Subgroups — #11773702

Order 1152.

461 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₄×#3711	C₂	1	1	yes	maximal, characteristic
3	C₄×#3711	C₂	1	1	yes	maximal, characteristic
4	#10589320	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4454	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589078	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈²	C₂	1	1	yes	maximal, characteristic, complemented
8	#69849	—	3	1	no	maximal
9	#68075	—	3	1	no	maximal
10	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
11	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
12	C₂×#3711	C₄	1	2	yes	—
13	C₂×#3711	C₄	1	2	yes	—
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₃×#815	C₄	1	2	yes	—
18	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	#63554	C₄	1	2	yes	—
22	#63554	C₄	1	2	yes	—
23	C₃×C₄×M₁₆	—	3	1	no	—
24	C₄×#208	—	3	1	no	—
25	C₄×#208	—	3	1	no	—
26	C₄×ΓL(1,𝔽₂₅)	—	3	1	no	—
27	C₄×ΓL(1,𝔽₂₅)	—	3	1	no	—
28	C₄×ΓL(1,𝔽₂₅)	D₃	1	1	yes	characteristic
29	C₃×#322	—	3	1	no	—
30	#4455	—	3	1	no	—
31	#4454	—	3	1	no	—
32	#4454	—	3	1	no	—
33	#4454	—	3	1	no	—
34	#4454	—	6	1	no	—
35	#4454	D₃	1	1	yes	characteristic, complemented
36	C₈×C₃⋊C₈	—	3	1	no	—
37	C₈×C₃⋊C₈	—	3	1	no	—
38	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
42	C₃²×C₄²	C₂³	1	1	yes	characteristic
43	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
44	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
45	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
49	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
50	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
51	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
52	#3711	—	2	4	no	—
53	#3711	—	2	4	no	—
54	#1455	—	9	1	no	Sylow
55	C₂×C₃×M₁₆	—	3	2	no	—
56	C₂×C₃×C₄²	—	3	1	no	—
57	C₂×#208	—	3	2	no	—
58	C₂×#208	—	3	2	no	—
59	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
60	C₂×ΓL(1,𝔽₂₅)	—	3	2	no	—
61	C₂×ΓL(1,𝔽₂₅)	Dic₃	1	2	yes	—
62	C₄²×D₃	C₂×D₃	1	1	yes	characteristic
63	C₃×#97	—	3	2	no	—
64	C₃×C₄×C₈	—	2	1	no	—
65	C₃×C₄×C₈	—	3	1	no	—
66	C₃×C₄×C₈	—	3	1	no	—
67	C₃×C₄×C₈	—	3	1	no	—
68	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
69	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
70	#815	—	3	2	no	—
71	#815	—	3	2	no	—
72	#815	Dic₃	1	2	yes	—
73	#803	—	3	2	no	—
74	#803	—	3	2	no	—
75	C₄×C₃⋊C₈	—	3	1	no	—
76	C₄×C₃⋊C₈	—	3	1	no	—
77	C₄×C₃⋊C₈	—	3	1	no	—
78	C₄×C₃⋊C₈	—	3	1	no	—
79	C₄×C₃⋊C₈	—	6	1	no	—
80	C₄×C₃⋊C₈	—	6	1	no	—
81	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
82	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
83	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
84	C₂²×C₃×D₃	C₄²	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
89	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
90	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
91	C₃×C₄×D₃	—	2	2	no	—
92	C₃×C₄×D₃	—	2	2	no	—
93	C₃×C₄×D₃	—	2	2	no	—
94	C₃²×C₈	—	2	2	no	—
95	C₃⋊(C₃⋊C₈)	—	2	2	no	—
96	C₃⋊(C₃⋊C₈)	M₁₆	1	4	yes	complemented
97	C₃×C₃⋊C₈	—	2	2	no	—
98	C₃×C₃⋊C₈	M₁₆	1	4	yes	complemented
99	C₃×C₃⋊C₈	M₁₆	1	4	yes	complemented
