Subgroups — #72255106

Order 3072.

At least 6774 conjugacy classes of subgroups; only the 319 normal subgroups are shown.

The subgroup lattice has 268 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72255063	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#496225	C₂	1	1	yes	maximal, characteristic, complemented
3	#72249307	C₂	1	1	yes	maximal, characteristic, complemented
4	#72249306	C₂	1	1	yes	maximal, characteristic, complemented
5	#72248674	C₂	1	1	yes	maximal, characteristic, complemented
6	#72248352	C₂	1	1	yes	maximal, characteristic, complemented
7	#72248257	C₂	1	1	yes	maximal, characteristic
8	C₂⁴×D₃×D₄	C₂²	1	1	yes	characteristic
9	C₂³×#937	C₄	1	1	yes	characteristic, complemented
10	C₂×#84052	C₄	1	1	yes	characteristic, complemented
11	C₂×#72129	C₂²	1	1	yes	characteristic
12	C₂×#72130	C₂²	1	1	yes	characteristic
13	#10726342	C₄	1	1	yes	characteristic, complemented
14	#10717058	C₄	1	1	yes	characteristic, complemented
15	#10717057	C₂²	1	1	yes	characteristic
16	C₂×C₃×#2403	C₂²	1	1	yes	characteristic
17	C₂×C₃×#1783	C₂²	1	1	yes	characteristic
18	C₂×C₃×#1781	C₂²	1	1	yes	characteristic
19	#496225	D₃	1	1	yes	characteristic, complemented
20	C₂⁶×D₃	C₂×C₄	1	1	yes	characteristic
21	C₂⁶×D₃	D₄	1	1	yes	characteristic
22	C₂⁴×C₃×D₄	C₂×C₄	1	1	yes	characteristic
23	C₂⁴×C₃×D₄	C₂³	1	1	yes	characteristic
24	C₂⁴×#67	C₂×C₄	1	1	yes	characteristic
25	C₂⁴×#67	D₄	1	1	yes	characteristic
26	C₂⁴×D₁₂	D₄	1	1	yes	characteristic
27	C₂⁴×C₄×D₃	D₄	1	1	yes	characteristic
28	C₂²×C₃×#120	D₄	1	1	yes	characteristic
29	C₂²×C₃×#120	D₄	1	1	yes	characteristic
30	C₂³×#217	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×#100	D₄	1	2	yes	complemented
32	C₂²×C₃×#99	D₄	1	2	yes	complemented
33	C₂×C₃×#386	D₄	1	1	yes	characteristic
34	C₂×C₃×#386	D₄	1	1	yes	characteristic
35	C₂×C₃×#379	C₂×C₄	1	1	yes	characteristic
36	C₂×#5247	C₂×C₄	1	1	yes	characteristic
37	C₂²×#834	D₄	1	2	yes	complemented
38	C₂²×#833	D₄	1	2	yes	complemented
39	#72131	D₄	1	2	yes	—
40	#72131	D₄	1	2	yes	—
41	C₂×#2403	C₂×D₃	1	1	yes	characteristic
42	C₂×#1783	C₂×D₃	1	1	yes	characteristic
43	C₂×#1781	C₂×D₃	1	1	yes	characteristic
44	C₂⁶×C₃	C₂²⋊C₄	1	1	yes	characteristic, complemented
45	C₂⁶×C₃	C₂²×C₄	1	1	yes	characteristic
46	C₂⁶×C₃	C₂×D₄	1	1	yes	characteristic
47	C₂⁵×D₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂⁵×D₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂⁵×D₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
51	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
52	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
53	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
54	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
55	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
56	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
57	C₂³×C₃×D₄	C₂×D₄	1	1	yes	characteristic
58	C₂³×C₃×D₄	#41	1	1	yes	characteristic
