Subgroups — C₂×#20245420 #26370499

Order 12032.

870 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093965	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245420	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688924	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₁₈₈	—	2	1	no	—
9	C₂×#13093965	—	2	8	no	—
10	C₂×#13093965	C₂²	1	8	yes	complemented
11	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
12	C₄×D₃₇₆	—	2	2	no	—
13	C₄×D₃₇₆	C₂²	1	4	yes	complemented
14	C₂×#13093958	C₂²	1	1	yes	characteristic
15	C₂×#13093946	—	2	1	no	—
16	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	complemented
17	#15688924	C₂²	1	8	yes	complemented
18	C₂²×D₈×C₄₇	—	2	1	no	—
19	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
20	C₂×#127×C₄₇	—	2	1	no	—
21	C₄₇×#493	—	2	2	no	—
22	C₄₇×#493	C₂²	1	4	yes	complemented
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂²×D₁₈₈	—	2	1	no	—
25	C₂²×C₄×D₄₇	—	4	1	no	—
26	C₂×D₃₇₆	—	2	1	no	—
27	C₂×D₃₇₆	—	2	2	no	—
28	C₂×D₃₇₆	D₄	1	4	yes	complemented
29	C₂×D₃₇₆	C₂³	1	2	yes	—
30	C₄×D₁₈₈	—	2	4	no	—
31	C₄×D₁₈₈	—	4	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	1	no	—
33	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
34	#13093965	—	2	16	no	—
35	C₂×C₄₇⋊C₁₆	—	2	4	no	—
36	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
37	#13093958	—	2	1	no	—
38	#13093958	C₂³	1	2	yes	—
39	#13093946	—	2	4	no	—
40	C₂²×D₄×C₄₇	—	2	2	no	—
41	C₂²×D₄×C₄₇	—	4	1	no	—
42	C₂²×D₄×C₄₇	—	4	1	no	—
43	C₂×D₈×C₄₇	—	2	4	no	—
44	C₂×D₈×C₄₇	—	2	2	no	—
45	C₂×D₈×C₄₇	—	2	4	no	—
46	C₂×D₈×C₄₇	—	2	4	no	—
47	C₂×D₈×C₄₇	—	4	4	no	—
48	C₂×D₈×C₄₇	C₂³	1	4	yes	—
49	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	#127×C₄₇	—	2	4	no	—
52	#127×C₄₇	—	4	2	no	—
53	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₄₇	D₄	1	2	yes	—
55	C₄×C₈×C₄₇	C₂³	1	2	yes	—
56	C₂⁴×C₄₇	—	4	2	no	—
57	C₂³×D₄₇	—	4	1	no	—
58	C₂×D₄×C₄₇	—	2	4	no	—
59	C₂×D₄×C₄₇	—	4	4	no	—
60	C₂×D₄×C₄₇	—	4	4	no	—
61	C₂×D₄×C₄₇	—	4	4	no	—
62	C₂×D₄×C₄₇	—	4	2	no	—
63	C₂×D₄×C₄₇	—	4	4	no	—
64	C₂×D₄×C₄₇	—	8	4	no	—
65	C₂×D₄×C₄₇	—	8	4	no	—
66	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	4	1	no	—
70	C₂×D₁₈₈	—	2	4	no	—
71	C₂×D₁₈₈	—	2	2	no	—
72	C₂×D₁₈₈	—	4	1	no	—
73	C₂×D₁₈₈	—	4	2	no	—
74	C₂×C₄×D₄₇	—	4	4	no	—
75	C₂×C₄×D₄₇	—	8	4	no	—
76	D₈×C₄₇	—	2	4	no	—
77	D₈×C₄₇	—	2	4	no	—
78	D₈×C₄₇	—	4	8	no	—
79	D₈×C₄₇	—	4	4	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₄₇	#41	1	2	yes	—
85	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₄²×C₄₇	D₈	1	2	yes	—
87	C₄²×C₄₇	C₂×D₄	1	2	yes	—
88	(C₂×D₄₇)⋊C₄	—	4	4	no	—
89	C₄⋊Dic₄₇	—	2	2	no	—
90	C₄⋊Dic₄₇	—	4	1	no	—
91	D₃₇₆	—	2	4	no	—
92	D₃₇₆	—	2	2	no	—
93	D₃₇₆	C₂×D₄	1	4	yes	complemented
94	C₄₇⋊C₁₆	—	2	4	no	—
95	C₂³×C₄₇	—	4	4	no	—
96	C₂³×C₄₇	—	8	4	no	—
97	C₂³×C₄₇	—	8	4	no	—
98	C₂³×C₄₇	—	8	4	no	—
99	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₄₇	—	4	2	no	—
101	C₂²×D₄₇	—	4	4	no	—
102	C₂²×D₄₇	—	8	2	no	—
103	C₂²×D₄₇	—	8	2	no	—
104	D₄×C₄₇	—	4	4	no	—
105	D₄×C₄₇	—	4	4	no	—
106	D₄×C₄₇	—	8	8	no	—
107	D₄×C₄₇	—	8	4	no	—
108	D₄×C₄₇	—	8	8	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	D₁₆	1	4	yes	complemented
111	C₂×C₄×C₄₇	#121	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₄₇	—	4	2	no	—
118	C₂×Dic₄₇	—	8	2	no	—
