Subgroups — C₂×#91575 #11755430

Order 960.

644 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×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#1204	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#6719	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic
5	#91575	C₂	1	8	yes	maximal, direct product factor
6	C₂×Dic₃⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#6669	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#6668	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#4070	—	3	1	no	maximal, Hall
10	C₂⁴×C₃×C₅	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₅	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₅	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Dic₃	—	2	4	no	—
14	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂²×C₅×Dic₃	C₂²	1	4	yes	—
16	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
18	C₂²×C₃×Dic₅	C₂²	1	1	yes	characteristic
19	C₂×#1204	—	2	4	no	—
20	C₂×#1204	C₂²	1	4	yes	complemented
21	#6719	C₂²	1	4	yes	—
22	C₃×C₂²⋊Dic₅	C₂²	1	4	yes	—
23	Dic₃⋊Dic₅	C₂²	1	4	yes	complemented
24	#6669	C₂²	1	4	yes	complemented
25	#6668	C₂²	1	4	yes	complemented
26	C₂×#878	—	5	1	no	maximal, Hall
27	C₂³×C₄×C₅	—	3	1	no	—
28	C₂²×Dic₁₀	—	3	1	no	—
29	C₂×C₂²⋊Dic₅	—	3	1	no	—
30	C₂×C₂²⋊Dic₅	D₃	1	1	yes	characteristic
31	C₂×C₄⋊Dic₅	—	3	1	no	—
32	#4070	—	3	8	no	—
33	C₂×#683	—	3	1	no	—
34	C₂×#683	—	3	1	no	—
35	C₂³×C₃×C₅	—	2	2	no	—
36	C₂³×C₃×C₅	—	2	2	no	—
37	C₂³×C₃×C₅	Q₈	1	4	yes	complemented
38	C₂³×C₃×C₅	C₂³	1	2	yes	—
39	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
40	C₂×Dic₁₅	—	2	2	no	—
41	C₂×Dic₁₅	—	2	2	no	—
42	C₂×Dic₁₅	C₂³	1	2	yes	—
43	C₂×Dic₁₅	C₂³	1	2	yes	—
44	C₂×C₅×Dic₃	—	2	2	no	—
45	C₂×C₅×Dic₃	—	2	8	no	—
46	C₂×C₅×Dic₃	D₄	1	4	yes	—
47	C₂×C₅×Dic₃	C₂³	1	2	yes	—
48	C₂×C₅×Dic₃	C₂³	1	2	yes	—
49	C₂×C₃×Dic₅	—	2	2	no	—
50	C₂×C₃×Dic₅	—	2	2	no	—
51	C₂×C₃×Dic₅	C₂³	1	2	yes	—
52	C₂×C₃×Dic₅	C₂³	1	2	yes	—
53	#1204	—	2	8	no	—
54	C₂³×Dic₃	D₅	1	1	yes	characteristic
55	C₂²×Dic₆	—	5	1	no	—
56	C₂×C₃×C₂²⋊C₄	—	5	1	no	—
57	C₂×#217	—	5	1	no	—
58	C₂×C₄⋊Dic₃	—	5	1	no	—
59	C₂×Dic₃⋊C₄	—	5	1	no	—
60	C₂×Dic₃⋊C₄	—	5	1	no	—
61	#878	—	5	8	no	—
62	C₂⁴×C₅	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₄×C₅	—	3	1	no	—
