Subgroups — C₂×#92029 #11757038

Order 960.

1588 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	C₂×#6821	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×D₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×Q₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#92029	C₂	1	16	yes	maximal, direct product factor
5	C₂×#6773	C₂	1	3	yes	maximal, complemented
6	C₂×#6771	C₂	1	3	yes	maximal, complemented
7	C₂×D₁₅⋊Q₈	C₂	1	3	yes	maximal, complemented
8	C₂×#6769	C₂	1	3	yes	maximal, complemented
9	C₂×#4147	—	3	1	no	maximal, Hall
10	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
11	#6821	C₂²	1	8	yes	complemented
12	C₂×D₆₀	C₂²	1	3	yes	—
13	C₂×C₄×D₁₅	C₂²	1	3	yes	—
14	C₅×D₃×Q₈	C₂²	1	8	yes	complemented
15	C₂×C₄×C₅×D₃	C₂²	1	3	yes	—
16	C₂×C₅×Dic₆	C₂²	1	3	yes	—
17	C₃×Q₈×D₅	C₂²	1	8	yes	complemented
18	C₂×C₃×C₄×D₅	C₂²	1	3	yes	—
19	C₂×C₃×Dic₁₀	C₂²	1	3	yes	—
20	C₂×#1204	C₂²	1	3	yes	—
21	C₂×#1203	C₂²	1	3	yes	—
22	C₂×#1202	C₂²	1	3	yes	—
23	C₂×#1201	C₂²	1	1	yes	characteristic
24	C₂×#1200	C₂²	1	6	yes	—
25	#6773	C₂²	1	24	yes	complemented
26	#6771	C₂²	1	24	yes	complemented
27	D₁₅⋊Q₈	C₂²	1	24	yes	complemented
28	#6769	C₂²	1	24	yes	complemented
29	C₂×#1007	—	5	1	no	maximal, Hall
30	C₂²×C₅×Q₈	—	3	1	no	—
31	#4147	—	3	16	no	—
32	C₂×#713	—	3	1	no	—
33	C₂×#713	—	3	3	no	—
34	C₂×Q₈×D₅	—	3	3	no	—
35	C₂×Q₈×D₅	D₃	1	1	yes	characteristic, complemented
36	C₂×#709	—	3	3	no	—
37	C₂×#709	—	3	3	no	—
38	C₂²×D₁₅	C₂³	1	3	yes	—
39	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
40	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
41	C₃×C₅×Q₈	C₂³	1	4	yes	—
42	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
43	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
44	D₆₀	C₂³	1	12	yes	—
45	C₄×D₁₅	C₂³	1	12	yes	—
46	C₂×C₅×Dic₃	C₂³	1	3	yes	—
47	C₄×C₅×D₃	C₂³	1	12	yes	—
48	C₅×Dic₆	C₂³	1	12	yes	—
49	C₂×C₃×Dic₅	C₂³	1	3	yes	—
50	C₃×C₄×D₅	C₂³	1	12	yes	—
51	C₃×Dic₁₀	C₂³	1	12	yes	—
52	#1204	C₂³	1	12	yes	—
53	#1203	C₂³	1	12	yes	—
54	#1202	C₂³	1	12	yes	—
55	#1201	C₂³	1	4	yes	—
56	#1200	C₂³	1	24	yes	—
57	C₂²×C₃×Q₈	—	5	1	no	—
58	#1007	—	5	16	no	—
59	C₂×#239	—	5	1	no	—