119	D₁₈₈	—	2	4	no	—
120	D₁₈₈	—	4	2	no	—
121	D₁₈₈	—	4	4	no	—
122	C₄×D₄₇	—	8	8	no	—
123	C₈×C₄₇	—	2	2	no	—
124	C₈×C₄₇	#121	1	2	yes	—
125	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
127	C₂×#2206	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	—	8	4	no	—
129	C₂²×C₄₇	—	8	4	no	—
130	C₂²×C₄₇	—	8	4	no	—
131	C₂²×C₄₇	—	16	8	no	—
132	C₂²×C₄₇	#459	1	2	yes	—
133	C₂²×C₄₇	C₂×D₁₆	1	2	yes	—
134	C₂²×C₄₇	#509	1	2	yes	—
135	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
136	C₂×D₄₇	—	4	4	no	—
137	C₂×D₄₇	—	8	2	no	—
138	C₂×D₄₇	—	8	2	no	—
139	C₂×D₄₇	—	8	8	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	#459	1	2	yes	—
142	C₄×C₄₇	C₂×D₁₆	1	4	yes	complemented
143	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
144	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
145	Dic₄₇	—	8	2	no	—
146	C₂²×D₁₆	—	47	2	no	—
147	C₂×#493	D₄₇	1	1	yes	characteristic, complemented
148	C₂×C₄×D₈	—	47	1	no	—
149	#2206	—	47	8	no	—
150	C₂×C₄⋊C₁₆	—	47	1	no	—
151	C₂×#357	—	47	2	no	—
152	C₂×C₄₇	—	16	4	no	—
153	C₂×C₄₇	#2206	1	4	yes	complemented
154	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
155	C₂×C₄₇	C₂²×D₁₆	1	1	yes	characteristic
156	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
157	D₄₇	—	8	4	no	—
158	C₂²×D₈	—	2	1	no	—
159	C₂²×D₈	—	47	1	no	—
160	C₂²×D₈	C₂×D₄₇	1	2	yes	—
161	C₂×#127	—	2	1	no	—
162	C₂×C₄×D₄	—	94	1	no	—
163	C₂×D₁₆	—	47	8	no	—
164	C₂×D₁₆	—	94	8	no	—
165	C₂²×C₁₆	—	47	2	no	—
166	#493	—	2	2	no	—
167	#493	C₂×D₄₇	1	4	yes	complemented
168	C₄×D₈	—	47	4	no	—
169	C₄×D₈	—	94	2	no	—
170	C₂×#107	—	47	1	no	—
171	C₂×#102	—	94	1	no	—
172	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
173	C₄⋊C₁₆	—	47	4	no	—
174	#357	—	47	8	no	—
175	C₄₇	C₂×#2206	1	1	yes	characteristic, complemented, Sylow
176	C₂²×D₄	—	2	2	no	—
177	C₂²×D₄	—	4	1	no	—
178	C₂²×D₄	—	4	1	no	—
179	C₂²×D₄	—	94	1	no	—
180	C₂³×C₄	—	188	1	no	—
181	C₂×D₈	—	2	4	no	—
182	C₂×D₈	—	2	4	no	—
183	C₂×D₈	—	2	4	no	—
184	C₂×D₈	—	2	2	no	—
185	C₂×D₈	—	4	4	no	—
186	C₂×D₈	—	47	4	no	—
187	C₂×D₈	—	47	2	no	—
188	C₂×D₈	—	94	2	no	—
189	C₂×D₈	—	94	1	no	—
190	C₂×D₈	C₂²×D₄₇	1	4	yes	—
191	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
193	#127	—	2	4	no	—
194	#127	—	4	2	no	—
195	C₄×D₄	—	94	4	no	—
196	C₄×D₄	—	188	2	no	—
197	C₂×C₄⋊C₄	—	94	1	no	—
198	C₂×C₂²⋊C₄	—	188	1	no	—
199	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
200	D₁₆	—	94	16	no	—
201	C₂×C₁₆	—	47	4	no	—
202	C₂×C₁₆	—	94	4	no	—
203	#107	—	47	2	no	—
204	#107	—	94	1	no	—
205	#102	—	94	4	no	—
206	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
207	C₄×C₈	C₂²×D₄₇	1	2	yes	—
208	C₂⁴	—	4	2	no	—
209	C₂⁴	—	188	1	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	4	4	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	—	4	4	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	4	no	—
216	C₂×D₄	—	8	4	no	—
217	C₂×D₄	—	8	4	no	—
218	C₂×D₄	—	94	4	no	—
219	C₂×D₄	—	94	2	no	—
220	C₂×D₄	—	188	1	no	—
221	C₂×D₄	—	188	2	no	—
222	C₂²×C₄	—	376	4	no	—
223	C₂²×C₄	—	188	1	no	—
224	C₂²×C₄	—	188	4	no	—
225	C₂²×C₄	#10625334	1	1	yes	characteristic
226	C₂²×C₄	#10625334	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
228	D₈	—	2	4	no	—
229	D₈	—	2	4	no	—
230	D₈	—	4	8	no	—
231	D₈	—	4	4	no	—
232	D₈	—	47	4	no	—
