Subgroups — C₂×#11746385 #14313816

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#91102	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊#672	C₂	1	1	yes	maximal, characteristic
4	#11746385	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64784	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×Q₈	—	2	1	no	—
7	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
8	C₂×C₃×C₅×Q₁₆	—	2	2	no	—
9	C₂×C₃×C₅×Q₁₆	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₅	—	2	1	no	—
13	#91102	—	2	1	no	—
14	#91102	C₂²	1	2	yes	—
15	C₂×C₃⋊#672	—	2	2	no	—
16	C₂×C₃⋊#672	C₂²	1	2	yes	—
17	C₂×#4564	—	5	1	no	maximal, Hall
18	C₂²×C₅×Q₁₆	D₃	1	1	yes	characteristic
19	C₂×#3950	—	3	1	no	—
20	C₂²×#672	—	3	1	no	—
21	#64784	—	3	4	no	—
22	C₂×C₃×C₅×Q₈	—	2	4	no	—
23	C₂×C₃×C₅×Q₈	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	—	4	1	no	—
25	C₂×C₃×C₅×Q₈	—	4	2	no	—
26	C₂²×C₃×C₄×C₅	—	4	1	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×Dic₁₅	—	4	1	no	—
29	C₃×C₅×Q₁₆	—	2	4	no	—
30	C₃×C₅×Q₁₆	—	2	2	no	—
31	C₃×C₅×Q₁₆	C₂×C₄	1	4	yes	complemented
32	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₄⋊Dic₁₅	—	2	2	no	—
37	C₄⋊Dic₁₅	—	4	1	no	—
38	C₃⋊#672	—	2	2	no	—
39	C₂²×C₃×Q₁₆	D₅	1	1	yes	characteristic
40	C₂×#818	—	5	1	no	—
41	C₂²×C₃⋊C₁₆	—	5	1	no	—
42	#4564	—	5	4	no	—
43	C₂²×C₅×Q₈	—	2	1	no	—
44	C₂×C₅×Q₁₆	—	2	1	no	—
45	C₂×C₅×Q₁₆	—	2	2	no	—
46	C₂×C₅×Q₁₆	Dic₃	1	4	yes	complemented
47	C₂×C₅×Q₁₆	C₂×D₃	1	2	yes	—
48	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂×C₄⋊Dic₅	—	6	1	no	—
50	#3950	—	3	2	no	—
51	#3950	—	6	1	no	—
52	C₂×#672	—	3	2	no	—
53	C₂×#672	—	6	2	no	—
54	C₂×#358	—	15	1	no	Sylow
55	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
56	C₃×C₅×Q₈	—	2	4	no	—
57	C₃×C₅×Q₈	—	4	4	no	—
58	C₃×C₅×Q₈	—	4	2	no	—
59	C₂×C₃×C₄×C₅	—	4	4	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×Dic₁₅	—	4	2	no	—
66	C₂×Dic₁₅	—	8	2	no	—
67	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
68	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×Q₈	—	2	1	no	—
71	C₂×C₃×Q₁₆	—	2	1	no	—
72	C₂×C₃×Q₁₆	—	2	2	no	—
73	C₂×C₃×Q₁₆	Dic₅	1	4	yes	complemented
74	C₂×C₃×Q₁₆	C₂×D₅	1	2	yes	—
75	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
76	C₂×C₄⋊Dic₃	—	10	1	no	—
77	#818	—	5	2	no	—
78	#818	—	10	1	no	—
79	C₂×C₃⋊C₁₆	—	5	2	no	—
80	C₂×C₃⋊C₁₆	—	10	2	no	—
81	C₂×C₅×Q₈	—	2	4	no	—
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	4	1	no	—
84	C₂×C₅×Q₈	—	4	2	no	—
85	C₂²×C₄×C₅	—	4	1	no	—
86	C₂²×C₄×C₅	#67	1	1	yes	characteristic
87	C₂²×Dic₅	—	12	1	no	—
88	C₅×Q₁₆	—	2	4	no	—
89	C₅×Q₁₆	—	2	2	no	—
90	C₅×Q₁₆	C₂×Dic₃	1	4	yes	complemented
91	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
92	C₂×C₅×C₈	#67	1	1	yes	characteristic
93	C₂×C₅×C₈	#67	1	2	yes	—
94	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
95	C₄⋊Dic₅	—	6	2	no	—
96	C₄⋊Dic₅	—	12	1	no	—
97	#672	—	6	2	no	—
98	C₂²×Q₁₆	D₁₅	1	1	yes	characteristic
99	C₂²×C₁₆	—	15	1	no	—
