Subgroups — C₂²×C₅×#362 #11956110

Order 1280.

1134 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#362	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₅×Q₈	—	2	1	no	—
6	C₂²×C₅×Q₁₆	—	2	3	no	—
7	C₂²×C₅×Q₁₆	—	2	1	no	—
8	C₂²×C₅×Q₁₆	—	2	3	no	—
9	C₂²×C₅×Q₁₆	C₄	1	6	yes	—
10	C₂²×C₅×Q₁₆	C₄	1	2	yes	—
11	C₂²×C₅×Q₁₆	C₂²	1	6	yes	—
12	C₂²×C₅×M₁₆	—	2	1	no	—
13	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×M₃₂	—	2	3	no	—
15	C₂×C₅×M₃₂	C₂²	1	6	yes	—
16	C₂²×C₅×C₁₆	—	2	1	no	—
17	C₂×C₅×#108	—	2	3	no	—
18	C₂×C₅×#108	C₂²	1	6	yes	—
19	C₅×#362	C₂²	1	16	yes	direct product factor
20	C₂²×#362	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	—	2	6	no	—
22	C₂²×C₅×Q₈	—	2	2	no	—
23	C₂²×C₅×Q₈	—	2	6	no	—
24	C₂²×C₅×Q₈	—	4	3	no	—
25	C₂²×C₅×Q₈	—	4	1	no	—
26	C₂²×C₅×Q₈	—	4	3	no	—
27	C₂³×C₄×C₅	—	4	1	no	—
28	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×C₅×Q₁₆	—	2	6	no	—
30	C₂×C₅×Q₁₆	—	2	12	no	—
31	C₂×C₅×Q₁₆	—	2	6	no	—
32	C₂×C₅×Q₁₆	—	2	6	no	—
33	C₂×C₅×Q₁₆	—	2	6	no	—
34	C₂×C₅×Q₁₆	—	2	6	no	—
35	C₂×C₅×Q₁₆	C₂×C₄	1	12	yes	—
36	C₂×C₅×Q₁₆	C₂×C₄	1	12	yes	—
37	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
38	C₂×C₅×M₁₆	—	2	6	no	—
39	C₂×C₅×M₁₆	—	4	3	no	—
40	C₂²×C₅×C₈	—	4	1	no	—
41	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
42	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	D₄	1	3	yes	—
44	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
45	C₂²×C₅×C₈	D₄	1	3	yes	—
46	C₂²×C₅×C₈	C₂³	1	3	yes	—
47	C₅×M₃₂	—	2	6	no	—
48	C₅×M₃₂	C₂³	1	4	yes	—
49	C₂×C₅×C₁₆	—	2	6	no	—
50	C₅×#108	—	2	6	no	—
51	C₅×#108	C₂³	1	4	yes	—
52	C₂³×Q₁₆	C₂×C₅	1	1	yes	characteristic
53	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic
54	C₂²×#108	C₂×C₅	1	1	yes	characteristic
55	C₂×#362	C₂×C₅	1	12	yes	direct product factor
56	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
57	C₂×C₅×Q₈	—	2	12	no	—
58	C₂×C₅×Q₈	—	2	12	no	—
59	C₂×C₅×Q₈	—	2	4	no	—
60	C₂×C₅×Q₈	—	4	12	no	—
61	C₂×C₅×Q₈	—	4	6	no	—
62	C₂×C₅×Q₈	—	4	6	no	—
63	C₂×C₅×Q₈	—	4	6	no	—
64	C₂×C₅×Q₈	—	4	6	no	—
65	C₂×C₅×Q₈	—	4	6	no	—
66	C₂²×C₄×C₅	—	4	6	no	—
67	C₂²×C₄×C₅	—	4	2	no	—
68	C₂²×C₄×C₅	—	4	6	no	—
69	C₂²×C₄×C₅	C₂²⋊C₄	1	3	yes	—
70	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	D₈	1	3	yes	—
72	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	SD₁₆	1	3	yes	—
74	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
75	C₅×Q₁₆	—	2	12	no	—
76	C₅×Q₁₆	—	2	12	no	—
77	C₅×Q₁₆	—	2	4	no	—
78	C₅×Q₁₆	C₂²×C₄	1	8	yes	—
79	C₅×M₁₆	—	2	4	no	—
80	C₅×M₁₆	—	4	6	no	—
81	C₂×C₅×C₈	—	4	6	no	—
82	C₂×C₅×C₈	C₂²⋊C₄	1	3	yes	—
83	C₂×C₅×C₈	C₂²⋊C₄	1	6	yes	—
84	C₂×C₅×C₈	C₂²⋊C₄	1	3	yes	—
85	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
86	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
87	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
88	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
90	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
91	C₅×C₁₆	—	2	4	no	—
92	C₂³×Q₈	—	2	1	no	—
93	C₂²×Q₁₆	—	2	3	no	—