60	C₂×#239	—	5	3	no	—
61	C₂×D₃×Q₈	—	5	3	no	—
62	C₂×D₃×Q₈	D₅	1	1	yes	characteristic, complemented
63	C₂×#235	—	5	3	no	—
64	C₂×#235	—	5	3	no	—
65	C₂×C₅×Q₈	—	3	8	no	—
66	C₂×C₅×Q₈	—	3	3	no	—
67	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
68	C₂²×C₄×C₅	—	3	3	no	—
69	C₂×C₂²⋊D₅	—	3	1	no	—
70	C₂×C₂²⋊D₅	—	3	3	no	—
71	#713	—	3	8	no	—
72	#713	—	3	24	no	—
73	Q₈×D₅	—	3	24	no	—
74	Q₈×D₅	C₂×D₃	1	8	yes	complemented
75	#709	—	3	24	no	—
76	#709	—	3	24	no	—
77	C₂×D₂₀	—	3	3	no	—
78	C₂×D₂₀	—	3	3	no	—
79	C₂×C₄×D₅	—	3	3	no	—
80	C₂×C₄×D₅	—	3	6	no	—
81	C₂×C₄×D₅	C₂×D₃	1	3	yes	—
82	C₂×Dic₁₀	—	3	3	no	—
83	C₂×Dic₁₀	C₂×D₃	1	3	yes	—
84	C₂×2₋¹⁺⁴	—	15	1	no	Sylow
85	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
86	C₂×D₁₅	—	2	6	no	—
87	C₂×D₁₅	C₂⁴	1	6	yes	—
88	C₂×C₅×D₃	—	2	2	no	—
89	C₂×C₅×D₃	C₂⁴	1	2	yes	—
90	C₂×C₃×D₅	—	2	2	no	—
91	C₂×C₃×D₅	C₂⁴	1	2	yes	—
92	C₃×C₄×C₅	C₂⁴	1	6	yes	—
93	Dic₁₅	C₂⁴	1	2	yes	—
94	C₃×Dic₅	C₂⁴	1	6	yes	—
95	C₅×Dic₃	C₂⁴	1	6	yes	—
96	C₂×C₃×Q₈	—	5	8	no	—
97	C₂×C₃×Q₈	—	5	3	no	—
98	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
99	C₂²×C₃×C₄	—	5	3	no	—
100	C₂×#67	—	5	1	no	—
101	C₂×#67	—	5	3	no	—
102	#239	—	5	8	no	—
103	#239	—	5	24	no	—
104	D₃×Q₈	—	5	24	no	—
105	D₃×Q₈	C₂×D₅	1	8	yes	complemented
106	#235	—	5	24	no	—
107	#235	—	5	24	no	—
108	C₂×D₁₂	—	5	3	no	—
109	C₂×D₁₂	—	5	3	no	—
110	C₂×C₄×D₃	—	5	3	no	—
111	C₂×C₄×D₃	—	5	6	no	—
112	C₂×C₄×D₃	C₂×D₅	1	3	yes	—
113	C₂×Dic₆	—	5	3	no	—
114	C₂×Dic₆	C₂×D₅	1	3	yes	—
115	C₂³×C₅	—	3	1	no	—
116	C₂²×D₅	—	3	3	no	—
117	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
118	C₅×Q₈	—	3	12	no	—
119	C₅×Q₈	C₂²×D₃	1	4	yes	—
120	C₂×C₄×C₅	—	3	3	no	—
121	C₂×C₄×C₅	—	3	12	no	—
122	C₂×C₄×C₅	C₂²×D₃	1	3	yes	—
123	C₂²⋊D₅	—	3	4	no	—
124	C₂²⋊D₅	—	3	12	no	—
125	C₂×Dic₅	—	3	1	no	—
126	C₂×Dic₅	C₂²×D₃	1	3	yes	—
127	D₂₀	—	3	12	no	—
128	D₂₀	—	3	12	no	—
129	C₄×D₅	—	3	12	no	—
130	C₄×D₅	—	3	24	no	—
131	C₄×D₅	C₂²×D₃	1	12	yes	—
132	Dic₁₀	—	3	12	no	—
133	Dic₁₀	C₂²×D₃	1	12	yes	—
134	2₋¹⁺⁴	—	15	16	no	—
135	C₂×#41	—	15	3	no	—
136	C₂×#41	—	15	3	no	—
137	C₂×#41	—	15	3	no	—
138	C₂×#41	—	15	1	no	—
139	C₂²×Q₈	—	3	1	no	—
