Subgroups — #11859471

Order 1152.

750 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	C₂×#63464	C₂	1	2	yes	maximal, complemented
2	#10589579	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588791	C₂	1	2	yes	maximal, complemented
5	#10588604	C₂	1	1	yes	maximal, characteristic, complemented
6	#10588370	C₂	1	4	yes	maximal, complemented
7	C₈×D₃₆	C₂	1	4	yes	maximal, complemented
8	#78458	—	3	1	no	maximal
9	C₂×#3697	C₂²	1	2	yes	—
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
13	#63492	C₄	1	4	yes	—
14	C₂×#3667	C₂²	1	2	yes	—
15	#63464	—	2	8	no	—
16	C₂×#3663	C₂²	1	4	yes	—
17	C₂×C₈×D₉	C₂²	1	4	yes	—
18	#63437	C₄	1	2	yes	—
19	#63436	C₄	1	1	yes	characteristic
20	C₄×D₃₆	C₂²	1	4	yes	—
21	#63431	C₄	1	1	yes	characteristic
22	C₄×C₈×C₉	C₂²	1	2	yes	—
23	C₄×C₈×C₉	C₂²	1	2	yes	—
24	(C₂×D₉)⋊C₈	C₂²	1	8	yes	—
25	C₄⋊(C₉⋊C₈)	C₂²	1	4	yes	—
26	C₂×#901	—	3	2	no	—
27	#5477	—	3	1	no	—
28	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
29	#5110	—	3	2	no	—
30	#4924	—	3	1	no	—
31	#4688	—	3	4	no	—
32	C₈×D₁₂	—	3	4	no	—
33	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂×#611	C₂×C₄	1	4	yes	—
36	#3697	—	2	4	no	—
37	#3697	—	2	4	no	—
38	C₂×D₃₆	C₂×C₄	1	4	yes	—
39	C₂×D₃₆	C₂×C₄	1	2	yes	—
40	C₂×C₄×D₉	C₂³	1	4	yes	—
41	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₉	—	2	2	no	—
43	C₂×C₈×C₉	D₄	1	4	yes	complemented
44	C₂×C₈×C₉	C₂³	1	2	yes	—
45	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
47	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₉	C₂³	1	2	yes	—
50	(C₂×D₉)⋊C₄	C₂×C₄	1	8	yes	—
51	C₄⋊Dic₉	C₂×C₄	1	4	yes	—
52	#3667	—	2	4	no	—
53	C₂×C₉⋊C₈	C₂³	1	4	yes	—
54	#3663	—	2	8	no	—
55	C₈×D₉	—	2	8	no	—
56	#2931	—	9	1	no	Sylow
57	C₂×#235	—	3	2	no	—
58	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
61	#930	—	3	4	no	—
62	C₂×#208	—	3	2	no	—
63	#901	—	6	8	no	—
64	C₂×ΓL(1,𝔽₂₅)	—	3	4	no	—
65	C₂×D₃×C₈	—	3	4	no	—
66	#875	—	3	2	no	—
67	#874	—	3	1	no	—
68	C₄×D₁₂	—	3	4	no	—
69	#869	—	3	1	no	—
70	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
71	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
72	(C₂×D₃)⋊C₈	—	3	8	no	—
73	C₄⋊(C₃⋊C₈)	—	3	4	no	—
74	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×D₉	C₂²×C₄	1	4	yes	—
76	C₂×C₄×C₉	—	2	4	no	—
77	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₉	#41	1	2	yes	—
82	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
83	#611	—	2	8	no	—
84	C₂×Dic₉	C₂²×C₄	1	4	yes	—
85	D₃₆	—	2	4	no	—
86	D₃₆	—	2	8	no	—
87	C₄×D₉	—	2	4	no	—
88	C₄×D₉	—	2	4	no	—
89	Dic₁₈	—	2	4	no	—
90	C₈×C₉	—	2	2	no	—
91	C₈×C₉	C₂×D₄	1	2	yes	—
92	C₈×C₉	C₂×D₄	1	2	yes	—
93	C₉⋊C₈	—	2	4	no	—
94	C₂×#131	—	9	2	no	—
95	#532	—	9	1	no	—
96	#436	—	9	4	no	—
97	C₈×D₄	—	9	4	no	—
98	#423	—	9	1	no	—
99	#408	—	9	2	no	—
100	C₂×C₄×C₈	D₉	1	1	yes	characteristic, complemented
101	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
102	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
103	C₂×#67	—	3	4	no	—
104	#235	—	6	4	no	—
105	#235	—	6	4	no	—
106	C₂×D₁₂	—	3	4	no	—
107	C₂×D₁₂	—	3	2	no	—
108	C₂×C₄×D₃	—	3	4	no	—
109	C₂×Dic₆	—	3	2	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	D₁₂	1	4	yes	—
112	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
113	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
114	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
115	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
116	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
117	C₃×C₄²	C₂²×D₃	1	2	yes	—
118	(C₂×D₃)⋊C₄	—	3	8	no	—
119	C₄⋊Dic₃	—	3	4	no	—
120	#208	—	6	4	no	—
121	C₂×C₃⋊C₈	—	3	4	no	—
122	ΓL(1,𝔽₂₅)	—	6	8	no	—
123	D₃×C₈	—	6	8	no	—
124	C₂²×C₉	—	2	2	no	—
125	C₂²×C₉	C₄×D₄	1	2	yes	—
126	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×D₉	—	2	4	no	—
128	C₂×D₉	—	2	8	no	—
129	C₄×C₉	—	2	2	no	—
130	C₄×C₉	C₄×D₄	1	2	yes	—
131	C₄×C₉	#131	1	4	yes	complemented
132	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₉	C₂×#41	1	1	yes	characteristic
134	Dic₉	—	2	4	no	—
135	C₂×#41	—	9	2	no	—
136	#131	—	18	8	no	—
137	C₂×M₁₆	—	9	2	no	—
138	C₂×M₁₆	—	9	4	no	—
139	C₂²×C₈	—	9	4	no	—
140	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
