Subgroups — D₄×3₋¹⁺²⋊AGL(1,𝔽₅) #72247442

Order 4320.

618 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72247404	C₂	1	2	yes	maximal, complemented
2	#72247390	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×D₅×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
4	C₄×3₋¹⁺²⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×3₋¹⁺²⋊AGL(1,𝔽₅)	C₂	1	2	yes	maximal, complemented
6	C₃×D₄×ΓL(1,𝔽₁₆)	—	3	1	no	maximal
7	D₄×C₉⋊AGL(1,𝔽₅)	C₃	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₅×3₋¹⁺²	C₂²	1	2	yes	—
9	C₂×3₋¹⁺²⋊AGL(1,𝔽₅)	—	2	4	no	—
10	C₂×3₋¹⁺²⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
11	C₂×3₋¹⁺²⋊AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
12	C₂×3₋¹⁺²⋊AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
13	C₅×D₄×3₋¹⁺²	C₄	1	1	yes	characteristic, complemented
14	3₋¹⁺²×C₂²⋊D₅	C₄	1	2	yes	complemented
15	3₋¹⁺²×D₂₀	C₄	1	1	yes	characteristic, complemented
16	C₄×D₅×3₋¹⁺²	C₂²	1	1	yes	characteristic
17	D₄×3₋¹⁺²⋊C₄	—	5	1	no	maximal, Hall
18	C₂²×C₃×ΓL(1,𝔽₁₆)	—	3	2	no	—
19	C₃²×D₄×D₅	D₃	1	1	yes	characteristic
20	C₃×#6763	—	3	2	no	—
21	C₃×C₄⋊ΓL(1,𝔽₁₆)	—	3	1	no	—
22	C₃×C₄×ΓL(1,𝔽₁₆)	—	3	1	no	—
23	C₂²×C₉⋊AGL(1,𝔽₅)	C₂×C₃	1	2	yes	complemented
24	D₄×C₉×D₅	—	2	1	no	—
25	D₄×C₉×D₅	C₂×C₃	1	1	yes	characteristic
26	#10623655	C₂×C₃	1	2	yes	complemented
27	C₄⋊(C₉⋊AGL(1,𝔽₅))	C₂×C₃	1	1	yes	characteristic, complemented
28	C₄×C₉⋊AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×C₅×3₋¹⁺²	C₂×C₄	1	2	yes	complemented
30	C₂×D₅×3₋¹⁺²	—	2	2	no	—
31	C₂×D₅×3₋¹⁺²	—	2	2	no	—
32	C₂×D₅×3₋¹⁺²	C₂×C₄	1	2	yes	complemented
33	C₂×D₅×3₋¹⁺²	C₂³	1	1	yes	characteristic
34	3₋¹⁺²⋊AGL(1,𝔽₅)	—	2	2	no	—
35	3₋¹⁺²⋊AGL(1,𝔽₅)	—	2	1	no	—
36	3₋¹⁺²⋊AGL(1,𝔽₅)	D₄	1	2	yes	direct product factor
37	C₄×C₅×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic, complemented
38	Dic₅×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic, complemented
39	D₄×ΓL(1,𝔽₁₆)	—	3	1	no	—
40	C₃×D₄×AGL(1,𝔽₅)	—	9	1	no	—
