Subgroups — C₂²×C₅×#332 #11940414

Order 1280.

1334 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#332	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₅×Q₈	—	2	1	no	—
6	C₂³×C₅×C₈	—	2	1	no	—
7	C₂³×C₅×C₈	—	2	1	no	—
8	C₂×C₅×C₄⋊Q₈	—	2	3	no	—
9	C₂×C₅×C₄⋊Q₈	—	2	3	no	—
10	C₂×C₅×C₄⋊Q₈	C₄	1	6	yes	—
11	C₂×C₅×C₄⋊Q₈	C₂²	1	6	yes	—
12	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
13	C₂²×C₅×C₄⋊C₄	C₄	1	2	yes	—
14	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₄⋊C₈	—	2	3	no	—
16	C₂×C₅×C₄⋊C₈	—	2	3	no	—
17	C₂×C₅×C₄⋊C₈	C₂²	1	6	yes	—
18	C₂×C₅×C₄⋊C₈	C₂²	1	6	yes	—
19	C₅×#332	C₂²	1	16	yes	direct product factor
20	C₂²×#332	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	—	2	6	no	—
22	C₂²×C₅×Q₈	—	2	6	no	—
23	C₂²×C₅×Q₈	—	4	4	no	—
24	C₂²×C₅×Q₈	—	4	4	no	—
25	C₂³×C₄×C₅	—	2	2	no	—
26	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	—	2	6	no	—
30	C₂²×C₅×C₈	—	2	6	no	—
31	C₂²×C₅×C₈	—	4	4	no	—
32	C₂²×C₅×C₈	—	4	4	no	—
33	C₅×C₄⋊Q₈	—	2	12	no	—
34	C₅×C₄⋊Q₈	—	2	6	no	—
35	C₅×C₄⋊Q₈	—	2	6	no	—
36	C₅×C₄⋊Q₈	C₂×C₄	1	12	yes	—
37	C₅×C₄⋊Q₈	C₂³	1	4	yes	—
38	C₂×C₅×C₄⋊C₄	—	2	6	no	—
39	C₂×C₅×C₄⋊C₄	—	2	6	no	—
40	C₂×C₅×C₄⋊C₄	—	2	6	no	—
41	C₂×C₅×C₄⋊C₄	C₂×C₄	1	12	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
43	C₂×C₄²×C₅	D₄	1	3	yes	—
44	C₂×C₄²×C₅	D₄	1	3	yes	—
45	C₂×C₄²×C₅	C₂³	1	3	yes	—
46	C₅×C₄⋊C₈	—	2	6	no	—
47	C₅×C₄⋊C₈	—	2	6	no	—
48	C₅×C₄⋊C₈	C₂³	1	4	yes	—
49	C₅×C₄⋊C₈	C₂³	1	4	yes	—
50	C₂²×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
51	C₂²×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
52	C₂²×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
53	C₂×#332	C₂×C₅	1	12	yes	direct product factor
54	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₅×Q₈	—	2	12	no	—
56	C₂×C₅×Q₈	—	2	4	no	—
57	C₂×C₅×Q₈	—	4	24	no	—
58	C₂×C₅×Q₈	—	4	12	no	—
59	C₂×C₅×Q₈	—	4	12	no	—
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	—	2	12	no	—
62	C₂²×C₄×C₅	—	4	8	no	—
63	C₂²×C₄×C₅	C₂²⋊C₄	1	3	yes	—
64	C₂²×C₄×C₅	C₂²⋊C₄	1	3	yes	—
65	C₂²×C₄×C₅	C₂²⋊C₄	1	3	yes	—
66	C₂²×C₄×C₅	D₈	1	4	yes	—
67	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
68	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
69	C₂²×C₄×C₅	Q₁₆	1	4	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
71	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
73	C₂×C₅×C₈	—	2	4	no	—
74	C₂×C₅×C₈	—	2	4	no	—
75	C₂×C₅×C₈	—	4	12	no	—
76	C₂×C₅×C₈	—	4	12	no	—
77	C₅×C₄⋊C₄	—	2	12	no	—
78	C₅×C₄⋊C₄	—	2	12	no	—
79	C₅×C₄⋊C₄	—	2	4	no	—
80	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
81	C₄²×C₅	C₂²⋊C₄	1	6	yes	—
82	C₄²×C₅	C₂²×C₄	1	3	yes	—
83	C₄²×C₅	C₂×D₄	1	3	yes	—
84	C₄²×C₅	C₂×D₄	1	3	yes	—
85	C₄²×C₅	C₂⁴	1	1	yes	characteristic
86	C₂³×Q₈	—	2	1	no	—
87	C₂³×C₈	—	2	1	no	—
88	C₂³×C₈	—	2	1	no	—
89	C₂×C₄⋊Q₈	—	2	3	no	—
90	C₂×C₄⋊Q₈	—	2	3	no	—
91	C₂×C₄⋊Q₈	C₄×C₅	1	6	yes	—
92	C₂×C₄⋊Q₈	C₂²×C₅	1	6	yes	—
