Subgroups — C₂×#20245970 #26374833

Order 12032.

7710 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093983	C₂	1	2	yes	maximal, complemented
2	#20245970	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689026	C₂	1	8	yes	maximal, complemented
4	C₂×#15688979	C₂	1	4	yes	maximal, complemented
5	C₂⁴×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093983	—	2	16	no	—
7	C₂×#13093983	C₂²	1	16	yes	complemented
8	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	complemented
9	C₂²×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
10	C₂²×C₂²⋊Dic₄₇	C₂²	1	16	yes	complemented
11	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	complemented
12	#15689026	C₂²	1	64	yes	complemented
13	#15688979	C₂²	1	32	yes	complemented
14	C₂⁶×C₄₇	C₂²	1	2	yes	—
15	C₂³×D₄×C₄₇	—	2	16	no	—
16	C₂³×D₄×C₄₇	C₄	1	16	yes	complemented
17	C₂³×D₄×C₄₇	C₂²	1	4	yes	—
18	C₂³×D₄×C₄₇	C₂²	1	8	yes	—
19	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂³×Dic₄₇	C₂³	1	8	yes	—
21	#13093983	—	2	64	no	—
22	C₂×C₄⋊Dic₄₇	—	2	16	no	—
23	C₂×C₄⋊Dic₄₇	C₂³	1	16	yes	complemented
24	C₂×C₂²⋊Dic₄₇	—	2	16	no	—
25	C₂×C₂²⋊Dic₄₇	—	4	32	no	—
26	C₂×C₂²⋊Dic₄₇	C₂³	1	16	yes	—
27	C₂×C₂²⋊Dic₄₇	C₂³	1	64	yes	complemented
28	C₂×C₄×Dic₄₇	—	2	16	no	—
29	C₂×C₄×Dic₄₇	C₂³	1	16	yes	complemented
30	C₂⁵×C₄₇	—	2	16	no	—
31	C₂⁵×C₄₇	—	4	16	no	—
32	C₂⁵×C₄₇	C₂×C₄	1	16	yes	—
33	C₂⁵×C₄₇	C₂³	1	4	yes	—
34	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
35	C₂⁵×C₄₇	C₂³	1	8	yes	—
36	C₂²×D₄×C₄₇	—	2	128	no	—
37	C₂²×D₄×C₄₇	—	2	16	no	—
38	C₂²×D₄×C₄₇	—	2	16	no	—
39	C₂²×D₄×C₄₇	—	2	64	no	—
40	C₂²×D₄×C₄₇	C₂×C₄	1	32	yes	—
41	C₂²×D₄×C₄₇	C₂×C₄	1	64	yes	complemented
42	C₂²×D₄×C₄₇	C₂³	1	16	yes	—
43	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
44	C₂³×C₄×C₄₇	D₄	1	16	yes	—
45	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
47	C₂⁴×C₄₇	—	2	8	no	—
48	C₂⁴×C₄₇	—	2	64	no	—
49	C₂⁴×C₄₇	—	2	32	no	—
50	C₂⁴×C₄₇	—	2	16	no	—
51	C₂⁴×C₄₇	—	2	32	no	—
52	C₂⁴×C₄₇	—	4	64	no	—
53	C₂⁴×C₄₇	—	4	32	no	—
54	C₂⁴×C₄₇	—	4	128	no	—
55	C₂⁴×C₄₇	C₂²×C₄	1	16	yes	—
56	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
57	C₂⁴×C₄₇	C₂²×C₄	1	32	yes	—
58	C₂⁴×C₄₇	C₂×D₄	1	8	yes	—
59	C₂⁴×C₄₇	C₂⁴	1	8	yes	—
60	C₂⁴×C₄₇	C₂⁴	1	2	yes	—
61	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂×D₄×C₄₇	—	2	128	no	—
63	C₂×D₄×C₄₇	—	2	256	no	—
64	C₂×D₄×C₄₇	—	2	32	no	—
65	C₂×D₄×C₄₇	—	2	32	no	—
66	C₂×D₄×C₄₇	C₂²×C₄	1	64	yes	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	64	yes	complemented
