Subgroups — C₂×#20243556 #26216696

Order 12032.

1230 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093958	C₂	1	2	yes	maximal
2	C₂²×#13093956	C₂	1	2	yes	maximal
3	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688989	C₂	1	1	yes	maximal, characteristic
5	#20243556	C₂	1	8	yes	maximal, direct product factor
6	C₂²×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₄₇	—	2	2	no	—
8	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂×#13093958	—	2	8	no	—
12	C₂×#13093958	C₂²	1	8	yes	—
13	C₂×#13093956	—	2	8	no	—
14	C₂×#13093956	C₂²	1	8	yes	—
15	#15688993	—	2	2	no	—
16	#15688993	C₂²	1	4	yes	—
17	#15688989	—	2	2	no	—
18	#15688989	C₂²	1	4	yes	—
19	C₂×C₄₇⋊#95	—	2	2	no	—
20	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#97×C₄₇	C₄	1	8	yes	complemented
23	C₂×#97×C₄₇	C₂²	1	4	yes	—
24	C₂³×Dic₄₇	—	2	1	no	—
25	C₂³×Dic₄₇	—	2	1	no	—
26	C₂³×Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	4	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	—	2	4	no	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	—	2	1	no	—
32	C₂×C₄⋊Dic₄₇	—	2	1	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	—	2	8	no	—
35	C₂×C₄⋊Dic₄₇	—	4	8	no	—
36	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
39	#13093958	—	2	16	no	—
40	#13093956	—	2	16	no	—
41	C₄₇⋊#95	—	2	8	no	—
42	C₂³×C₄×C₄₇	D₄	1	2	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₄₇	D₄	1	8	yes	—
46	C₂²×C₈×C₄₇	Q₈	1	8	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₇	D₄	1	2	yes	—
50	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
51	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
52	#97×C₄₇	C₂×C₄	1	16	yes	complemented
53	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₄₇	—	2	2	no	—
65	C₂²×Dic₄₇	—	2	2	no	—
66	C₂²×Dic₄₇	—	2	4	no	—
67	C₂²×Dic₄₇	—	4	4	no	—
68	C₂²×Dic₄₇	—	4	2	no	—
69	C₂²×Dic₄₇	—	4	2	no	—
70	C₂²×Dic₄₇	—	4	8	no	—
71	C₂²×Dic₄₇	—	4	4	no	—
72	C₂²×Dic₄₇	—	4	4	no	—
73	C₂×C₈×C₄₇	C₄⋊C₄	1	32	yes	complemented
74	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
75	C₂×C₈×C₄₇	C₂×D₄	1	8	yes	—
76	C₂×C₈×C₄₇	C₂×Q₈	1	8	yes	—
77	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
79	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
80	C₄⋊Dic₄₇	—	2	4	no	—
81	C₄⋊Dic₄₇	—	2	4	no	—
82	C₄⋊Dic₄₇	—	2	4	no	—
83	C₄⋊Dic₄₇	—	2	4	no	—
84	C₄⋊Dic₄₇	—	4	16	no	—
85	C₄⋊Dic₄₇	—	4	2	no	—
86	C₄⋊Dic₄₇	—	4	4	no	—
87	C₄⋊Dic₄₇	—	4	2	no	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₄₇	#127	1	2	yes	—
90	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
91	C₂³×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
92	C₂³×C₄₇	#137	1	4	yes	—
93	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	16	no	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₇	#127	1	2	yes	—
101	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
102	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
106	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
107	C₂×Dic₄₇	—	4	4	no	—
108	C₂×Dic₄₇	—	4	2	no	—
109	C₂×Dic₄₇	—	4	2	no	—
110	C₂×Dic₄₇	—	4	8	no	—
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	4	4	no	—
113	C₂×Dic₄₇	—	8	8	no	—
114	C₂×Dic₄₇	—	8	4	no	—