64	C₂²×C₄×C₅	—	3	1	no	—
65	C₂²×C₄×C₅	—	3	4	no	—
66	C₂²×C₄×C₅	—	6	4	no	—
67	C₂²×Dic₅	—	3	1	no	—
68	C₂²×Dic₅	—	3	1	no	—
69	C₂²×Dic₅	C₂×D₃	1	1	yes	characteristic
70	C₂²×Dic₅	C₂×D₃	1	1	yes	characteristic
71	C₂×Dic₁₀	—	3	4	no	—
72	C₂×Dic₁₀	—	6	4	no	—
73	C₂²⋊Dic₅	—	3	4	no	—
74	C₂²⋊Dic₅	C₂×D₃	1	4	yes	—
75	C₄⋊Dic₅	—	3	4	no	—
76	#683	—	3	4	no	—
77	#683	—	3	4	no	—
78	C₂×C₂²⋊Q₈	—	15	1	no	Sylow
79	C₂²×C₃×C₅	—	2	8	no	—
80	C₂²×C₃×C₅	—	2	2	no	—
81	C₂²×C₃×C₅	—	2	2	no	—
82	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
83	C₂²×C₃×C₅	C₂×Q₈	1	2	yes	—
84	C₂²×C₃×C₅	C₂×Q₈	1	4	yes	complemented
85	C₂²×C₃×C₅	#41	1	2	yes	—
86	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
87	Dic₁₅	—	2	2	no	—
88	Dic₁₅	—	2	2	no	—
89	C₃×Dic₅	—	2	2	no	—
90	C₃×Dic₅	—	2	2	no	—
91	C₅×Dic₃	—	2	2	no	—
92	C₅×Dic₃	C₂×D₄	1	4	yes	—
93	C₂⁴×C₃	C₂×D₅	1	1	yes	characteristic
94	C₂²×C₃×C₄	—	5	1	no	—
95	C₂²×C₃×C₄	—	5	1	no	—
96	C₂²×Dic₃	—	2	4	no	—
97	C₂²×Dic₃	—	5	1	no	—
98	C₂²×Dic₃	—	5	1	no	—
99	C₂²×Dic₃	C₂×D₅	1	4	yes	—
100	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
101	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
102	C₂×Dic₆	—	5	4	no	—
103	C₂×Dic₆	—	10	4	no	—
104	C₃×C₂²⋊C₄	—	5	4	no	—
105	#217	—	5	4	no	—
106	C₄⋊Dic₃	—	5	4	no	—
107	Dic₃⋊C₄	—	5	4	no	—
108	Dic₃⋊C₄	—	5	4	no	—
109	C₂³×C₅	—	2	2	no	—
110	C₂³×C₅	—	2	2	no	—
111	C₂³×C₅	Dic₆	1	4	yes	complemented
112	C₂³×C₅	C₂²×D₃	1	2	yes	—
113	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
114	C₂×C₄×C₅	—	3	4	no	—
115	C₂×C₄×C₅	—	3	2	no	—
116	C₂×C₄×C₅	—	3	2	no	—
117	C₂×C₄×C₅	—	6	8	no	—
118	C₂×C₄×C₅	—	6	2	no	—
119	C₂×Dic₅	—	2	2	no	—
120	C₂×Dic₅	—	2	2	no	—
121	C₂×Dic₅	—	3	2	no	—
122	C₂×Dic₅	—	3	2	no	—
123	C₂×Dic₅	—	6	2	no	—
124	C₂×Dic₅	—	6	2	no	—
125	C₂×Dic₅	C₂²×D₃	1	2	yes	—
126	C₂×Dic₅	C₂²×D₃	1	2	yes	—
127	Dic₁₀	—	6	8	no	—
128	C₂²×Q₈	—	15	1	no	—
129	C₂³×C₄	—	3	1	no	—
130	C₂²⋊Q₈	—	15	8	no	—
131	C₂×C₄⋊C₄	—	15	1	no	—
132	C₂×C₄⋊C₄	—	15	1	no	—
133	C₂×C₄⋊C₄	—	15	1	no	—
134	C₂×C₂²⋊C₄	—	5	1	no	—
135	C₂×C₂²⋊C₄	—	15	1	no	—
136	C₂×C₃×C₅	—	2	4	no	—
137	C₂×C₃×C₅	C₂²⋊Q₈	1	4	yes	complemented
138	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
141	C₂³×C₃	—	2	2	no	—
142	C₂³×C₃	—	2	2	no	—
