Subgroups — C₂×#20241691 #26181984

Order 12032.

994 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625329	C₂	1	2	yes	maximal
2	#20241691	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15688845	C₂	1	2	yes	maximal, complemented
4	C₂×#15688844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₄	1	4	yes	—
8	C₄×#10625329	C₄	1	8	yes	—
9	C₄×#10625329	C₂²	1	8	yes	—
10	C₂×#13093935	C₄	1	4	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
13	#15688845	C₂²	1	8	yes	complemented
14	#15688844	C₂²	1	4	yes	complemented
15	C₈×C₄₇⋊C₈	C₂²	1	4	yes	complemented
16	C₂²×M₁₆×C₄₇	C₄	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄×M₁₆×C₄₇	C₄	1	4	yes	—
19	C₄×M₁₆×C₄₇	C₂²	1	4	yes	—
20	C₂×#97×C₄₇	C₄	1	2	yes	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#10625329	—	2	16	no	—
24	C₂×#10625329	C₂×C₄	1	16	yes	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
27	#13093935	C₂×C₄	1	16	yes	—
28	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
30	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
31	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
32	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₄₇	—	2	8	no	—
35	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	#97×C₄₇	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₄₇	C₂³	1	2	yes	—
48	C₄×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂⁴×C₄₇	C₄²	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	2	8	no	—
52	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
54	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
55	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	M₁₆×C₄₇	—	2	16	no	—
61	C₂×C₈×C₄₇	—	2	4	no	—
62	C₂×C₈×C₄₇	M₁₆	1	8	yes	complemented
63	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
65	C₄²×C₄₇	C₄²	1	4	yes	—
66	C₄²×C₄₇	C₄²	1	2	yes	—
67	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
71	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
72	#10625329	—	2	32	no	—
73	C₂×C₄₇⋊C₈	—	2	8	no	—
74	C₂×C₄₇⋊C₈	M₁₆	1	16	yes	complemented
75	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
76	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
77	C₂³×C₄₇	—	2	4	no	—
78	C₂³×C₄₇	—	2	2	no	—
79	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
80	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₄₇	—	2	4	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	8	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	2	8	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
88	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
89	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
90	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
92	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
93	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
95	C₈×C₄₇	—	2	4	no	—
96	C₈×C₄₇	C₂×M₁₆	1	8	yes	complemented
97	C₄₇⋊C₈	—	2	8	no	—
98	C₄₇⋊C₈	C₂×M₁₆	1	16	yes	complemented
99	C₂×#1455	—	47	1	no	maximal, Sylow
100	C₂²×C₄₇	—	2	4	no	—
101	C₂²×C₄₇	—	2	2	no	—
102	C₂²×C₄₇	—	4	4	no	—
103	C₂²×C₄₇	C₄×M₁₆	1	4	yes	—
104	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
105	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
106	C₄×C₄₇	—	2	4	no	—
107	C₄×C₄₇	—	2	2	no	—
108	C₄×C₄₇	C₄×M₁₆	1	4	yes	—
109	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
110	C₄×C₄₇	C₂²×M₁₆	1	4	yes	—
111	C₄×C₄₇	C₂²×M₁₆	1	2	yes	—
112	C₂×C₄×M₁₆	—	47	2	no	—
113	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
114	#1455	—	47	8	no	—
115	C₂×#322	—	47	2	no	—
116	C₂×#322	—	47	1	no	—
117	C₂×C₈²	—	47	1	no	—
118	C₂×C₄₇	—	4	2	no	—
119	C₂×C₄₇	#1455	1	4	yes	complemented
120	C₂×C₄₇	C₂×C₄×M₁₆	1	2	yes	—
121	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
122	C₂²×M₁₆	—	47	4	no	—
123	C₂²×M₁₆	Dic₄₇	1	2	yes	—
124	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
125	C₄×M₁₆	—	47	8	no	—
126	C₄×M₁₆	—	47	8	no	—
127	C₄×M₁₆	Dic₄₇	1	4	yes	—
128	C₄×M₁₆	C₂×D₄₇	1	4	yes	—
129	C₂×#97	—	47	4	no	—
130	C₂×#97	Dic₄₇	1	2	yes	—
131	C₂×C₄×C₈	—	47	2	no	—
132	C₂×C₄×C₈	—	47	2	no	—
133	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
