	9	1	no	—
89	D₄⋊C₈	—	9	4	no	—
90	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
91	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
92	C₂²×C₃×C₄	#67	1	1	yes	characteristic
93	C₂×#67	—	6	2	no	—
94	#235	—	12	2	no	—
95	#235	—	12	2	no	—
96	C₂×D₁₂	—	3	2	no	—
97	C₂×D₁₂	—	6	1	no	—
98	C₂×D₁₂	—	6	1	no	—
99	C₂×C₄×D₃	—	6	2	no	—
100	C₂×Dic₆	—	6	1	no	—
101	C₃×M₁₆	—	4	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
104	C₃×C₄²	D₁₂	1	1	yes	characteristic
105	C₃×C₄²	#67	1	1	yes	characteristic
106	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
107	(C₂×D₃)⋊C₄	—	6	2	no	—
108	(C₂×D₃)⋊C₄	—	6	2	no	—
109	C₄⋊Dic₃	—	3	2	no	—
110	C₄⋊Dic₃	—	6	1	no	—
111	C₂×C₃⋊C₈	—	6	4	no	—
112	C₂×C₃⋊C₈	—	6	2	no	—
113	C₂²×C₉	—	2	1	no	—
114	C₂²×C₉	—	2	1	no	—
115	C₂²×C₉	#102	1	2	yes	—
116	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×D₉	—	4	2	no	—
118	C₂×D₉	—	4	2	no	—
119	C₂×D₉	—	4	2	no	—
120	C₄×C₉	—	2	1	no	—
121	C₄×C₉	—	2	1	no	—
122	C₄×C₉	#102	1	2	yes	—
123	C₄×C₉	#104	1	2	yes	—
124	C₄×C₉	#131	1	2	yes	—
125	C₄×C₉	C₂×D₈	1	1	yes	characteristic
126	C₄×C₉	C₂×SD₁₆	1	1	yes	characteristic
127	Dic₉	—	4	2	no	—
128	C₂×#41	—	18	1	no	—
129	C₂×M₁₆	—	2	1	no	—
130	C₂²×C₈	—	18	1	no	—
131	C₄⋊Q₈	—	9	1	no	—
132	#127	—	9	1	no	—
133	#124	—	18	1	no	—
134	#121	—	9	2	no	—
135	#121	—	18	1	no	—
136	C₄×D₄	—	9	2	no	—
137	C₄×D₄	—	18	1	no	—
138	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
139	C₄⋊C₈	—	2	1	no	—
140	C₄⋊C₈	C₂×D₉	1	2	yes	—
141	C₄×C₈	—	9	2	no	—
142	C₄×C₈	—	18	1	no	—
143	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
144	C₂²×D₃	—	6	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
148	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
149	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₄	#213	1	2	yes	—
151	C₂×C₃×C₄	#215	1	2	yes	—
152	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
155	#67	—	12	4	no	—
156	C₂×Dic₃	—	6	2	no	—
157	D₁₂	—	6	4	no	—
158	D₁₂	—	12	2	no	—
159	D₁₂	—	12	2	no	—
160	C₄×D₃	—	12	2	no	—
161	C₄×D₃	—	12	2	no	—
162	Dic₆	—	12	2	no	—
163	C₃×C₈	—	4	2	no	—
164	C₃⋊C₈	—	6	4	no	—
165	C₂×C₉	—	2	2	no	—
166	C₂×C₉	#408	1	1	yes	characteristic
167	C₂×C₉	C₂×#102	1	1	yes	characteristic
168	C₂×C₉	C₂×#104	1	1	yes	characteristic
169	D₉	—	8	2	no	—
170	#41	—	36	2	no	—
171	#41	—	36	2	no	—
172	C₂×Q₈	—	18	1	no	—
173	C₂×D₄	—	9	2	no	—
174	C₂×D₄	—	18	2	no	—
175	C₂×D₄	—	18	1	no	—
176	C₂×D₄	—	18	1	no	—
177	C₂²×C₄	—	18	2	no	—
178	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
179	C₂²×C₄	D₃₆	1	1	yes	characteristic
180	C₂²×C₄	#611	1	1	yes	characteristic
181	M₁₆	—	4	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	18	4	no	—
184	C₂×C₈	—	18	2	no	—
185	C₄⋊C₄	—	9	2	no	—