143	C₂³×C₃	Dic₁₀	1	4	yes	complemented
144	C₂³×C₃	C₂²×D₅	1	2	yes	—
145	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
146	C₂×C₃×C₄	—	5	2	no	—
147	C₂×C₃×C₄	—	5	2	no	—
148	C₂×C₃×C₄	—	10	2	no	—
149	C₂×C₃×C₄	—	10	2	no	—
150	C₂×Dic₃	—	2	2	no	—
151	C₂×Dic₃	—	2	8	no	—
152	C₂×Dic₃	—	5	2	no	—
153	C₂×Dic₃	—	5	2	no	—
154	C₂×Dic₃	—	10	2	no	—
155	C₂×Dic₃	—	10	2	no	—
156	C₂×Dic₃	C₂²⋊D₅	1	4	yes	—
157	C₂×Dic₃	C₂²×D₅	1	2	yes	—
158	C₂×Dic₃	C₂²×D₅	1	2	yes	—
159	Dic₆	—	10	8	no	—
160	C₂²×C₅	—	2	8	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	C₂×Dic₆	1	2	yes	—
164	C₂²×C₅	C₂×Dic₆	1	4	yes	complemented
165	C₂²×C₅	D₃×D₄	1	2	yes	—
166	C₂²×C₅	#237	1	2	yes	—
167	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
168	C₄×C₅	—	3	4	no	—
169	C₄×C₅	—	6	2	no	—
170	Dic₅	—	2	2	no	—
171	Dic₅	—	2	2	no	—
172	Dic₅	—	6	2	no	—
173	Dic₅	—	6	2	no	—
174	C₂⁴	D₃×D₅	1	1	yes	characteristic
175	C₂×Q₈	—	15	4	no	—
176	C₂×Q₈	—	30	4	no	—
177	C₂²×C₄	—	3	1	no	—
178	C₂²×C₄	—	3	4	no	—
179	C₂²×C₄	—	3	1	no	—
180	C₂²×C₄	—	5	1	no	—
181	C₂²×C₄	—	5	1	no	—
182	C₂²×C₄	—	6	4	no	—
183	C₂²×C₄	—	15	1	no	—
184	C₂²×C₄	—	15	1	no	—
185	C₄⋊C₄	—	15	4	no	—
186	C₄⋊C₄	—	15	4	no	—
187	C₄⋊C₄	—	15	4	no	—
188	C₂²⋊C₄	—	5	4	no	—
189	C₂²⋊C₄	—	15	4	no	—
190	C₃×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	—	2	2	no	—
193	C₂²×C₃	—	2	8	no	—
194	C₂²×C₃	C₂×Dic₁₀	1	2	yes	—
195	C₂²×C₃	C₂×Dic₁₀	1	4	yes	complemented
196	C₂²×C₃	#709	1	2	yes	—
197	C₂²×C₃	C₂×C₂²⋊D₅	1	2	yes	—
198	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
199	C₃×C₄	—	10	2	no	—
200	C₃×C₄	—	10	2	no	—
201	Dic₃	—	2	2	no	—
202	Dic₃	—	10	2	no	—
203	Dic₃	—	10	2	no	—
204	Dic₃	C₂×C₂²⋊D₅	1	4	yes	—
205	C₂×C₅	—	2	4	no	—
206	C₂×C₅	#878	1	4	yes	complemented
207	C₂×C₅	C₂²×Dic₆	1	1	yes	characteristic
208	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
209	C₂×C₅	C₂×#237	1	1	yes	characteristic
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	#1204	1	4	yes	complemented
213	C₂³	C₂×D₃×D₅	1	2	yes	—
214	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
215	Q₈	—	30	8	no	—
216	C₂×C₄	—	3	4	no	—
217	C₂×C₄	—	3	2	no	—
218	C₂×C₄	—	3	2	no	—
219	C₂×C₄	—	5	2	no	—
220	C₂×C₄	—	5	2	no	—
221	C₂×C₄	—	6	8	no	—
222	C₂×C₄	—	6	2	no	—
