Subgroups — #11860670

Order 1152.

792 conjugacy classes of subgroups.

The subgroup lattice has 234 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	#10589623	C₂	1	2	yes	maximal, complemented
3	C₉×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588782	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588682	C₂	1	2	yes	maximal, complemented
6	#10588552	C₂	1	4	yes	maximal, complemented
7	#10588549	C₂	1	4	yes	maximal, complemented
8	#79167	—	3	1	no	maximal
9	C₂²×D₃₆	C₂²	1	2	yes	—
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₉×#117	C₂²	1	2	yes	—
12	#63488	C₂²	1	1	yes	characteristic
13	C₂×#3667	C₂²	1	1	yes	characteristic
14	C₂×D₇₂	C₄	1	8	yes	complemented
15	C₂×D₇₂	C₂²	1	2	yes	—
16	C₂×D₇₂	C₂²	1	2	yes	—
17	#63455	C₂²	1	8	yes	complemented
18	C₂²⋊C₄×D₉	—	2	4	no	—
19	C₄×D₃₆	—	2	4	no	—
20	C₉×#107	C₂²	1	2	yes	complemented
21	C₉×#106	C₂²	1	2	yes	complemented
22	#63375	C₂²	1	2	yes	complemented
23	C₈×Dic₉	C₂²	1	2	yes	complemented
24	#63369	C₂²	1	8	yes	complemented
25	C₂²×D₂₄	—	3	1	no	—
26	#5521	—	3	2	no	—
27	C₃×#427	D₃	1	1	yes	characteristic
28	#5101	—	3	1	no	—
29	#5001	—	3	2	no	—
30	#4872	—	3	4	no	—
31	#4869	—	3	4	no	—
32	C₂³×D₉	—	2	2	no	—
33	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂×D₃₆	—	2	2	no	—
35	C₂×D₃₆	—	2	4	no	—
36	C₂×D₃₆	C₂×C₄	1	8	yes	complemented
37	C₂×D₃₆	C₂³	1	4	yes	—
38	C₂×C₄×D₉	—	2	8	no	—
39	C₂×C₈×C₉	C₂×C₄	1	4	yes	complemented
40	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
41	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
42	C₉×C₄⋊C₄	C₂³	1	4	yes	—
43	C₉×C₂²⋊C₄	—	2	2	no	—
44	C₄²×C₉	—	2	2	no	—
45	#3676	D₄	1	2	yes	—
46	(C₂×D₉)⋊C₄	—	2	8	no	—
47	C₄⋊Dic₉	D₄	1	2	yes	—
48	C₄×Dic₉	C₂³	1	2	yes	—
49	#3667	—	2	2	no	—
50	C₂×C₉⋊C₈	C₂³	1	2	yes	—
51	D₇₂	C₂×C₄	1	8	yes	complemented
52	D₇₂	C₂×C₄	1	8	yes	complemented
53	#2948	—	9	1	no	Sylow
54	C₂²×D₁₂	—	3	2	no	—
55	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
56	C₃×#117	C₂×D₃	1	2	yes	—
57	#926	—	3	1	no	—
58	C₂×#208	—	3	1	no	—
59	C₂×D₂₄	—	3	8	no	—
60	C₂×D₂₄	—	3	2	no	—
61	C₂×D₂₄	—	3	2	no	—
62	#893	—	3	8	no	—
63	D₃×C₂²⋊C₄	—	6	4	no	—
64	C₄×D₁₂	—	6	4	no	—
65	C₃×#107	C₂×D₃	1	2	yes	—
66	C₃×#106	C₂×D₃	1	2	yes	—
67	#815	—	3	2	no	—
68	C₈×Dic₃	—	3	2	no	—
69	#809	—	3	8	no	—
70	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×D₉	—	2	4	no	—
72	C₂²×D₉	—	2	8	no	—
73	C₂²×D₉	—	8	4	no	—
74	C₂×C₄×C₉	—	2	4	no	—
75	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₉	#41	1	2	yes	—
78	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
79	C₂×Dic₉	C₂×D₄	1	4	yes	—
80	D₃₆	—	2	8	no	—
81	D₃₆	—	2	4	no	—
82	D₃₆	C₂²×C₄	1	8	yes	—
83	C₄×D₉	—	8	8	no	—