59	C₂³×C₃×D₄	#41	1	1	yes	characteristic
60	C₂³×C₃×D₄	#41	1	1	yes	characteristic
61	C₂³×C₃×D₄	#41	1	1	yes	characteristic
62	C₂⁴×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
63	C₂⁴×C₃×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂⁴×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₃×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
66	C₂²×C₃×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic, complemented
67	C₂²×#217	C₂²⋊C₄	1	2	yes	complemented
68	C₂²×#217	C₂²⋊C₄	1	2	yes	complemented
69	C₂⁴×D₄	C₄×D₃	1	1	yes	characteristic
70	C₂⁴×D₄	C₂²×D₃	1	1	yes	characteristic
71	C₂²×#120	D₁₂	1	1	yes	characteristic
72	C₂²×#120	#67	1	1	yes	characteristic
73	C₂²×#100	#67	1	2	yes	complemented
74	C₂²×#99	#67	1	2	yes	complemented
75	C₂×#386	D₁₂	1	1	yes	characteristic
76	C₂×#386	#67	1	1	yes	characteristic
77	C₂×#379	C₄×D₃	1	1	yes	characteristic
78	C₂⁵×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂⁵×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂⁵×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂⁵×C₃	C₄×D₄	1	1	yes	characteristic
82	C₂⁵×C₃	C₄×D₄	1	1	yes	characteristic
83	C₂⁵×C₃	C₄×D₄	1	1	yes	characteristic
84	C₂⁵×C₃	C₄×D₄	1	1	yes	characteristic
85	C₂⁵×C₃	#120	1	1	yes	characteristic
86	C₂⁵×C₃	#120	1	1	yes	characteristic
87	C₂⁵×C₃	#120	1	1	yes	characteristic
88	C₂⁵×C₃	#121	1	1	yes	characteristic
89	C₂⁵×C₃	#121	1	1	yes	characteristic
90	C₂⁵×C₃	#123	1	1	yes	characteristic
91	C₂⁴×D₃	#99	1	2	yes	—
92	C₂⁴×D₃	#99	1	2	yes	—
93	C₂⁴×D₃	#99	1	2	yes	—
94	C₂²×C₃×D₄	#120	1	1	yes	characteristic
95	C₂²×C₃×D₄	#120	1	1	yes	characteristic
96	C₂²×C₃×D₄	#121	1	1	yes	characteristic
97	C₂²×C₃×D₄	#121	1	1	yes	characteristic
98	C₂²×C₃×D₄	#121	1	1	yes	characteristic
99	C₂²×C₃×D₄	#121	1	1	yes	characteristic
100	C₂²×C₃×D₄	#121	1	1	yes	characteristic
101	C₂²×C₃×D₄	#121	1	1	yes	characteristic
102	C₂²×C₃×D₄	#121	1	1	yes	characteristic
103	C₂²×C₃×D₄	#121	1	1	yes	characteristic
104	C₂²×C₃×D₄	#123	1	1	yes	characteristic
105	C₂²×C₃×D₄	#123	1	1	yes	characteristic
106	C₂²×C₃×D₄	#124	1	1	yes	characteristic
107	C₂²×C₃×D₄	#124	1	1	yes	characteristic
108	C₂²×C₃×D₄	#127	1	1	yes	characteristic
109	C₂²×C₃×D₄	#127	1	1	yes	characteristic
110	C₂³×C₃×C₄	#120	1	1	yes	characteristic
111	C₂³×C₃×C₄	#120	1	1	yes	characteristic
112	C₂³×C₃×C₄	#120	1	1	yes	characteristic
113	C₂³×C₃×C₄	#121	1	1	yes	characteristic
114	C₂³×C₃×C₄	#121	1	1	yes	characteristic
115	C₂³×C₃×C₄	#123	1	1	yes	characteristic
116	C₂³×Dic₃	#99	1	2	yes	—
117	C₂⁶	(C₂×D₃)⋊C₄	1	1	yes	characteristic, complemented
118	C₂⁶	C₂×C₄×D₃	1	1	yes	characteristic
119	C₂⁶	D₃×D₄	1	1	yes	characteristic
120	C₂³×D₄	C₂×D₁₂	1	1	yes	characteristic
121	C₂³×D₄	#235	1	1	yes	characteristic
122	C₂³×D₄	#235	1	1	yes	characteristic
123	C₂³×D₄	D₃×D₄	1	1	yes	characteristic
124	C₂³×D₄	D₃×D₄	1	1	yes	characteristic
125	C₂³×D₄	D₃×D₄	1	1	yes	characteristic
126	C₂³×D₄	D₃×D₄	1	1	yes	characteristic
