Subgroups — #11791028

Order 1152.

576 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	C₂²×D₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63380	C₂	1	2	yes	maximal, complemented
4	C₂×#63378	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588619	C₂	1	2	yes	maximal, complemented
6	#69776	—	3	1	no	maximal
7	C₂²×D₃₆	C₂²	1	2	yes	—
8	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₂×(C₂×D₉)⋊C₄	—	2	4	no	—
12	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
13	C₂×D₇₂	—	2	2	no	—
14	C₂×D₇₂	—	2	2	no	—
15	C₂×D₇₂	C₄	1	4	yes	complemented
16	C₄×C₈×C₉	—	2	2	no	—
17	#63380	—	2	4	no	—
18	#63378	—	2	2	no	—
19	C₂²×D₂₄	—	3	1	no	—
20	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
21	C₂×#821	—	3	2	no	—
22	C₂×#818	—	3	1	no	—
23	#4939	—	3	2	no	—
24	C₂³×D₉	—	2	2	no	—
25	C₂²×C₄×C₉	D₄	1	2	yes	—
26	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
27	C₂²×Dic₉	—	2	2	no	—
28	C₂×D₃₆	—	2	2	no	—
29	C₂×D₃₆	—	2	2	no	—
30	C₂×D₃₆	—	2	2	no	—
31	C₂×D₃₆	—	2	2	no	—
32	C₂×D₃₆	—	2	2	no	—
33	C₂×D₃₆	C₂×C₄	1	4	yes	—
34	C₂×C₈×C₉	—	2	1	no	—
35	C₂×C₈×C₉	—	2	1	no	—
36	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₉	D₄	1	2	yes	—
38	C₂×C₈×C₉	D₄	1	2	yes	—
39	C₂×C₈×C₉	D₄	1	2	yes	—
40	C₄²×C₉	—	2	1	no	—
41	C₄²×C₉	—	2	1	no	—
42	(C₂×D₉)⋊C₄	—	4	8	no	—
43	C₄⋊Dic₉	—	2	2	no	—
44	C₄⋊Dic₉	—	2	2	no	—
45	D₇₂	—	2	8	no	—
46	#1970	—	9	1	no	Sylow
47	C₂²×D₁₂	—	3	2	no	—
48	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
51	C₂×(C₂×D₃)⋊C₄	—	6	4	no	—
52	C₂×C₄⋊Dic₃	—	3	2	no	—
53	C₂×D₂₄	—	3	4	no	—
54	C₂×D₂₄	—	6	2	no	—
55	C₂×D₂₄	—	6	2	no	—
56	C₃×C₄×C₈	—	2	2	no	—
57	#821	—	6	4	no	—
58	#818	—	6	2	no	—
59	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×D₉	—	2	4	no	—
61	C₂²×D₉	—	4	2	no	—
62	C₂²×D₉	—	4	2	no	—
63	C₂²×D₉	—	4	4	no	—
64	C₂²×D₉	—	4	4	no	—
65	C₂×C₄×C₉	—	2	2	no	—
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	D₈	1	4	yes	complemented
68	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₉	#41	1	1	yes	characteristic
73	C₂×C₄×C₉	#41	1	1	yes	characteristic
74	C₂×Dic₉	—	4	2	no	—
75	C₂×Dic₉	—	4	4	no	—
76	D₃₆	—	2	4	no	—
77	D₃₆	—	2	4	no	—
78	D₃₆	—	4	2	no	—
79	D₃₆	—	4	2	no	—
80	C₈×C₉	—	2	2	no	—
81	C₈×C₉	C₂²⋊C₄	1	4	yes	complemented
82	C₂²×D₈	—	9	1	no	—
83	C₂×#107	—	9	1	no	—
84	C₂×#102	—	9	2	no	—
85	C₂×C₄×C₈	D₉	1	1	yes	characteristic, complemented
86	#390	—	9	2	no	—
87	C₂³×D₃	—	6	2	no	—
88	C₂²×C₃×C₄	D₁₂	1	2	yes	—
89	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
90	C₂²×Dic₃	—	6	2	no	—
91	C₂×D₁₂	—	3	4	no	—
92	C₂×D₁₂	—	6	2	no	—
93	C₂×D₁₂	—	6	2	no	—
94	C₂×D₁₂	—	6	2	no	—
95	C₂×D₁₂	—	6	2	no	—
96	C₂×D₁₂	—	6	2	no	—
97	C₂×C₃×C₈	—	2	1	no	—
98	C₂×C₃×C₈	—	2	1	no	—