68	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	16	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	8	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	32	yes	—
73	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
74	C₂²×Dic₄₇	—	2	16	no	—
75	C₂²×Dic₄₇	—	2	32	no	—
76	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
77	C₂²⋊Dic₄₇	—	2	32	no	—
78	C₂²⋊Dic₄₇	—	4	64	no	—
79	C₄⋊Dic₄₇	—	2	32	no	—
80	C₄×Dic₄₇	—	2	32	no	—
81	C₂³×C₄₇	—	2	16	no	—
82	C₂³×C₄₇	—	2	32	no	—
83	C₂³×C₄₇	—	2	8	no	—
84	C₂³×C₄₇	—	2	64	no	—
85	C₂³×C₄₇	—	2	32	no	—
86	C₂³×C₄₇	—	2	64	no	—
87	C₂³×C₄₇	—	2	16	no	—
88	C₂³×C₄₇	—	2	32	no	—
89	C₂³×C₄₇	—	4	64	no	—
90	C₂³×C₄₇	—	4	256	no	—
91	C₂³×C₄₇	—	4	128	no	—
92	C₂³×C₄₇	—	4	64	no	—
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	16	yes	—
94	C₂³×C₄₇	C₂³×C₄	1	16	yes	—
95	C₂³×C₄₇	C₂³×C₄	1	4	yes	—
96	C₂³×C₄₇	C₂³×C₄	1	2	yes	—
97	C₂³×C₄₇	C₂²×D₄	1	8	yes	—
98	C₂³×C₄₇	C₂²×D₄	1	4	yes	—
99	C₂³×C₄₇	H(5,𝔽₂)	1	8	yes	—
100	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
101	D₄×C₄₇	—	2	128	no	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	32	yes	—
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	64	yes	complemented
104	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
105	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
106	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
107	C₂×Dic₄₇	—	2	16	no	—
108	C₂×Dic₄₇	—	2	32	no	—
109	C₂×Dic₄₇	—	4	32	no	—
110	C₂×#2309	—	47	1	no	maximal, Sylow
111	C₂²×C₄₇	—	2	32	no	—
112	C₂²×C₄₇	—	2	8	no	—
113	C₂²×C₄₇	—	2	16	no	—
114	C₂²×C₄₇	—	2	16	no	—
115	C₂²×C₄₇	—	2	32	no	—
116	C₂²×C₄₇	—	4	128	no	—
117	C₂²×C₄₇	—	4	64	no	—
118	C₂²×C₄₇	—	4	32	no	—
119	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	8	yes	—
120	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	4	yes	—
121	C₂²×C₄₇	#518	1	8	yes	—
122	C₂²×C₄₇	#535	1	16	yes	—
123	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
124	C₂²×C₄₇	C₂³×D₄	1	2	yes	—
125	C₂²×C₄₇	C₂×H(5,𝔽₂)	1	4	yes	—
126	C₄×C₄₇	C₂²×C₂²⋊C₄	1	16	yes	—
127	Dic₄₇	—	4	16	no	—
128	C₂⁴×D₄	D₄₇	1	1	yes	characteristic
129	C₂²×#117	—	47	2	no	—
130	#2309	—	47	16	no	—
131	C₂×#390	—	47	8	no	—
132	C₂×#379	—	47	4	no	—
133	C₂×C₄₇	—	2	8	no	—
134	C₂×C₄₇	—	4	16	no	—
135	C₂×C₄₇	#2309	1	8	yes	complemented
136	C₂×C₄₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄₇	C₂×#518	1	2	yes	—
138	C₂×C₄₇	C₂×#535	1	4	yes	—
139	C₂⁶	C₂×D₄₇	1	2	yes	—
140	C₂³×D₄	—	2	16	no	—
141	C₂³×D₄	Dic₄₇	1	16	yes	complemented
142	C₂³×D₄	C₂×D₄₇	1	4	yes	—
143	C₂³×D₄	C₂×D₄₇	1	8	yes	—
