Subgroups — C₂×#20245465 #26370659

Order 12032.

790 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689561	C₂	1	2	yes	maximal, complemented
2	C₂×#15689500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15689241	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245465	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688937	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13094046	C₂²	1	1	yes	characteristic
8	C₂²×D₃₇₆	—	2	1	no	—
9	C₂²×D₃₇₆	C₂²	1	2	yes	—
10	C₂×#13094030	—	2	1	no	—
11	C₂×#13094021	—	2	1	no	—
12	#15689561	—	2	4	no	—
13	#15689561	C₂²	1	8	yes	complemented
14	C₂×#13093969	—	2	2	no	—
15	C₂×D₇₅₂	—	2	2	no	—
16	#15689500	—	2	2	no	—
17	#15689500	C₂²	1	4	yes	complemented
18	#15689241	—	2	2	no	—
19	#15689241	C₂²	1	4	yes	complemented
20	C₂×#13093960	C₂²	1	1	yes	characteristic
21	C₂×#13093952	—	2	1	no	—
22	#15688937	C₂²	1	8	yes	complemented
23	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₄₇×#364	C₂²	1	4	yes	complemented
26	C₂×#10625350	—	2	1	no	—
27	C₂²×D₁₈₈	—	2	2	no	—
28	C₂²×D₁₈₈	—	4	2	no	—
29	#13094046	—	2	1	no	—
30	#13094046	—	2	1	no	—
31	#13094046	D₄	1	2	yes	complemented
32	#13094046	C₂³	1	2	yes	—
33	C₂×Dic₁₈₈	D₄	1	1	yes	characteristic, complemented
34	C₂×#10625325	—	2	1	no	—
35	C₂×D₃₇₆	—	2	2	no	—
36	C₂×D₃₇₆	—	2	2	no	—
37	C₂×D₃₇₆	—	2	2	no	—
38	C₂×D₃₇₆	—	2	2	no	—
39	C₂×D₃₇₆	—	2	2	no	—
40	C₂×D₃₇₆	—	2	4	no	—
41	C₂×D₃₇₆	—	4	1	no	—
42	C₂×D₃₇₆	—	4	1	no	—
43	C₂×D₃₇₆	—	4	2	no	—
44	C₂×D₃₇₆	D₄	1	1	yes	characteristic, complemented
45	C₂×D₃₇₆	C₂³	1	4	yes	—
46	#13094030	—	2	4	no	—
47	#13094030	—	4	2	no	—
48	C₂×#10625329	—	2	1	no	—
49	C₂×#10625324	—	4	1	no	—
50	C₂×C₈×D₄₇	—	4	1	no	—
51	#13094021	—	2	4	no	—
52	#13094021	—	4	2	no	—
53	#13093969	—	2	8	no	—
54	D₇₅₂	—	2	8	no	—
55	#13093960	—	2	1	no	—
56	#13093960	C₂³	1	2	yes	—
57	#13093952	—	2	4	no	—
58	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
61	M₃₂×C₄₇	—	2	2	no	—
62	M₃₂×C₄₇	C₂³	1	4	yes	—
63	C₂×C₁₆×C₄₇	—	2	2	no	—
64	C₄×C₈×C₄₇	D₄	1	2	yes	—
65	C₄×C₈×C₄₇	C₂³	1	2	yes	—
66	C₂³×D₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	—	2	1	no	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₄₇	—	4	1	no	—
70	#10625350	—	2	2	no	—
71	#10625350	—	2	2	no	—
72	#10625350	—	4	1	no	—
73	#10625350	—	4	1	no	—
74	C₂×D₁₈₈	—	2	1	no	—
75	C₂×D₁₈₈	—	2	4	no	—
76	C₂×D₁₈₈	—	4	2	no	—
77	C₂×D₁₈₈	—	4	2	no	—
78	C₂×D₁₈₈	—	4	2	no	—
79	C₂×D₁₈₈	—	4	2	no	—
80	C₂×D₁₈₈	—	4	8	no	—
81	C₂×D₁₈₈	—	4	2	no	—
82	C₂×D₁₈₈	—	8	8	no	—
83	C₂×C₄×D₄₇	—	4	1	no	—
84	C₂×Dic₉₄	—	2	1	no	—
85	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
86	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
87	C₂×C₈×C₄₇	D₈	1	2	yes	—