41	C₂×D₄×3₋¹⁺²	—	5	1	no	—
42	C₂²×3₋¹⁺²⋊C₄	—	5	2	no	—
43	#3676⋊C₃	—	5	2	no	—
44	C₄⋊(3₋¹⁺²⋊C₄)	—	5	1	no	—
45	C₄×3₋¹⁺²⋊C₄	—	5	1	no	—
46	C₂²×C₃²×D₅	C₂×D₃	1	2	yes	—
47	C₂×C₃×ΓL(1,𝔽₁₆)	—	3	1	no	—
48	C₂×C₃×ΓL(1,𝔽₁₆)	—	3	1	no	—
49	C₂×C₃×ΓL(1,𝔽₁₆)	—	3	2	no	—
50	C₂×C₃×ΓL(1,𝔽₁₆)	—	6	4	no	—
51	C₃²×C₅×D₄	Dic₃	1	1	yes	characteristic
52	C₃²×C₂²⋊D₅	Dic₃	1	2	yes	—
53	C₃²×D₂₀	Dic₃	1	1	yes	characteristic
54	C₃²×C₄×D₅	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₉×D₅	—	2	2	no	—
56	C₂²×C₉×D₅	C₂²×C₃	1	2	yes	—
57	C₂×C₉⋊AGL(1,𝔽₅)	—	2	4	no	—
58	C₂×C₉⋊AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₉⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
60	C₂×C₉⋊AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
61	C₅×D₄×C₉	—	2	1	no	—
62	C₅×D₄×C₉	C₃×C₄	1	1	yes	characteristic, complemented
63	C₉×C₂²⋊D₅	—	2	2	no	—
64	C₉×C₂²⋊D₅	C₃×C₄	1	2	yes	complemented
65	C₉×D₂₀	—	2	1	no	—
66	C₉×D₂₀	C₃×C₄	1	1	yes	characteristic, complemented
67	C₄×C₉×D₅	—	2	1	no	—
68	C₄×C₉×D₅	C₂²×C₃	1	1	yes	characteristic
69	C₃×D₄×Dic₃	—	15	1	no	—
70	D₄×Dic₉	—	5	1	no	—
71	C₂×C₅×3₋¹⁺²	—	2	2	no	—
72	C₂×C₅×3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic
73	D₅×3₋¹⁺²	—	2	2	no	—
74	D₅×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
75	D₅×3₋¹⁺²	#41	1	1	yes	characteristic
76	C₂²×ΓL(1,𝔽₁₆)	—	3	2	no	—
77	C₂²×C₃×AGL(1,𝔽₅)	—	9	2	no	—
78	C₃×D₄×D₅	—	3	1	no	—
79	C₃×D₄×D₅	C₃×D₃	1	1	yes	characteristic
80	#6763	—	3	2	no	—
81	C₄⋊ΓL(1,𝔽₁₆)	—	3	1	no	—
82	C₄×ΓL(1,𝔽₁₆)	—	3	1	no	—
83	C₃×C₂²⋊AGL(1,𝔽₅)	—	9	2	no	—
84	C₃×C₄⋊AGL(1,𝔽₅)	—	9	1	no	—
85	C₃×C₄×AGL(1,𝔽₅)	—	9	1	no	—
86	C₂³×3₋¹⁺²	—	5	2	no	—
87	D₄×3₋¹⁺²	—	5	2	no	—
88	D₄×3₋¹⁺²	—	5	1	no	—
89	D₄×3₋¹⁺²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
90	C₂×C₄×3₋¹⁺²	—	5	1	no	—
91	C₂×3₋¹⁺²⋊C₄	—	5	1	no	—
92	C₂×3₋¹⁺²⋊C₄	—	5	2	no	—
93	C₂×3₋¹⁺²⋊C₄	—	5	1	no	—
94	C₂×3₋¹⁺²⋊C₄	—	10	4	no	—
95	C₂²×C₃²×C₅	C₂×Dic₃	1	2	yes	—
96	C₂×C₃²×D₅	—	2	2	no	—
97	C₂×C₃²×D₅	—	2	2	no	—
98	C₂×C₃²×D₅	C₂×Dic₃	1	2	yes	—