100	C₄×M₁₆	—	3	1	no	—
101	C₄×M₁₆	—	9	1	no	—
102	C₄×M₁₆	—	9	1	no	—
103	#322	—	3	1	no	—
104	#322	—	9	1	no	—
105	#322	—	9	1	no	—
106	C₈²	—	9	1	no	—
107	C₂²×C₃×C₄	—	3	1	no	—
108	C₂²×C₃×C₄	—	3	1	no	—
109	C₂²×C₃×C₄	—	3	1	no	—
110	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
111	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
112	C₂×C₄×D₃	C₂×Dic₃	1	1	yes	characteristic
113	C₃×M₁₆	—	6	4	no	—
114	C₂×C₃×C₈	—	2	2	no	—
115	C₂×C₃×C₈	—	3	2	no	—
116	C₂×C₃×C₈	—	3	2	no	—
117	C₂×C₃×C₈	—	3	2	no	—
118	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
119	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
120	C₃×C₄²	—	2	1	no	—
121	C₃×C₄²	—	3	1	no	—
122	C₃×C₄²	—	3	1	no	—
123	C₃×C₄²	—	3	1	no	—
124	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
125	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
126	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
127	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
128	C₄×Dic₃	C₂×Dic₃	1	1	yes	characteristic
129	#208	—	6	4	no	—
130	#208	—	6	4	no	—
131	C₂×C₃⋊C₈	—	3	2	no	—
132	C₂×C₃⋊C₈	—	3	2	no	—
133	C₂×C₃⋊C₈	—	3	2	no	—
134	C₂×C₃⋊C₈	—	3	2	no	—
135	C₂×C₃⋊C₈	—	6	2	no	—
136	C₂×C₃⋊C₈	—	6	2	no	—
137	C₂×C₃⋊C₈	C₄×D₃	1	2	yes	—
138	C₂×C₃⋊C₈	C₄×D₃	1	2	yes	—
139	C₂×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
140	ΓL(1,𝔽₂₅)	—	2	4	no	—
141	ΓL(1,𝔽₂₅)	—	6	4	no	—
142	ΓL(1,𝔽₂₅)	—	6	4	no	—
143	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₃×D₃	—	2	1	no	—
145	C₂×C₃×D₃	—	2	1	no	—
146	C₂×C₃×D₃	—	2	1	no	—
147	C₃²×C₄	C₂×M₁₆	1	2	yes	—
148	C₃²×C₄	C₂×M₁₆	1	2	yes	—
149	C₃²×C₄	C₂×M₁₆	1	2	yes	—
150	C₃×Dic₃	—	2	1	no	—
151	C₃×Dic₃	—	2	1	no	—
152	C₃×Dic₃	—	2	1	no	—
153	C₂×M₁₆	—	3	2	no	—
154	C₂×M₁₆	—	9	2	no	—
155	C₂×M₁₆	—	9	2	no	—
156	C₂×C₄²	—	3	1	no	—
157	#97	—	3	2	no	—
158	#97	—	9	2	no	—
159	#97	—	9	2	no	—
160	C₄×C₈	—	3	1	no	—
161	C₄×C₈	—	3	1	no	—
162	C₄×C₈	—	3	1	no	—
163	C₄×C₈	—	9	1	no	—
164	C₄×C₈	—	9	1	no	—
165	C₄×C₈	D₃²	1	1	yes	characteristic
166	C₂³×C₃	—	3	1	no	—
167	C₂²×D₃	C₄×Dic₃	1	1	yes	characteristic
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	3	1	no	—
172	C₂×C₃×C₄	—	3	1	no	—
173	C₂×C₃×C₄	—	3	1	no	—
174	C₂×C₃×C₄	—	6	2	no	—
175	C₂×C₃×C₄	—	6	2	no	—
176	C₂×C₃×C₄	—	6	2	no	—
177	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
183	C₂×Dic₃	C₄×Dic₃	1	1	yes	characteristic
184	C₂×Dic₃	C₄×Dic₃	1	1	yes	characteristic
185	C₂×Dic₃	C₄×Dic₃	1	1	yes	characteristic
186	C₄×D₃	—	2	2	no	—
187	C₄×D₃	—	2	2	no	—
188	C₄×D₃	—	2	2	no	—
189	C₃×C₈	—	2	2	no	—
190	C₃×C₈	—	2	2	no	—
191	C₃×C₈	—	3	4	no	—
192	C₃×C₈	—	3	4	no	—
193	C₃×C₈	—	4	2	no	—
194	C₃×C₈	—	6	2	no	—
195	C₃⋊C₈	—	2	2	no	—
196	C₃⋊C₈	—	3	4	no	—
197	C₃⋊C₈	—	3	4	no	—
198	C₃⋊C₈	—	6	4	no	—
199	C₃⋊C₈	—	6	2	no	—
200	C₃⋊C₈	—	6	2	no	—
201	C₃⋊C₈	—	12	2	no	—
202	C₃⋊C₈	ΓL(1,𝔽₂₅)	1	4	yes	complemented
203	C₃⋊C₈	#208	1	4	yes	complemented
204	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
205	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
206	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
207	C₃×D₃	—	4	1	no	—
208	C₂²×C₄	—	3	1	no	—
209	C₂²×C₄	—	3	1	no	—
210	C₂²×C₄	—	3	1	no	—