115	C₂×Dic₄₇	—	8	4	no	—
116	C₈×C₄₇	C₂×C₄⋊C₄	1	16	yes	complemented
117	C₂×#1847	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	#384	1	4	yes	—
119	C₂²×C₄₇	#428	1	8	yes	—
120	C₂²×C₄₇	#496	1	4	yes	—
121	C₂²×C₄₇	#497	1	4	yes	—
122	C₂²×C₄₇	#501	1	8	yes	—
123	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
125	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
126	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
127	C₂²×C₄₇	C₂×#137	1	2	yes	—
128	C₄×C₄₇	—	2	8	no	—
129	C₄×C₄₇	#384	1	4	yes	—
130	C₄×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
131	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
132	Dic₄₇	—	8	4	no	—
133	Dic₄₇	—	8	2	no	—
134	Dic₄₇	—	8	2	no	—
135	C₂²×#107	—	47	2	no	—
136	C₂²×#106	—	47	2	no	—
137	C₂²×#97	D₄₇	1	1	yes	characteristic
138	C₂×#384	—	47	1	no	—
139	C₂×#382	—	47	1	no	—
140	#1847	—	47	8	no	—
141	C₂×C₄₇	#1847	1	8	yes	complemented
142	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
143	C₂×C₄₇	C₂×#428	1	2	yes	—
144	C₂×C₄₇	C₂×#496	1	1	yes	characteristic
145	C₂×C₄₇	C₂×#497	1	1	yes	characteristic
146	C₂×C₄₇	C₂×#501	1	2	yes	—
147	C₂³×C₈	C₂×D₄₇	1	2	yes	—
148	C₂²×C₄⋊C₄	—	47	1	no	—
149	C₂²×C₄⋊C₄	—	47	2	no	—
150	C₂²×C₄⋊C₄	—	47	1	no	—
151	C₂²×C₄⋊C₄	—	94	2	no	—
152	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
153	C₂×#107	—	47	8	no	—
154	C₂×#107	—	94	8	no	—
155	C₂×#106	—	47	8	no	—
156	C₂×#106	—	94	8	no	—
157	C₂×#97	Dic₄₇	1	8	yes	complemented
158	C₂×#97	C₂×D₄₇	1	4	yes	—
159	#384	—	47	4	no	—
160	#384	—	94	2	no	—
161	#382	—	47	4	no	—
162	#382	—	94	2	no	—
163	C₂×#95	—	94	2	no	—
164	C₄₇	C₂×#1847	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	—	94	2	no	—
166	C₂³×C₄	—	94	1	no	—
167	C₂³×C₄	—	94	1	no	—
168	C₂³×C₄	D₁₈₈	1	2	yes	—
169	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
170	C₂²×C₈	Dic₉₄	1	8	yes	—
171	C₂²×C₈	D₁₈₈	1	8	yes	—
172	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
173	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
174	C₂×C₄⋊C₄	—	47	2	no	—
175	C₂×C₄⋊C₄	—	47	2	no	—
176	C₂×C₄⋊C₄	—	47	4	no	—
177	C₂×C₄⋊C₄	—	94	4	no	—
178	C₂×C₄⋊C₄	—	94	4	no	—
179	C₂×C₄⋊C₄	—	94	4	no	—
180	C₂×C₄⋊C₄	—	94	4	no	—
181	C₂×C₄⋊C₄	—	94	1	no	—
182	C₂×C₄⋊C₄	—	94	1	no	—
183	C₂×C₄⋊C₄	—	94	2	no	—
184	C₂×C₄⋊C₄	—	94	8	no	—
185	C₂×C₄⋊C₄	—	188	8	no	—
186	C₂×C₄²	Dic₉₄	1	4	yes	—
187	C₂×C₄²	D₁₈₈	1	2	yes	—
188	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
189	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
190	#107	—	94	16	no	—
191	#106	—	94	16	no	—
192	#97	C₂×Dic₄₇	1	16	yes	complemented
193	#95	—	94	8	no	—
194	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	94	2	no	—
198	C₂²×C₄	—	94	2	no	—
199	C₂²×C₄	—	94	4	no	—
200	C₂²×C₄	—	188	4	no	—
201	C₂²×C₄	—	188	2	no	—
202	C₂²×C₄	—	188	2	no	—
203	C₂²×C₄	—	188	8	no	—
204	C₂²×C₄	—	188	4	no	—
205	C₂²×C₄	—	188	4	no	—
206	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
207	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
208	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
209	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
210	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