144	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
145	C₂×#117	—	47	16	no	—
146	C₂×#117	—	94	16	no	—
147	C₂²×C₄⋊C₄	—	47	4	no	—
148	C₂²×C₂²⋊C₄	—	47	4	no	—
149	C₂²×C₂²⋊C₄	—	47	16	no	—
150	C₂²×C₄²	—	47	4	no	—
151	#390	—	47	64	no	—
152	#379	—	47	32	no	—
153	C₄₇	C₂×#2309	1	1	yes	characteristic, complemented, Sylow
154	C₂⁵	—	2	16	no	—
155	C₂⁵	—	4	16	no	—
156	C₂⁵	C₂×Dic₄₇	1	16	yes	—
157	C₂⁵	C₂²×D₄₇	1	4	yes	—
158	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
159	C₂⁵	C₂²×D₄₇	1	8	yes	—
160	C₂²×D₄	—	2	128	no	—
161	C₂²×D₄	—	2	16	no	—
162	C₂²×D₄	—	2	16	no	—
163	C₂²×D₄	—	2	64	no	—
164	C₂²×D₄	C₂×Dic₄₇	1	32	yes	—
165	C₂²×D₄	C₂×Dic₄₇	1	64	yes	complemented
166	C₂²×D₄	C₂²×D₄₇	1	16	yes	—
167	C₂³×C₄	—	47	8	no	—
168	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
169	C₂³×C₄	C₂²⋊D₄₇	1	16	yes	—
170	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
171	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
172	#117	—	94	64	no	—
173	C₂×C₄⋊C₄	—	47	16	no	—
174	C₂×C₄⋊C₄	—	94	16	no	—
175	C₂×C₂²⋊C₄	—	47	16	no	—
176	C₂×C₂²⋊C₄	—	47	64	no	—
177	C₂×C₂²⋊C₄	—	94	16	no	—
178	C₂×C₂²⋊C₄	—	188	32	no	—
179	C₂×C₄²	—	47	16	no	—
180	C₂×C₄²	—	94	16	no	—
181	C₂⁴	—	2	8	no	—
182	C₂⁴	—	2	64	no	—
183	C₂⁴	—	2	32	no	—
184	C₂⁴	—	2	16	no	—
185	C₂⁴	—	2	32	no	—
186	C₂⁴	—	4	64	no	—
187	C₂⁴	—	4	32	no	—
188	C₂⁴	—	4	128	no	—
189	C₂⁴	C₂²×Dic₄₇	1	16	yes	—
190	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
191	C₂⁴	C₂²×Dic₄₇	1	32	yes	—
192	C₂⁴	C₂×C₂²⋊D₄₇	1	8	yes	—
193	C₂⁴	C₂³×D₄₇	1	8	yes	—
194	C₂⁴	C₂³×D₄₇	1	2	yes	—
195	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
196	C₂×D₄	—	2	128	no	—
197	C₂×D₄	—	2	256	no	—
198	C₂×D₄	—	2	32	no	—
199	C₂×D₄	—	2	32	no	—
200	C₂×D₄	C₂²×Dic₄₇	1	64	yes	—
201	C₂²×C₄	—	47	16	no	—
202	C₂²×C₄	—	94	16	no	—
203	C₂²×C₄	—	94	32	no	—
204	C₂²×C₄	C₂²⋊Dic₄₇	1	64	yes	complemented
205	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
206	C₂²×C₄	C₂²×Dic₄₇	1	16	yes	—
207	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₄₇	1	8	yes	—
209	C₂²×C₄	C₂×C₂²⋊D₄₇	1	32	yes	—
210	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
211	C₄⋊C₄	—	94	32	no	—
212	C₂²⋊C₄	—	94	32	no	—
213	C₂²⋊C₄	—	188	64	no	—
214	C₄²	—	94	32	no	—
215	C₂³	—	2	16	no	—
216	C₂³	—	2	32	no	—
217	C₂³	—	2	8	no	—
218	C₂³	—	2	64	no	—
219	C₂³	—	2	32	no	—
220	C₂³	—	2	64	no	—
221	C₂³	—	2	16	no	—
222	C₂³	—	2	32	no	—
223	C₂³	—	4	64	no	—
224	C₂³	—	4	128	no	—
225	C₂³	—	4	256	no	—
226	C₂³	—	4	64	no	—
227	C₂³	C₂×C₂²⋊Dic₄₇	1	16	yes	—
228	C₂³	C₂³×Dic₄₇	1	16	yes	—
