Subgroups — C₂×#11746332 #14314401

Order 1920.

336 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₅	C₂	1	1	yes	maximal, characteristic
3	C₂×#91096	C₂	1	1	yes	maximal, characteristic
4	#11746332	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64731	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₅	C₄	1	1	yes	characteristic
8	C₂²×C₃⋊#168	C₄	1	1	yes	characteristic
9	C₃×C₅×M₃₂	—	2	1	no	—
10	C₃×C₅×M₃₂	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₁₆	—	2	1	no	—
12	C₈×Dic₁₅	C₄	1	2	yes	—
13	C₈×Dic₁₅	C₂²	1	2	yes	—
14	#91096	—	2	1	no	—
15	#91096	C₂²	1	2	yes	—
16	C₂×C₃⋊#672	—	2	1	no	—
17	C₂×#4511	—	5	1	no	maximal, Hall
18	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic
19	C₂×C₈×Dic₅	—	3	1	no	—
20	C₂×#3944	—	3	1	no	—
21	#64731	—	3	4	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₅	—	2	1	no	—
24	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₄×Dic₁₅	—	2	1	no	—
31	C₄×Dic₁₅	C₂×C₄	1	2	yes	—
32	C₂×C₃⋊#168	C₈	1	2	yes	—
33	C₂×C₃⋊#168	C₈	1	2	yes	—
34	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
35	C₃×C₅×C₁₆	—	2	2	no	—
36	C₃⋊#672	—	2	2	no	—
37	C₂×C₃×M₃₂	D₅	1	1	yes	characteristic
38	C₂×C₈×Dic₃	—	5	1	no	—
39	C₂×#812	—	5	1	no	—
40	#4511	—	5	4	no	—
41	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂×C₄×Dic₅	—	3	1	no	—
43	C₂²×#168	—	3	1	no	—
44	C₅×M₃₂	—	2	1	no	—
45	C₅×M₃₂	C₂×D₃	1	2	yes	—
46	C₂×C₅×C₁₆	—	2	1	no	—
47	C₈×Dic₅	—	3	2	no	—
48	C₈×Dic₅	—	3	2	no	—
49	#3944	—	3	2	no	—
50	#3944	—	6	1	no	—
51	C₂×#672	—	6	1	no	—
52	C₂×#364	—	15	1	no	Sylow
53	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×Dic₁₅	—	2	4	no	—
61	C₂×Dic₁₅	—	2	2	no	—
62	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
64	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
65	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
66	C₃⋊#168	C₂×C₈	1	4	yes	—
67	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
68	C₂×C₄×Dic₃	—	5	1	no	—
69	C₂²×C₃⋊C₈	—	5	1	no	—
70	C₃×M₃₂	—	2	1	no	—
71	C₃×M₃₂	C₂×D₅	1	2	yes	—
72	C₂×C₃×C₁₆	—	2	1	no	—
73	C₈×Dic₃	—	5	2	no	—
74	C₈×Dic₃	—	5	2	no	—
75	#812	—	5	2	no	—
76	#812	—	10	1	no	—
77	C₂×C₃⋊C₁₆	—	10	1	no	—
78	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
79	C₂²×Dic₅	—	6	1	no	—
80	C₂×C₅×C₈	Dic₆	1	1	yes	characteristic
81	C₂×C₅×C₈	Dic₆	1	1	yes	characteristic
82	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
83	C₂×C₅×C₈	#67	1	1	yes	characteristic
84	C₂×C₅×C₈	#67	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
86	C₄×Dic₅	—	3	2	no	—
87	C₄×Dic₅	—	6	1	no	—
88	C₂×#168	—	3	2	no	—
89	C₂×#168	—	3	2	no	—
90	C₂×#168	—	3	2	no	—
91	C₅×C₁₆	—	2	2	no	—
92	#672	—	6	2	no	—
93	C₂×M₃₂	—	15	1	no	—
94	C₂×M₃₂	D₁₅	1	1	yes	characteristic
95	C₂×C₄×C₈	—	15	1	no	—
96	#364	—	15	4	no	—
97	C₂²×C₃×C₅	—	2	2	no	—
98	C₂²×C₃×C₅	C₄⋊C₈	1	1	yes	characteristic
99	C₂²×C₃×C₅	C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
101	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
102	C₃×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
103	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
105	Dic₁₅	—	2	4	no	—
106	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
107	C₂²×Dic₃	—	10	1	no	—
108	C₂×C₃×C₈	Dic₁₀	1	1	yes	characteristic
109	C₂×C₃×C₈	Dic₁₀	1	1	yes	characteristic
110	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
112	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
113	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
114	C₄×Dic₃	—	5	2	no	—
115	C₄×Dic₃	—	10	1	no	—
116	C₂×C₃⋊C₈	—	5	2	no	—
117	C₂×C₃⋊C₈	—	5	2	no	—
118	C₂×C₃⋊C₈	—	5	2	no	—
119	C₃×C₁₆	—	2	2	no	—
120	C₃⋊C₁₆	—	10	2	no	—
121	C₂³×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
122	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
125	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
128	C₂×Dic₅	—	6	4	no	—
129	C₂×Dic₅	—	6	2	no	—
130	C₅×C₈	Dic₃⋊C₄	1	1	yes	characteristic
131	C₅×C₈	Dic₃⋊C₄	1	1	yes	characteristic
132	C₅×C₈	C₂×Dic₆	1	1	yes	characteristic
133	C₅×C₈	C₂×#67	1	1	yes	characteristic
134	#168	—	3	4	no	—
135	C₂²×C₈	—	15	1	no	—
136	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
137	C₂×C₄²	—	15	1	no	—
138	M₃₂	—	2	1	no	—
139	M₃₂	—	15	2	no	—
140	M₃₂	—	30	1	no	—
141	M₃₂	C₂×D₁₅	1	2	yes	—
142	C₂×C₁₆	—	2	1	no	—
