Subgroups — C₂×#20245618 #26371570

Order 12032.

610 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689524	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689506	C₂	1	1	yes	maximal, characteristic
3	#20245618	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688928	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688926	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688924	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#357	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	C₂²×#10625335	—	2	1	no	—
11	C₂×#13094015	—	2	1	no	—
12	C₂×#13094011	—	2	1	no	—
13	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	#15689524	—	2	2	no	—
15	#15689524	C₂²	1	4	yes	complemented
16	#15689506	—	2	2	no	—
17	#15689506	C₂²	1	4	yes	—
18	C₂×#13093958	C₂²	1	1	yes	characteristic
19	C₂×#13093955	—	2	1	no	—
20	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
21	#15688928	C₂²	1	4	yes	complemented
22	#15688926	C₂²	1	4	yes	complemented
23	#15688924	C₂²	1	4	yes	—
24	C₁₆×Dic₄₇	C₂²	1	4	yes	complemented
25	C₂²×D₈×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×#357	C₂²	1	4	yes	—
29	C₂²×Dic₉₄	—	2	1	no	—
30	C₂×Dic₁₈₈	—	2	2	no	—
31	C₂×Dic₁₈₈	—	2	2	no	—
32	C₂×Dic₁₈₈	—	2	1	no	—
33	C₂×Dic₁₈₈	C₂³	1	2	yes	—
34	C₂×#10625335	—	2	4	no	—
35	C₂×#10625335	—	4	4	no	—
36	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
37	#13094015	—	2	4	no	—
38	#13094015	—	4	2	no	—
39	#13094011	—	2	4	no	—
40	#13094011	—	4	2	no	—
41	C₂×C₄⋊Dic₄₇	—	2	1	no	—
42	C₂×C₄⋊Dic₄₇	—	4	1	no	—
43	C₂×Dic₄₇⋊C₄	—	4	1	no	—
44	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
45	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
46	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄₇⋊C₁₆	—	2	2	no	—
48	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
49	#13093958	—	2	1	no	—
50	#13093958	C₂³	1	2	yes	—
51	#13093955	—	2	4	no	—
52	C₈×Dic₄₇	D₄	1	2	yes	—
53	C₈×Dic₄₇	C₂³	1	2	yes	—
54	C₂²×D₄×C₄₇	—	2	1	no	—
55	C₂×D₈×C₄₇	—	2	1	no	—
56	C₂×D₈×C₄₇	—	2	2	no	—
57	C₂×D₈×C₄₇	—	2	2	no	—
58	C₂×D₈×C₄₇	C₂³	1	2	yes	—
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
61	C₂×C₁₆×C₄₇	—	2	2	no	—
62	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
63	#107×C₄₇	—	2	1	no	—
64	#107×C₄₇	C₂³	1	2	yes	—
65	C₂⁴×C₄₇	—	4	1	no	—
66	C₂×D₄×C₄₇	—	2	2	no	—
67	C₂×D₄×C₄₇	—	4	2	no	—
68	C₂×D₄×C₄₇	—	4	1	no	—
69	C₂×D₄×C₄₇	—	4	2	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	C₂²×Dic₄₇	D₈	1	1	yes	characteristic
76	C₂×Dic₉₄	—	2	2	no	—
77	C₂×Dic₉₄	—	4	2	no	—
78	C₂×Dic₉₄	—	4	2	no	—
79	C₂×Dic₉₄	—	4	1	no	—
80	D₈×C₄₇	—	2	2	no	—
81	D₈×C₄₇	—	2	2	no	—
82	D₈×C₄₇	—	4	2	no	—
83	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₄₇	#41	1	2	yes	—
85	C₂×C₈×C₄₇	#41	1	2	yes	—
86	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₄₇	—	2	2	no	—
88	C₄⋊C₄×C₄₇	—	4	1	no	—
89	C₂²⋊Dic₄₇	—	4	4	no	—
90	C₂²⋊Dic₄₇	—	4	4	no	—
91	#10625335	—	4	8	no	—
92	C₄⋊Dic₄₇	—	2	2	no	—
93	C₄⋊Dic₄₇	—	4	1	no	—
94	C₄⋊Dic₄₇	—	4	4	no	—
95	Dic₄₇⋊C₄	—	4	4	no	—
96	C₄×Dic₄₇	D₈	1	2	yes	—
97	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
98	C₂×C₄₇⋊C₈	SD₁₆	1	4	yes	—
99	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
100	Dic₁₈₈	—	2	2	no	—
101	Dic₁₈₈	—	2	2	no	—
102	Dic₁₈₈	—	4	2	no	—
103	C₁₆×C₄₇	—	2	2	no	—
104	C₄₇⋊C₁₆	—	2	2	no	—
105	C₂³×C₄₇	—	4	2	no	—