186	C₄⋊C₄	—	18	1	no	—
187	C₂²⋊C₄	—	18	2	no	—
188	C₂²⋊C₄	—	18	2	no	—
189	C₄²	C₄×D₉	1	1	yes	characteristic
190	C₄²	D₃₆	1	1	yes	characteristic
191	C₄²	#611	1	1	yes	characteristic
192	C₄²	C₂²×D₉	1	1	yes	characteristic
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	#809	1	2	yes	—
196	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
197	C₂×D₃	—	12	2	no	—
198	C₂×D₃	—	12	2	no	—
199	C₂×D₃	—	12	2	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	#809	1	2	yes	—
203	C₃×C₄	#825	1	2	yes	—
204	C₃×C₄	#907	1	2	yes	—
205	C₃×C₄	C₂×#213	1	1	yes	characteristic
206	C₃×C₄	C₂×#215	1	1	yes	characteristic
207	Dic₃	—	12	2	no	—
208	C₉	#1492	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	18	2	no	—
210	C₂³	(C₂×D₉)⋊C₄	1	1	yes	characteristic
211	Q₈	—	36	2	no	—
212	D₄	—	18	4	no	—
213	D₄	—	36	4	no	—
214	D₄	—	36	2	no	—
215	D₄	—	36	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	18	2	no	—
219	C₂×C₄	—	36	2	no	—
220	C₂×C₄	—	36	2	no	—
221	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
222	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
223	C₂×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
224	C₂×C₄	#3673	1	2	yes	—
225	C₂×C₄	#3675	1	2	yes	—
226	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
227	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
228	C₂×C₄	C₂×#611	1	1	yes	characteristic
229	C₈	—	4	2	no	—
230	C₈	—	18	4	no	—
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	C₂×#809	1	1	yes	characteristic
233	C₂×C₃	#5129	1	1	yes	characteristic
234	C₂×C₃	C₂×#825	1	1	yes	characteristic
235	D₃	—	24	2	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	36	2	no	—
239	C₂²	—	36	2	no	—
240	C₂²	—	36	2	no	—
241	C₂²	#63369	1	2	yes	—
242	C₂²	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	36	2	no	—
246	C₄	#63369	1	2	yes	—
247	C₄	#63384	1	2	yes	—
248	C₄	#63469	1	2	yes	—
249	C₄	C₂×#3673	1	1	yes	characteristic
250	C₄	C₂×#3675	1	1	yes	characteristic
251	C₃	#70498	1	1	yes	characteristic
252	C₂	—	2	2	no	—
253	C₂	—	72	2	no	—
254	C₂	C₂×#63369	1	1	yes	characteristic
255	C₂	#10588811	1	1	yes	characteristic
256	C₂	C₂×#63384	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₃×C₄²	C₂²×D₃
Fitting	C₉×#423	C₂
socle	C₂²×C₃	C₂×(C₂×D₃)⋊C₄
prime radical	#10589579	C₂
cosocle	C₄²×C₉	C₂³
hypercenter	#423	D₉
hypocenter	C₉	#1492
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
2. C₉×#423	C₂	semidirect
3. C₂×C₄×C₉⋊C₈	C₂	semidirect
4. #10588162	C₂	semidirect
87. #423	D₉	semidirect
208. C₉	#1492	semidirect

Maximal subgroups

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

subgroup	index
#10589579	2
C₉×#423	2
C₂×C₄×C₉⋊C₈	2
#10588162	2
#70498	3