99	C₂×C₃²×D₅	C₂²×D₃	1	1	yes	characteristic
100	C₃×ΓL(1,𝔽₁₆)	—	3	2	no	—
101	C₃×ΓL(1,𝔽₁₆)	—	6	2	no	—
102	C₃×ΓL(1,𝔽₁₆)	—	6	1	no	—
103	C₃²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
104	C₃²×Dic₅	C₂×Dic₃	1	1	yes	characteristic
105	C₂²×C₅×C₉	—	2	2	no	—
106	C₂²×C₅×C₉	C₂×C₃×C₄	1	2	yes	complemented
107	C₂×C₉×D₅	—	2	1	no	—
108	C₂×C₉×D₅	—	2	2	no	—
109	C₂×C₉×D₅	—	2	2	no	—
110	C₂×C₉×D₅	—	2	2	no	—
111	C₂×C₉×D₅	—	4	2	no	—
112	C₂×C₉×D₅	—	4	2	no	—
113	C₂×C₉×D₅	C₂×C₃×C₄	1	2	yes	complemented
114	C₂×C₉×D₅	C₂³×C₃	1	1	yes	characteristic
115	C₉⋊AGL(1,𝔽₅)	—	2	2	no	—
116	C₉⋊AGL(1,𝔽₅)	—	2	1	no	—
117	C₉⋊AGL(1,𝔽₅)	C₃×D₄	1	2	yes	complemented
118	C₄×C₅×C₉	—	2	1	no	—
119	C₄×C₅×C₉	C₂×C₃×C₄	1	1	yes	characteristic, complemented
120	C₉×Dic₅	—	2	1	no	—
121	C₉×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
122	D₄×AGL(1,𝔽₅)	—	9	1	no	Hall
123	C₂×C₃²×D₄	—	5	1	no	—
124	C₂²×C₃×Dic₃	—	15	2	no	—
125	C₃×#217	—	15	2	no	—
126	C₃×C₄⋊Dic₃	—	15	1	no	—
127	C₃×C₄×Dic₃	—	15	1	no	—
128	C₂×D₄×C₉	—	5	1	no	—
129	C₂×D₄×C₉	—	10	1	no	—
130	C₂²×Dic₉	—	5	2	no	—
131	#3676	—	5	2	no	—
132	C₄⋊Dic₉	—	5	1	no	—
133	C₄×Dic₉	—	5	1	no	—
134	C₅×3₋¹⁺²	C₄×D₄	1	1	yes	characteristic, complemented, Hall
135	C₂²×C₃×D₅	—	3	2	no	—
136	C₂²×C₃×D₅	C₂×C₃×D₃	1	2	yes	—
137	C₂×ΓL(1,𝔽₁₆)	—	3	1	no	—
138	C₂×ΓL(1,𝔽₁₆)	—	3	1	no	—
139	C₂×ΓL(1,𝔽₁₆)	—	3	2	no	—
140	C₂×ΓL(1,𝔽₁₆)	—	6	4	no	—
141	C₂×C₃×AGL(1,𝔽₅)	—	9	1	no	—
142	C₂×C₃×AGL(1,𝔽₅)	—	9	1	no	—
143	C₂×C₃×AGL(1,𝔽₅)	—	9	2	no	—
144	C₂×C₃×AGL(1,𝔽₅)	—	18	4	no	—
145	C₃×C₅×D₄	—	3	1	no	—
146	C₃×C₅×D₄	C₃×Dic₃	1	1	yes	characteristic
147	C₃×C₂²⋊D₅	—	3	2	no	—
148	C₃×C₂²⋊D₅	C₃×Dic₃	1	2	yes	—
149	C₃×D₂₀	—	3	1	no	—
150	C₃×D₂₀	C₃×Dic₃	1	1	yes	characteristic
151	C₃×C₄×D₅	—	3	1	no	—
152	C₃×C₄×D₅	C₂×C₃×D₃	1	1	yes	characteristic
153	C₂²×3₋¹⁺²	—	5	1	no	—
154	C₂²×3₋¹⁺²	—	5	2	no	—
155	C₂²×3₋¹⁺²	—	10	2	no	—
156	C₂²×3₋¹⁺²	—	10	2	no	—
157	C₂²×3₋¹⁺²	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
158	C₄×3₋¹⁺²	—	5	1	no	—