140	C₂²×Q₈	—	5	1	no	—
141	C₂²×Q₈	—	15	3	no	—
142	C₂×C₃×C₅	2₋¹⁺⁴	1	2	yes	complemented
143	C₂×C₃×C₅	C₂⁵	1	1	yes	characteristic
144	D₁₅	—	2	6	no	—
145	C₃×D₅	—	2	2	no	—
146	C₅×D₃	—	2	2	no	—
147	C₂³×C₃	—	5	1	no	—
148	C₂²×D₃	—	5	3	no	—
149	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
150	C₃×Q₈	—	5	12	no	—
151	C₃×Q₈	C₂²×D₅	1	4	yes	—
152	C₂×C₃×C₄	—	5	3	no	—
153	C₂×C₃×C₄	—	5	12	no	—
154	C₂×C₃×C₄	C₂²×D₅	1	3	yes	—
155	#67	—	5	4	no	—
156	#67	—	5	12	no	—
157	C₂×Dic₃	—	5	1	no	—
158	C₂×Dic₃	C₂²×D₅	1	3	yes	—
159	D₁₂	—	5	12	no	—
160	D₁₂	—	5	12	no	—
161	C₄×D₃	—	5	12	no	—
162	C₄×D₃	—	5	24	no	—
163	C₄×D₃	C₂²×D₅	1	12	yes	—
164	Dic₆	—	5	12	no	—
165	Dic₆	C₂²×D₅	1	12	yes	—
166	C₂²×C₅	—	3	2	no	—
167	C₂²×C₅	—	6	2	no	—
168	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
169	C₂×D₅	—	2	2	no	—
170	C₂×D₅	—	3	6	no	—
171	C₂×D₅	—	6	6	no	—
172	C₂×D₅	C₂³×D₃	1	2	yes	—
173	C₄×C₅	—	3	6	no	—
174	C₄×C₅	C₂³×D₃	1	6	yes	—
175	Dic₅	—	3	2	no	—
176	Dic₅	C₂³×D₃	1	6	yes	—
177	#41	—	15	24	no	—
178	#41	—	15	24	no	—
179	#41	—	15	24	no	—
180	#41	—	15	8	no	—
181	C₂×Q₈	—	3	8	no	—
182	C₂×Q₈	—	3	3	no	—
183	C₂×Q₈	—	5	8	no	—
184	C₂×Q₈	—	5	3	no	—
185	C₂×Q₈	—	15	3	no	—
186	C₂×Q₈	—	15	24	no	—
187	C₂×Q₈	D₃×D₅	1	1	yes	characteristic
188	C₂×D₄	—	15	3	no	—
189	C₂×D₄	—	15	3	no	—
190	C₂×D₄	—	15	1	no	—
191	C₂×D₄	—	15	3	no	—
192	C₂²×C₄	—	3	3	no	—
193	C₂²×C₄	—	5	3	no	—
194	C₂²×C₄	—	15	6	no	—
195	C₂²×C₄	—	15	3	no	—
196	C₃×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Hall
197	C₂²×C₃	—	5	2	no	—
198	C₂²×C₃	—	10	2	no	—
199	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
200	C₂×D₃	—	2	2	no	—
201	C₂×D₃	—	5	6	no	—
202	C₂×D₃	—	10	6	no	—
203	C₂×D₃	C₂³×D₅	1	2	yes	—
204	C₃×C₄	—	5	6	no	—
205	C₃×C₄	C₂³×D₅	1	6	yes	—
206	Dic₃	—	5	2	no	—
207	Dic₃	C₂³×D₅	1	6	yes	—
208	C₂×C₅	—	6	2	no	—
209	C₂×C₅	#1007	1	2	yes	complemented
210	C₂×C₅	C₂⁴×D₃	1	1	yes	characteristic
211	D₅	—	2	2	no	—
212	D₅	—	6	6	no	—
213	C₂³	—	3	1	no	—
214	C₂³	—	5	1	no	—
215	C₂³	—	15	3	no	—
216	Q₈	—	3	12	no	—
217	Q₈	—	5	12	no	—
218	Q₈	—	15	12	no	—
219	Q₈	C₂×D₃×D₅	1	4	yes	—
220	D₄	—	15	12	no	—
221	D₄	—	15	12	no	—
222	D₄	—	15	4	no	—
