Subgroups — C₄×#65313 #12097927

Order 1280.

958 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#65313	C₂	1	2	yes	maximal
3	#10600271	C₂	1	4	yes	maximal
4	#10600270	C₂	1	2	yes	maximal
5	C₄×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊#168	C₂	1	2	yes	maximal
7	C₄×#3935	C₂	1	2	yes	maximal
8	C₄²×#168	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₅×Q₈	C₄	1	2	yes	—
10	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₄×C₅×C₄⋊C₄	C₄	1	4	yes	—
12	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
13	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₄³×C₅	C₂²	1	1	yes	characteristic
16	C₄³×C₅	C₂²	1	2	yes	—
17	#65313	C₄	1	16	yes	direct product factor
18	C₂×C₄⋊#168	C₂²	1	2	yes	—
19	C₂×C₄⋊#168	C₂²	1	4	yes	—
20	C₂×#3935	C₂²	1	4	yes	—
21	C₂×#3935	C₂²	1	4	yes	—
22	C₂×C₄×#168	C₂²	1	2	yes	—
23	C₂×C₄×#168	C₂²	1	1	yes	characteristic
24	C₂×C₄×#168	C₂²	1	2	yes	—
25	C₂×C₄×#168	C₂²	1	2	yes	—
26	C₂×C₄×#168	C₂²	1	1	yes	characteristic
27	#64744	C₂²	1	4	yes	—
28	#64744	C₂²	1	2	yes	—
29	C₄×#446	—	5	1	no	maximal, Sylow
30	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₅×Q₈	—	2	8	no	—
32	C₄×C₅×Q₈	C₂×C₄	1	8	yes	—
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
35	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂³	1	2	yes	—
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂³	1	2	yes	—
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂²×#168	C₂³	1	2	yes	—
47	C₂²×#168	C₂³	1	4	yes	—
48	C₂²×#168	C₂³	1	2	yes	—
49	C₅×#95	C₂×C₄	1	2	yes	—
50	C₅×#95	C₂×C₄	1	4	yes	—
51	C₄⋊#168	C₂×C₄	1	16	yes	—
52	C₄⋊#168	C₂×C₄	1	8	yes	—
53	#3935	—	2	8	no	—
54	#3935	C₂×C₄	1	16	yes	—
55	C₄×#168	—	2	2	no	—
56	C₄×#168	—	2	4	no	—
57	C₄×#168	—	2	4	no	—
58	C₄×#168	C₂×C₄	1	8	yes	—
59	C₄×#168	Q₈	1	4	yes	complemented
60	C₂×#446	—	5	2	no	—
61	C₄²×Q₈	D₅	1	1	yes	characteristic
62	#1771	—	5	4	no	—
63	#1770	—	5	2	no	—
64	C₄×C₄⋊C₈	—	5	2	no	—
65	C₄×C₄⋊C₈	—	5	1	no	—
66	C₄×#97	—	5	2	no	—
67	C₄²×C₈	—	5	1	no	—
68	C₂×C₅×Q₈	—	2	4	no	—
69	C₂×C₅×Q₈	C₄²	1	4	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
78	C₅×C₄⋊C₄	—	2	8	no	—
79	C₅×C₄⋊C₄	—	2	4	no	—
80	C₅×C₄⋊C₄	C₄²	1	8	yes	—
81	C₅×C₄⋊C₄	C₄²	1	4	yes	—
82	C₄²×C₅	—	2	8	no	—
83	C₄²×C₅	—	2	4	no	—
84	C₄²×C₅	—	2	8	no	—
85	C₄²×C₅	—	2	2	no	—
86	C₄²×C₅	—	2	4	no	—
87	C₄²×C₅	M₁₆	1	8	yes	—
88	C₄²×C₅	C₂²×C₄	1	4	yes	—
89	C₄²×C₅	C₂²×C₄	1	8	yes	—
90	C₄²×C₅	C₂×Q₈	1	2	yes	—
91	C₄²×C₅	#41	1	2	yes	—
92	C₂×#168	—	2	2	no	—
93	C₂×#168	—	2	4	no	—
94	C₂×#168	—	2	2	no	—
95	C₂×#168	—	2	4	no	—
96	C₂×#168	C₂²×C₄	1	4	yes	—
97	C₂×#168	C₂²×C₄	1	8	yes	—
98	C₂×#168	C₂²×C₄	1	4	yes	—
99	C₂×#168	C₂×Q₈	1	4	yes	—
100	C₂×#168	#41	1	4	yes	—
101	C₂×C₄×Q₈	Dic₅	1	2	yes	—
102	C₂×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
103	#446	—	5	16	no	—
104	C₂×C₄⋊C₈	—	5	2	no	—
105	C₂×C₄⋊C₈	—	5	4	no	—
106	C₂×#97	—	5	4	no	—
