Subgroups — #11791094

Order 1152.

564 conjugacy classes of subgroups.

The subgroup lattice has 249 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×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63380	C₂	1	2	yes	maximal, complemented
4	C₂×#63377	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588619	C₂	1	2	yes	maximal, complemented
6	#69825	—	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	C₉×#97	—	2	2	no	—
17	#63380	—	2	4	no	—
18	#63377	—	2	2	no	—
19	C₂²×D₂₄	—	3	1	no	—
20	C₂×C₃×#97	D₃	1	1	yes	characteristic
21	C₂×#821	—	3	2	no	—
22	C₂×#817	—	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	#1972	—	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×#97	—	2	2	no	—
57	#821	—	6	4	no	—
58	#817	—	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	4	no	—
67	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₉	C₂×D₄	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	#41	1	1	yes	characteristic
72	C₂×C₄×C₉	#41	1	1	yes	characteristic
73	C₂×Dic₉	—	4	2	no	—
74	C₂×Dic₉	—	4	4	no	—
75	D₃₆	—	2	4	no	—
76	D₃₆	—	2	4	no	—
77	D₃₆	—	4	2	no	—
78	D₃₆	—	4	2	no	—
79	C₈×C₉	—	2	2	no	—
80	C₈×C₉	C₂²⋊C₄	1	4	yes	complemented
81	C₂²×D₈	—	9	1	no	—
82	C₂×#106	—	9	1	no	—
83	C₂×#102	—	9	2	no	—
84	C₂×#97	D₉	1	1	yes	characteristic, complemented
85	#390	—	9	2	no	—
86	C₂³×D₃	—	6	2	no	—
87	C₂²×C₃×C₄	D₁₂	1	2	yes	—
88	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₂²×Dic₃	—	6	2	no	—
90	C₂×D₁₂	—	3	4	no	—
91	C₂×D₁₂	—	6	2	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×C₃×C₈	—	2	1	no	—
97	C₂×C₃×C₈	—	2	1	no	—
98	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
99	C₂×C₃×C₈	D₁₂	1	2	yes	—
100	C₂×C₃×C₈	D₁₂	1	2	yes	—
101	C₂×C₃×C₈	#67	1	2	yes	—
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	—	2	1	no	—
104	(C₂×D₃)⋊C₄	—	12	8	no	—
105	C₄⋊Dic₃	—	6	2	no	—
106	C₄⋊Dic₃	—	6	2	no	—
107	D₂₄	—	6	8	no	—
108	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
109	C₂²×C₉	#121	1	1	yes	characteristic
110	C₂²×C₉	#127	1	1	yes	characteristic
111	C₂²×C₉	AGL(1,ℤ/8)	1	4	yes	—
112	C₂×D₉	—	4	4	no	—
113	C₂×D₉	—	4	8	no	—
114	C₂×D₉	—	8	4	no	—
115	C₂×D₉	—	8	2	no	—
116	C₂×D₉	—	8	2	no	—
117	C₄×C₉	—	4	2	no	—
118	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₉	C₄×D₄	1	1	yes	characteristic
120	C₄×C₉	#121	1	1	yes	characteristic
121	C₄×C₉	#124	1	1	yes	characteristic
122	Dic₉	—	8	2	no	—
123	C₂²×D₄	—	9	2	no	—
124	C₂×D₈	—	9	4	no	—
125	C₂×D₈	—	18	2	no	—
126	C₂×D₈	—	18	2	no	—
127	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
128	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
129	C₂×C₄⋊C₄	—	9	2	no	—
130	C₂×C₂²⋊C₄	—	18	4	no	—
131	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
132	#106	—	18	2	no	—
133	#102	—	18	4	no	—
134	#97	—	2	2	no	—
135	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
136	C₂²×D₃	—	6	4	no	—
137	C₂²×D₃	—	12	2	no	—
138	C₂²×D₃	—	12	4	no	—
139	C₂²×D₃	—	12	4	no	—
140	C₂²×D₃	—	12	2	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
146	C₂×C₃×C₄	#235	1	1	yes	characteristic
147	C₂×C₃×C₄	#235	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
149	C₂×Dic₃	—	12	2	no	—
150	C₂×Dic₃	—	12	4	no	—
151	D₁₂	—	6	4	no	—
152	D₁₂	—	6	4	no	—
153	D₁₂	—	12	2	no	—
154	D₁₂	—	12	2	no	—
155	C₃×C₈	—	2	2	no	—