159	C₄×3₋¹⁺²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
160	3₋¹⁺²⋊C₄	—	5	2	no	—
161	3₋¹⁺²⋊C₄	—	10	2	no	—
162	3₋¹⁺²⋊C₄	—	10	1	no	—
163	C₃×C₄×D₄	—	45	1	no	—
164	D₄×Dic₃	—	15	1	no	—
165	C₂×C₃²×C₅	—	2	2	no	—
166	C₂×C₃²×C₅	C₂²×Dic₃	1	1	yes	characteristic
167	C₃²×D₅	—	2	2	no	—
168	C₃²×D₅	D₃×D₄	1	1	yes	characteristic
169	C₃²×D₅	#237	1	1	yes	characteristic
170	C₂×C₅×C₉	—	2	2	no	—
171	C₂×C₅×C₉	—	2	1	no	—
172	C₂×C₅×C₉	—	4	2	no	—
173	C₂×C₅×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₉×D₅	—	2	2	no	—
175	C₉×D₅	—	2	1	no	—
176	C₉×D₅	—	2	1	no	—
177	C₉×D₅	—	4	2	no	—
178	C₉×D₅	C₂×C₃×D₄	1	1	yes	characteristic
179	C₉×D₅	C₃×#41	1	1	yes	characteristic
180	C₂²×AGL(1,𝔽₅)	—	9	2	no	—
181	D₄×D₅	AGL(1,ℤ/9)	1	1	yes	characteristic
182	C₂²⋊AGL(1,𝔽₅)	—	9	2	no	—
183	C₄⋊AGL(1,𝔽₅)	—	9	1	no	—
184	C₄×AGL(1,𝔽₅)	—	9	1	no	—
185	C₂³×C₃²	—	5	2	no	—
186	C₃²×D₄	—	5	2	no	—
187	C₃²×D₄	—	5	1	no	—
188	C₃²×D₄	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
189	C₂×C₃²×C₄	—	5	1	no	—
190	C₂×C₃×Dic₃	—	15	1	no	—
191	C₂×C₃×Dic₃	—	15	1	no	—
192	C₂×C₃×Dic₃	—	15	2	no	—
193	C₂×C₃×Dic₃	—	30	4	no	—
194	C₂³×C₉	—	5	2	no	—
195	C₂³×C₉	—	10	2	no	—
196	D₄×C₉	—	2	1	no	—
197	D₄×C₉	—	5	2	no	—
198	D₄×C₉	—	5	1	no	—
199	D₄×C₉	—	10	1	no	—
200	D₄×C₉	—	10	2	no	—
201	D₄×C₉	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
202	C₂×C₄×C₉	—	5	1	no	—
203	C₂×C₄×C₉	—	10	1	no	—
204	C₂×Dic₉	—	5	1	no	—
205	C₂×Dic₉	—	5	1	no	—
206	C₂×Dic₉	—	5	2	no	—
207	C₂×Dic₉	—	10	4	no	—
208	C₂²×C₃×C₅	—	3	2	no	—
209	C₂²×C₃×C₅	C₂×C₃×Dic₃	1	2	yes	—
210	C₂×C₃×D₅	—	2	2	no	—
211	C₂×C₃×D₅	—	2	2	no	—
212	C₂×C₃×D₅	—	3	2	no	—
213	C₂×C₃×D₅	—	3	1	no	—
214	C₂×C₃×D₅	—	6	2	no	—
215	C₂×C₃×D₅	—	6	2	no	—
216	C₂×C₃×D₅	C₂×C₃×Dic₃	1	2	yes	—
217	C₂×C₃×D₅	C₂²×C₃×D₃	1	1	yes	characteristic
218	ΓL(1,𝔽₁₆)	—	3	2	no	—
219	ΓL(1,𝔽₁₆)	—	6	2	no	—
220	ΓL(1,𝔽₁₆)	—	6	1	no	—
221	C₃×AGL(1,𝔽₅)	—	9	2	no	—
222	C₃×AGL(1,𝔽₅)	—	18	1	no	—
223	C₃×AGL(1,𝔽₅)	—	18	2	no	—
224	C₃×C₄×C₅	—	3	1	no	—
