Subgroups — C₅×#7638 #11930358

Order 1280.

468 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1745	C₂	1	1	yes	maximal, characteristic
3	C₅×#1745	C₂	1	1	yes	maximal, characteristic
4	C₅×#1271	C₂	1	4	yes	maximal
5	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	—	2	1	no	—
7	C₂×C₅×#97	—	2	1	no	—
8	C₂×C₅×#97	—	2	1	no	—
9	C₅×#391	—	2	1	no	—
10	C₅×#389	—	2	1	no	—
11	C₅×#389	C₂²	1	2	yes	—
12	C₅×#384	C₄	1	1	yes	characteristic
13	C₄×C₅×C₄⋊C₄	—	2	1	no	—
14	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
15	C₄³×C₅	C₂²	1	1	yes	characteristic
16	C₅×#336	—	2	1	no	—
17	C₅×#336	—	2	1	no	—
18	C₅×#336	C₂²	1	2	yes	—
19	C₅×#336	C₂²	1	2	yes	—
20	#7638	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	—	2	2	no	—
24	C₅×C₄⋊Q₈	—	2	4	no	—
25	C₄×C₅×Q₈	—	4	4	no	—
26	C₂×C₅×C₄⋊C₄	—	2	1	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	1	no	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	—	2	2	no	—
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₅×#97	—	2	4	no	—
41	C₅×#97	—	2	4	no	—
42	C₅×#95	—	2	2	no	—
43	C₅×#95	—	2	2	no	—
44	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
45	#1745	C₂×C₅	1	1	yes	characteristic
46	#1745	C₂×C₅	1	1	yes	characteristic
47	#1271	C₂×C₅	1	4	yes	—
48	C₂×C₅×Q₈	—	2	4	no	—
49	C₂×C₅×Q₈	—	4	2	no	—
50	C₂×C₅×Q₈	—	4	2	no	—
51	C₂²×C₄×C₅	—	2	2	no	—
52	C₂²×C₄×C₅	—	2	2	no	—
53	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₅×C₈	—	2	4	no	—
61	C₂×C₅×C₈	—	2	4	no	—
62	C₂×C₅×C₈	—	4	2	no	—
63	C₂×C₅×C₈	—	4	2	no	—
64	C₂×C₅×C₈	—	4	2	no	—
65	C₂×C₅×C₈	—	4	2	no	—
66	C₅×C₄⋊C₄	—	2	4	no	—
67	C₅×C₄⋊C₄	—	2	2	no	—
68	C₅×C₄⋊C₄	—	4	2	no	—
69	C₅×C₄⋊C₄	—	4	2	no	—
70	C₅×C₄⋊C₄	—	4	2	no	—
71	C₅×C₄⋊C₄	—	4	4	no	—
72	C₅×C₄⋊C₄	M₁₆	1	4	yes	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	—	2	2	no	—
75	C₄²×C₅	—	2	2	no	—
76	C₄²×C₅	—	2	2	no	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	—	2	2	no	—
79	C₄²×C₅	—	4	1	no	—
80	C₄²×C₅	—	4	4	no	—
81	C₂×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
82	C₂×C₄×Q₈	—	2	1	no	—
83	C₂×#97	—	2	1	no	—
84	C₂×#97	—	2	1	no	—
85	#391	—	2	1	no	—
86	#389	—	2	1	no	—
87	#389	C₂²×C₅	1	2	yes	—
88	#384	C₄×C₅	1	1	yes	characteristic
89	C₄×C₄⋊C₄	—	2	1	no	—
90	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
91	C₄³	C₂²×C₅	1	1	yes	characteristic
92	#336	—	2	1	no	—
93	#336	—	2	1	no	—
94	#336	C₂²×C₅	1	2	yes	—
95	#336	C₂²×C₅	1	2	yes	—
96	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₅×Q₈	—	4	8	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	2	1	no	—
100	C₂×C₄×C₅	—	2	1	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	#101	1	2	yes	—
120	C₂×C₄×C₅	#104	1	2	yes	—
121	C₂×C₄×C₅	#104	1	2	yes	—
122	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₅	#137	1	2	yes	—
125	C₅×C₈	—	4	4	no	—
126	C₅×C₈	—	4	4	no	—
127	C₂²×Q₈	—	2	1	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	4	no	—
131	C₄×Q₈	—	4	4	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄⋊C₄	—	2	1	no	—