93	C₂²×C₄⋊C₄	—	2	1	no	—
94	C₂²×C₄⋊C₄	C₄×C₅	1	2	yes	—
95	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₄⋊C₈	—	2	3	no	—
97	C₂×C₄⋊C₈	—	2	3	no	—
98	C₂×C₄⋊C₈	C₂²×C₅	1	6	yes	—
99	C₂×C₄⋊C₈	C₂²×C₅	1	6	yes	—
100	#332	C₂²×C₅	1	16	yes	direct product factor
101	C₂³×C₅	#101	1	4	yes	—
102	C₂³×C₅	#102	1	4	yes	—
103	C₂³×C₅	#103	1	4	yes	—
104	C₂³×C₅	C₂×C₂²⋊C₄	1	3	yes	—
105	C₅×Q₈	—	4	4	no	—
106	C₅×Q₈	—	4	4	no	—
107	C₅×Q₈	—	4	24	no	—
108	C₂×C₄×C₅	—	2	6	no	—
109	C₂×C₄×C₅	—	2	6	no	—
110	C₂×C₄×C₅	—	2	8	no	—
111	C₂×C₄×C₅	—	4	24	no	—
112	C₂×C₄×C₅	#102	1	12	yes	—
113	C₂×C₄×C₅	#103	1	12	yes	—
114	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
115	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
116	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
117	C₂×C₄×C₅	C₂×D₈	1	6	yes	—
118	C₂×C₄×C₅	C₂×SD₁₆	1	6	yes	—
119	C₂×C₄×C₅	C₂×SD₁₆	1	6	yes	—
120	C₂×C₄×C₅	C₂×Q₁₆	1	6	yes	—
121	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
124	C₅×C₈	—	4	4	no	—
125	C₅×C₈	—	4	4	no	—
126	C₂²×Q₈	—	2	6	no	—
127	C₂²×Q₈	—	2	6	no	—
128	C₂²×Q₈	—	4	4	no	—
129	C₂²×Q₈	—	4	4	no	—
130	C₂³×C₄	—	2	2	no	—
131	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
132	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
133	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
134	C₂²×C₈	—	2	6	no	—
135	C₂²×C₈	—	2	6	no	—
136	C₂²×C₈	—	4	4	no	—
137	C₂²×C₈	—	4	4	no	—
138	C₄⋊Q₈	—	2	12	no	—
139	C₄⋊Q₈	—	2	6	no	—
140	C₄⋊Q₈	—	2	6	no	—
141	C₄⋊Q₈	C₂×C₄×C₅	1	12	yes	—
142	C₄⋊Q₈	C₂³×C₅	1	4	yes	—
143	C₂×C₄⋊C₄	—	2	6	no	—
144	C₂×C₄⋊C₄	—	2	6	no	—
145	C₂×C₄⋊C₄	—	2	6	no	—
146	C₂×C₄⋊C₄	C₂×C₄×C₅	1	12	yes	—
147	C₂×C₄²	C₂×C₄×C₅	1	3	yes	—
148	C₂×C₄²	C₅×D₄	1	3	yes	—
149	C₂×C₄²	C₅×D₄	1	3	yes	—
150	C₂×C₄²	C₂³×C₅	1	3	yes	—
151	C₄⋊C₈	—	2	6	no	—
152	C₄⋊C₈	—	2	6	no	—
153	C₄⋊C₈	C₂³×C₅	1	4	yes	—
154	C₄⋊C₈	C₂³×C₅	1	4	yes	—
155	C₂²×C₅	#332	1	16	yes	direct product factor
156	C₂²×C₅	C₂×#101	1	6	yes	—
157	C₂²×C₅	C₂×#102	1	6	yes	—
158	C₂²×C₅	C₂×#103	1	6	yes	—
159	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
160	C₄×C₅	—	2	3	no	—
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	4	8	no	—
163	C₄×C₅	C₂×#102	1	6	yes	—
164	C₄×C₅	C₂×#103	1	6	yes	—
165	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
166	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
167	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
168	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
169	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×Q₈	—	2	12	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	4	12	no	—
173	C₂×Q₈	—	4	12	no	—
174	C₂×Q₈	—	4	24	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	12	no	—
177	C₂²×C₄	—	4	8	no	—
178	C₂²×C₄	C₅×C₂²⋊C₄	1	3	yes	—
179	C₂²×C₄	C₅×C₂²⋊C₄	1	3	yes	—
180	C₂²×C₄	C₅×C₂²⋊C₄	1	3	yes	—
181	C₂²×C₄	C₅×D₈	1	4	yes	—
182	C₂²×C₄	C₅×SD₁₆	1	4	yes	—