225	C₃×C₄×C₅	C₂×C₃×Dic₃	1	1	yes	characteristic
226	C₃×Dic₅	—	3	1	no	—
227	C₃×Dic₅	C₂×C₃×Dic₃	1	1	yes	characteristic
228	C₂×3₋¹⁺²	—	2	2	no	—
229	C₂×3₋¹⁺²	—	5	1	no	—
230	C₂×3₋¹⁺²	—	5	1	no	—
231	C₂×3₋¹⁺²	—	10	2	no	—
232	C₂×3₋¹⁺²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
233	C₂×C₃×D₄	—	5	1	no	—
234	C₂×C₃×D₄	—	30	1	no	—
235	C₂²×C₃×C₄	—	45	2	no	—
236	C₂²×Dic₃	—	15	2	no	—
237	C₃×C₄⋊C₄	—	45	1	no	—
238	C₃×C₂²⋊C₄	—	45	2	no	—
239	C₃×C₄²	—	45	1	no	—
240	#217	—	15	2	no	—
241	C₄⋊Dic₃	—	15	1	no	—
242	C₄×Dic₃	—	15	1	no	—
243	C₃²×C₅	D₄×Dic₃	1	1	yes	characteristic
244	C₅×C₉	—	2	1	no	—
245	C₅×C₉	C₃×C₄×D₄	1	1	yes	characteristic, complemented
246	C₂²×D₅	C₂×AGL(1,ℤ/9)	1	2	yes	—
247	C₂×AGL(1,𝔽₅)	—	9	1	no	—
248	C₂×AGL(1,𝔽₅)	—	9	1	no	—
249	C₂×AGL(1,𝔽₅)	—	9	2	no	—
250	C₂×AGL(1,𝔽₅)	—	18	4	no	—
251	C₅×D₄	3₋¹⁺²⋊C₄	1	1	yes	characteristic, complemented
252	C₂²⋊D₅	3₋¹⁺²⋊C₄	1	2	yes	complemented
253	D₂₀	3₋¹⁺²⋊C₄	1	1	yes	characteristic, complemented
254	C₄×D₅	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
255	C₂²×C₃²	—	5	1	no	—
256	C₂²×C₃²	—	5	2	no	—
257	C₂²×C₃²	—	10	2	no	—
258	C₂²×C₃²	—	10	2	no	—
259	C₂²×C₃²	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
260	C₃²×C₄	—	5	1	no	—
261	C₃²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
262	C₃×Dic₃	—	15	2	no	—
263	C₃×Dic₃	—	30	1	no	—
264	C₃×Dic₃	—	30	2	no	—
265	C₂²×C₉	—	2	2	no	—
266	C₂²×C₉	—	5	1	no	—
267	C₂²×C₉	—	5	2	no	—
268	C₂²×C₉	—	10	1	no	—
269	C₂²×C₉	—	10	2	no	—
270	C₂²×C₉	—	10	2	no	—
271	C₂²×C₉	—	10	2	no	—
272	C₂²×C₉	—	20	2	no	—
273	C₂²×C₉	—	20	2	no	—
274	C₂²×C₉	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
275	C₄×C₉	—	2	1	no	—
276	C₄×C₉	—	5	1	no	—
277	C₄×C₉	—	10	1	no	—
278	C₄×C₉	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
279	Dic₉	—	5	2	no	—
280	Dic₉	—	10	1	no	—
281	Dic₉	—	10	2	no	—
282	C₄×D₄	—	45	1	no	Sylow
283	C₂×C₃×C₅	—	2	2	no	—
284	C₂×C₃×C₅	—	3	1	no	—
285	C₂×C₃×C₅	—	6	2	no	—
286	C₂×C₃×C₅	C₂²×C₃×Dic₃	1	1	yes	characteristic
