Subgroups — C₂×#20245599 #26371059

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×#15689365	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689160	C₂	1	1	yes	maximal, characteristic
3	#20245599	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688948	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688928	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688926	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×(C₂×D₄₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#367	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
11	C₂×#13094004	—	2	1	no	—
12	C₂×#13093982	—	2	1	no	—
13	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	#15689365	—	2	2	no	—
15	#15689365	C₂²	1	4	yes	—
16	#15689160	—	2	2	no	—
17	#15689160	C₂²	1	4	yes	—
18	C₂×#13093958	C₂²	1	1	yes	characteristic
19	C₂×#13093955	—	2	1	no	—
20	C₂×#13093947	—	2	1	no	—
21	#15688948	C₂²	1	4	yes	complemented
22	#15688928	C₂²	1	4	yes	complemented
23	#15688926	C₂²	1	4	yes	complemented
24	(C₂×D₄₇)⋊C₁₆	C₂²	1	4	yes	—
25	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#367	C₂²	1	4	yes	complemented
29	C₂²×Dic₉₄	—	2	1	no	—
30	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
31	C₂×Dic₁₈₈	—	2	2	no	—
32	C₂×Dic₁₈₈	—	2	2	no	—
33	C₂×Dic₁₈₈	—	2	1	no	—
34	C₂×Dic₁₈₈	C₂³	1	2	yes	—
35	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
36	C₂×C₈×D₄₇	—	2	2	no	—
37	C₂×C₈×D₄₇	—	2	2	no	—
38	C₂×C₈×D₄₇	D₄	1	2	yes	—
39	C₂×C₈×D₄₇	C₂³	1	2	yes	—
40	#13094004	—	2	4	no	—
41	#13094004	—	4	2	no	—
42	#13093982	—	2	4	no	—
43	#13093982	—	4	2	no	—
44	C₂×C₄⋊Dic₄₇	—	2	1	no	—
45	C₂×C₄⋊Dic₄₇	—	4	1	no	—
46	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
47	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
48	C₂×C₄×Dic₄₇	—	4	1	no	—
49	C₂×C₄₇⋊C₁₆	—	2	2	no	—
50	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
51	#13093958	—	2	1	no	—
52	#13093958	C₂³	1	2	yes	—
53	#13093955	—	2	4	no	—
54	#13093947	—	2	4	no	—
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
57	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
58	C₂×C₁₆×C₄₇	—	2	2	no	—
59	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
60	#107×C₄₇	—	2	1	no	—
61	#107×C₄₇	—	2	1	no	—
62	#107×C₄₇	C₂³	1	2	yes	—
63	#107×C₄₇	C₂³	1	2	yes	—
64	C₂³×D₄₇	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	—	4	1	no	—
66	C₂²×C₄×C₄₇	—	4	1	no	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	4	1	no	—
69	C₂²×Dic₄₇	—	4	1	no	—
70	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
71	C₂×C₄×D₄₇	—	2	2	no	—
72	C₂×C₄×D₄₇	—	2	2	no	—
73	C₂×C₄×D₄₇	D₈	1	2	yes	—
74	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
75	C₂×Dic₉₄	—	2	2	no	—
76	C₂×Dic₉₄	—	4	2	no	—
77	C₂×Dic₉₄	—	4	2	no	—
78	C₂×Dic₉₄	—	4	1	no	—
79	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	#41	1	2	yes	—
81	C₂×C₈×C₄₇	#41	1	2	yes	—
82	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₄₇	—	2	2	no	—
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	—	4	1	no	—
86	C₄⋊C₄×C₄₇	—	4	1	no	—
87	(C₂×D₄₇)⋊C₄	—	4	4	no	—
88	(C₂×D₄₇)⋊C₄	—	4	4	no	—
89	C₄⋊Dic₄₇	—	2	2	no	—
90	C₄⋊Dic₄₇	—	4	1	no	—
91	C₄⋊Dic₄₇	—	4	4	no	—
92	C₄×Dic₄₇	—	4	4	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
95	Dic₁₈₈	—	2	2	no	—
96	Dic₁₈₈	—	2	2	no	—
97	Dic₁₈₈	—	4	2	no	—
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	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	8	2	no	—
111	C₂×C₄×C₄₇	—	8	2	no	—
112	C₂×C₄×C₄₇	#123	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#137	1	2	yes	—