94	C₂²×Q₁₆	—	2	1	no	—
95	C₂²×Q₁₆	—	2	3	no	—
96	C₂²×Q₁₆	C₄×C₅	1	6	yes	—
97	C₂²×Q₁₆	C₄×C₅	1	2	yes	—
98	C₂²×Q₁₆	C₂²×C₅	1	6	yes	—
99	C₂²×M₁₆	—	2	1	no	—
100	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
101	C₂×M₃₂	—	2	3	no	—
102	C₂×M₃₂	C₂²×C₅	1	6	yes	—
103	C₂²×C₁₆	—	2	1	no	—
104	C₂×#108	—	2	3	no	—
105	C₂×#108	C₂²×C₅	1	6	yes	—
106	#362	C₂²×C₅	1	16	yes	direct product factor
107	C₂³×C₅	—	2	4	no	—
108	C₂³×C₅	#102	1	3	yes	—
109	C₂³×C₅	#102	1	1	yes	characteristic
110	C₂³×C₅	C₂×SD₁₆	1	3	yes	—
111	C₅×Q₈	—	2	8	no	—
112	C₅×Q₈	—	4	4	no	—
113	C₅×Q₈	—	4	12	no	—
114	C₅×Q₈	—	4	12	no	—
115	C₂×C₄×C₅	—	4	4	no	—
116	C₂×C₄×C₅	—	4	12	no	—
117	C₂×C₄×C₅	—	4	12	no	—
118	C₂×C₄×C₅	#102	1	6	yes	—
119	C₂×C₄×C₅	#102	1	3	yes	—
120	C₂×C₄×C₅	#102	1	3	yes	—
121	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
122	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
123	C₂×C₄×C₅	C₂×D₈	1	3	yes	—
124	C₂×C₄×C₅	C₂×D₈	1	3	yes	—
125	C₂×C₄×C₅	C₂×SD₁₆	1	3	yes	—
126	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₅×C₈	—	4	4	no	—
128	C₅×C₈	C₂×C₂²⋊C₄	1	3	yes	—
129	C₅×C₈	C₂×C₂²⋊C₄	1	3	yes	—
130	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
131	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
132	C₂²×Q₈	—	2	6	no	—
133	C₂²×Q₈	—	2	2	no	—
134	C₂²×Q₈	—	2	6	no	—
135	C₂²×Q₈	—	4	1	no	—
136	C₂²×Q₈	—	4	3	no	—
137	C₂²×Q₈	—	4	3	no	—
138	C₂³×C₄	—	4	1	no	—
139	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
140	C₂×Q₁₆	—	2	6	no	—
141	C₂×Q₁₆	—	2	12	no	—
142	C₂×Q₁₆	—	2	6	no	—
143	C₂×Q₁₆	—	2	6	no	—
144	C₂×Q₁₆	—	2	6	no	—
145	C₂×Q₁₆	—	2	6	no	—
146	C₂×Q₁₆	C₂×C₄×C₅	1	12	yes	—
147	C₂×Q₁₆	C₂×C₄×C₅	1	12	yes	—
148	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
149	C₂×M₁₆	—	2	6	no	—
150	C₂×M₁₆	—	4	3	no	—
151	C₂²×C₈	—	4	1	no	—
152	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
153	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
154	C₂²×C₈	C₅×D₄	1	3	yes	—
155	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
156	C₂²×C₈	C₅×D₄	1	3	yes	—
157	C₂²×C₈	C₂³×C₅	1	3	yes	—
158	M₃₂	—	2	6	no	—
159	M₃₂	C₂³×C₅	1	4	yes	—
160	C₂×C₁₆	—	2	6	no	—
161	#108	—	2	6	no	—
162	#108	C₂³×C₅	1	4	yes	—
163	C₂²×C₅	—	2	6	no	—
164	C₂²×C₅	—	2	6	no	—
165	C₂²×C₅	#362	1	4	yes	direct product factor
166	C₂²×C₅	C₂×#102	1	3	yes	—
167	C₂²×C₅	C₂×#102	1	3	yes	—
168	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
169	C₄×C₅	—	4	8	no	—
170	C₄×C₅	C₂×#102	1	3	yes	—
171	C₄×C₅	C₂×#102	1	3	yes	—
172	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
173	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
174	C₂⁴	C₅×SD₁₆	1	1	yes	characteristic
175	C₂×Q₈	—	2	12	no	—
176	C₂×Q₈	—	2	12	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	4	6	no	—
179	C₂×Q₈	—	4	6	no	—
180	C₂×Q₈	—	4	6	no	—
181	C₂×Q₈	—	4	12	no	—
182	C₂×Q₈	—	4	6	no	—
183	C₂×Q₈	—	4	6	no	—
184	C₂²×C₄	—	4	6	no	—
185	C₂²×C₄	—	4	6	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	C₅×C₂²⋊C₄	1	3	yes	—
188	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