287	C₃×D₅	—	2	2	no	—
288	C₃×D₅	—	3	1	no	—
289	C₃×D₅	—	3	1	no	—
290	C₃×D₅	—	6	2	no	—
291	C₃×D₅	C₃×D₃×D₄	1	1	yes	characteristic
292	C₃×D₅	C₃×#237	1	1	yes	characteristic
293	3₋¹⁺²	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
294	C₂³×C₃	—	5	2	no	—
295	C₂³×C₃	—	30	2	no	—
296	C₃×D₄	—	5	2	no	—
297	C₃×D₄	—	5	1	no	—
298	C₃×D₄	—	6	1	no	—
299	C₃×D₄	—	30	1	no	—
300	C₃×D₄	—	30	2	no	—
301	C₃×D₄	C₃×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
302	C₂×C₃×C₄	—	5	1	no	—
303	C₂×C₃×C₄	—	30	1	no	—
304	C₂×C₃×C₄	—	45	1	no	—
305	C₂×C₃×C₄	—	45	1	no	—
306	C₂×C₃×C₄	—	45	2	no	—
307	C₂×C₃×C₄	—	90	4	no	—
308	C₂×Dic₃	—	15	1	no	—
309	C₂×Dic₃	—	15	1	no	—
310	C₂×Dic₃	—	15	2	no	—
311	C₂×Dic₃	—	30	4	no	—
312	C₂²×C₅	C₂×3₋¹⁺²⋊C₄	1	2	yes	complemented
313	C₂×D₅	—	2	2	no	—
314	C₂×D₅	—	2	2	no	—
315	C₂×D₅	C₂×3₋¹⁺²⋊C₄	1	2	yes	complemented
316	C₂×D₅	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
317	AGL(1,𝔽₅)	—	9	2	no	—
318	AGL(1,𝔽₅)	—	18	2	no	—
319	AGL(1,𝔽₅)	—	18	1	no	—
320	C₄×C₅	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic, complemented
321	Dic₅	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic, complemented
322	C₂×C₃²	—	2	2	no	—
323	C₂×C₃²	—	5	1	no	—
324	C₂×C₃²	—	5	1	no	—
325	C₂×C₃²	—	10	2	no	—
326	C₂×C₃²	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
327	C₂×C₉	—	2	2	no	—
328	C₂×C₉	—	2	1	no	—
329	C₂×C₉	—	4	2	no	—
330	C₂×C₉	—	5	1	no	—
331	C₂×C₉	—	5	1	no	—
332	C₂×C₉	—	10	1	no	—
333	C₂×C₉	—	10	1	no	—
334	C₂×C₉	—	10	2	no	—
335	C₂×C₉	—	20	2	no	—
336	C₂×C₉	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
337	C₂×D₄	—	5	1	no	—
338	C₂²×C₄	—	45	2	no	—
339	C₄⋊C₄	—	45	1	no	—
340	C₂²⋊C₄	—	45	2	no	—
341	C₄²	—	45	1	no	—
342	C₃×C₅	—	3	1	no	—
343	C₃×C₅	C₃×D₄×Dic₃	1	1	yes	characteristic
344	C₂²×C₃	—	5	1	no	—
345	C₂²×C₃	—	5	2	no	—
346	C₂²×C₃	—	6	2	no	—
347	C₂²×C₃	—	10	2	no	—
348	C₂²×C₃	—	10	2	no	—
349	C₂²×C₃	—	30	1	no	—
350	C₂²×C₃	—	30	2	no	—
351	C₂²×C₃	—	60	2	no	—
352	C₂²×C₃	—	60	2	no	—
