Subgroups — C₂×#20245594 #26371054

Order 12032.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689361	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689160	C₂	1	1	yes	maximal, characteristic
3	#20245594	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688951	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688927	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×(C₂×D₄₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#358	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
10	C₂×#13094000	—	2	1	no	—
11	C₂×#13093982	—	2	1	no	—
12	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#15689361	—	2	2	no	—
14	#15689361	C₂²	1	4	yes	—
15	#15689160	—	2	2	no	—
16	#15689160	C₂²	1	4	yes	—
17	C₂×#13093958	C₂²	1	1	yes	characteristic
18	C₂×#13093958	C₂²	1	1	yes	characteristic
19	C₂×#13093955	—	2	1	no	—
20	C₂×#13093948	—	2	1	no	—
21	#15688951	C₂²	1	4	yes	complemented
22	#15688948	C₂²	1	4	yes	complemented
23	#15688927	C₂²	1	4	yes	complemented
24	(C₂×D₄₇)⋊C₁₆	C₂²	1	4	yes	—
25	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#358	C₂²	1	4	yes	complemented
29	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
30	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
31	C₂×C₈×D₄₇	—	2	2	no	—
32	C₂×C₈×D₄₇	—	2	2	no	—
33	C₂×C₈×D₄₇	D₄	1	2	yes	—
34	C₂×C₈×D₄₇	C₂³	1	2	yes	—
35	#13094000	—	2	4	no	—
36	#13094000	—	4	2	no	—
37	#13093982	—	2	4	no	—
38	#13093982	—	4	2	no	—
39	C₂×C₄⋊Dic₄₇	—	2	1	no	—
40	C₂×C₄⋊Dic₄₇	—	2	1	no	—
41	C₂×Dic₄₇⋊C₄	—	4	1	no	—
42	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
43	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
44	C₂×C₄×Dic₄₇	—	4	1	no	—
45	C₂×C₄₇⋊C₁₆	—	2	2	no	—
46	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
47	#13093958	—	2	1	no	—
48	#13093958	—	2	1	no	—
49	#13093958	C₂³	1	2	yes	—
50	#13093958	C₂³	1	2	yes	—
51	#13093955	—	2	4	no	—
52	#13093948	—	2	4	no	—
53	C₂²×Q₈×C₄₇	—	2	1	no	—
54	C₂×Q₁₆×C₄₇	—	2	1	no	—
55	C₂×Q₁₆×C₄₇	—	2	2	no	—
56	C₂×Q₁₆×C₄₇	—	2	2	no	—
57	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
60	C₂×C₁₆×C₄₇	—	2	2	no	—
61	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
62	#107×C₄₇	—	2	1	no	—
63	#107×C₄₇	C₂³	1	2	yes	—
64	C₂³×D₄₇	D₈	1	1	yes	characteristic
65	C₂×Q₈×C₄₇	—	2	2	no	—
66	C₂×Q₈×C₄₇	—	4	1	no	—
67	C₂×Q₈×C₄₇	—	4	2	no	—
68	C₂×Q₈×C₄₇	—	4	2	no	—
69	C₂²×C₄×C₄₇	—	4	1	no	—
70	C₂²×C₄×C₄₇	—	4	1	no	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	4	1	no	—
73	C₂²×Dic₄₇	—	4	1	no	—
74	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
75	C₂×C₄×D₄₇	—	2	2	no	—
76	C₂×C₄×D₄₇	—	2	2	no	—
77	C₂×C₄×D₄₇	D₈	1	2	yes	—
78	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
79	Q₁₆×C₄₇	—	2	2	no	—
80	Q₁₆×C₄₇	—	2	2	no	—
81	Q₁₆×C₄₇	—	4	2	no	—
82	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	#41	1	2	yes	—
84	C₂×C₈×C₄₇	#41	1	2	yes	—
85	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₄₇	—	2	2	no	—
87	C₄⋊C₄×C₄₇	—	4	1	no	—
88	(C₂×D₄₇)⋊C₄	—	4	4	no	—
89	(C₂×D₄₇)⋊C₄	—	4	4	no	—
90	C₄⋊Dic₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	—	2	2	no	—
92	C₄⋊Dic₄₇	—	4	1	no	—
93	C₄⋊Dic₄₇	—	4	1	no	—
94	Dic₄₇⋊C₄	—	4	4	no	—
95	C₄×Dic₄₇	—	4	4	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
98	C₈×D₄₇	—	2	4	no	—
99	C₈×D₄₇	—	2	2	no	—
100	C₈×D₄₇	—	2	2	no	—
101	C₁₆×C₄₇	—	2	2	no	—
102	C₄₇⋊C₁₆	—	2	2	no	—