100	C₂×#107	—	15	1	no	—
101	#358	—	15	4	no	—
102	C₂²×C₃×C₅	#102	1	2	yes	—
103	C₂²×C₃×C₅	SD₃₂	1	2	yes	—
104	C₂²×C₃×C₅	Q₃₂	1	2	yes	—
105	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
106	C₃×C₄×C₅	—	4	4	no	—
107	C₃×C₄×C₅	#102	1	2	yes	—
108	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
110	Dic₁₅	—	8	2	no	—
111	C₂×C₃×Q₈	—	2	4	no	—
112	C₂×C₃×Q₈	—	2	2	no	—
113	C₂×C₃×Q₈	—	4	1	no	—
114	C₂×C₃×Q₈	—	4	2	no	—
115	C₂²×C₃×C₄	—	4	1	no	—
116	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
117	C₂²×Dic₃	—	20	1	no	—
118	C₃×Q₁₆	—	2	4	no	—
119	C₃×Q₁₆	—	2	2	no	—
120	C₃×Q₁₆	C₂×Dic₅	1	4	yes	complemented
121	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
122	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
124	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
125	C₄⋊Dic₃	—	10	2	no	—
126	C₄⋊Dic₃	—	20	1	no	—
127	C₃⋊C₁₆	—	10	2	no	—
128	C₂³×C₅	#213	1	1	yes	characteristic
129	C₅×Q₈	—	2	4	no	—
130	C₅×Q₈	—	4	4	no	—
131	C₅×Q₈	—	4	2	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	#213	1	1	yes	characteristic
135	C₂×C₄×C₅	#214	1	2	yes	—
136	C₂×C₄×C₅	#217	1	2	yes	—
137	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
138	C₂×Dic₅	—	12	2	no	—
139	C₂×Dic₅	—	24	2	no	—
140	C₅×C₈	#217	1	2	yes	—
141	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
142	C₅×C₈	C₂×#67	1	1	yes	characteristic
143	C₂²×Q₈	—	2	1	no	—
144	C₂×Q₁₆	—	2	1	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	Dic₁₅	1	4	yes	complemented
147	C₂×Q₁₆	C₂×D₁₅	1	2	yes	—
148	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
149	C₂×C₄⋊C₄	—	30	1	no	—
150	C₂×C₁₆	—	15	2	no	—
151	C₂×C₁₆	—	30	2	no	—
152	#107	—	15	2	no	—
153	#107	—	30	1	no	—
154	C₂×C₃×C₅	#358	1	4	yes	complemented
155	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
156	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
157	C₂×C₃×C₅	C₂×Q₃₂	1	1	yes	characteristic
158	C₂³×C₃	#686	1	1	yes	characteristic
159	C₃×Q₈	—	2	4	no	—
160	C₃×Q₈	—	4	4	no	—
161	C₃×Q₈	—	4	2	no	—
162	C₂×C₃×C₄	—	4	4	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	#686	1	1	yes	characteristic
165	C₂×C₃×C₄	#687	1	2	yes	—
166	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
167	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
168	C₂×Dic₃	—	20	2	no	—
169	C₂×Dic₃	—	40	2	no	—
170	C₃×C₈	C₂²⋊Dic₅	1	2	yes	—
171	C₃×C₈	C₂²×Dic₅	1	1	yes	characteristic
172	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
173	C₂²×C₅	#828	1	2	yes	—
174	C₂²×C₅	#829	1	2	yes	—
175	C₂²×C₅	#832	1	2	yes	—
176	C₂²×C₅	C₂×#213	1	1	yes	characteristic
177	C₄×C₅	—	4	4	no	—
178	C₄×C₅	#832	1	2	yes	—
179	C₄×C₅	C₂×#214	1	1	yes	characteristic
180	C₄×C₅	C₂×#217	1	1	yes	characteristic
181	Dic₅	—	24	2	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	4	1	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	60	1	no	—
188	C₂²×C₄	#1220	1	1	yes	characteristic
189	Q₁₆	—	2	4	no	—
190	Q₁₆	—	2	2	no	—
