Subgroups — C₂×#20245950 #26374789

Order 12032.

5694 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094021	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093986	C₂	1	6	yes	maximal, complemented
3	#20245950	C₂	1	16	yes	maximal, direct product factor
4	C₂×#15689024	C₂	1	6	yes	maximal, complemented
5	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₈₈	C₂²	1	6	yes	—
8	C₂³×C₄×D₄₇	C₂²	1	4	yes	—
9	C₂×#13094021	C₄	1	16	yes	complemented
10	C₂×#13094021	C₂²	1	8	yes	—
11	C₂×#13093986	—	2	48	no	—
12	C₂×#13093986	C₂²	1	48	yes	complemented
13	C₂²×(C₂×D₄₇)⋊C₄	C₂²	1	12	yes	—
14	C₂²×C₄×Dic₄₇	C₂²	1	3	yes	—
15	C₂²×C₄×Dic₄₇	C₂²	1	3	yes	—
16	#15689024	C₂²	1	48	yes	complemented
17	C₄²×Dic₄₇	C₂²	1	8	yes	complemented
18	C₂²×C₄⋊C₄×C₄₇	C₂²	1	6	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×#384	C₂²	1	8	yes	complemented
21	C₂⁴×D₄₇	C₂³	1	4	yes	—
22	C₂²×D₁₈₈	—	2	24	no	—
23	C₂²×D₁₈₈	—	2	24	no	—
24	C₂²×D₁₈₈	C₂×C₄	1	48	yes	complemented
25	C₂²×D₁₈₈	C₂³	1	24	yes	—
26	C₂³×Dic₄₇	C₂³	1	3	yes	—
27	C₂³×Dic₄₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×D₄₇	—	2	48	no	—
29	C₂²×C₄×D₄₇	C₂³	1	16	yes	—
30	#13094021	C₂×C₄	1	64	yes	complemented
31	#13093986	—	2	192	no	—
32	C₂×(C₂×D₄₇)⋊C₄	—	2	48	no	—
33	C₂×(C₂×D₄₇)⋊C₄	C₂³	1	48	yes	—
34	C₂×C₄×Dic₄₇	—	2	12	no	—
35	C₂×C₄×Dic₄₇	D₄	1	24	yes	complemented
36	C₂×C₄×Dic₄₇	C₂³	1	12	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	12	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
40	C₂×C₄⋊C₄×C₄₇	—	2	24	no	—
41	C₂×C₄⋊C₄×C₄₇	C₂³	1	24	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	complemented
43	C₂×C₄²×C₄₇	C₂³	1	4	yes	—
44	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
45	C₂³×D₄₇	—	2	24	no	—
46	C₂³×D₄₇	—	2	24	no	—
47	C₂³×D₄₇	C₂²×C₄	1	16	yes	—
48	C₂³×D₄₇	C₂⁴	1	8	yes	—
49	C₂²×C₄×C₄₇	—	2	24	no	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
52	C₂²×C₄×C₄₇	#41	1	12	yes	—
53	C₂²×C₄×C₄₇	C₂⁴	1	8	yes	—
54	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
55	C₂²×Dic₄₇	—	2	12	no	—
56	C₂²×Dic₄₇	C₂×D₄	1	12	yes	—
57	C₂²×Dic₄₇	C₂×D₄	1	12	yes	—
58	C₂²×Dic₄₇	C₂⁴	1	6	yes	—
59	C₂²×Dic₄₇	C₂⁴	1	2	yes	—
60	C₂×D₁₈₈	—	2	192	no	—
61	C₂×D₁₈₈	—	2	48	no	—
62	C₂×D₁₈₈	—	2	48	no	—
63	C₂×D₁₈₈	C₂²×C₄	1	96	yes	complemented
64	C₂×C₄×D₄₇	—	2	96	no	—
65	C₂×C₄×D₄₇	—	4	64	no	—
66	C₄⋊C₄×C₄₇	—	2	48	no	—
67	C₄²×C₄₇	C₂²×C₄	1	16	yes	complemented
68	(C₂×D₄₇)⋊C₄	—	2	96	no	—
69	C₄×Dic₄₇	—	2	24	no	—
70	C₄×Dic₄₇	C₂×D₄	1	48	yes	complemented
71	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
72	C₂³×C₄₇	C₂²×D₄	1	6	yes	—
73	C₂³×C₄₇	C₂×#41	1	6	yes	—
74	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
75	C₂²×D₄₇	—	2	96	no	—
76	C₂²×D₄₇	—	2	24	no	—
77	C₂²×D₄₇	—	2	24	no	—
78	C₂²×D₄₇	—	4	16	no	—
79	C₂²×D₄₇	—	4	48	no	—
80	C₂²×D₄₇	C₂³×C₄	1	16	yes	—
81	C₂×C₄×C₄₇	—	2	24	no	—
82	C₂×C₄×C₄₇	—	4	16	no	—
