Subgroups — C₂×#20245753 #26374502

Order 12032.

3654 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×Dic₄₇	C₂	1	2	yes	maximal
2	#20245753	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄⋊C₄×Dic₄₇	C₂	1	6	yes	maximal
4	C₂×C₄×C₄⋊Dic₄₇	C₂	1	3	yes	maximal
5	C₂×#15688969	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×Dic₄₇	C₄	1	32	yes	complemented
8	C₂×Q₈×Dic₄₇	C₂²	1	16	yes	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	6	yes	—
10	C₂²×C₄×Dic₄₇	C₂²	1	6	yes	—
11	C₂²×C₄×Dic₄₇	C₂²	1	6	yes	—
12	C₄⋊C₄×Dic₄₇	C₂²	1	48	yes	—
13	C₄×C₄⋊Dic₄₇	C₂²	1	24	yes	—
14	#15688969	C₂²	1	24	yes	—
15	C₂×C₄₇⋊#95	C₂²	1	1	yes	characteristic
16	C₂×C₄₇⋊#95	C₂²	1	3	yes	—
17	C₂×C₄₇⋊#95	C₂²	1	6	yes	—
18	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₄₇	C₄	1	16	yes	complemented
20	C₂×C₄×Q₈×C₄₇	C₂²	1	8	yes	—
21	C₂²×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
23	C₂³×Dic₄₇	C₂³	1	2	yes	—
24	C₂³×Dic₄₇	C₂³	1	6	yes	—
25	Q₈×Dic₄₇	C₂×C₄	1	128	yes	complemented
26	C₂×C₄⋊Dic₄₇	C₂×C₄	1	48	yes	—
27	C₂×C₄⋊Dic₄₇	C₂³	1	24	yes	—
28	C₂×C₄×Dic₄₇	—	4	12	no	—
29	C₂×C₄×Dic₄₇	C₂×C₄	1	48	yes	—
30	C₂×C₄×Dic₄₇	C₂³	1	24	yes	—
31	C₂×C₄×Dic₄₇	C₂³	1	24	yes	—
32	C₄₇⋊#95	C₂³	1	4	yes	—
33	C₄₇⋊#95	C₂³	1	12	yes	—
34	C₄₇⋊#95	C₂³	1	24	yes	—
35	C₂²×Q₈×C₄₇	C₂×C₄	1	16	yes	—
36	C₂²×Q₈×C₄₇	C₂×C₄	1	8	yes	—
37	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
40	C₂³×C₄×C₄₇	C₂³	1	3	yes	—
41	C₄×Q₈×C₄₇	C₂×C₄	1	64	yes	complemented
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	24	yes	—
43	C₂×C₄⋊C₄×C₄₇	C₂³	1	12	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	24	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	12	yes	—
46	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
47	C₂×Q₈×C₄₇	C₄²	1	64	yes	complemented
48	C₂×Q₈×C₄₇	C₂²×C₄	1	32	yes	—
49	C₂×Q₈×C₄₇	C₂²×C₄	1	16	yes	—
50	C₂²×C₄×C₄₇	—	2	6	no	—
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	6	no	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	12	yes	—
57	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	6	yes	—
60	C₂²×Dic₄₇	—	2	12	no	—
61	C₂²×Dic₄₇	—	2	4	no	—
62	C₂²×Dic₄₇	—	2	4	no	—
63	C₂²×Dic₄₇	—	2	12	no	—
64	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
65	C₂²×Dic₄₇	C₂²×C₄	1	24	yes	—
66	C₂²×Dic₄₇	C₂⁴	1	4	yes	—
67	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
68	C₄⋊C₄×C₄₇	C₂²×C₄	1	48	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	48	yes	—
70	C₄⋊Dic₄₇	C₂²×C₄	1	96	yes	—
71	C₄×Dic₄₇	—	4	24	no	—
72	C₄×Dic₄₇	C₂²×C₄	1	96	yes	—
73	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
74	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
75	C₂³×C₄₇	H(5,𝔽₂)	1	2	yes	—
76	C₂³×C₄₇	2₋¹⁺⁴	1	2	yes	—
77	C₂³×C₄₇	2₋¹⁺⁴	1	4	yes	—
78	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
79	Q₈×C₄₇	C₂×C₄²	1	64	yes	complemented
80	C₂×C₄×C₄₇	—	2	8	no	—
81	C₂×C₄×C₄₇	—	2	24	no	—
82	C₂×C₄×C₄₇	—	2	6	no	—
83	C₂×C₄×C₄₇	—	2	6	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	C₂×C₄²	1	48	yes	—
86	C₂×C₄×C₄₇	C₂³×C₄	1	24	yes	—