191	Q₁₆	C₂×Dic₁₅	1	4	yes	complemented
192	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
193	C₂×C₈	#1220	1	1	yes	characteristic
194	C₂×C₈	#1220	1	2	yes	—
195	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
196	C₄⋊C₄	—	30	2	no	—
197	C₄⋊C₄	—	60	1	no	—
198	C₁₆	—	30	2	no	—
199	C₃×C₅	C₂×#358	1	1	yes	characteristic, complemented, Hall
200	C₂²×C₃	#3960	1	2	yes	—
201	C₂²×C₃	#3961	1	2	yes	—
202	C₂²×C₃	#3964	1	2	yes	—
203	C₂²×C₃	C₂×#686	1	1	yes	characteristic
204	C₃×C₄	—	4	4	no	—
205	C₃×C₄	#3964	1	2	yes	—
206	C₃×C₄	C₂×#687	1	1	yes	characteristic
207	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
208	Dic₃	—	40	2	no	—
209	C₂×C₅	#4564	1	4	yes	complemented
210	C₂×C₅	C₂×#828	1	1	yes	characteristic
211	C₂×C₅	C₂×#829	1	1	yes	characteristic
212	C₂×C₅	C₂×#832	1	1	yes	characteristic
213	C₂³	#6715	1	1	yes	characteristic
214	Q₈	—	2	4	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	60	2	no	—
220	C₂×C₄	—	120	2	no	—
221	C₂×C₄	#6715	1	1	yes	characteristic
222	C₂×C₄	#6716	1	2	yes	—
223	C₂×C₄	#6719	1	2	yes	—
224	C₂×C₄	C₂×#1220	1	1	yes	characteristic
225	C₈	#6719	1	2	yes	—
226	C₈	C₂²×Dic₁₅	1	1	yes	characteristic
227	C₈	C₂×#1220	1	1	yes	characteristic
228	C₂×C₃	#64784	1	4	yes	complemented
229	C₂×C₃	C₂×#3960	1	1	yes	characteristic
230	C₂×C₃	C₂×#3961	1	1	yes	characteristic
231	C₂×C₃	C₂×#3964	1	1	yes	characteristic
232	C₅	C₂×#4564	1	1	yes	characteristic, complemented, Sylow
233	C₂²	#91112	1	2	yes	—
234	C₂²	#91113	1	2	yes	—
235	C₂²	#91116	1	2	yes	—
236	C₂²	C₂×#6715	1	1	yes	characteristic
237	C₄	—	4	4	no	—
238	C₄	—	120	2	no	—
239	C₄	#91116	1	2	yes	—
240	C₄	C₂×#6716	1	1	yes	characteristic
241	C₄	C₂×#6719	1	1	yes	characteristic
242	C₃	C₂×#64784	1	1	yes	characteristic, complemented, Sylow
243	C₂	#11746385	1	4	yes	direct product factor
244	C₂	C₂×#91112	1	1	yes	characteristic
245	C₂	C₂×#91113	1	1	yes	characteristic
246	C₂	C₂×#91116	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6715
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅
Fitting	C₂²×C₃×C₅×Q₁₆	C₂
socle, prime radical	C₂³×C₃×C₅	D₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×Q₁₆	D₁₅
hypocenter	C₃×C₅	C₂×#358
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
4. #11746385	C₂	direct (N × H)
9. C₂×C₃×C₅×Q₁₆	C₄	semidirect
31. C₃×C₅×Q₁₆	C₂×C₄	semidirect
46. C₂×C₅×Q₁₆	Dic₃	semidirect
73. C₂×C₃×Q₁₆	Dic₅	semidirect
90. C₅×Q₁₆	C₂×Dic₃	semidirect
120. C₃×Q₁₆	C₂×Dic₅	semidirect
146. C₂×Q₁₆	Dic₁₅	semidirect
154. C₂×C₃×C₅	#358	semidirect
191. Q₁₆	C₂×Dic₁₅	semidirect
199. C₃×C₅	C₂×#358	semidirect
209. C₂×C₅	#4564	semidirect
228. C₂×C₃	#64784	semidirect
232. C₅	C₂×#4564	semidirect
242. C₃	C₂×#64784	semidirect
243. C₂	#11746385	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×C₃×C₅×Q₁₆	2
C₂×#91102	2
C₂²×C₃⋊#672	2
#11746385	2
C₂×#64784	3
C₂×#4564	5

Subgroups — C2×#11746385 #14313816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746385 #14313816