107	C₂×#97	—	5	4	no	—
108	C₂×C₄×C₈	—	5	2	no	—
109	C₂×C₄×C₈	—	5	1	no	—
110	C₂×C₄×C₈	—	5	2	no	—
111	C₂×C₄×C₈	—	5	1	no	—
112	C₂×C₄×C₈	—	5	2	no	—
113	C₄×C₄⋊C₄	Dic₅	1	4	yes	—
114	C₄×C₄⋊C₄	Dic₅	1	2	yes	—
115	C₄×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
116	C₄×C₄⋊C₄	C₂×D₅	1	2	yes	—
117	C₄³	C₂×D₅	1	1	yes	characteristic
118	C₄³	C₂×D₅	1	2	yes	—
119	#336	—	5	4	no	—
120	#336	—	5	2	no	—
121	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
122	C₅×Q₈	—	2	8	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
133	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
134	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
135	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
136	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
137	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
138	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
139	C₂×C₄×C₅	#131	1	4	yes	—
140	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
141	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×#41	1	2	yes	—
144	#168	—	2	4	no	—
145	#168	—	2	8	no	—
146	#168	C₄×Q₈	1	8	yes	complemented
147	C₂²×Q₈	C₂×Dic₅	1	1	yes	characteristic
148	C₂²×C₈	—	5	2	no	—
149	C₂²×C₈	—	5	2	no	—
150	C₂²×C₈	—	5	4	no	—
151	C₄×Q₈	—	2	8	no	—
152	C₄×Q₈	C₄×D₅	1	8	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
154	C₂×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
155	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
156	C₂×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
157	C₂×C₄²	C₂×Dic₅	1	4	yes	—
158	C₂×C₄²	C₂×Dic₅	1	4	yes	—
159	C₂×C₄²	C₂×Dic₅	1	2	yes	—
160	C₂×C₄²	C₂×Dic₅	1	2	yes	—
161	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
162	C₂×C₄²	C₂²×D₅	1	2	yes	—
163	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
164	C₂×C₄²	C₂²×D₅	1	2	yes	—
165	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
166	C₄⋊C₈	—	5	8	no	—
167	C₄⋊C₈	—	5	16	no	—
168	#97	—	5	16	no	—
169	#97	—	10	8	no	—
170	C₄×C₈	—	5	8	no	—
171	C₄×C₈	—	5	4	no	—
172	C₄×C₈	—	10	4	no	—
173	C₄×C₈	—	10	2	no	—
174	C₄×C₈	—	10	4	no	—
175	#95	C₂×Dic₅	1	2	yes	—
176	#95	C₂×Dic₅	1	4	yes	—
177	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
178	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
179	C₂²×C₅	C₄×#41	1	1	yes	characteristic
180	C₂²×C₅	C₂²×M₁₆	1	2	yes	—
181	C₂²×C₅	C₂×#131	1	2	yes	—
182	C₄×C₅	—	2	4	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	2	4	no	—
185	C₄×C₅	—	2	4	no	—
186	C₄×C₅	—	4	4	no	—
187	C₄×C₅	C₄×M₁₆	1	4	yes	—
188	C₄×C₅	#446	1	8	yes	complemented
189	C₄×C₅	C₂×C₄×Q₈	1	2	yes	—
190	C₄×C₅	C₄×#41	1	2	yes	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	C₄×Dic₅	1	4	yes	—
193	C₂²×C₄	C₂²×Dic₅	1	4	yes	—
194	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
195	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