83	C₂×C₄×C₄₇	C₄×D₄	1	48	yes	complemented
84	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
85	C₂×C₄×C₄₇	C₂²×D₄	1	12	yes	—
86	C₂×C₄×C₄₇	C₂×#41	1	12	yes	—
87	C₂×Dic₄₇	—	2	24	no	—
88	C₂×Dic₄₇	—	2	12	no	—
89	C₂×Dic₄₇	#127	1	16	yes	complemented
90	C₂×Dic₄₇	C₂²×D₄	1	12	yes	—
91	C₂×Dic₄₇	C₂²×D₄	1	12	yes	—
92	D₁₈₈	—	2	192	no	—
93	C₄×D₄₇	—	4	64	no	—
94	C₂×C₄×#127	—	47	1	no	maximal, Sylow
95	C₂²×C₄₇	C₂×C₄×D₄	1	12	yes	—
96	C₂²×C₄₇	C₂×#127	1	4	yes	—
97	C₂²×C₄₇	#532	1	12	yes	—
98	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
99	C₂²×C₄₇	C₂³×D₄	1	3	yes	—
100	C₂²×C₄₇	C₂²×#41	1	3	yes	—
101	C₂×D₄₇	—	2	48	no	—
102	C₂×D₄₇	—	4	64	no	—
103	C₂×D₄₇	—	4	8	no	—
104	C₂×D₄₇	—	4	24	no	—
105	C₄×C₄₇	—	4	8	no	—
106	C₄×C₄₇	C₂×C₄×D₄	1	24	yes	complemented
107	Dic₄₇	—	2	12	no	—
108	Dic₄₇	C₂×#127	1	8	yes	complemented
109	C₂²×#127	—	47	1	no	—
110	C₂²×C₄×D₄	—	47	6	no	—
111	C₄×#127	—	47	16	no	—
112	C₂×#390	—	47	6	no	—
113	C₂×#384	D₄₇	1	1	yes	characteristic, complemented
114	C₂×C₄³	—	47	1	no	—
115	C₂×C₄₇	C₄×#127	1	8	yes	complemented
116	C₂×C₄₇	C₂²×C₄×D₄	1	3	yes	—
117	C₂×C₄₇	C₂²×#127	1	1	yes	characteristic
118	C₂×C₄₇	C₂×#532	1	3	yes	—
119	D₄₇	—	4	16	no	—
120	C₂³×D₄	—	47	6	no	—
121	C₂⁴×C₄	—	47	4	no	—
122	C₂×#127	—	47	16	no	—
123	C₂×#127	—	47	8	no	—
124	C₂×C₄×D₄	—	47	48	no	—
125	C₂×C₄×D₄	—	94	48	no	—
126	C₂²×C₄⋊C₄	C₂×D₄₇	1	6	yes	—
127	C₂²×C₂²⋊C₄	—	47	12	no	—
128	C₂²×C₄²	—	47	3	no	—
129	C₂²×C₄²	—	47	3	no	—
130	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
131	#390	—	47	48	no	—
132	#384	C₂×D₄₇	1	8	yes	complemented
133	C₄³	—	47	8	no	—
134	C₄₇	C₂×C₄×#127	1	1	yes	characteristic, complemented, Sylow
135	C₂⁵	—	47	4	no	—
136	C₂²×D₄	—	47	48	no	—
137	C₂²×D₄	—	47	24	no	—
138	C₂²×D₄	—	94	24	no	—
139	C₂²×D₄	—	94	24	no	—
140	C₂³×C₄	—	47	1	no	—
141	C₂³×C₄	—	47	3	no	—
142	C₂³×C₄	—	47	16	no	—
143	C₂³×C₄	—	94	48	no	—
144	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
145	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
146	#127	—	47	64	no	—
147	C₄×D₄	—	94	192	no	—
148	C₂×C₄⋊C₄	—	2	24	no	—
149	C₂×C₄⋊C₄	C₂²×D₄₇	1	24	yes	—
150	C₂×C₂²⋊C₄	—	47	48	no	—
151	C₂×C₂²⋊C₄	—	94	48	no	—
152	C₂×C₄²	—	47	24	no	—
153	C₂×C₄²	—	47	12	no	—
154	C₂×C₄²	—	47	12	no	—
155	C₂×C₄²	—	94	12	no	—
156	C₂×C₄²	C₄×D₄₇	1	8	yes	complemented
157	C₂×C₄²	C₂²×D₄₇	1	4	yes	—
158	C₂⁴	—	47	16	no	—
159	C₂⁴	—	47	8	no	—
160	C₂⁴	—	94	24	no	—
161	C₂⁴	—	94	24	no	—
162	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
163	C₂×D₄	—	47	96	no	—
164	C₂×D₄	—	94	192	no	—
165	C₂×D₄	—	94	48	no	—
166	C₂×D₄	—	94	48	no	—
167	C₂²×C₄	—	2	24	no	—
168	C₂²×C₄	—	47	12	no	—
169	C₂²×C₄	—	47	2	no	—
170	C₂²×C₄	—	47	12	no	—
171	C₂²×C₄	—	47	6	no	—
172	C₂²×C₄	—	94	96	no	—
173	C₂²×C₄	—	94	12	no	—
174	C₂²×C₄	—	188	64	no	—
175	C₂²×C₄	C₂×C₄×D₄₇	1	12	yes	—
176	C₂²×C₄	D₄×D₄₇	1	12	yes	—
177	C₂²×C₄	#10625354	1	12	yes	—