190	C₂²×C₄	C₅×D₈	1	3	yes	—
191	C₂²×C₄	C₅×SD₁₆	1	3	yes	—
192	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
193	Q₁₆	—	2	12	no	—
194	Q₁₆	—	2	12	no	—
195	Q₁₆	—	2	4	no	—
196	Q₁₆	C₂²×C₄×C₅	1	8	yes	—
197	M₁₆	—	2	4	no	—
198	M₁₆	—	4	6	no	—
199	C₂×C₈	—	4	6	no	—
200	C₂×C₈	C₅×C₂²⋊C₄	1	3	yes	—
201	C₂×C₈	C₅×C₂²⋊C₄	1	6	yes	—
202	C₂×C₈	C₅×C₂²⋊C₄	1	3	yes	—
203	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
204	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
205	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
206	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
207	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
208	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
209	C₁₆	—	2	4	no	—
210	C₂×C₅	—	2	1	no	—
211	C₂×C₅	—	2	3	no	—
212	C₂×C₅	C₂×#362	1	6	yes	direct product factor
213	C₂×C₅	C₂²×#102	1	1	yes	characteristic
214	C₂³	—	2	4	no	—
215	C₂³	C₅×#102	1	3	yes	—
216	C₂³	C₅×#102	1	1	yes	characteristic
217	C₂³	C₂×C₅×SD₁₆	1	3	yes	—
218	Q₈	—	2	8	no	—
219	Q₈	—	4	12	no	—
220	Q₈	—	4	12	no	—
221	Q₈	—	4	4	no	—
222	C₂×C₄	—	4	12	no	—
223	C₂×C₄	—	4	12	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	C₅×#102	1	6	yes	—
226	C₂×C₄	C₅×#102	1	3	yes	—
227	C₂×C₄	C₅×#102	1	3	yes	—
228	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
229	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
230	C₂×C₄	C₂×C₅×D₈	1	3	yes	—
231	C₂×C₄	C₂×C₅×D₈	1	3	yes	—
232	C₂×C₄	C₂×C₅×SD₁₆	1	3	yes	—
233	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
234	C₈	—	4	4	no	—
235	C₈	C₂×C₅×C₂²⋊C₄	1	3	yes	—
236	C₈	C₂×C₅×C₂²⋊C₄	1	3	yes	—
237	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
238	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
239	C₅	C₂²×#362	1	1	yes	characteristic, direct product factor, Sylow
240	C₂²	—	2	6	no	—
241	C₂²	—	2	6	no	—
242	C₂²	C₅×#362	1	4	yes	direct product factor
243	C₂²	C₂×C₅×#102	1	3	yes	—
244	C₂²	C₂×C₅×#102	1	3	yes	—
245	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
246	C₄	—	4	8	no	—
247	C₄	C₂×C₅×#102	1	3	yes	—
248	C₄	C₂×C₅×#102	1	3	yes	—
249	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
250	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
251	C₂	—	2	3	no	—
252	C₂	—	2	1	no	—
253	C₂	C₂×C₅×#362	1	6	yes	direct product factor
254	C₂	C₂²×C₅×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	#102
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂⁴×C₅	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#362	C₂	direct (N × H)
19. C₅×#362	C₂²	direct (N × H)
20. C₂²×#362	C₅	direct (N × H)
55. C₂×#362	C₂×C₅	direct (N × H)
106. #362	C₂²×C₅	direct (N × H)
165. C₂²×C₅	#362	direct (N × H)
212. C₂×C₅	C₂×#362	direct (N × H)
239. C₅	C₂²×#362	direct (N × H)
242. C₂²	C₅×#362	direct (N × H)
253. C₂	C₂×C₅×#362	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×Q₁₆	2
C₂²×C₅×M₃₂	2
C₂²×C₅×#108	2
C₂×C₅×#362	2
C₂²×#362	5

Subgroups — C22×C5×#362 #11956110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#362 #11956110