223	C₂×C₄	—	10	2	no	—
224	C₂×C₄	—	10	2	no	—
225	C₂×C₄	—	15	2	no	—
226	C₂×C₄	—	15	2	no	—
227	C₂×C₄	—	30	2	no	—
228	C₂×C₄	—	30	2	no	—
229	C₂×C₃	—	2	4	no	—
230	C₂×C₃	#4070	1	4	yes	complemented
231	C₂×C₃	C₂²×Dic₁₀	1	1	yes	characteristic
232	C₂×C₃	C₂×#709	1	1	yes	characteristic
233	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
234	C₅	C₂×#878	1	1	yes	characteristic, complemented, Sylow
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	8	no	—
238	C₂²	#6779	1	2	yes	—
239	C₂²	C₂×#1204	1	2	yes	—
240	C₂²	C₂×#1204	1	4	yes	complemented
241	C₂²	D₃×C₂²⋊D₅	1	2	yes	—
242	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
243	C₄	—	3	4	no	—
244	C₄	—	6	2	no	—
245	C₄	—	10	2	no	—
246	C₄	—	10	2	no	—
247	C₄	—	30	2	no	—
248	C₄	—	30	2	no	—
249	C₃	C₂×#4070	1	1	yes	characteristic, complemented, Sylow
250	C₂	—	2	4	no	—
251	C₂	#91575	1	4	yes	direct product factor
252	C₂	C₂×#6779	1	1	yes	characteristic
253	C₂	C₂²×#1204	1	1	yes	characteristic
254	C₂	C₂×D₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃×D₅
commutator, cosocle	C₂²×C₃×C₅	C₂⁴
Frattini	C₂²	C₂²×D₃×D₅
Fitting, prime radical	C₂⁴×C₃×C₅	C₂²
socle	C₂³×C₃×C₅	C₂³
hypercenter	C₂⁴	D₃×D₅
hypocenter	C₃×C₅	C₂×C₂²⋊Q₈
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂²×#1204	C₂	semidirect
5. #91575	C₂	direct (N × H)
6. C₂×Dic₃⋊Dic₅	C₂	semidirect
7. C₂×#6669	C₂	semidirect
8. C₂×#6668	C₂	semidirect
20. C₂×#1204	C₂²	semidirect
23. Dic₃⋊Dic₅	C₂²	semidirect
24. #6669	C₂²	semidirect
25. #6668	C₂²	semidirect
37. C₂³×C₃×C₅	Q₈	semidirect
84. C₂²×C₃×C₅	C₂×Q₈	semidirect
111. C₂³×C₅	Dic₆	semidirect
137. C₂×C₃×C₅	C₂²⋊Q₈	semidirect
143. C₂³×C₃	Dic₁₀	semidirect
164. C₂²×C₅	C₂×Dic₆	semidirect
190. C₃×C₅	C₂×C₂²⋊Q₈	semidirect
195. C₂²×C₃	C₂×Dic₁₀	semidirect
206. C₂×C₅	#878	semidirect
212. C₂³	#1204	semidirect
230. C₂×C₃	#4070	semidirect
234. C₅	C₂×#878	semidirect
240. C₂²	C₂×#1204	semidirect
249. C₃	C₂×#4070	semidirect
251. C₂	#91575	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×Dic₃	2
C₂²×#1204	2
C₂×#6719	2
C₂×C₃×C₂²⋊Dic₅	2
#91575	2
C₂×Dic₃⋊Dic₅	2
C₂×#6669	2
C₂×#6668	2
C₂×#4070	3
C₂×#878	5

Subgroups — C2×#91575 #11755430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#91575 #11755430