103	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
104	C₂²×D₄₇	—	2	2	no	—
105	C₂²×D₄₇	—	2	2	no	—
106	C₂²×D₄₇	—	2	2	no	—
107	C₂²×D₄₇	C₂×D₈	1	2	yes	—
108	Q₈×C₄₇	—	4	2	no	—
109	Q₈×C₄₇	—	4	2	no	—
110	Q₈×C₄₇	—	8	2	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	8	2	no	—
114	C₂×C₄×C₄₇	—	8	2	no	—
115	C₂×C₄×C₄₇	#123	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#137	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₄₇	—	2	2	no	—
120	C₂×Dic₄₇	—	4	2	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	8	2	no	—
123	C₂×Dic₄₇	—	8	2	no	—
124	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
125	C₄×D₄₇	—	2	2	no	—
126	C₄×D₄₇	—	2	2	no	—
127	C₄×D₄₇	—	2	4	no	—
128	C₈×C₄₇	#123	1	2	yes	—
129	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
130	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
131	C₄₇⋊C₈	—	2	2	no	—
132	C₂×#2237	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	#480	1	2	yes	—
134	C₂²×C₄₇	#508	1	2	yes	—
135	C₂²×C₄₇	#510	1	2	yes	—
136	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
137	C₂×D₄₇	—	2	2	no	—
138	C₂×D₄₇	—	2	2	no	—
139	C₂×D₄₇	—	2	2	no	—
140	C₂×D₄₇	—	4	4	no	—
141	C₄×C₄₇	—	8	2	no	—
142	C₄×C₄₇	—	8	2	no	—
143	C₄×C₄₇	#480	1	2	yes	—
144	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
145	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
146	Dic₄₇	—	2	2	no	—
147	Dic₄₇	—	8	2	no	—
148	Dic₄₇	—	8	2	no	—
149	C₂×#467	—	47	1	no	—
150	C₂×#427	—	47	1	no	—
151	#2237	—	47	8	no	—
152	C₂×#367	—	47	1	no	—
153	C₂×#366	—	47	1	no	—
154	C₂×#358	—	47	1	no	—
155	C₂×#358	D₄₇	1	1	yes	characteristic, complemented
156	C₂×C₂²⋊C₁₆	—	47	1	no	—
157	C₂×C₄₇	#2237	1	4	yes	complemented
158	C₂×C₄₇	C₂×#480	1	1	yes	characteristic
159	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
160	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
161	D₄₇	—	4	2	no	—
162	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
163	C₂³×C₈	—	47	1	no	—
164	C₂×C₂²⋊Q₈	—	94	1	no	—
165	C₂×#117	—	94	1	no	—
166	C₂²×C₁₆	—	47	1	no	—
167	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
168	#467	—	47	4	no	—
169	#467	—	94	2	no	—
170	#427	—	47	4	no	—
171	#427	—	94	2	no	—
172	C₂×#107	—	47	1	no	—
173	C₂×#107	—	47	1	no	—
174	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
175	C₂×#106	—	94	1	no	—
176	C₂×#103	—	94	1	no	—
177	#367	—	47	4	no	—
178	#366	—	47	4	no	—
179	#358	—	47	4	no	—
180	#358	C₂×D₄₇	1	4	yes	complemented
181	C₂²⋊C₁₆	—	47	4	no	—
182	C₄₇	C₂×#2237	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	—	2	1	no	—
184	C₂³×C₄	—	47	1	no	—
185	C₂×Q₁₆	—	2	2	no	—
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×Q₁₆	—	2	2	no	—
188	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
189	C₂²×C₈	—	47	1	no	—
190	C₂²×C₈	—	47	2	no	—
191	C₂²×C₈	—	47	2	no	—
192	C₂²×C₈	—	94	2	no	—
193	C₂²×C₈	—	94	2	no	—
194	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²⋊Q₈	—	94	4	no	—
196	C₂²⋊Q₈	—	188	2	no	—
197	#117	—	94	4	no	—
198	#117	—	188	2	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	94	1	no	—
201	C₂×C₄⋊C₄	—	94	1	no	—
202	C₂×C₄⋊C₄	—	188	1	no	—
203	C₂×C₂²⋊C₄	—	188	1	no	—
204	C₂×C₂²⋊C₄	—	188	1	no	—
205	C₂×C₄²	—	188	1	no	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	47	2	no	—
208	C₂×C₁₆	—	94	2	no	—
209	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
210	#107	—	2	1	no	—
211	#107	—	47	2	no	—