156	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
157	C₂×C₉	#390	1	1	yes	characteristic
158	C₂×C₉	#442	1	2	yes	—
159	C₂×C₉	#469	1	2	yes	—
160	C₂×C₉	#496	1	2	yes	—
161	D₉	—	8	4	no	—
162	C₂⁴	—	18	2	no	—
163	C₂×D₄	—	9	4	no	—
164	C₂×D₄	—	18	2	no	—
165	C₂×D₄	—	18	2	no	—
166	C₂×D₄	—	18	2	no	—
167	C₂×D₄	—	18	2	no	—
168	C₂×D₄	—	18	2	no	—
169	C₂²×C₄	—	18	2	no	—
170	C₂²×C₄	D₃₆	1	2	yes	—
171	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
172	D₈	—	18	8	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	C₄×D₉	1	2	yes	—
176	C₂×C₈	D₃₆	1	2	yes	—
177	C₂×C₈	D₃₆	1	2	yes	—
178	C₂×C₈	#611	1	2	yes	—
179	C₄⋊C₄	—	18	2	no	—
180	C₄⋊C₄	—	18	2	no	—
181	C₂²⋊C₄	—	36	8	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
185	C₂²×C₃	#874	1	1	yes	characteristic
186	C₂²×C₃	#908	1	4	yes	—
187	C₂²×C₃	#930	1	1	yes	characteristic
188	C₂×D₃	—	12	4	no	—
189	C₂×D₃	—	12	8	no	—
190	C₂×D₃	—	24	4	no	—
191	C₂×D₃	—	24	2	no	—
192	C₂×D₃	—	24	2	no	—
193	C₃×C₄	—	4	2	no	—
194	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
195	C₃×C₄	#875	1	1	yes	characteristic
196	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
197	C₃×C₄	#930	1	1	yes	characteristic
198	Dic₃	—	24	2	no	—
199	C₉	#1972	1	1	yes	characteristic, complemented, Sylow
200	C₂³	—	18	4	no	—
201	C₂³	—	36	2	no	—
202	C₂³	—	36	2	no	—
203	C₂³	—	36	4	no	—
204	C₂³	—	36	4	no	—
205	C₂³	C₂×D₃₆	1	1	yes	characteristic
206	D₄	—	18	4	no	—
207	D₄	—	18	4	no	—
208	D₄	—	36	2	no	—
209	D₄	—	36	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	36	2	no	—
213	C₂×C₄	—	36	4	no	—
214	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
215	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
216	C₂×C₄	C₂×D₃₆	1	1	yes	characteristic
217	C₂×C₄	#3697	1	1	yes	characteristic
218	C₂×C₄	#3697	1	1	yes	characteristic
219	C₂×C₄	C₂×#611	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	(C₂×D₉)⋊C₄	1	4	yes	complemented
222	C₂×C₃	#4711	1	2	yes	—
223	C₂×C₃	#4712	1	2	yes	—
224	C₂×C₃	#4939	1	1	yes	characteristic
225	C₂×C₃	#5135	1	2	yes	—
226	D₃	—	24	4	no	—
227	C₂²	—	36	4	no	—
228	C₂²	—	36	8	no	—
229	C₂²	—	72	4	no	—
230	C₂²	—	72	2	no	—
231	C₂²	—	72	2	no	—
232	C₂²	C₄×D₃₆	1	1	yes	characteristic
233	C₂²	#63436	1	1	yes	characteristic
234	C₂²	#63470	1	4	yes	—
235	C₂²	#63492	1	1	yes	characteristic
236	C₄	—	4	2	no	—
237	C₄	—	72	2	no	—
238	C₄	C₄×D₃₆	1	1	yes	characteristic
239	C₄	#63437	1	1	yes	characteristic
240	C₄	C₂×(C₂×D₉)⋊C₄	1	1	yes	characteristic
241	C₄	#63492	1	1	yes	characteristic
242	C₃	#69825	1	1	yes	characteristic
243	C₂	—	72	4	no	—
244	C₂	#10588392	1	2	yes	—
245	C₂	#10588393	1	2	yes	—
246	C₂	#10588619	1	1	yes	characteristic
247	C₂	#10588817	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₃₆
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#97	C₂
socle	C₂³×C₃	C₂×D₁₂
prime radical	C₂²×D₇₂	C₂
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#97	D₉
hypocenter	C₉	#1972
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#97	C₂	semidirect
3. C₂×#63380	C₂	semidirect
4. C₂×#63377	C₂	semidirect
5. #10588619	C₂	semidirect
15. C₂×D₇₂	C₄	semidirect
80. C₈×C₉	C₂²⋊C₄	semidirect
84. C₂×#97	D₉	semidirect
199. C₉	#1972	semidirect
221. 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×C₉×#97	2
C₂×#63380	2
C₂×#63377	2
#10588619	2
#69825	3