353	C₂²×C₃	C₂×C₃×ΓL(1,𝔽₁₆)	1	2	yes	—
354	C₃×C₄	—	5	1	no	—
355	C₃×C₄	—	6	1	no	—
356	C₃×C₄	—	30	1	no	—
357	C₃×C₄	—	45	2	no	—
358	C₃×C₄	—	90	2	no	—
359	C₃×C₄	—	90	1	no	—
360	C₃×C₄	C₂×C₃×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
361	Dic₃	—	15	2	no	—
362	Dic₃	—	30	2	no	—
363	Dic₃	—	30	1	no	—
364	C₂×C₅	—	2	2	no	—
365	C₂×C₅	C₂²×3₋¹⁺²⋊C₄	1	1	yes	characteristic
366	D₅	—	2	2	no	—
367	D₅	D₄×AGL(1,ℤ/9)	1	1	yes	characteristic
368	D₅	#3699⋊C₃	1	1	yes	characteristic
369	C₃²	D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
370	C₉	—	2	1	no	—
371	C₉	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
372	C₂³	—	5	2	no	—
373	D₄	—	5	2	no	—
374	D₄	—	5	1	no	—
375	D₄	3₋¹⁺²⋊AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
376	C₂×C₄	—	5	1	no	—
377	C₂×C₄	—	45	1	no	—
378	C₂×C₄	—	45	1	no	—
379	C₂×C₄	—	45	2	no	—
380	C₂×C₄	—	90	4	no	—
381	C₂×C₃	—	2	2	no	—
382	C₂×C₃	—	5	1	no	—
383	C₂×C₃	—	5	1	no	—
384	C₂×C₃	—	6	1	no	—
385	C₂×C₃	—	10	2	no	—
386	C₂×C₃	—	12	2	no	—
387	C₂×C₃	—	30	1	no	—
388	C₂×C₃	—	30	1	no	—
389	C₂×C₃	—	60	2	no	—
390	C₂×C₃	C₂²×C₃×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
391	C₅	D₄×3₋¹⁺²⋊C₄	1	1	yes	characteristic, complemented, Sylow
392	C₂²	—	5	1	no	—
393	C₂²	—	5	2	no	—
394	C₂²	—	10	2	no	—
395	C₂²	—	10	2	no	—
396	C₂²	C₂×3₋¹⁺²⋊AGL(1,𝔽₅)	1	2	yes	complemented
397	C₄	—	5	1	no	—
398	C₄	—	45	2	no	—
399	C₄	—	90	2	no	—
400	C₄	—	90	1	no	—
401	C₄	C₂×3₋¹⁺²⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
402	C₃	—	6	1	no	—
403	C₃	C₃×D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
404	C₂	—	2	2	no	—
405	C₂	—	5	1	no	—
406	C₂	—	5	1	no	—
407	C₂	—	10	2	no	—
408	C₂	C₂²×3₋¹⁺²⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×3₋¹⁺²⋊AGL(1,𝔽₅)
commutator	C₂×C₅×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃	C₂²×C₃×ΓL(1,𝔽₁₆)
Fitting	C₅×D₄×3₋¹⁺²	C₄
socle	C₂×C₃×C₅	C₂²×C₃×Dic₃
prime radical	C₃²×D₄×D₅	D₃
cosocle	C₂×C₉×D₅	C₂³×C₃
hypercenter	D₄	3₋¹⁺²⋊AGL(1,𝔽₅)
hypocenter	C₅×C₉	C₃×C₄×D₄