141	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
142	C₄⋊Q₈	—	9	1	no	—
143	#127	—	9	1	no	—
144	#124	—	9	2	no	—
145	#121	—	9	4	no	—
146	C₄×D₄	—	9	4	no	—
147	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
148	C₄⋊C₈	—	9	4	no	—
149	C₂²⋊C₈	—	9	8	no	—
150	C₄×C₈	C₂×D₉	1	2	yes	—
151	C₄×C₈	C₂×D₉	1	2	yes	—
152	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
153	C₂²×D₃	—	3	4	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
156	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
157	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
158	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
159	C₂×C₃×C₄	#235	1	2	yes	—
160	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
161	#67	—	6	8	no	—
162	C₂×Dic₃	—	3	4	no	—
163	D₁₂	—	6	4	no	—
164	D₁₂	—	6	8	no	—
165	C₄×D₃	—	6	4	no	—
166	C₄×D₃	—	6	4	no	—
167	Dic₆	—	6	4	no	—
168	C₃×C₈	—	2	2	no	—
169	C₃×C₈	C₂×D₁₂	1	2	yes	—
170	C₃×C₈	C₂×D₁₂	1	2	yes	—
171	C₃⋊C₈	—	6	4	no	—
172	C₂×C₉	—	2	2	no	—
173	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×C₉	C₂×#131	1	2	yes	—
175	D₉	—	4	4	no	—
176	#41	—	18	4	no	—
177	#41	—	18	4	no	—
178	C₂×Q₈	—	9	2	no	—
179	C₂×D₄	—	9	4	no	—
180	C₂×D₄	—	9	4	no	—
181	C₂×D₄	—	9	2	no	—
182	C₂²×C₄	—	9	4	no	—
183	C₂²×C₄	C₄×D₉	1	2	yes	—
184	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
185	M₁₆	—	18	4	no	—
186	M₁₆	—	18	8	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	9	4	no	—
189	C₂×C₈	—	18	8	no	—
190	C₂×C₈	D₃₆	1	4	yes	complemented
191	C₂×C₈	C₂²×D₉	1	2	yes	—
192	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
194	C₄⋊C₄	—	9	4	no	—
195	C₂²⋊C₄	—	9	8	no	—
196	C₄²	C₄×D₉	1	1	yes	characteristic
197	C₄²	C₄×D₉	1	1	yes	characteristic
198	C₄²	C₂²×D₉	1	2	yes	—
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	C₄×D₁₂	1	2	yes	—
201	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
202	C₂×D₃	—	6	4	no	—
203	C₂×D₃	—	6	8	no	—
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	C₄×D₁₂	1	2	yes	—
206	C₃×C₄	#901	1	4	yes	—
207	C₃×C₄	C₂²×D₁₂	1	1	yes	characteristic
208	C₃×C₄	C₂×#235	1	1	yes	characteristic
209	Dic₃	—	6	4	no	—
210	C₉	#2931	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	9	4	no	—
212	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
213	Q₈	—	18	4	no	—
214	D₄	—	18	8	no	—
215	D₄	—	18	4	no	—
216	D₄	—	18	8	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	9	4	no	—
219	C₂×C₄	—	18	4	no	—
220	C₂×C₄	—	18	4	no	—
221	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
222	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
224	C₂×C₄	C₂×D₃₆	1	2	yes	—
225	C₂×C₄	#3697	1	2	yes	—
226	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	18	4	no	—
229	C₈	C₂×D₃₆	1	2	yes	—
230	C₈	C₂×D₃₆	1	2	yes	—
231	C₂×C₃	—	2	2	no	—
232	C₂×C₃	C₂×C₄×D₁₂	1	1	yes	characteristic
233	C₂×C₃	C₂×#901	1	2	yes	—
234	D₃	—	12	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	18	4	no	—
237	C₂²	—	18	8	no	—
238	C₂²	C₄×D₃₆	1	2	yes	—
239	C₂²	C₂²×C₄×D₉	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	18	4	no	—
242	C₄	C₄×D₃₆	1	2	yes	—
243	C₄	#63464	1	4	yes	complemented
244	C₄	C₂²×D₃₆	1	1	yes	characteristic
245	C₄	C₂×#3697	1	1	yes	characteristic
246	C₃	#78458	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	36	4	no	—
249	C₂	C₂×C₄×D₃₆	1	1	yes	characteristic
250	C₂	C₂×#63464	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₉
commutator	C₂²×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂×C₄×C₈×C₉	C₂
socle	C₂²×C₃	C₂²×C₄×D₃
prime radical	#10589579	C₂
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂×C₄×C₈	D₉
hypocenter	C₉	#2931
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#63464	C₂	semidirect
3. C₂×C₄×C₈×C₉	C₂	semidirect
4. #10588791	C₂	semidirect
5. #10588604	C₂	semidirect
6. #10588370	C₂	semidirect
7. C₈×D₃₆	C₂	semidirect
43. C₂×C₈×C₉	D₄	semidirect
100. C₂×C₄×C₈	D₉	semidirect
131. C₄×C₉	#131	semidirect
190. C₂×C₈	D₃₆	semidirect
210. C₉	#2931	semidirect
243. C₄	#63464	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63464	2
#10589579	2
C₂×C₄×C₈×C₉	2
#10588791	2
#10588604	2
#10588370	2
C₈×D₃₆	2
#78458	3