211	M₁₆	—	6	4	no	—
212	M₁₆	—	18	4	no	—
213	M₁₆	—	18	4	no	—
214	C₂×C₈	—	3	2	no	—
215	C₂×C₈	—	3	2	no	—
216	C₂×C₈	—	3	2	no	—
217	C₂×C₈	—	9	2	no	—
218	C₂×C₈	—	9	2	no	—
219	C₂×C₈	D₃×Dic₃	1	2	yes	—
220	C₄²	—	3	1	no	—
221	C₄²	—	3	1	no	—
222	C₄²	—	3	1	no	—
223	C₄²	C₂×D₃²	1	1	yes	characteristic
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	6	1	no	—
226	C₂²×C₃	—	6	1	no	—
227	C₂²×C₃	—	6	1	no	—
228	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
229	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
230	C₂×D₃	—	2	1	no	—
231	C₂×D₃	—	2	1	no	—
232	C₂×D₃	—	2	1	no	—
233	C₃×C₄	—	2	2	no	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	6	1	no	—
237	C₃×C₄	—	6	1	no	—
238	C₃×C₄	—	6	1	no	—
239	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
240	C₃×C₄	D₃×M₁₆	1	2	yes	—
241	C₃×C₄	D₃×M₁₆	1	2	yes	—
242	C₃×C₄	D₃×M₁₆	1	2	yes	—
243	C₃×C₄	D₃×M₁₆	1	2	yes	—
244	C₃×C₄	C₂×#208	1	2	yes	—
245	Dic₃	—	2	1	no	—
246	Dic₃	—	2	1	no	—
247	Dic₃	—	2	1	no	—
248	C₃²	#1455	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	3	1	no	—
250	C₂×C₄	—	3	1	no	—
251	C₂×C₄	—	3	1	no	—
252	C₂×C₄	—	3	1	no	—
253	C₂×C₄	—	6	2	no	—
254	C₂×C₄	—	6	2	no	—
255	C₂×C₄	—	6	2	no	—
256	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
257	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
258	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	3	4	no	—
261	C₈	—	3	4	no	—
262	C₈	—	6	2	no	—
263	C₈	—	9	4	no	—
264	C₈	—	18	2	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	12	1	no	—
269	C₂×C₃	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
270	C₂×C₃	#4707	1	1	yes	characteristic
271	C₂×C₃	#4707	1	1	yes	characteristic
272	C₂×C₃	C₄×#208	1	1	yes	characteristic
273	C₂×C₃	Dic₃×M₁₆	1	1	yes	characteristic
274	C₂×C₃	Dic₃×M₁₆	1	1	yes	characteristic
275	D₃	—	4	1	no	—
276	C₂²	—	6	1	no	—
277	C₂²	—	6	1	no	—
278	C₂²	—	6	1	no	—
279	C₂²	C₄×D₃×Dic₃	1	1	yes	characteristic
280	C₄	—	6	1	no	—
281	C₄	—	6	1	no	—
282	C₄	—	6	1	no	—
283	C₄	D₃×ΓL(1,𝔽₂₅)	1	2	yes	—
284	C₄	#63791	1	2	yes	—
285	C₄	D₃×#208	1	2	yes	—
286	C₃	—	2	1	no	—
287	C₃	#68075	1	1	yes	characteristic, complemented
288	C₃	#69849	1	1	yes	characteristic, complemented
289	C₂	—	12	1	no	—
290	C₂	#10590334	1	1	yes	characteristic
291	C₂	Dic₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
292	C₂	#10590667	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×D₃²
commutator, socle	C₂²×C₃²	C₂×C₄²
Fitting	C₃²×C₄×C₈	C₂²
prime radical	C₂²×C₃×D₃	C₄²
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄×C₈	D₃²
hypocenter	C₃²	#1455
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #10589320	C₂	semidirect
5. C₃×#4454	C₂	semidirect
6. #10589078	C₂	semidirect
7. C₃⋊C₈²	C₂	semidirect
35. #4454	D₃	semidirect
96. C₃⋊(C₃⋊C₈)	M₁₆	semidirect
98. C₃×C₃⋊C₈	M₁₆	semidirect
99. C₃×C₃⋊C₈	M₁₆	semidirect
202. C₃⋊C₈	ΓL(1,𝔽₂₅)	semidirect
203. C₃⋊C₈	#208	semidirect
248. C₃²	#1455	semidirect
287. C₃	#68075	semidirect
288. C₃	#69849	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×ΓL(1,𝔽₂₅)	2
C₄×#3711	2
#10589320	2
C₃×#4454	2
#10589078	2
C₃⋊C₈²	2
#69849	3
#68075	3