LCS	G ▷ C₂×C₅×C₉ ▷ C₅×C₉
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. #72247404	C₂	semidirect
2. #72247390	C₂	semidirect
4. C₄×3₋¹⁺²⋊AGL(1,𝔽₅)	C₂	semidirect
5. C₂²×3₋¹⁺²⋊AGL(1,𝔽₅)	C₂	semidirect
7. D₄×C₉⋊AGL(1,𝔽₅)	C₃	semidirect
13. C₅×D₄×3₋¹⁺²	C₄	semidirect
14. 3₋¹⁺²×C₂²⋊D₅	C₄	semidirect
15. 3₋¹⁺²×D₂₀	C₄	semidirect
23. C₂²×C₉⋊AGL(1,𝔽₅)	C₂×C₃	semidirect
26. #10623655	C₂×C₃	semidirect
27. C₄⋊(C₉⋊AGL(1,𝔽₅))	C₂×C₃	semidirect
28. C₄×C₉⋊AGL(1,𝔽₅)	C₂×C₃	semidirect
29. C₂²×C₅×3₋¹⁺²	C₂×C₄	semidirect
32. C₂×D₅×3₋¹⁺²	C₂×C₄	semidirect
36. 3₋¹⁺²⋊AGL(1,𝔽₅)	D₄	direct (N × H)
37. C₄×C₅×3₋¹⁺²	C₂×C₄	semidirect
38. Dic₅×3₋¹⁺²	C₂×C₄	semidirect
62. C₅×D₄×C₉	C₃×C₄	semidirect
64. C₉×C₂²⋊D₅	C₃×C₄	semidirect
66. C₉×D₂₀	C₃×C₄	semidirect
89. D₄×3₋¹⁺²	AGL(1,𝔽₅)	semidirect
106. C₂²×C₅×C₉	C₂×C₃×C₄	semidirect
113. C₂×C₉×D₅	C₂×C₃×C₄	semidirect
117. C₉⋊AGL(1,𝔽₅)	C₃×D₄	semidirect
119. C₄×C₅×C₉	C₂×C₃×C₄	semidirect
121. C₉×Dic₅	C₂×C₃×C₄	semidirect
134. C₅×3₋¹⁺²	C₄×D₄	semidirect
157. C₂²×3₋¹⁺²	C₂×AGL(1,𝔽₅)	semidirect
159. C₄×3₋¹⁺²	C₂×AGL(1,𝔽₅)	semidirect
201. D₄×C₉	C₃×AGL(1,𝔽₅)	semidirect
245. C₅×C₉	C₃×C₄×D₄	semidirect
251. C₅×D₄	3₋¹⁺²⋊C₄	semidirect
252. C₂²⋊D₅	3₋¹⁺²⋊C₄	semidirect
253. D₂₀	3₋¹⁺²⋊C₄	semidirect
274. C₂²×C₉	C₂×C₃×AGL(1,𝔽₅)	semidirect
278. C₄×C₉	C₂×C₃×AGL(1,𝔽₅)	semidirect
293. 3₋¹⁺²	D₄×AGL(1,𝔽₅)	semidirect
312. C₂²×C₅	C₂×3₋¹⁺²⋊C₄	semidirect
315. C₂×D₅	C₂×3₋¹⁺²⋊C₄	semidirect
320. C₄×C₅	C₂×3₋¹⁺²⋊C₄	semidirect
321. Dic₅	C₂×3₋¹⁺²⋊C₄	semidirect
371. C₉	C₃×D₄×AGL(1,𝔽₅)	semidirect
375. D₄	3₋¹⁺²⋊AGL(1,𝔽₅)	direct (N × H)
391. C₅	D₄×3₋¹⁺²⋊C₄	semidirect
396. C₂²	C₂×3₋¹⁺²⋊AGL(1,𝔽₅)	semidirect
401. C₄	C₂×3₋¹⁺²⋊AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
#72247404	2
#72247390	2
D₄×D₅×3₋¹⁺²	2
C₄×3₋¹⁺²⋊AGL(1,𝔽₅)	2
C₂²×3₋¹⁺²⋊AGL(1,𝔽₅)	2
C₃×D₄×ΓL(1,𝔽₁₆)	3
D₄×C₉⋊AGL(1,𝔽₅)	3
D₄×3₋¹⁺²⋊C₄	5

Subgroups — D4×3−1+2⋊AGL(1,𝔽5) #72247442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×3₋¹⁺²⋊AGL(1,𝔽₅) #72247442