127	C₂³×D₄	#237	1	1	yes	characteristic
128	C₂³×D₄	#237	1	1	yes	characteristic
129	C₂³×D₄	C₂×#67	1	1	yes	characteristic
130	C₂³×D₄	C₂×#67	1	1	yes	characteristic
131	C₂³×D₄	C₂×#67	1	1	yes	characteristic
132	C₂⁴×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic, complemented
133	C₂⁴×C₄	D₃×D₄	1	1	yes	characteristic
134	C₂²×C₂²⋊C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic, complemented
135	C₂²×C₂²⋊C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic, complemented
136	C₂⁴×C₃	#379	1	1	yes	characteristic
137	C₂⁴×C₃	#386	1	1	yes	characteristic
138	C₂⁴×C₃	#386	1	1	yes	characteristic
139	C₂⁴×C₃	#386	1	1	yes	characteristic
140	C₂⁴×C₃	#386	1	1	yes	characteristic
141	C₂⁴×C₃	#390	1	1	yes	characteristic
142	C₂⁴×C₃	#390	1	1	yes	characteristic
143	C₂⁴×C₃	#392	1	1	yes	characteristic
144	C₂⁴×C₃	#392	1	1	yes	characteristic
145	C₂⁴×C₃	#394	1	1	yes	characteristic
146	C₂⁴×C₃	#394	1	1	yes	characteristic
147	C₂⁴×C₃	C₂×#99	1	1	yes	characteristic
148	C₂⁴×C₃	C₂×#99	1	1	yes	characteristic
149	C₂⁴×C₃	C₂×#99	1	1	yes	characteristic
150	C₂⁴×C₃	C₂×#99	1	1	yes	characteristic
151	C₂³×D₃	AΣL(1,𝔽₁₆)	1	4	yes	complemented
152	C₂×C₃×D₄	B(4,𝔽₂)	1	4	yes	—
153	C₂×C₃×D₄	B(4,𝔽₂)	1	4	yes	—
154	C₂²×C₃×C₄	#392	1	1	yes	characteristic
155	C₂²×C₃×C₄	#392	1	1	yes	characteristic
156	C₂²×C₃×C₄	#394	1	1	yes	characteristic
157	C₂²×C₃×C₄	#394	1	1	yes	characteristic
158	C₂⁵	D₃×C₂²⋊C₄	1	1	yes	characteristic
159	C₂⁵	D₃×C₂²⋊C₄	1	1	yes	characteristic
160	C₂⁵	#67⋊C₄	1	1	yes	characteristic
161	C₂⁵	#67⋊C₄	1	1	yes	characteristic
162	C₂⁵	#882	1	1	yes	characteristic
163	C₂⁵	#883	1	1	yes	characteristic
164	C₂⁵	#884	1	1	yes	characteristic
165	C₂⁵	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
166	C₂⁵	C₄×#67	1	1	yes	characteristic
167	C₂⁵	C₄×#67	1	1	yes	characteristic
168	C₂⁵	#937	1	1	yes	characteristic
169	C₂⁵	#937	1	1	yes	characteristic
170	C₂⁵	#938	1	1	yes	characteristic
171	C₂²×D₄	#882	1	1	yes	characteristic
172	C₂²×D₄	#884	1	1	yes	characteristic
173	C₂²×D₄	#884	1	1	yes	characteristic
174	C₂²×D₄	#885	1	1	yes	characteristic
175	C₂²×D₄	#885	1	1	yes	characteristic
176	C₂²×D₄	#886	1	1	yes	characteristic
177	C₂²×D₄	#929	1	1	yes	characteristic
178	C₂²×D₄	#930	1	1	yes	characteristic
179	C₂²×D₄	#930	1	1	yes	characteristic
180	C₂²×D₄	#939	1	1	yes	characteristic
181	C₂²×D₄	#939	1	1	yes	characteristic
182	C₂²×D₄	#939	1	1	yes	characteristic
183	C₂²×D₄	#939	1	1	yes	characteristic
184	C₂²×D₄	#940	1	1	yes	characteristic
185	C₂²×D₄	#940	1	1	yes	characteristic
186	C₂²×D₄	#953	1	1	yes	characteristic
187	C₂³×C₄	#882	1	1	yes	characteristic
188	C₂³×C₄	#883	1	1	yes	characteristic
189	C₂³×C₄	#884	1	1	yes	characteristic
190	C₂³×C₄	#937	1	1	yes	characteristic
191	C₂³×C₄	#937	1	1	yes	characteristic
192	C₂³×C₄	#938	1	1	yes	characteristic
193	C₂³×C₃	#1438	1	1	yes	characteristic
194	C₂³×C₃	#1781	1	1	yes	characteristic
195	C₂³×C₃	#1781	1	1	yes	characteristic
196	C₂³×C₃	#1889	1	2	yes	—