87	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
88	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
89	C₂×C₄×C₄₇	C₂³×C₄	1	12	yes	—
90	C₂×Dic₄₇	—	2	16	no	—
91	C₂×Dic₄₇	—	2	48	no	—
92	C₂×Dic₄₇	—	2	12	no	—
93	C₂×Dic₄₇	—	2	4	no	—
94	C₂×Dic₄₇	—	2	4	no	—
95	C₂×Dic₄₇	—	2	12	no	—
96	C₂×Dic₄₇	C₂³×C₄	1	8	yes	—
97	C₂×Dic₄₇	C₂³×C₄	1	24	yes	—
98	C₂×#2276	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	C₂²×C₄²	1	4	yes	—
100	C₂²×C₄₇	#519	1	8	yes	—
101	C₂²×C₄₇	#519	1	4	yes	—
102	C₂²×C₄₇	#520	1	8	yes	—
103	C₂²×C₄₇	#520	1	4	yes	—
104	C₂²×C₄₇	C₂⁴×C₄	1	2	yes	—
105	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
106	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
107	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
108	C₂²×C₄₇	C₂×2₋¹⁺⁴	1	2	yes	—
109	C₄×C₄₇	—	2	4	no	—
110	C₄×C₄₇	—	2	12	no	—
111	C₄×C₄₇	C₂²×C₄²	1	24	yes	—
112	Dic₄₇	—	2	8	no	—
113	Dic₄₇	—	2	24	no	—
114	C₂²×C₄×Q₈	—	47	2	no	—
115	C₂²×C₄×Q₈	D₄₇	1	1	yes	characteristic
116	#2276	—	47	16	no	—
117	C₂×C₄×C₄⋊C₄	—	47	3	no	—
118	C₂×C₄×C₄⋊C₄	—	47	6	no	—
119	C₂×#376	—	47	3	no	—
120	C₂×C₄₇	#2276	1	8	yes	complemented
121	C₂×C₄₇	C₂³×C₄²	1	1	yes	characteristic
122	C₂×C₄₇	C₂×#519	1	2	yes	—
123	C₂×C₄₇	C₂×#519	1	1	yes	characteristic
124	C₂×C₄₇	C₂×#520	1	2	yes	—
125	C₂×C₄₇	C₂×#520	1	1	yes	characteristic
126	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
127	C₂×C₄×Q₈	—	47	32	no	—
128	C₂×C₄×Q₈	—	47	16	no	—
129	C₂×C₄×Q₈	Dic₄₇	1	16	yes	complemented
130	C₂×C₄×Q₈	C₂×D₄₇	1	8	yes	—
131	C₂²×C₄⋊C₄	—	47	6	no	—
132	C₂²×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
133	C₂²×C₄²	—	47	6	no	—
134	C₂²×C₄²	—	47	6	no	—
135	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
136	C₄×C₄⋊C₄	—	47	48	no	—
137	C₄×C₄⋊C₄	—	47	24	no	—
138	#376	—	47	24	no	—
139	C₂×#95	—	47	1	no	—
140	C₂×#95	—	47	3	no	—
141	C₂×#95	—	47	6	no	—
142	C₄₇	C₂×#2276	1	1	yes	characteristic, complemented, Sylow
143	C₂²×Q₈	C₄×D₄₇	1	16	yes	—
144	C₂²×Q₈	C₂×Dic₄₇	1	8	yes	—
145	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
146	C₂³×C₄	—	47	6	no	—
147	C₂³×C₄	—	47	2	no	—
148	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
149	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
150	C₂³×C₄	C₂²×D₄₇	1	3	yes	—
151	C₄×Q₈	—	47	128	no	—
152	C₄×Q₈	C₂×Dic₄₇	1	64	yes	complemented
153	C₂×C₄⋊C₄	—	47	48	no	—
154	C₂×C₄⋊C₄	—	47	24	no	—
155	C₂×C₄⋊C₄	C₂×Dic₄₇	1	24	yes	—
156	C₂×C₄⋊C₄	C₂²×D₄₇	1	12	yes	—
157	C₂×C₄²	—	47	48	no	—
158	C₂×C₄²	—	47	24	no	—
159	C₂×C₄²	—	47	24	no	—
160	C₂×C₄²	—	188	12	no	—
161	C₂×C₄²	C₂×Dic₄₇	1	24	yes	—
162	C₂×C₄²	C₂²×D₄₇	1	12	yes	—
163	#95	—	47	4	no	—
164	#95	—	47	24	no	—
165	#95	—	47	12	no	—
166	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
167	C₂×Q₈	C₄×Dic₄₇	1	64	yes	complemented
168	C₂×Q₈	C₂×C₄×D₄₇	1	32	yes	—
169	C₂×Q₈	C₂²×Dic₄₇	1	16	yes	—
170	C₂²×C₄	—	2	6	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	47	8	no	—
174	C₂²×C₄	—	47	24	no	—
175	C₂²×C₄	—	47	4	no	—
176	C₂²×C₄	—	47	12	no	—
177	C₂²×C₄	—	94	12	no	—