106	C₂³×C₄₇	—	8	2	no	—
107	C₂³×C₄₇	—	8	2	no	—
108	C₂³×C₄₇	—	8	2	no	—
109	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
110	D₄×C₄₇	—	4	2	no	—
111	D₄×C₄₇	—	4	2	no	—
112	D₄×C₄₇	—	8	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	8	2	no	—
115	C₂×C₄×C₄₇	#124	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₄₇	—	2	2	no	—
120	C₂×Dic₄₇	—	2	2	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	4	2	no	—
123	C₂×Dic₄₇	—	8	2	no	—
124	C₂×Dic₄₇	—	8	2	no	—
125	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
126	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
127	Dic₉₄	—	4	2	no	—
128	Dic₉₄	—	4	2	no	—
129	Dic₉₄	—	8	2	no	—
130	C₈×C₄₇	#124	1	2	yes	—
131	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
132	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₄₇⋊C₈	C₂×SD₁₆	1	4	yes	—
134	C₂×#2242	—	47	1	no	maximal, Sylow
135	C₂²×C₄₇	—	8	2	no	—
136	C₂²×C₄₇	—	8	2	no	—
137	C₂²×C₄₇	—	8	2	no	—
138	C₂²×C₄₇	—	16	4	no	—
139	C₂²×C₄₇	#486	1	2	yes	—
140	C₂²×C₄₇	#508	1	2	yes	—
141	C₂²×C₄₇	#508	1	2	yes	—
142	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
143	C₄×C₄₇	—	8	2	no	—
144	C₄×C₄₇	#486	1	2	yes	—
145	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
146	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
147	Dic₄₇	—	2	2	no	—
148	Dic₄₇	—	2	2	no	—
149	Dic₄₇	—	8	2	no	—
150	Dic₄₇	—	8	2	no	—
151	C₂×#499	—	47	1	no	—
152	C₂×#495	—	47	1	no	—
153	#2242	—	47	8	no	—
154	C₂×#358	—	47	1	no	—
155	C₂×#358	—	47	1	no	—
156	C₂×#357	—	47	1	no	—
157	C₂×#357	D₄₇	1	1	yes	characteristic
158	C₂×C₄×C₁₆	—	47	1	no	—
159	C₂×C₄₇	—	16	2	no	—
160	C₂×C₄₇	#2242	1	4	yes	complemented
161	C₂×C₄₇	C₂×#486	1	1	yes	characteristic
162	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
163	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
164	C₂²×Q₁₆	—	47	1	no	—
165	C₂²×SD₁₆	—	94	1	no	—
166	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
167	C₂×#125	—	94	1	no	—
168	C₂×#124	—	94	1	no	—
169	C₂²×C₁₆	—	47	1	no	—
170	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
171	#499	—	47	4	no	—
172	#499	—	94	2	no	—
173	#495	—	47	4	no	—
174	#495	—	94	2	no	—
175	C₂×#107	—	47	1	no	—
176	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
177	C₂×#106	—	94	1	no	—
178	C₂×C₄×C₈	—	47	1	no	—
179	#358	—	47	4	no	—
180	#358	—	47	4	no	—
181	#357	—	47	4	no	—
182	#357	C₂×D₄₇	1	4	yes	—
183	C₄×C₁₆	—	47	4	no	—
184	C₄₇	C₂×#2242	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	94	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂×Q₁₆	—	47	2	no	—
188	C₂×Q₁₆	—	94	2	no	—
189	C₂×Q₁₆	—	94	1	no	—
190	C₂×Q₁₆	—	94	2	no	—
191	C₂×SD₁₆	—	94	4	no	—
192	C₂×SD₁₆	—	188	4	no	—
193	C₂×D₈	—	2	2	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	C₂²×D₄₇	1	2	yes	—
197	C₂²×C₈	—	47	1	no	—
198	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
199	#125	—	94	4	no	—
200	#125	—	188	2	no	—
201	#124	—	94	4	no	—
202	#124	—	188	2	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	94	1	no	—
205	C₂×C₄⋊C₄	—	188	1	no	—
206	C₂×C₄⋊C₄	—	188	1	no	—
207	C₂×C₂²⋊C₄	—	188	1	no	—
208	C₂×C₂²⋊C₄	—	188	1	no	—
209	C₂×C₄²	—	47	1	no	—
210	C₂×C₁₆	—	2	2	no	—
211	C₂×C₁₆	—	47	2	no	—
212	C₂×C₁₆	—	94	2	no	—
213	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
214	#107	—	2	1	no	—
215	#107	—	47	2	no	—
216	#107	—	94	1	no	—