134	C₂×C₄⋊C₄	—	2	1	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
137	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
138	C₂×C₄²	—	2	2	no	—
139	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
141	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
142	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
143	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
144	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
145	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
146	#97	—	2	4	no	—
147	#97	—	2	4	no	—
148	#95	—	2	2	no	—
149	#95	—	2	2	no	—
150	C₂²×C₅	#407	1	1	yes	characteristic
151	C₂²×C₅	#410	1	1	yes	characteristic
152	C₂²×C₅	C₂×#101	1	1	yes	characteristic
153	C₂²×C₅	#418	1	1	yes	characteristic
154	C₂²×C₅	#419	1	1	yes	characteristic
155	C₂²×C₅	C₂×#104	1	1	yes	characteristic
156	C₂²×C₅	C₂×#104	1	1	yes	characteristic
157	C₄×C₅	—	2	2	no	—
158	C₄×C₅	—	2	2	no	—
159	C₄×C₅	—	2	2	no	—
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	2	2	no	—
162	C₄×C₅	—	2	2	no	—
163	C₄×C₅	—	4	4	no	—
164	C₄×C₅	—	4	1	no	—
165	C₄×C₅	—	8	2	no	—
166	C₂×Q₈	—	2	4	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	C₅×M₁₆	1	4	yes	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	4	1	no	—
198	C₄²	—	4	4	no	—
199	C₂×C₅	#1467	1	1	yes	characteristic
200	C₂×C₅	#1490	1	1	yes	characteristic
201	C₂×C₅	#1491	1	1	yes	characteristic
202	C₂×C₅	#1505	1	1	yes	characteristic
203	C₂×C₅	#1506	1	1	yes	characteristic
204	C₂×C₅	#1526	1	1	yes	characteristic
205	C₂×C₅	#1556	1	1	yes	characteristic
206	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
207	Q₈	—	4	8	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	C₅×#101	1	2	yes	—
230	C₂×C₄	C₅×#104	1	2	yes	—
231	C₂×C₄	C₅×#104	1	2	yes	—
232	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
233	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
234	C₂×C₄	C₅×#137	1	2	yes	—
235	C₈	—	4	4	no	—
236	C₈	—	4	4	no	—
237	C₅	#7638	1	1	yes	characteristic, direct product factor, Sylow
238	C₂²	C₅×#407	1	1	yes	characteristic
239	C₂²	C₅×#410	1	1	yes	characteristic
240	C₂²	C₂×C₅×#101	1	1	yes	characteristic
241	C₂²	C₅×#418	1	1	yes	characteristic
242	C₂²	C₅×#419	1	1	yes	characteristic
243	C₂²	C₂×C₅×#104	1	1	yes	characteristic
244	C₂²	C₂×C₅×#104	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	4	1	no	—
252	C₄	—	4	4	no	—
253	C₄	—	8	2	no	—
254	C₂	C₅×#1467	1	1	yes	characteristic
255	C₂	C₅×#1490	1	1	yes	characteristic
256	C₂	C₅×#1491	1	1	yes	characteristic
257	C₂	C₅×#1505	1	1	yes	characteristic
258	C₂	C₅×#1506	1	1	yes	characteristic
259	C₂	C₅×#1526	1	1	yes	characteristic
260	C₂	C₅×#1556	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
20. #7638	C₅	direct (N × H)
237. C₅	#7638	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×C₄⋊Q₈	2
C₅×#1745	2
C₅×#1271	2
#7638	5

Subgroups — C5×#7638 #11930358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#7638 #11930358