Subgroups — C₂×#20241529 #26136484

Order 12032.

294 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689539	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688957	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241529	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13094030	C₄	1	1	yes	characteristic
6	#15689539	—	2	1	no	—
7	#15689539	—	2	1	no	—
8	#15689539	C₄	1	2	yes	—
9	#15689539	C₂²	1	2	yes	—
10	C₂×#13093964	C₄	1	1	yes	characteristic
11	C₂×#13093963	—	2	1	no	—
12	C₂×C₁₆×D₄₇	—	2	1	no	—
13	#15688957	—	2	1	no	—
14	#15688957	C₂²	1	2	yes	complemented
15	C₂×C₄₇⋊C₃₂	—	2	1	no	—
16	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₄₇	—	2	1	no	—
18	C₂×C₃₂×C₄₇	—	2	1	no	—
19	C₂×C₃₂×C₄₇	C₂²	1	2	yes	complemented
20	C₂×#10625350	C₈	1	1	yes	characteristic
21	#13094030	—	2	1	no	—
22	#13094030	—	2	1	no	—
23	#13094030	C₈	1	2	yes	—
24	#13094030	C₂×C₄	1	2	yes	—
25	C₂×#10625329	C₈	1	1	yes	characteristic
26	C₂×#10625324	—	2	1	no	—
27	C₂×C₈×D₄₇	—	2	1	no	—
28	#13093964	—	2	1	no	—
29	#13093964	C₂×C₄	1	2	yes	—
30	#13093963	—	2	2	no	—
31	#13093963	—	2	2	no	—
32	C₂×C₄₇⋊C₁₆	—	2	1	no	—
33	C₁₆×D₄₇	—	2	2	no	—
34	C₁₆×D₄₇	—	2	2	no	—
35	C₄₇⋊C₃₂	—	2	2	no	—
36	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
43	C₃₂×C₄₇	—	2	2	no	—
44	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₂²⋊D₄₇	—	2	1	no	—
46	#10625350	—	2	1	no	—
47	#10625350	—	2	1	no	—
48	#10625350	C₁₆	1	2	yes	—
49	#10625350	C₂×C₈	1	2	yes	—
50	C₂×D₁₈₈	C₁₆	1	1	yes	characteristic
51	C₂×C₄×D₄₇	—	2	1	no	—
52	C₂×Dic₉₄	C₁₆	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
57	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
58	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	#10625329	—	2	1	no	—
60	#10625329	C₂×C₈	1	2	yes	—
61	C₂×C₄₇⋊C₈	—	2	1	no	—
62	#10625324	—	2	2	no	—
63	#10625324	—	2	2	no	—
64	C₈×D₄₇	—	2	2	no	—
65	C₈×D₄₇	—	2	2	no	—
66	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₄₇⋊C₁₆	—	2	2	no	—
71	C₂³×C₄₇	M₃₂	1	1	yes	characteristic
72	C₂²×D₄₇	—	2	1	no	—
73	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₇	M₃₂	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
79	C₂²⋊D₄₇	—	2	2	no	—
80	C₂²⋊D₄₇	—	2	2	no	—
81	C₂×Dic₄₇	—	2	1	no	—
82	D₁₈₈	—	2	1	no	—
83	D₁₈₈	C₂×C₁₆	1	2	yes	—
84	C₄×D₄₇	—	2	2	no	—
85	C₄×D₄₇	—	2	2	no	—
86	Dic₉₄	—	2	1	no	—
87	Dic₉₄	C₂×C₁₆	1	2	yes	—
88	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
89	C₈×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
90	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₈×C₄₇	C₂×M₁₆	1	1	yes	characteristic
92	C₄₇⋊C₈	—	2	2	no	—
93	C₂×#1401	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	—	2	2	no	—
95	C₂²×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
96	C₂²×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
97	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
98	C₂×D₄₇	—	2	2	no	—
99	C₂×D₄₇	—	4	2	no	—
100	C₄×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
101	C₄×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
102	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₄×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
104	Dic₄₇	—	2	2	no	—
105	C₂×#504	—	47	1	no	—
106	C₂×M₆₄	—	47	1	no	—
107	C₂²×C₃₂	D₄₇	1	1	yes	characteristic, complemented
108	#1401	—	47	4	no	—
109	C₂×C₄₇	—	2	1	no	—
110	C₂×C₄₇	—	2	1	no	—
111	C₂×C₄₇	#1401	1	2	yes	complemented
112	C₂×C₄₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
113	D₄₇	—	4	2	no	—
114	C₂×#131	—	47	1	no	—
115	#504	—	47	2	no	—
116	#504	—	47	2	no	—
117	#504	—	94	1	no	—
118	#504	—	94	1	no	—
119	C₂×M₃₂	—	47	1	no	—
120	C₂×M₃₂	—	94	1	no	—
121	C₂²×C₁₆	—	94	1	no	—
122	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
123	M₆₄	—	47	2	no	—
124	M₆₄	—	94	1	no	—
125	C₂×C₃₂	—	2	1	no	—
126	C₂×C₃₂	—	2	1	no	—
127	C₂×C₃₂	—	94	1	no	—
128	C₂×C₃₂	C₂×D₄₇	1	2	yes	complemented
129	C₄₇	C₂×#1401	1	1	yes	characteristic, complemented, Sylow