84	C₈×C₉	C₂²×C₄	1	2	yes	—
85	C₈×C₉	C₂²×C₄	1	2	yes	—
86	C₉⋊C₈	—	2	2	no	—
87	C₂²×D₈	—	9	1	no	—
88	#518	—	9	2	no	—
89	#442	—	9	4	no	—
90	C₄×D₈	—	9	4	no	—
91	#427	D₉	1	1	yes	characteristic, complemented
92	#418	—	9	2	no	—
93	#405	—	9	1	no	—
94	C₂³×D₃	—	6	2	no	—
95	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
96	C₂×D₁₂	—	3	8	no	—
97	C₂×D₁₂	—	3	4	no	—
98	C₂×D₁₂	—	6	2	no	—
99	C₂×D₁₂	—	6	4	no	—
100	C₂×C₄×D₃	—	6	8	no	—
101	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
102	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
104	C₃×C₄⋊C₄	C₂²×D₃	1	4	yes	—
105	C₃×C₂²⋊C₄	—	2	2	no	—
106	C₃×C₄²	—	2	2	no	—
107	#217	—	3	2	no	—
108	(C₂×D₃)⋊C₄	—	6	8	no	—
109	C₄⋊Dic₃	—	3	2	no	—
110	C₄×Dic₃	—	3	2	no	—
111	#208	—	6	2	no	—
112	C₂×C₃⋊C₈	—	3	2	no	—
113	D₂₄	—	3	8	no	—
114	D₂₄	—	3	8	no	—
115	C₂²×C₉	—	2	2	no	—
116	C₂²×C₉	C₄×D₄	1	2	yes	—
117	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₉	—	2	8	no	—
119	C₂×D₉	—	4	8	no	—
120	C₂×D₉	—	8	8	no	—
121	C₄×C₉	—	4	4	no	—
122	C₄×C₉	C₄×D₄	1	2	yes	—
123	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₉	C₂×#41	1	1	yes	characteristic
125	Dic₉	—	2	4	no	—
126	C₂²×D₄	—	9	2	no	—
127	C₂×D₈	—	9	8	no	—
128	C₂×D₈	—	9	2	no	—
129	C₂×D₈	—	9	2	no	—
130	C₂×M₁₆	—	9	1	no	—
131	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
132	C₄×D₄	—	9	8	no	—
133	C₄×D₄	—	18	4	no	—
134	#117	—	9	1	no	—
135	#117	C₂×D₉	1	2	yes	—
136	C₂×C₂²⋊C₄	—	18	4	no	—
137	#107	C₂×D₉	1	2	yes	complemented
138	#106	C₂×D₉	1	2	yes	complemented
139	#102	—	9	8	no	—
140	#97	—	9	2	no	—
141	C₄×C₈	—	9	2	no	—
142	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
143	C₂²×D₃	—	6	4	no	—
144	C₂²×D₃	—	6	8	no	—
145	C₂²×D₃	—	24	4	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
148	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	#239	1	2	yes	—
150	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
151	C₂×Dic₃	—	3	4	no	—
152	D₁₂	—	3	8	no	—
153	D₁₂	—	6	8	no	—
154	D₁₂	—	6	4	no	—
155	C₄×D₃	—	24	8	no	—
156	C₃×C₈	C₂×C₄×D₃	1	2	yes	—
157	C₃×C₈	C₂×C₄×D₃	1	2	yes	—
158	C₃⋊C₈	—	6	2	no	—
159	C₂×C₉	—	2	2	no	—
160	C₂×C₉	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂×C₉	#576	1	2	yes	—
162	D₉	—	4	8	no	—
163	C₂⁴	—	18	2	no	—
164	C₂×D₄	—	9	8	no	—
165	C₂×D₄	—	9	4	no	—
166	C₂×D₄	—	18	2	no	—
167	C₂×D₄	—	18	4	no	—
168	C₂²×C₄	—	18	8	no	—
169	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
170	D₈	—	9	8	no	—
171	D₈	—	9	8	no	—
172	M₁₆	—	18	2	no	—
173	C₂×C₈	—	9	2	no	—
174	C₂×C₈	C₄×D₉	1	4	yes	complemented
175	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
176	C₂×C₈	C₂²×D₉	1	1	yes	characteristic
177	C₄⋊C₄	—	9	2	no	—