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×Dic₄₇	—	2	2	no	—
117	C₂×Dic₄₇	—	4	2	no	—
118	C₂×Dic₄₇	—	4	2	no	—
119	C₂×Dic₄₇	—	8	2	no	—
120	C₂×Dic₄₇	—	8	2	no	—
121	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
122	C₄×D₄₇	—	2	2	no	—
123	C₄×D₄₇	—	2	2	no	—
124	C₄×D₄₇	—	2	4	no	—
125	Dic₉₄	—	4	2	no	—
126	Dic₉₄	—	4	2	no	—
127	Dic₉₄	—	8	2	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	D₄₇	1	1	yes	characteristic, complemented
153	C₂×#366	—	47	1	no	—
154	C₂×#358	—	47	1	no	—
155	C₂×#358	—	47	1	no	—
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	—	47	1	no	—
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	C₂×D₄₇	1	1	yes	characteristic
174	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
175	C₂×#106	—	94	1	no	—
176	C₂×#103	—	94	1	no	—
177	#367	C₂×D₄₇	1	4	yes	complemented
178	#366	—	47	4	no	—
179	#358	—	47	4	no	—
180	#358	—	47	4	no	—
181	C₂²⋊C₁₆	—	47	4	no	—
182	C₄₇	C₂×#2237	1	1	yes	characteristic, complemented, Sylow
183	C₂²×Q₈	—	94	1	no	—
184	C₂³×C₄	—	47	1	no	—
185	C₂×Q₁₆	—	47	2	no	—
186	C₂×Q₁₆	—	94	2	no	—
187	C₂×Q₁₆	—	94	1	no	—
188	C₂×Q₁₆	—	94	2	no	—
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	—	2	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	—	2	1	no	—
212	#107	—	47	2	no	—
213	#107	—	94	1	no	—
214	#107	C₂²×D₄₇	1	2	yes	—
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	—	94	2	no	—
220	C₂×Q₈	—	188	2	no	—
221	C₂×Q₈	—	188	2	no	—
222	C₂×Q₈	—	188	1	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	—	94	2	no	—
234	Q₁₆	—	94	2	no	—
235	Q₁₆	—	188	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	#10625354	1	2	yes	—
244	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	94	2	no	—
250	C₄⋊C₄	—	188	4	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	—	376	2	no	—
263	Q₈	—	188	2	no	—
264	Q₈	—	188	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	#13094009	1	2	yes	—
279	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
280	C₂×C₄	#13094056	1	2	yes	—
281	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
282	C₈	—	94	2	no	—
283	C₈	#13094009	1	2	yes	—
284	C₈	C₂×#10625350	1	1	yes	characteristic
285	C₈	C₂×#10625354	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	#15689438	1	2	yes	—
291	C₂²	#15689554	1	2	yes	—
292	C₂²	#15689562	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	#15689438	1	2	yes	—
300	C₄	C₂×#13094009	1	1	yes	characteristic
301	C₄	C₂×#13094056	1	1	yes	characteristic
302	C₂	—	188	2	no	—
303	C₂	#20245599	1	4	yes	direct product factor
304	C₂	C₂×#15689438	1	1	yes	characteristic
305	C₂	C₂×#15689554	1	1	yes	characteristic
306	C₂	C₂×#15689562	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×#367	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂³×D₄₇	D₈
hypercenter	C₂×#367	D₄₇
hypocenter	C₄₇	C₂×#2237
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
3. #20245599	C₂	direct (N × H)
4. C₂×#15688948	C₂	semidirect
5. C₂×#15688928	C₂	semidirect
6. C₂×#15688926	C₂	semidirect
8. C₂×C₄₇×#367	C₂	semidirect
21. #15688948	C₂²	semidirect
22. #15688928	C₂²	semidirect
23. #15688926	C₂²	semidirect
28. C₄₇×#367	C₂²	semidirect
152. C₂×#367	D₄₇	semidirect
157. C₂×C₄₇	#2237	semidirect
177. #367	C₂×D₄₇	semidirect
182. C₄₇	C₂×#2237	semidirect
303. C₂	#20245599	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×#15689365	2
C₂×#15689160	2
#20245599	2
C₂×#15688948	2
C₂×#15688928	2
C₂×#15688926	2
C₂×(C₂×D₄₇)⋊C₁₆	2
C₂×C₄₇×#367	2
C₂×#2237	47

Subgroups — C2×#20245599 #26371059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245599 #26371059