212	#107	—	47	2	no	—
213	#107	—	94	1	no	—
214	#107	—	94	1	no	—
215	#107	C₂²×D₄₇	1	2	yes	—
216	#106	—	94	4	no	—
217	#103	—	94	4	no	—
218	C₂⁴	—	47	1	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×Q₈	—	4	2	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	47	1	no	—
226	C₂²×C₄	—	47	2	no	—
227	C₂²×C₄	—	47	2	no	—
228	C₂²×C₄	—	94	2	no	—
229	C₂²×C₄	—	94	2	no	—
230	C₂²×C₄	—	188	1	no	—
231	C₂²×C₄	—	188	1	no	—
232	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	4	2	no	—
236	C₂×C₈	—	47	2	no	—
237	C₂×C₈	—	94	2	no	—
238	C₂×C₈	—	94	4	no	—
239	C₂×C₈	—	94	2	no	—
240	C₂×C₈	—	94	2	no	—
241	C₂×C₈	#10625350	1	2	yes	—
242	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
243	C₂×C₈	#10625352	1	2	yes	—
244	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	94	2	no	—
248	C₄⋊C₄	—	94	2	no	—
249	C₄⋊C₄	—	188	4	no	—
250	C₄⋊C₄	—	188	1	no	—
251	C₄⋊C₄	—	188	1	no	—
252	C₂²⋊C₄	—	188	4	no	—
253	C₂²⋊C₄	—	188	4	no	—
254	C₄²	—	188	4	no	—
255	C₁₆	—	2	2	no	—
256	C₁₆	—	94	2	no	—
257	C₂³	—	47	2	no	—
258	C₂³	—	94	2	no	—
259	C₂³	—	94	2	no	—
260	C₂³	—	94	2	no	—
261	C₂³	D₈×D₄₇	1	1	yes	characteristic
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	8	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	—	47	2	no	—
270	C₂×C₄	—	94	2	no	—
271	C₂×C₄	—	94	2	no	—
272	C₂×C₄	—	94	2	no	—
273	C₂×C₄	—	94	4	no	—
274	C₂×C₄	—	376	2	no	—
275	C₂×C₄	—	376	2	no	—
276	C₂×C₄	—	188	2	no	—
277	C₂×C₄	—	188	2	no	—
278	C₂×C₄	#13094006	1	2	yes	—
279	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
280	C₂×C₄	#13094053	1	2	yes	—
281	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
282	C₈	—	94	2	no	—
283	C₈	#13094006	1	2	yes	—
284	C₈	C₂×#10625352	1	1	yes	characteristic
285	C₈	C₂×#10625350	1	1	yes	characteristic
286	C₂²	—	94	2	no	—
287	C₂²	—	94	2	no	—
288	C₂²	—	94	2	no	—
289	C₂²	—	188	4	no	—
290	C₂²	#15689435	1	2	yes	—
291	C₂²	#15689554	1	2	yes	—
292	C₂²	#15689565	1	2	yes	—
293	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	94	2	no	—
297	C₄	—	376	2	no	—
298	C₄	—	376	2	no	—
299	C₄	#15689435	1	2	yes	—
300	C₄	C₂×#13094006	1	1	yes	characteristic
301	C₄	C₂×#13094053	1	1	yes	characteristic
302	C₂	—	188	2	no	—
303	C₂	#20245594	1	4	yes	direct product factor
304	C₂	C₂×#15689435	1	1	yes	characteristic
305	C₂	C₂×#15689554	1	1	yes	characteristic
306	C₂	C₂×#15689565	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#358	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂³×D₄₇	D₈
hypercenter	C₂×#358	D₄₇
hypocenter	C₄₇	C₂×#2237
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#358

Complements

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

N	H ≅ G/N	type
3. #20245594	C₂	direct (N × H)
4. C₂×#15688951	C₂	semidirect
5. C₂×#15688948	C₂	semidirect
6. C₂×#15688927	C₂	semidirect
8. C₂×C₄₇×#358	C₂	semidirect
21. #15688951	C₂²	semidirect
22. #15688948	C₂²	semidirect
23. #15688927	C₂²	semidirect
28. C₄₇×#358	C₂²	semidirect
155. C₂×#358	D₄₇	semidirect
157. C₂×C₄₇	#2237	semidirect
180. #358	C₂×D₄₇	semidirect
182. C₄₇	C₂×#2237	semidirect
303. C₂	#20245594	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15689361	2
C₂×#15689160	2
#20245594	2
C₂×#15688951	2
C₂×#15688948	2
C₂×#15688927	2
C₂×(C₂×D₄₇)⋊C₁₆	2
C₂×C₄₇×#358	2
C₂×#2237	47

Subgroups — C2×#20245594 #26371054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245594 #26371054