135	#322	—	47	4	no	—
136	#322	—	47	8	no	—
137	C₈²	—	47	4	no	—
138	C₄₇	C₂×#1455	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₄	C₄×D₄₇	1	2	yes	—
140	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
141	C₂×M₁₆	—	2	8	no	—
142	C₂×M₁₆	—	47	16	no	—
143	C₂×M₁₆	—	94	16	no	—
144	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
145	C₂²×C₈	—	47	4	no	—
146	C₂²×C₈	—	47	4	no	—
147	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
148	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
149	C₂×C₄²	C₄×D₄₇	1	2	yes	—
150	C₂×C₄²	C₄×D₄₇	1	4	yes	—
151	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
153	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
154	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
155	#97	—	47	16	no	—
156	#97	C₂×Dic₄₇	1	8	yes	—
157	C₄×C₈	—	47	4	no	—
158	C₄×C₈	—	47	4	no	—
159	C₄×C₈	—	47	4	no	—
160	C₄×C₈	—	47	4	no	—
161	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
162	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
163	C₄×C₈	C₂²×D₄₇	1	2	yes	—
164	C₄×C₈	C₂²×D₄₇	1	2	yes	—
165	C₂⁴	C₄×Dic₄₇	1	1	yes	characteristic
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
169	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
170	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
171	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
172	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
173	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
176	M₁₆	—	2	16	no	—
177	M₁₆	—	94	32	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	47	16	no	—
180	C₂×C₈	—	47	8	no	—
181	C₂×C₈	—	47	8	no	—
182	C₂×C₈	—	94	8	no	—
183	C₂×C₈	#10625329	1	8	yes	complemented
184	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
185	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
186	C₄²	C₄×Dic₄₇	1	4	yes	—
187	C₄²	C₄×Dic₄₇	1	2	yes	—
188	C₄²	C₂×C₄×D₄₇	1	2	yes	—
189	C₄²	C₂×C₄×D₄₇	1	4	yes	—
190	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
191	C₄²	C₂²×Dic₄₇	1	2	yes	—
192	C₄²	C₂³×D₄₇	1	1	yes	characteristic
193	C₂³	—	2	4	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
196	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	8	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
204	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
205	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
206	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
207	C₂×C₄	M₁₆×D₄₇	1	8	yes	—
208	C₂×C₄	C₂×#10625329	1	4	yes	—
209	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
210	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
211	C₈	—	2	4	no	—
212	C₈	—	47	16	no	—
213	C₈	—	94	8	no	—
214	C₈	C₂×#10625329	1	8	yes	complemented
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	M₁₆×Dic₄₇	1	4	yes	—
219	C₂²	C₄×#10625329	1	2	yes	—
220	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	2	2	no	—
223	C₄	M₁₆×Dic₄₇	1	4	yes	—
224	C₄	C₄×#10625329	1	2	yes	—
225	C₄	C₂×M₁₆×D₄₇	1	4	yes	—
226	C₄	C₂²×#10625329	1	2	yes	—
227	C₂	—	4	2	no	—
228	C₂	#20241691	1	4	yes	direct product factor
229	C₂	C₂×M₁₆×Dic₄₇	1	2	yes	—
230	C₂	C₂×C₄×#10625329	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄²
prime radical	C₂⁴×C₄₇	C₄²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	C₂×#1455
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20241691	C₂	direct (N × H)
3. C₂×#15688845	C₂	semidirect
4. C₂×#15688844	C₂	semidirect
5. C₂×C₈×C₄₇⋊C₈	C₂	semidirect
13. #15688845	C₂²	semidirect
14. #15688844	C₂²	semidirect
15. C₈×C₄₇⋊C₈	C₂²	semidirect
62. C₂×C₈×C₄₇	M₁₆	semidirect
74. C₂×C₄₇⋊C₈	M₁₆	semidirect
96. C₈×C₄₇	C₂×M₁₆	semidirect
98. C₄₇⋊C₈	C₂×M₁₆	semidirect
119. C₂×C₄₇	#1455	semidirect
138. C₄₇	C₂×#1455	semidirect
183. C₂×C₈	#10625329	semidirect
214. C₈	C₂×#10625329	semidirect
228. C₂	#20241691	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×C₄×#10625329	2
#20241691	2
C₂×#15688845	2
C₂×#15688844	2
C₂×C₈×C₄₇⋊C₈	2
C₂×C₄×M₁₆×C₄₇	2
C₂×#1455	47

Subgroups — C2×#20241691 #26181984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241691 #26181984