233	D₈	—	94	4	no	—
234	D₈	—	94	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	D₄×D₄₇	1	2	yes	—
237	C₂×C₈	#10625352	1	2	yes	—
238	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
239	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
240	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
241	C₄⋊C₄	—	94	2	no	—
242	C₄⋊C₄	—	188	1	no	—
243	C₂²⋊C₄	—	188	4	no	—
244	C₄²	#10625334	1	2	yes	—
245	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
246	C₁₆	—	94	4	no	—
247	C₂³	—	4	4	no	—
248	C₂³	—	8	4	no	—
249	C₂³	—	8	4	no	—
250	C₂³	—	8	4	no	—
251	C₂³	—	376	2	no	—
252	C₂³	—	376	2	no	—
253	C₂³	—	188	4	no	—
254	C₂³	—	188	2	no	—
255	C₂³	C₂×#10625334	1	1	yes	characteristic
256	D₄	—	4	4	no	—
257	D₄	—	4	4	no	—
258	D₄	—	8	8	no	—
259	D₄	—	8	8	no	—
260	D₄	—	8	4	no	—
261	D₄	—	94	4	no	—
262	D₄	—	188	4	no	—
263	D₄	—	188	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	376	2	no	—
266	C₂×C₄	—	376	8	no	—
267	C₂×C₄	—	188	2	no	—
268	C₂×C₄	#13093965	1	4	yes	complemented
269	C₂×C₄	#13093995	1	2	yes	—
270	C₂×C₄	C₂×#10625334	1	2	yes	—
271	C₂×C₄	C₂×#10625334	1	2	yes	—
272	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
273	C₂×C₄	#13094050	1	2	yes	—
274	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	#13093995	1	2	yes	—
277	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
278	C₈	C₂×#10625352	1	1	yes	characteristic
279	C₂²	—	8	4	no	—
280	C₂²	—	8	4	no	—
281	C₂²	—	8	4	no	—
282	C₂²	—	16	8	no	—
283	C₂²	—	376	8	no	—
284	C₂²	—	376	2	no	—
285	C₂²	—	376	2	no	—
286	C₂²	—	188	4	no	—
287	C₂²	#15689315	1	2	yes	—
288	C₂²	C₂×#13093965	1	2	yes	—
289	C₂²	#15689559	1	2	yes	—
290	C₂²	C₂²×#10625334	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	376	2	no	—
293	C₄	#15689315	1	2	yes	—
294	C₄	C₂×#13093965	1	4	yes	complemented
295	C₄	C₂×#13093995	1	1	yes	characteristic
296	C₄	C₂×#13094050	1	1	yes	characteristic
297	C₂	—	16	4	no	—
298	C₂	—	376	4	no	—
299	C₂	#20245420	1	4	yes	direct product factor
300	C₂	C₂×#15689315	1	1	yes	characteristic
301	C₂	C₂²×#13093965	1	1	yes	characteristic
302	C₂	C₂×#15689559	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#493	C₂
socle	C₂³×C₄₇	C₂×D₈
hypercenter	C₂×#493	D₄₇
hypocenter	C₄₇	C₂×#2206
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093965	C₂	semidirect
2. C₂×C₄×D₃₇₆	C₂	semidirect
3. #20245420	C₂	direct (N × H)
4. C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
5. C₂×#15688924	C₂	semidirect
6. C₂×C₄₇×#493	C₂	semidirect
10. C₂×#13093965	C₂²	semidirect
13. C₄×D₃₇₆	C₂²	semidirect
16. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
17. #15688924	C₂²	semidirect
22. C₄₇×#493	C₂²	semidirect
28. C₂×D₃₇₆	D₄	semidirect
93. D₃₇₆	C₂×D₄	semidirect
110. C₂×C₄×C₄₇	D₁₆	semidirect
142. C₄×C₄₇	C₂×D₁₆	semidirect
147. C₂×#493	D₄₇	semidirect
153. C₂×C₄₇	#2206	semidirect
167. #493	C₂×D₄₇	semidirect
175. C₄₇	C₂×#2206	semidirect
268. C₂×C₄	#13093965	semidirect
294. C₄	C₂×#13093965	semidirect
299. C₂	#20245420	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×#13093965	2
C₂×C₄×D₃₇₆	2
#20245420	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
C₂×#15688924	2
C₂×C₄₇×#493	2
C₂×#2206	47

Subgroups — C2×#20245420 #26370499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245420 #26370499