183	C₂²×C₄	C₅×SD₁₆	1	4	yes	—
184	C₂²×C₄	C₅×Q₁₆	1	4	yes	—
185	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
186	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
187	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	4	12	no	—
191	C₂×C₈	—	4	12	no	—
192	C₄⋊C₄	—	2	12	no	—
193	C₄⋊C₄	—	2	12	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
196	C₄²	C₅×C₂²⋊C₄	1	6	yes	—
197	C₄²	C₂²×C₄×C₅	1	3	yes	—
198	C₄²	C₂×C₅×D₄	1	3	yes	—
199	C₄²	C₂×C₅×D₄	1	3	yes	—
200	C₄²	C₂⁴×C₅	1	1	yes	characteristic
201	C₂×C₅	C₂×#332	1	12	yes	direct product factor
202	C₂×C₅	C₂²×#101	1	1	yes	characteristic
203	C₂×C₅	C₂²×#102	1	1	yes	characteristic
204	C₂×C₅	C₂²×#103	1	1	yes	characteristic
205	C₂³	C₅×#101	1	4	yes	—
206	C₂³	C₅×#102	1	4	yes	—
207	C₂³	C₅×#103	1	4	yes	—
208	C₂³	C₂×C₅×C₂²⋊C₄	1	3	yes	—
209	Q₈	—	4	24	no	—
210	Q₈	—	4	4	no	—
211	Q₈	—	4	4	no	—
212	C₂×C₄	—	2	6	no	—
213	C₂×C₄	—	2	6	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	4	24	no	—
216	C₂×C₄	C₅×#102	1	12	yes	—
217	C₂×C₄	C₅×#103	1	12	yes	—
218	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
219	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
220	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
221	C₂×C₄	C₂×C₅×D₈	1	6	yes	—
222	C₂×C₄	C₂×C₅×SD₁₆	1	6	yes	—
223	C₂×C₄	C₂×C₅×SD₁₆	1	6	yes	—
224	C₂×C₄	C₂×C₅×Q₁₆	1	6	yes	—
225	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
226	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
228	C₈	—	4	4	no	—
229	C₈	—	4	4	no	—
230	C₅	C₂²×#332	1	1	yes	characteristic, direct product factor, Sylow
231	C₂²	C₅×#332	1	16	yes	direct product factor
232	C₂²	C₂×C₅×#101	1	6	yes	—
233	C₂²	C₂×C₅×#102	1	6	yes	—
234	C₂²	C₂×C₅×#103	1	6	yes	—
235	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
236	C₄	—	2	3	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	8	no	—
239	C₄	C₂×C₅×#102	1	6	yes	—
240	C₄	C₂×C₅×#103	1	6	yes	—
241	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
242	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
243	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
244	C₄	C₂²×C₅×Q₁₆	1	1	yes	characteristic
245	C₂	C₂×C₅×#332	1	12	yes	direct product factor
246	C₂	C₂²×C₅×#101	1	1	yes	characteristic
247	C₂	C₂²×C₅×#102	1	1	yes	characteristic
248	C₂	C₂²×C₅×#103	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#332	C₂	direct (N × H)
19. C₅×#332	C₂²	direct (N × H)
20. C₂²×#332	C₅	direct (N × H)
53. C₂×#332	C₂×C₅	direct (N × H)
100. #332	C₂²×C₅	direct (N × H)
155. C₂²×C₅	#332	direct (N × H)
201. C₂×C₅	C₂×#332	direct (N × H)
230. C₅	C₂²×#332	direct (N × H)
231. C₂²	C₅×#332	direct (N × H)
245. C₂	C₂×C₅×#332	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₄⋊Q₈	2
C₂²×C₅×C₄⋊C₈	2
C₂×C₅×#332	2
C₂²×#332	5

Subgroups — C22×C5×#332 #11940414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#332 #11940414