229	C₂³	C₂³×Dic₄₇	1	4	yes	—
230	C₂³	C₂³×Dic₄₇	1	2	yes	—
231	C₂³	C₂²×C₂²⋊D₄₇	1	8	yes	—
232	C₂³	C₂²×C₂²⋊D₄₇	1	4	yes	—
233	C₂³	#13094069	1	8	yes	—
234	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
235	D₄	—	2	128	no	—
236	C₂×C₄	—	94	16	no	—
237	C₂×C₄	—	94	32	no	—
238	C₂×C₄	—	188	32	no	—
239	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	32	yes	—
240	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	64	yes	complemented
241	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
242	C₂×C₄	C₂²×C₂²⋊D₄₇	1	8	yes	—
243	C₂×C₄	C₂²×C₂²⋊D₄₇	1	8	yes	—
244	C₂²	—	2	8	no	—
245	C₂²	—	2	16	no	—
246	C₂²	—	2	32	no	—
247	C₂²	—	2	16	no	—
248	C₂²	—	2	32	no	—
249	C₂²	—	4	128	no	—
250	C₂²	—	4	32	no	—
251	C₂²	—	4	64	no	—
252	C₂²	C₂²×C₂²⋊Dic₄₇	1	8	yes	—
253	C₂²	C₂²×C₂²⋊Dic₄₇	1	4	yes	—
254	C₂²	#15689598	1	8	yes	—
255	C₂²	#15689690	1	16	yes	—
256	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
257	C₂²	C₂³×C₂²⋊D₄₇	1	2	yes	—
258	C₂²	C₂×#13094069	1	4	yes	—
259	C₄	—	188	16	no	—
260	C₄	C₂²×C₂²⋊Dic₄₇	1	16	yes	—
261	C₂	—	2	8	no	—
262	C₂	—	4	16	no	—
263	C₂	#20245970	1	8	yes	direct product factor
264	C₂	C₂³×C₂²⋊Dic₄₇	1	1	yes	characteristic
265	C₂	C₂×#15689598	1	2	yes	—
266	C₂	C₂×#15689690	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₇
Fitting, prime radical	C₂⁴×D₄×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂⁴
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂⁴×D₄	D₄₇
hypocenter	C₄₇	C₂×#2309
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093983	C₂	semidirect
2. #20245970	C₂	direct (N × H)
3. C₂×#15689026	C₂	semidirect
4. C₂×#15688979	C₂	semidirect
7. C₂×#13093983	C₂²	semidirect
8. C₂²×C₄⋊Dic₄₇	C₂²	semidirect
10. C₂²×C₂²⋊Dic₄₇	C₂²	semidirect
11. C₂²×C₄×Dic₄₇	C₂²	semidirect
12. #15689026	C₂²	semidirect
13. #15688979	C₂²	semidirect
16. C₂³×D₄×C₄₇	C₄	semidirect
23. C₂×C₄⋊Dic₄₇	C₂³	semidirect
27. C₂×C₂²⋊Dic₄₇	C₂³	semidirect
29. C₂×C₄×Dic₄₇	C₂³	semidirect
41. C₂²×D₄×C₄₇	C₂×C₄	semidirect
67. C₂²×C₄×C₄₇	C₂²⋊C₄	semidirect
103. C₂×C₄×C₄₇	C₂×C₂²⋊C₄	semidirect
135. C₂×C₄₇	#2309	semidirect
141. C₂³×D₄	Dic₄₇	semidirect
153. C₄₇	C₂×#2309	semidirect
165. C₂²×D₄	C₂×Dic₄₇	semidirect
204. C₂²×C₄	C₂²⋊Dic₄₇	semidirect
240. C₂×C₄	C₂×C₂²⋊Dic₄₇	semidirect
263. C₂	#20245970	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×#13093983	2
#20245970	2
C₂×#15689026	2
C₂×#15688979	2
C₂⁴×D₄×C₄₇	2
C₂×#2309	47

Subgroups — C2×#20245970 #26374833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245970 #26374833