217	#107	C₂²×D₄₇	1	2	yes	—
218	#106	—	94	4	no	—
219	C₄×C₈	—	47	2	no	—
220	C₄×C₈	—	47	2	no	—
221	C₂⁴	—	4	1	no	—
222	C₂×Q₈	—	94	2	no	—
223	C₂×Q₈	—	188	2	no	—
224	C₂×Q₈	—	188	2	no	—
225	C₂×Q₈	—	188	1	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	47	1	no	—
232	C₂²×C₄	—	47	1	no	—
233	C₂²×C₄	—	188	1	no	—
234	C₂²×C₄	—	188	1	no	—
235	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
236	Q₁₆	—	94	2	no	—
237	Q₁₆	—	94	2	no	—
238	Q₁₆	—	188	2	no	—
239	SD₁₆	—	188	8	no	—
240	D₈	—	2	2	no	—
241	D₈	—	2	2	no	—
242	D₈	—	4	2	no	—
243	C₂×C₈	—	47	4	no	—
244	C₂×C₈	—	47	2	no	—
245	C₂×C₈	#10625350	1	2	yes	—
246	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
247	C₂×C₈	#10625352	1	2	yes	—
248	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	94	2	no	—
252	C₄⋊C₄	—	188	4	no	—
253	C₄⋊C₄	—	188	4	no	—
254	C₄⋊C₄	—	188	1	no	—
255	C₂²⋊C₄	—	188	4	no	—
256	C₂²⋊C₄	—	188	4	no	—
257	C₄²	—	47	2	no	—
258	C₄²	—	47	2	no	—
259	C₁₆	—	2	2	no	—
260	C₁₆	—	94	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	8	2	no	—
263	C₂³	—	8	2	no	—
264	C₂³	—	8	2	no	—
265	C₂³	D₈×D₄₇	1	1	yes	characteristic
266	Q₈	—	376	2	no	—
267	Q₈	—	188	2	no	—
268	Q₈	—	188	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	8	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	47	2	no	—
275	C₂×C₄	—	47	2	no	—
276	C₂×C₄	—	94	2	no	—
277	C₂×C₄	—	94	2	no	—
278	C₂×C₄	—	376	2	no	—
279	C₂×C₄	—	376	2	no	—
280	C₂×C₄	—	188	2	no	—
281	C₂×C₄	—	188	2	no	—
282	C₂×C₄	#13094010	1	2	yes	—
283	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
284	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
285	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
286	C₈	—	47	4	no	—
287	C₈	#13094010	1	2	yes	—
288	C₈	C₂×#10625352	1	1	yes	characteristic
289	C₈	C₂×#10625350	1	1	yes	characteristic
290	C₂²	—	8	2	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	16	4	no	—
294	C₂²	#15689468	1	2	yes	—
295	C₂²	#15689552	1	2	yes	—
296	C₂²	#15689554	1	2	yes	—
297	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
298	C₄	—	8	2	no	—
299	C₄	—	94	2	no	—
300	C₄	—	94	2	no	—
301	C₄	—	376	2	no	—
302	C₄	—	376	2	no	—
303	C₄	#15689468	1	2	yes	—
304	C₄	C₂×#13094010	1	1	yes	characteristic
305	C₄	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
306	C₂	—	16	2	no	—
307	C₂	#20245618	1	4	yes	direct product factor
308	C₂	C₂×#15689468	1	1	yes	characteristic
309	C₂	C₂×#15689552	1	1	yes	characteristic
310	C₂	C₂×#15689554	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×#357	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂²×D₈×C₄₇	C₂²
hypercenter	C₂×#357	D₄₇
hypocenter	C₄₇	C₂×#2242
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
1. C₂×#15689524	C₂	semidirect
3. #20245618	C₂	direct (N × H)
4. C₂×#15688928	C₂	semidirect
5. C₂×#15688926	C₂	semidirect
7. C₂×C₁₆×Dic₄₇	C₂	semidirect
15. #15689524	C₂²	semidirect
21. #15688928	C₂²	semidirect
22. #15688926	C₂²	semidirect
24. C₁₆×Dic₄₇	C₂²	semidirect
160. C₂×C₄₇	#2242	semidirect
184. C₄₇	C₂×#2242	semidirect
307. C₂	#20245618	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×#15689524	2
C₂×#15689506	2
#20245618	2
C₂×#15688928	2
C₂×#15688926	2
C₂×#15688924	2
C₂×C₁₆×Dic₄₇	2
C₂×C₄₇×#357	2
C₂×#2242	47

Subgroups — C2×#20245618 #26371570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245618 #26371570