212	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
213	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
214	C₂×C₈	C₄⋊Dic₄₇	1	32	yes	complemented
215	C₂×C₈	C₂×Dic₉₄	1	8	yes	—
216	C₂×C₈	C₂×D₁₈₈	1	8	yes	—
217	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
218	C₄⋊C₄	—	94	4	no	—
219	C₄⋊C₄	—	94	4	no	—
220	C₄⋊C₄	—	94	4	no	—
221	C₄⋊C₄	—	94	4	no	—
222	C₄⋊C₄	—	188	16	no	—
223	C₄⋊C₄	—	188	2	no	—
224	C₄⋊C₄	—	188	4	no	—
225	C₄⋊C₄	—	188	2	no	—
226	C₄²	C₄⋊Dic₄₇	1	8	yes	—
227	C₄²	C₂×Dic₉₄	1	4	yes	—
228	C₄²	C₂²×Dic₄₇	1	4	yes	—
229	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
230	C₂³	#13094021	1	2	yes	—
231	C₂³	#13094024	1	2	yes	—
232	C₂³	#13094052	1	4	yes	—
233	C₂³	#13094057	1	4	yes	—
234	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	16	no	—
238	C₂×C₄	—	376	8	no	—
239	C₂×C₄	—	376	4	no	—
240	C₂×C₄	—	376	4	no	—
241	C₂×C₄	—	188	8	no	—
242	C₂×C₄	—	188	4	no	—
243	C₂×C₄	—	188	4	no	—
244	C₂×C₄	—	188	4	no	—
245	C₂×C₄	—	188	2	no	—
246	C₂×C₄	—	188	2	no	—
247	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
248	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
249	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
250	C₂×C₄	#13094021	1	2	yes	—
251	C₂×C₄	#13094024	1	2	yes	—
252	C₂×C₄	#13094024	1	8	yes	—
253	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
254	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
255	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
256	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
257	C₈	C₂×C₄⋊Dic₄₇	1	16	yes	complemented
258	C₂²	#15688993	1	4	yes	—
259	C₂²	#15689165	1	8	yes	—
260	C₂²	#15689512	1	4	yes	—
261	C₂²	#15689517	1	4	yes	—
262	C₂²	#15689532	1	8	yes	—
263	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
264	C₂²	C₂×#13094021	1	1	yes	characteristic
265	C₂²	C₂×#13094024	1	1	yes	characteristic
266	C₂²	C₂×#13094052	1	2	yes	—
267	C₂²	C₂×#13094057	1	2	yes	—
268	C₄	—	2	8	no	—
269	C₄	—	376	4	no	—
270	C₄	—	376	2	no	—
271	C₄	—	376	2	no	—
272	C₄	#15688993	1	4	yes	—
273	C₄	C₂²×C₄⋊Dic₄₇	1	2	yes	—
274	C₄	C₂×#13094024	1	2	yes	—
275	C₂	#20243556	1	8	yes	direct product factor
276	C₂	C₂×#15688993	1	1	yes	characteristic
277	C₂	C₂×#15689165	1	2	yes	—
278	C₂	C₂×#15689512	1	1	yes	characteristic
279	C₂	C₂×#15689517	1	1	yes	characteristic
280	C₂	C₂×#15689532	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#97×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#97	D₄₇
hypocenter	C₄₇	C₂×#1847
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
5. #20243556	C₂	direct (N × H)
22. C₂×#97×C₄₇	C₄	semidirect
52. #97×C₄₇	C₂×C₄	semidirect
73. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
116. C₈×C₄₇	C₂×C₄⋊C₄	semidirect
141. C₂×C₄₇	#1847	semidirect
157. C₂×#97	Dic₄₇	semidirect
164. C₄₇	C₂×#1847	semidirect
192. #97	C₂×Dic₄₇	semidirect
214. C₂×C₈	C₄⋊Dic₄₇	semidirect
257. C₈	C₂×C₄⋊Dic₄₇	semidirect
275. C₂	#20243556	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×#13093958	2
C₂²×#13093956	2
C₂×#15688993	2
C₂×#15688989	2
#20243556	2
C₂²×#97×C₄₇	2
C₂×#1847	47

Subgroups — C2×#20243556 #26216696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243556 #26216696