223	D₄	—	15	12	no	—
224	C₂×C₄	—	3	3	no	—
225	C₂×C₄	—	3	12	no	—
226	C₂×C₄	—	5	3	no	—
227	C₂×C₄	—	5	12	no	—
228	C₂×C₄	—	15	24	no	—
229	C₂×C₄	—	15	1	no	—
230	C₂×C₄	—	15	12	no	—
231	C₂×C₄	C₂×D₃×D₅	1	3	yes	—
232	C₂×C₃	—	10	2	no	—
233	C₂×C₃	#4147	1	2	yes	complemented
234	C₂×C₃	C₂⁴×D₅	1	1	yes	characteristic
235	D₃	—	2	2	no	—
236	D₃	—	10	6	no	—
237	C₅	C₂×#1007	1	1	yes	characteristic, complemented, Sylow
238	C₂²	—	3	2	no	—
239	C₂²	—	5	2	no	—
240	C₂²	—	6	2	no	—
241	C₂²	—	10	2	no	—
242	C₂²	—	15	6	no	—
243	C₂²	—	30	6	no	—
244	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
245	C₄	—	3	6	no	—
246	C₄	—	5	6	no	—
247	C₄	—	15	2	no	—
248	C₄	C₂²×D₃×D₅	1	6	yes	—
249	C₃	C₂×#4147	1	1	yes	characteristic, complemented, Sylow
250	C₂	—	6	2	no	—
251	C₂	—	10	2	no	—
252	C₂	—	30	6	no	—
253	C₂	#92029	1	2	yes	direct product factor
254	C₂	C₂³×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃×D₅
commutator, cosocle	C₂×C₃×C₅	C₂⁵
Frattini	C₂	C₂³×D₃×D₅
Fitting	C₂×C₃×C₅×Q₈	C₂²
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₂×Q₈	D₃×D₅
hypocenter	C₃×C₅	C₂×2₋¹⁺⁴
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#6821	C₂	semidirect
2. C₂×C₅×D₃×Q₈	C₂	semidirect
3. C₂×C₃×Q₈×D₅	C₂	semidirect
4. #92029	C₂	direct (N × H)
5. C₂×#6773	C₂	semidirect
6. C₂×#6771	C₂	semidirect
7. C₂×D₁₅⋊Q₈	C₂	semidirect
8. C₂×#6769	C₂	semidirect
11. #6821	C₂²	semidirect
14. C₅×D₃×Q₈	C₂²	semidirect
17. C₃×Q₈×D₅	C₂²	semidirect
25. #6773	C₂²	semidirect
26. #6771	C₂²	semidirect
27. D₁₅⋊Q₈	C₂²	semidirect
28. #6769	C₂²	semidirect
35. C₂×Q₈×D₅	D₃	semidirect
62. C₂×D₃×Q₈	D₅	semidirect
74. Q₈×D₅	C₂×D₃	semidirect
105. D₃×Q₈	C₂×D₅	semidirect
142. C₂×C₃×C₅	2₋¹⁺⁴	semidirect
196. C₃×C₅	C₂×2₋¹⁺⁴	semidirect
209. C₂×C₅	#1007	semidirect
233. C₂×C₃	#4147	semidirect
237. C₅	C₂×#1007	semidirect
249. C₃	C₂×#4147	semidirect
253. C₂	#92029	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#6821	2
C₂×C₅×D₃×Q₈	2
C₂×C₃×Q₈×D₅	2
#92029	2
C₂×#6773	2
C₂×#6771	2
C₂×D₁₅⋊Q₈	2
C₂×#6769	2
C₂×#4147	3
C₂×#1007	5

Subgroups — C2×#92029 #11757038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#92029 #11757038