197	C₂³×C₃	#1889	1	2	yes	—
198	C₂³×C₃	#1898	1	1	yes	characteristic
199	C₂³×C₃	#1898	1	1	yes	characteristic
200	C₂³×C₃	#1898	1	1	yes	characteristic
201	C₂³×C₃	#1898	1	1	yes	characteristic
202	C₂³×C₃	#2021	1	2	yes	—
203	C₂³×C₃	#2021	1	2	yes	—
204	C₂³×C₃	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
205	C₂³×C₃	#2121	1	2	yes	—
206	C₂×C₃×C₄	#2021	1	2	yes	—
207	C₂×C₃×C₄	#2021	1	2	yes	—
208	C₂⁴	D₃×#99	1	1	yes	characteristic
209	C₂⁴	D₃×#99	1	1	yes	characteristic
210	C₂⁴	D₃×#99	1	1	yes	characteristic
211	C₂⁴	D₃×#99	1	1	yes	characteristic
212	C₂⁴	#4955	1	1	yes	characteristic
213	C₂⁴	#4956	1	1	yes	characteristic
214	C₂⁴	#4956	1	1	yes	characteristic
215	C₂⁴	#4957	1	1	yes	characteristic
216	C₂⁴	#4957	1	1	yes	characteristic
217	C₂⁴	#4958	1	1	yes	characteristic
218	C₂⁴	#4959	1	1	yes	characteristic
219	C₂⁴	#4960	1	1	yes	characteristic
220	C₂⁴	#4961	1	1	yes	characteristic
221	C₂⁴	#5213	1	1	yes	characteristic
222	C₂⁴	#5223	1	1	yes	characteristic
223	C₂×D₄	#4741	1	4	yes	—
224	C₂×D₄	#5244	1	4	yes	—
225	C₂²×C₄	#4959	1	1	yes	characteristic
226	C₂²×C₄	#4960	1	1	yes	characteristic
227	C₂²×C₄	#4961	1	1	yes	characteristic
228	C₂²×C₄	#5223	1	1	yes	characteristic
229	C₂²×C₃	#10965	1	1	yes	characteristic
230	C₂²×C₃	#10991	1	1	yes	characteristic
231	C₂²×C₃	#10991	1	1	yes	characteristic
232	C₂²×C₃	#12687	1	1	yes	characteristic
233	C₂²×C₃	#12690	1	1	yes	characteristic
234	C₂²×C₃	#12705	1	2	yes	—
235	C₂²×C₃	#13082	1	2	yes	—
236	C₂²×C₃	#13086	1	2	yes	—
237	C₂³	D₃×AΣL(1,𝔽₁₆)	1	2	yes	—
238	C₂³	#69135	1	2	yes	—
239	C₂³	#70022	1	2	yes	—
240	C₂³	#70035	1	1	yes	characteristic
241	C₂³	#70035	1	1	yes	characteristic
242	C₂³	#70036	1	1	yes	characteristic
243	C₂³	#70036	1	1	yes	characteristic
244	C₂³	#70036	1	1	yes	characteristic
245	C₂³	#70036	1	1	yes	characteristic
246	C₂³	#70037	1	2	yes	—
247	C₂³	#71550	1	1	yes	characteristic
248	C₂³	#71654	1	2	yes	—
249	C₂³	#71851	1	2	yes	—
250	C₂×C₄	#70037	1	2	yes	—
251	C₂×C₄	#71851	1	2	yes	—
252	C₂×C₃	#115690	1	1	yes	characteristic
253	C₂×C₃	#147555	1	1	yes	characteristic
254	C₂×C₃	#147567	1	1	yes	characteristic
255	C₂²	#10701676	1	1	yes	characteristic
256	C₂²	#10701795	1	1	yes	characteristic
257	C₂²	#10701799	1	1	yes	characteristic
258	C₂²	#10710645	1	1	yes	characteristic
259	C₂²	#10710658	1	1	yes	characteristic
260	C₂²	#10710711	1	2	yes	—
261	C₂²	#10712673	1	2	yes	—
262	C₂²	#10712693	1	2	yes	—
263	C₃	#72213256	1	1	yes	characteristic, complemented, Sylow
264	C₂	#72248033	1	1	yes	characteristic
265	C₂	#72248093	1	1	yes	characteristic
266	C₂	#72248175	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10701676
commutator	C₂⁶×C₃	C₂²×C₄
Frattini	C₂⁴×D₄	C₂²×D₃
Fitting	C₃×#496225	C₂
socle	C₂²×C₃	#10965
prime radical	#72248257	C₂
cosocle	C₂⁴×C₃×D₄	C₂³
hypercenter	#496225	D₃
hypocenter	C₃	#72213256
LCS	G ▷ C₂⁶×C₃ ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×D₄ ◁ #496225