99	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
100	C₂×C₃×C₈	D₁₂	1	2	yes	—
101	C₂×C₃×C₈	D₁₂	1	2	yes	—
102	C₂×C₃×C₈	#67	1	2	yes	—
103	C₃×C₄²	—	2	1	no	—
104	C₃×C₄²	—	2	1	no	—
105	(C₂×D₃)⋊C₄	—	12	8	no	—
106	C₄⋊Dic₃	—	6	2	no	—
107	C₄⋊Dic₃	—	6	2	no	—
108	D₂₄	—	6	8	no	—
109	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
110	C₂²×C₉	#121	1	1	yes	characteristic
111	C₂²×C₉	#127	1	1	yes	characteristic
112	C₂²×C₉	C₂×D₈	1	2	yes	—
113	C₂²×C₉	#135	1	2	yes	—
114	C₂×D₉	—	4	4	no	—
115	C₂×D₉	—	4	8	no	—
116	C₂×D₉	—	8	4	no	—
117	C₂×D₉	—	8	2	no	—
118	C₂×D₉	—	8	2	no	—
119	C₄×C₉	—	2	4	no	—
120	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₄×C₉	C₄×D₄	1	1	yes	characteristic
122	C₄×C₉	#121	1	1	yes	characteristic
123	C₄×C₉	#124	1	1	yes	characteristic
124	Dic₉	—	8	2	no	—
125	C₂²×D₄	—	9	2	no	—
126	C₂×D₈	—	9	4	no	—
127	C₂×D₈	—	18	2	no	—
128	C₂×D₈	—	18	2	no	—
129	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
130	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	9	2	no	—
132	C₂×C₂²⋊C₄	—	18	4	no	—
133	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
134	#107	—	18	2	no	—
135	#102	—	18	4	no	—
136	C₄×C₈	—	2	2	no	—
137	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
138	C₂²×D₃	—	6	4	no	—
139	C₂²×D₃	—	12	2	no	—
140	C₂²×D₃	—	12	4	no	—
141	C₂²×D₃	—	12	4	no	—
142	C₂²×D₃	—	12	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	D₂₄	1	4	yes	—
146	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
149	C₂×C₃×C₄	#235	1	1	yes	characteristic
150	C₂×C₃×C₄	#235	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
152	C₂×Dic₃	—	12	2	no	—
153	C₂×Dic₃	—	12	4	no	—
154	D₁₂	—	6	4	no	—
155	D₁₂	—	6	4	no	—
156	D₁₂	—	12	2	no	—
157	D₁₂	—	12	2	no	—
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
160	C₂×C₉	#390	1	1	yes	characteristic
161	C₂×C₉	C₄×D₈	1	2	yes	—
162	C₂×C₉	#466	1	2	yes	—
163	C₂×C₉	#493	1	1	yes	characteristic
164	C₂×C₉	#495	1	1	yes	characteristic
165	D₉	—	8	4	no	—
166	C₂⁴	—	18	2	no	—
167	C₂×D₄	—	9	4	no	—
168	C₂×D₄	—	18	2	no	—
169	C₂×D₄	—	18	2	no	—
170	C₂×D₄	—	18	2	no	—
171	C₂×D₄	—	18	2	no	—
172	C₂×D₄	—	18	2	no	—
173	C₂²×C₄	—	18	2	no	—
174	C₂²×C₄	D₃₆	1	2	yes	—
175	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
176	D₈	—	18	8	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	C₄×D₉	1	2	yes	—
180	C₂×C₈	D₃₆	1	2	yes	—
181	C₂×C₈	D₃₆	1	2	yes	—
182	C₂×C₈	#611	1	2	yes	—
183	C₄⋊C₄	—	18	2	no	—
184	C₄⋊C₄	—	18	2	no	—
185	C₂²⋊C₄	—	36	8	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
189	C₂²×C₃	#874	1	1	yes	characteristic
190	C₂²×C₃	C₂×D₂₄	1	2	yes	—
191	C₂²×C₃	#904	1	2	yes	—
192	C₂²×C₃	#930	1	1	yes	characteristic
193	C₂×D₃	—	12	4	no	—
194	C₂×D₃	—	12	8	no	—