178	C₄⋊C₄	C₂²×D₉	1	4	yes	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	9	2	no	—
181	C₂²⋊C₄	—	18	8	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	9	2	no	—
184	C₂²×C₃	—	2	2	no	—
185	C₂²×C₃	#893	1	2	yes	—
186	C₂²×C₃	C₂×D₃×D₄	1	1	yes	characteristic
187	C₂×D₃	—	6	8	no	—
188	C₂×D₃	—	12	8	no	—
189	C₂×D₃	—	24	8	no	—
190	C₃×C₄	—	4	4	no	—
191	C₃×C₄	#893	1	2	yes	—
192	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
193	C₃×C₄	C₂×#239	1	1	yes	characteristic
194	Dic₃	—	6	4	no	—
195	C₉	#2948	1	1	yes	characteristic, complemented, Sylow
196	C₂³	—	18	4	no	—
197	C₂³	—	18	8	no	—
198	C₂³	—	72	4	no	—
199	C₂³	D₄×D₉	1	1	yes	characteristic
200	D₄	—	9	8	no	—
201	D₄	—	18	8	no	—
202	D₄	—	18	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	9	4	no	—
205	C₂×C₄	—	72	8	no	—
206	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
207	C₂×C₄	D₄×D₉	1	1	yes	characteristic
208	C₂×C₄	#3701	1	2	yes	—
209	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
210	C₈	—	18	2	no	—
211	C₈	C₂×C₄×D₉	1	2	yes	—
212	C₈	C₂×C₄×D₉	1	2	yes	—
213	C₂×C₃	—	2	2	no	—
214	C₂×C₃	C₂×#893	1	1	yes	characteristic
215	C₂×C₃	#5769	1	2	yes	—
216	D₃	—	12	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	18	8	no	—
219	C₂²	—	36	8	no	—
220	C₂²	—	72	8	no	—
221	C₂²	#63455	1	2	yes	—
222	C₂²	C₂×D₄×D₉	1	1	yes	characteristic
223	C₄	—	4	4	no	—
224	C₄	—	18	4	no	—
225	C₄	#63455	1	2	yes	—
226	C₄	C₂²×C₄×D₉	1	1	yes	characteristic
227	C₄	C₂×#3701	1	1	yes	characteristic
228	C₃	#79167	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	36	8	no	—
231	C₂	C₂×#63455	1	1	yes	characteristic
232	C₂	#10589871	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₉
commutator	C₄×C₉	C₂³×C₄
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₉×#427	C₂
socle	C₂²×C₃	C₂×D₃×D₄
prime radical	C₂²×D₇₂	C₂
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	#427	D₉
hypocenter	C₉	#2948
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
2. #10589623	C₂	semidirect
3. C₉×#427	C₂	semidirect
4. #10588782	C₂	semidirect
5. #10588682	C₂	semidirect
6. #10588552	C₂	semidirect
7. #10588549	C₂	semidirect
14. C₂×D₇₂	C₄	semidirect
17. #63455	C₂²	semidirect
20. C₉×#107	C₂²	semidirect
21. C₉×#106	C₂²	semidirect
22. #63375	C₂²	semidirect
23. C₈×Dic₉	C₂²	semidirect
24. #63369	C₂²	semidirect
36. C₂×D₃₆	C₂×C₄	semidirect
39. C₂×C₈×C₉	C₂×C₄	semidirect
51. D₇₂	C₂×C₄	semidirect
52. D₇₂	C₂×C₄	semidirect
91. #427	D₉	semidirect
137. #107	C₂×D₉	semidirect
138. #106	C₂×D₉	semidirect
174. C₂×C₈	C₄×D₉	semidirect
195. C₉	#2948	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
#10589623	2
C₉×#427	2
#10588782	2
#10588682	2
#10588552	2
#10588549	2
#79167	3