Complements

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

N	H ≅ G/N	type
1. #72255063	C₂	semidirect
2. C₃×#496225	C₂	semidirect
3. #72249307	C₂	semidirect
4. #72249306	C₂	semidirect
5. #72248674	C₂	semidirect
6. #72248352	C₂	semidirect
9. C₂³×#937	C₄	semidirect
10. C₂×#84052	C₄	semidirect
13. #10726342	C₄	semidirect
14. #10717058	C₄	semidirect
19. #496225	D₃	semidirect
31. C₂²×C₃×#100	D₄	semidirect
32. C₂²×C₃×#99	D₄	semidirect
37. C₂²×#834	D₄	semidirect
38. C₂²×#833	D₄	semidirect
44. C₂⁶×C₃	C₂²⋊C₄	semidirect
62. C₂⁴×C₃×C₄	C₂²⋊C₄	semidirect
65. C₂²×C₃×C₂²⋊C₄	C₂²⋊C₄	semidirect
66. C₂²×C₃×C₂²⋊C₄	C₂²⋊C₄	semidirect
67. C₂²×#217	C₂²⋊C₄	semidirect
68. C₂²×#217	C₂²⋊C₄	semidirect
73. C₂²×#100	#67	semidirect
74. C₂²×#99	#67	semidirect
117. C₂⁶	(C₂×D₃)⋊C₄	semidirect
132. C₂⁴×C₄	(C₂×D₃)⋊C₄	semidirect
134. C₂²×C₂²⋊C₄	(C₂×D₃)⋊C₄	semidirect
135. C₂²×C₂²⋊C₄	(C₂×D₃)⋊C₄	semidirect
151. C₂³×D₃	AΣL(1,𝔽₁₆)	semidirect
263. C₃	#72213256	semidirect

Maximal subgroups

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

subgroup	index
#72255063	2
C₃×#496225	2
#72249307	2
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