130	C₂×#41	—	47	1	no	—
131	#131	—	47	2	no	—
132	#131	—	47	2	no	—
133	#131	—	94	1	no	—
134	#131	—	94	1	no	—
135	C₂×M₁₆	—	47	1	no	—
136	C₂×M₁₆	—	94	1	no	—
137	C₂²×C₈	—	94	1	no	—
138	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
139	M₃₂	—	47	2	no	—
140	M₃₂	—	94	1	no	—
141	M₃₂	—	94	2	no	—
142	M₃₂	—	94	2	no	—
143	C₂×C₁₆	—	94	1	no	—
144	C₂×C₁₆	—	94	2	no	—
145	C₂×C₁₆	—	94	2	no	—
146	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
147	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
148	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
149	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
150	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
152	C₃₂	—	2	2	no	—
153	C₃₂	—	94	2	no	—
154	#41	—	47	2	no	—
155	#41	—	47	2	no	—
156	#41	—	94	1	no	—
157	#41	—	94	1	no	—
158	C₂×Q₈	—	47	1	no	—
159	C₂×D₄	—	47	1	no	—
160	C₂×D₄	—	94	1	no	—
161	C₂²×C₄	—	94	1	no	—
162	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
163	M₁₆	—	47	2	no	—
164	M₁₆	—	94	1	no	—
165	M₁₆	—	94	2	no	—
166	M₁₆	—	94	2	no	—
167	C₂×C₈	—	94	1	no	—
168	C₂×C₈	—	94	2	no	—
169	C₂×C₈	—	94	2	no	—
170	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
171	C₂×C₈	#10625324	1	1	yes	characteristic
172	C₂×C₈	#10625324	1	1	yes	characteristic
173	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
174	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
176	C₁₆	—	94	2	no	—
177	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
178	C₁₆	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
179	C₁₆	C₂×D₁₈₈	1	1	yes	characteristic
180	C₁₆	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂³	—	94	1	no	—
182	C₂³	#13093963	1	1	yes	characteristic
183	Q₈	—	47	2	no	—
184	Q₈	—	94	1	no	—
185	D₄	—	47	2	no	—
186	D₄	—	94	2	no	—
187	D₄	—	94	2	no	—
188	D₄	—	94	1	no	—
189	C₂×C₄	—	94	1	no	—
190	C₂×C₄	—	94	2	no	—
191	C₂×C₄	—	94	2	no	—
192	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
193	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
194	C₂×C₄	C₁₆×D₄₇	1	1	yes	characteristic
195	C₂×C₄	C₁₆×D₄₇	1	1	yes	characteristic
196	C₂×C₄	#13093963	1	1	yes	characteristic
197	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
198	C₈	—	94	2	no	—
199	C₈	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
200	C₈	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
201	C₈	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
202	C₈	C₂×#10625324	1	1	yes	characteristic
203	C₂²	—	2	2	no	—
204	C₂²	—	94	2	no	—
205	C₂²	—	188	2	no	—
206	C₂²	(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic
207	C₂²	(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic
208	C₂²	C₂×#13093963	1	1	yes	characteristic
209	C₄	—	94	2	no	—
210	C₄	(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic
211	C₄	(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic
212	C₄	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
213	C₄	C₂×C₁₆×D₄₇	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	188	2	no	—
217	C₂	#20241529	1	2	yes	direct product factor
218	C₂	C₂×(C₂×D₄₇)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	D₁₈₈
commutator	C₄×C₄₇	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₂²×C₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂²⋊C₁₆
prime radical	C₂×#10625350	C₈
cosocle	C₂×C₁₆×C₄₇	C₂³
hypercenter	C₂²×C₃₂	D₄₇
hypocenter	C₄₇	C₂×#1401
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#15688957	C₂	semidirect
3. #20241529	C₂	direct (N × H)
4. C₂²×C₃₂×C₄₇	C₂	semidirect
14. #15688957	C₂²	semidirect
19. C₂×C₃₂×C₄₇	C₂²	semidirect
107. C₂²×C₃₂	D₄₇	semidirect
111. C₂×C₄₇	#1401	semidirect
128. C₂×C₃₂	C₂×D₄₇	semidirect
129. C₄₇	C₂×#1401	semidirect
217. C₂	#20241529	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×#15689539	2
C₂×#15688957	2
#20241529	2
C₂²×C₃₂×C₄₇	2
C₂×#1401	47

Subgroups — C2×#20241529 #26136484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241529 #26136484