178	C₂²×C₄	C₂³×D₄₇	1	8	yes	—
179	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
180	C₄⋊C₄	—	2	48	no	—
181	C₂²⋊C₄	—	94	96	no	—
182	C₄²	—	47	48	no	—
183	C₄²	—	94	24	no	—
184	C₄²	C₂×C₄×D₄₇	1	16	yes	complemented
185	C₂³	—	47	16	no	—
186	C₂³	—	94	96	no	—
187	C₂³	—	94	24	no	—
188	C₂³	—	94	24	no	—
189	C₂³	—	188	16	no	—
190	C₂³	—	188	48	no	—
191	C₂³	C₂²×C₄×D₄₇	1	2	yes	—
192	C₂³	C₂×D₄×D₄₇	1	6	yes	—
193	C₂³	C₂×#10625354	1	6	yes	—
194	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
195	D₄	—	94	192	no	—
196	C₂×C₄	—	2	24	no	—
197	C₂×C₄	—	4	16	no	—
198	C₂×C₄	—	47	16	no	—
199	C₂×C₄	—	47	12	no	—
200	C₂×C₄	—	47	12	no	—
201	C₂×C₄	—	94	24	no	—
202	C₂×C₄	—	94	12	no	—
203	C₂×C₄	—	188	64	no	—
204	C₂×C₄	#13093986	1	48	yes	complemented
205	C₂×C₄	C₂²×C₄×D₄₇	1	12	yes	—
206	C₂×C₄	C₂×D₄×D₄₇	1	12	yes	—
207	C₂×C₄	C₂×#10625354	1	12	yes	—
208	C₂²	—	94	48	no	—
209	C₂²	—	188	8	no	—
210	C₂²	—	188	24	no	—
211	C₂²	—	188	64	no	—
212	C₂²	C₂×#13093986	1	12	yes	—
213	C₂²	#127×D₄₇	1	4	yes	—
214	C₂²	#15689674	1	12	yes	—
215	C₂²	C₂³×C₄×D₄₇	1	1	yes	characteristic
216	C₂²	C₂²×D₄×D₄₇	1	3	yes	—
217	C₂²	C₂²×#10625354	1	3	yes	—
218	C₄	—	4	8	no	—
219	C₄	—	47	8	no	—
220	C₄	—	94	12	no	—
221	C₄	C₂×#13093986	1	24	yes	complemented
222	C₂	—	188	16	no	—
223	C₂	#20245950	1	8	yes	direct product factor
224	C₂	C₂²×#13093986	1	3	yes	—
225	C₂	C₂×#127×D₄₇	1	1	yes	characteristic
226	C₂	C₂×#15689674	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#384	C₂
socle	C₂⁴×C₄₇	C₂⁴
prime radical	C₂²×#13094021	C₂
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂×#384	D₄₇
hypocenter	C₄₇	C₂×C₄×#127
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
2. C₂²×#13093986	C₂	semidirect
3. #20245950	C₂	direct (N × H)
4. C₂×#15689024	C₂	semidirect
5. C₂×C₄²×Dic₄₇	C₂	semidirect
6. C₂×C₄₇×#384	C₂	semidirect
9. C₂×#13094021	C₄	semidirect
12. C₂×#13093986	C₂²	semidirect
16. #15689024	C₂²	semidirect
17. C₄²×Dic₄₇	C₂²	semidirect
20. C₄₇×#384	C₂²	semidirect
24. C₂²×D₁₈₈	C₂×C₄	semidirect
30. #13094021	C₂×C₄	semidirect
35. C₂×C₄×Dic₄₇	D₄	semidirect
42. C₂×C₄²×C₄₇	C₂×C₄	semidirect
63. C₂×D₁₈₈	C₂²×C₄	semidirect
67. C₄²×C₄₇	C₂²×C₄	semidirect
70. C₄×Dic₄₇	C₂×D₄	semidirect
83. C₂×C₄×C₄₇	C₄×D₄	semidirect
89. C₂×Dic₄₇	#127	semidirect
106. C₄×C₄₇	C₂×C₄×D₄	semidirect
108. Dic₄₇	C₂×#127	semidirect
113. C₂×#384	D₄₇	semidirect
115. C₂×C₄₇	C₄×#127	semidirect
132. #384	C₂×D₄₇	semidirect
134. C₄₇	C₂×C₄×#127	semidirect
156. C₂×C₄²	C₄×D₄₇	semidirect
184. C₄²	C₂×C₄×D₄₇	semidirect
204. C₂×C₄	#13093986	semidirect
221. C₄	C₂×#13093986	semidirect
223. C₂	#20245950	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×#13094021	2
C₂²×#13093986	2
#20245950	2
C₂×#15689024	2
C₂×C₄²×Dic₄₇	2
C₂×C₄₇×#384	2
C₂×C₄×#127	47

Subgroups — C2×#20245950 #26374789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245950 #26374789