197	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
198	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
199	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
200	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
201	C₂×C₈	—	5	4	no	—
202	C₂×C₈	—	5	4	no	—
203	C₂×C₈	—	5	4	no	—
204	C₂×C₈	—	5	4	no	—
205	C₂×C₈	—	5	8	no	—
206	C₂×C₈	—	10	2	no	—
207	C₂×C₈	—	10	4	no	—
208	C₂×C₈	—	10	2	no	—
209	C₂×C₈	—	10	4	no	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	C₄×Dic₅	1	8	yes	—
213	C₄⋊C₄	C₄×Dic₅	1	4	yes	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	2	8	no	—
217	C₄²	—	2	4	no	—
218	C₄²	—	2	8	no	—
219	C₄²	#681	1	8	yes	—
220	C₄²	C₂×C₄×D₅	1	4	yes	—
221	C₄²	C₂×C₄×D₅	1	8	yes	—
222	C₄²	Q₈×D₅	1	2	yes	—
223	C₄²	#713	1	2	yes	—
224	C₂×C₅	C₄²×Q₈	1	1	yes	characteristic
225	C₂×C₅	C₂×C₄×M₁₆	1	1	yes	characteristic
226	C₂×C₅	C₄×#131	1	1	yes	characteristic
227	C₂×C₅	C₂×#446	1	2	yes	—
228	C₂×C₅	#2965	1	2	yes	—
229	C₂³	C₂³×Dic₅	1	1	yes	characteristic
230	Q₈	—	2	8	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	C₂×#681	1	4	yes	—
240	C₂×C₄	C₂×#681	1	4	yes	—
241	C₂×C₄	C₂×#681	1	4	yes	—
242	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
244	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
245	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
246	C₂×C₄	Q₈×Dic₅	1	4	yes	—
247	C₂×C₄	#4094	1	4	yes	—
248	C₂×C₄	C₂²×C₄×D₅	1	2	yes	—
249	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
250	C₂×C₄	C₂×#713	1	1	yes	characteristic
251	C₂×C₄	D₅×#41	1	2	yes	—
252	C₈	—	5	8	no	—
253	C₈	—	10	4	no	—
254	C₈	—	10	8	no	—
255	C₅	C₄×#446	1	1	yes	characteristic, complemented, Sylow
256	C₂²	C₂²×#681	1	2	yes	—
257	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
258	C₂²	C₂×Q₈×Dic₅	1	1	yes	characteristic
259	C₂²	C₂×#4094	1	2	yes	—
260	C₂²	#41×Dic₅	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	4	4	no	—
266	C₄	C₄×#681	1	4	yes	—
267	C₄	#65313	1	8	yes	direct product factor
268	C₄	C₄×Q₈×D₅	1	2	yes	—
269	C₄	C₄×#713	1	2	yes	—
270	C₂	C₂×C₄×#681	1	1	yes	characteristic
271	C₂	C₂×#65313	1	2	yes	—
272	C₂	C₄×Q₈×Dic₅	1	1	yes	characteristic
273	C₂	C₄×#4094	1	1	yes	characteristic
274	C₂	#10614248	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₄²×C₅×Q₈	C₂
socle, prime radical	C₂³×C₅	C₂³×C₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₄²×Q₈	D₅
hypocenter	C₅	C₄×#446
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
17. #65313	C₄	direct (N × H)
59. C₄×#168	Q₈	semidirect
146. #168	C₄×Q₈	semidirect
188. C₄×C₅	#446	semidirect
255. C₅	C₄×#446	semidirect
267. C₄	#65313	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₅×Q₈	2
C₂×#65313	2
#10600271	2
#10600270	2
C₄×C₄⋊#168	2
C₄×#3935	2
C₄²×#168	2
C₄×#446	5

Subgroups — C4×#65313 #12097927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#65313 #12097927