143	C₂×C₁₆	—	30	1	no	—
144	C₄×C₈	—	15	2	no	—
145	C₄×C₈	—	15	2	no	—
146	C₂×C₃×C₅	—	2	1	no	—
147	C₂×C₃×C₅	—	2	1	no	—
148	C₂×C₃×C₅	#364	1	2	yes	complemented
149	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
150	C₂³×C₃	#676	1	1	yes	characteristic
151	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
152	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
153	C₂×C₃×C₄	#676	1	1	yes	characteristic
154	C₂×C₃×C₄	#683	1	1	yes	characteristic
155	C₂×C₃×C₄	#683	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
157	C₂×Dic₃	—	10	4	no	—
158	C₂×Dic₃	—	10	2	no	—
159	C₃×C₈	#683	1	1	yes	characteristic
160	C₃×C₈	#683	1	1	yes	characteristic
161	C₃×C₈	C₂×Dic₁₀	1	1	yes	characteristic
162	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
163	C₃⋊C₈	—	5	4	no	—
164	C₂²×C₅	—	2	2	no	—
165	C₂²×C₅	Dic₃⋊C₈	1	1	yes	characteristic
166	C₂²×C₅	Dic₃⋊C₈	1	1	yes	characteristic
167	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
168	C₄×C₅	Dic₃⋊C₈	1	1	yes	characteristic
169	C₄×C₅	Dic₃⋊C₈	1	1	yes	characteristic
170	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
171	C₄×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
172	Dic₅	—	6	4	no	—
173	C₂²×C₄	—	30	1	no	—
174	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
175	C₂×C₈	—	15	2	no	—
176	C₂×C₈	—	15	2	no	—
177	C₂×C₈	—	15	2	no	—
178	C₂×C₈	Dic₃₀	1	1	yes	characteristic
179	C₂×C₈	Dic₃₀	1	1	yes	characteristic
180	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
181	C₂×C₈	#1220	1	1	yes	characteristic
182	C₂×C₈	#1220	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
184	C₄²	—	15	2	no	—
185	C₄²	—	30	1	no	—
186	C₁₆	—	2	2	no	—
187	C₁₆	—	30	2	no	—
188	C₃×C₅	C₂×#364	1	1	yes	characteristic, complemented, Hall
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	#3946	1	1	yes	characteristic
191	C₂²×C₃	#3946	1	1	yes	characteristic
192	C₂²×C₃	C₂×#676	1	1	yes	characteristic
193	C₃×C₄	#3946	1	1	yes	characteristic
194	C₃×C₄	#3946	1	1	yes	characteristic
195	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
196	C₃×C₄	C₂×#683	1	1	yes	characteristic
197	Dic₃	—	10	4	no	—
198	C₂×C₅	—	2	1	no	—
199	C₂×C₅	—	2	1	no	—
200	C₂×C₅	#4511	1	2	yes	complemented
201	C₂×C₅	C₂×Dic₃⋊C₈	1	1	yes	characteristic
202	C₂³	#6705	1	1	yes	characteristic
203	C₂×C₄	—	30	4	no	—
204	C₂×C₄	—	30	2	no	—
205	C₂×C₄	C₈×D₁₅	1	1	yes	characteristic
206	C₂×C₄	C₈×D₁₅	1	1	yes	characteristic
207	C₂×C₄	#6705	1	1	yes	characteristic
208	C₂×C₄	#6712	1	1	yes	characteristic
209	C₂×C₄	#6712	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
211	C₈	—	15	4	no	—
212	C₈	#6712	1	1	yes	characteristic
213	C₈	#6712	1	1	yes	characteristic
214	C₈	C₂×Dic₃₀	1	1	yes	characteristic
215	C₈	C₂×#1220	1	1	yes	characteristic
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	#64731	1	2	yes	complemented
219	C₂×C₃	C₂×#3946	1	1	yes	characteristic
220	C₅	C₂×#4511	1	1	yes	characteristic, complemented, Sylow
221	C₂²	—	2	2	no	—
222	C₂²	#91098	1	1	yes	characteristic
223	C₂²	#91098	1	1	yes	characteristic
224	C₂²	C₂×#6705	1	1	yes	characteristic
225	C₄	—	30	4	no	—
226	C₄	#91098	1	1	yes	characteristic
227	C₄	#91098	1	1	yes	characteristic
228	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
229	C₄	C₂×#6712	1	1	yes	characteristic
230	C₃	C₂×#64731	1	1	yes	characteristic, complemented, Sylow
231	C₂	—	2	1	no	—
232	C₂	—	2	1	no	—
233	C₂	#11746332	1	2	yes	direct product factor
234	C₂	C₂×#91098	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#1220
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₁₅
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂²×C₃×C₅	C₄⋊C₈
prime radical	C₂³×C₃×C₅	M₁₆
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂×M₃₂	D₁₅
hypocenter	C₃×C₅	C₂×#364
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
4. #11746332	C₂	direct (N × H)
148. C₂×C₃×C₅	#364	semidirect
188. C₃×C₅	C₂×#364	semidirect
200. C₂×C₅	#4511	semidirect
218. C₂×C₃	#64731	semidirect
220. C₅	C₂×#4511	semidirect
230. C₃	C₂×#64731	semidirect
233. C₂	#11746332	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×C₃×C₅×M₃₂	2
C₂×C₈×Dic₁₅	2
C₂×#91096	2
#11746332	2
C₂×#64731	3
C₂×#4511	5

Subgroups — C2×#11746332 #14314401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746332 #14314401