178	C₂²×C₄	—	94	4	no	—
179	C₂²×C₄	—	94	4	no	—
180	C₂²×C₄	—	94	12	no	—
181	C₂²×C₄	C₂×C₄×D₄₇	1	24	yes	—
182	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
183	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
184	C₂²×C₄	C₂²×Dic₄₇	1	12	yes	—
185	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
186	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
187	C₂²×C₄	C₂³×D₄₇	1	6	yes	—
188	C₄⋊C₄	—	47	96	no	—
189	C₄⋊C₄	C₂²×Dic₄₇	1	48	yes	—
190	C₄²	—	47	96	no	—
191	C₄²	—	188	24	no	—
192	C₄²	C₂²×Dic₄₇	1	48	yes	—
193	C₂³	C₂²×C₄×D₄₇	1	4	yes	—
194	C₂³	C₂³×Dic₄₇	1	2	yes	—
195	C₂³	#13094070	1	2	yes	—
196	C₂³	#13094071	1	2	yes	—
197	C₂³	#13094072	1	4	yes	—
198	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
199	Q₈	C₂×C₄×Dic₄₇	1	64	yes	complemented
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	2	24	no	—
202	C₂×C₄	—	2	6	no	—
203	C₂×C₄	—	2	6	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	47	8	no	—
206	C₂×C₄	—	47	24	no	—
207	C₂×C₄	—	94	16	no	—
208	C₂×C₄	—	94	48	no	—
209	C₂×C₄	—	94	12	no	—
210	C₂×C₄	—	94	4	no	—
211	C₂×C₄	—	94	4	no	—
212	C₂×C₄	—	94	12	no	—
213	C₂×C₄	C₂×C₄×Dic₄₇	1	48	yes	—
214	C₂×C₄	C₂²×C₄×D₄₇	1	24	yes	—
215	C₂×C₄	C₂³×Dic₄₇	1	4	yes	—
216	C₂×C₄	C₂³×Dic₄₇	1	12	yes	—
217	C₂×C₄	C₂³×Dic₄₇	1	12	yes	—
218	C₂²	C₂²×C₄×Dic₄₇	1	4	yes	—
219	C₂²	#15689599	1	8	yes	—
220	C₂²	#15689601	1	4	yes	—
221	C₂²	#15689605	1	8	yes	—
222	C₂²	#15689606	1	4	yes	—
223	C₂²	C₂³×C₄×D₄₇	1	2	yes	—
224	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
225	C₂²	C₂×#13094070	1	1	yes	characteristic
226	C₂²	C₂×#13094071	1	1	yes	characteristic
227	C₂²	C₂×#13094072	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	2	12	no	—
230	C₄	—	94	8	no	—
231	C₄	—	94	24	no	—
232	C₄	C₂²×C₄×Dic₄₇	1	24	yes	—
233	C₂	#20245753	1	8	yes	direct product factor
234	C₂	C₂³×C₄×Dic₄₇	1	1	yes	characteristic
235	C₂	C₂×#15689599	1	2	yes	—
236	C₂	C₂×#15689601	1	1	yes	characteristic
237	C₂	C₂×#15689605	1	2	yes	—
238	C₂	C₂×#15689606	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂×C₄₇	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂²×C₄×Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂⁴
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₄₇
hypocenter	C₄₇	C₂×#2276
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #20245753	C₂	direct (N × H)
7. C₂×Q₈×Dic₄₇	C₄	semidirect
19. C₂×C₄×Q₈×C₄₇	C₄	semidirect
25. Q₈×Dic₄₇	C₂×C₄	semidirect
41. C₄×Q₈×C₄₇	C₂×C₄	semidirect
47. C₂×Q₈×C₄₇	C₄²	semidirect
79. Q₈×C₄₇	C₂×C₄²	semidirect
120. C₂×C₄₇	#2276	semidirect
129. C₂×C₄×Q₈	Dic₄₇	semidirect
142. C₄₇	C₂×#2276	semidirect
152. C₄×Q₈	C₂×Dic₄₇	semidirect
167. C₂×Q₈	C₄×Dic₄₇	semidirect
199. Q₈	C₂×C₄×Dic₄₇	semidirect
233. C₂	#20245753	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×Q₈×Dic₄₇	2
#20245753	2
C₂×C₄⋊C₄×Dic₄₇	2
C₂×C₄×C₄⋊Dic₄₇	2
C₂×#15688969	2
C₂²×C₄×Q₈×C₄₇	2
C₂×#2276	47

Subgroups — C2×#20245753 #26374502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245753 #26374502