195	C₂×D₃	—	24	2	no	—
196	C₂×D₃	—	24	2	no	—
197	C₂×D₃	—	24	4	no	—
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
200	C₃×C₄	#875	1	1	yes	characteristic
201	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
202	C₃×C₄	#930	1	1	yes	characteristic
203	Dic₃	—	24	2	no	—
204	C₉	#1970	1	1	yes	characteristic, complemented, Sylow
205	C₂³	—	18	4	no	—
206	C₂³	—	36	2	no	—
207	C₂³	—	36	2	no	—
208	C₂³	—	36	4	no	—
209	C₂³	—	36	4	no	—
210	C₂³	C₂×D₃₆	1	1	yes	characteristic
211	D₄	—	18	4	no	—
212	D₄	—	18	4	no	—
213	D₄	—	36	2	no	—
214	D₄	—	36	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	36	2	no	—
218	C₂×C₄	—	36	4	no	—
219	C₂×C₄	D₇₂	1	4	yes	complemented
220	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
221	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
222	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
223	C₂×C₄	#3697	1	1	yes	characteristic
224	C₂×C₄	#3697	1	1	yes	characteristic
225	C₂×C₄	C₂×#611	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	(C₂×D₉)⋊C₄	1	4	yes	complemented
228	C₂×C₃	C₄×D₂₄	1	2	yes	—
229	C₂×C₃	#4695	1	1	yes	characteristic
230	C₂×C₃	#4696	1	1	yes	characteristic
231	C₂×C₃	#4939	1	1	yes	characteristic
232	C₂×C₃	#5115	1	2	yes	—
233	D₃	—	24	4	no	—
234	C₂²	—	36	4	no	—
235	C₂²	—	36	8	no	—
236	C₂²	—	72	4	no	—
237	C₂²	—	72	2	no	—
238	C₂²	—	72	2	no	—
239	C₂²	C₄×D₃₆	1	1	yes	characteristic
240	C₂²	#63436	1	1	yes	characteristic
241	C₂²	C₂×D₇₂	1	2	yes	—
242	C₂²	#63467	1	2	yes	—
243	C₂²	#63492	1	1	yes	characteristic
244	C₄	—	2	4	no	—
245	C₄	—	72	2	no	—
246	C₄	C₄×D₃₆	1	1	yes	characteristic
247	C₄	#63437	1	1	yes	characteristic
248	C₄	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
249	C₄	#63492	1	1	yes	characteristic
250	C₃	#69776	1	1	yes	characteristic
251	C₂	—	72	4	no	—
252	C₂	C₄×D₇₂	1	2	yes	—
253	C₂	#10588377	1	1	yes	characteristic
254	C₂	#10588378	1	1	yes	characteristic
255	C₂	#10588619	1	1	yes	characteristic
256	C₂	#10588796	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₆
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₄×C₈×C₉	C₂
socle	C₂³×C₃	C₂×D₁₂
prime radical	C₂²×D₇₂	C₂
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×C₄×C₈	D₉
hypocenter	C₉	#1970
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₈×C₉	C₂	semidirect
3. C₂×#63380	C₂	semidirect
4. C₂×#63378	C₂	semidirect
5. #10588619	C₂	semidirect
15. C₂×D₇₂	C₄	semidirect
67. C₂×C₄×C₉	D₈	semidirect
81. C₈×C₉	C₂²⋊C₄	semidirect
85. C₂×C₄×C₈	D₉	semidirect
204. C₉	#1970	semidirect
219. C₂×C₄	D₇₂	semidirect
227. C₈	(C₂×D₉)⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₇₂	2
C₂×C₄×C₈×C₉	2
C₂×#63380	2
C₂×#63378	2
#10588619	2
#69776	3