88	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
93	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
95	C₄²×C₄₇	—	2	1	no	—
96	C₄²×C₄₇	C₂×D₄	1	2	yes	—
97	#10625329	—	2	2	no	—
98	#10625329	—	4	1	no	—
99	C₂×C₄₇⋊C₈	—	4	1	no	—
100	Dic₁₈₈	—	2	1	no	—
101	Dic₁₈₈	C₂×D₄	1	2	yes	complemented
102	D₃₇₆	—	2	1	no	—
103	D₃₇₆	—	2	4	no	—
104	D₃₇₆	—	2	4	no	—
105	D₃₇₆	—	4	2	no	—
106	D₃₇₆	—	4	2	no	—
107	D₃₇₆	—	4	4	no	—
108	D₃₇₆	—	4	4	no	—
109	D₃₇₆	C₂×D₄	1	2	yes	complemented
110	#10625325	—	2	2	no	—
111	#10625325	—	2	2	no	—
112	#10625324	—	4	4	no	—
113	C₈×D₄₇	—	4	4	no	—
114	C₁₆×C₄₇	—	2	4	no	—
115	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂²×D₄₇	—	4	1	no	—
117	C₂²×D₄₇	—	4	4	no	—
118	C₂²×D₄₇	—	8	8	no	—
119	C₂²×D₄₇	—	8	2	no	—
120	C₂²×D₄₇	—	8	2	no	—
121	C₂×C₄×C₄₇	—	2	4	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₄₇	—	4	2	no	—
130	C₂²⋊D₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	1	no	—
132	D₁₈₈	—	2	2	no	—
133	D₁₈₈	—	4	4	no	—
134	D₁₈₈	—	4	4	no	—
135	D₁₈₈	—	4	1	no	—
136	D₁₈₈	—	8	16	no	—
137	D₁₈₈	—	8	2	no	—
138	D₁₈₈	—	8	2	no	—
139	C₄×D₄₇	—	4	2	no	—
140	C₄×D₄₇	—	4	2	no	—
141	Dic₉₄	—	2	2	no	—
142	Dic₉₄	—	4	1	no	—
143	C₈×C₄₇	#121	1	1	yes	characteristic
144	C₈×C₄₇	#121	1	1	yes	characteristic
145	C₈×C₄₇	C₂×D₈	1	2	yes	—
146	C₈×C₄₇	C₂×D₈	1	2	yes	—
147	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
149	C₄₇⋊C₈	—	4	2	no	—
150	C₂×#2213	—	47	1	no	maximal, Sylow
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	#459	1	1	yes	characteristic
153	C₂²×C₄₇	#459	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×D₄₇	—	4	2	no	—
156	C₂×D₄₇	—	8	2	no	—
157	C₂×D₄₇	—	8	8	no	—
158	C₂×D₄₇	—	8	4	no	—
159	C₂×D₄₇	—	16	4	no	—
160	C₂×D₄₇	—	16	4	no	—
161	C₄×C₄₇	—	2	4	no	—
162	C₄×C₄₇	#459	1	1	yes	characteristic
163	C₄×C₄₇	#459	1	1	yes	characteristic
164	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
165	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
166	Dic₄₇	—	4	2	no	—
167	C₂×#509	—	47	2	no	—
168	C₂×#493	—	47	1	no	—
169	C₂×#443	—	47	1	no	—
170	#2213	—	47	8	no	—
171	C₂×#364	D₄₇	1	1	yes	characteristic, complemented
172	C₂×#361	—	47	2	no	—
173	C₂×C₄₇	—	2	1	no	—
174	C₂×C₄₇	—	2	1	no	—
175	C₂×C₄₇	#2213	1	2	yes	complemented
176	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
177	D₄₇	—	8	2	no	—
178	D₄₇	—	16	4	no	—
179	C₂×#135	—	47	1	no	—
180	C₂²×D₈	—	47	2	no	—
181	C₂²×D₈	—	94	1	no	—
182	C₂×#131	—	94	1	no	—
183	C₂×#127	—	94	1	no	—
184	#509	—	47	8	no	—
185	#509	—	94	4	no	—
186	C₂×SD₃₂	—	94	2	no	—
187	C₂×D₁₆	—	94	2	no	—
188	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
189	#493	—	47	4	no	—
190	#493	—	94	2	no	—
191	#443	—	47	4	no	—
192	#443	—	94	2	no	—
193	C₂×#108	—	47	1	no	—
194	C₂×#104	—	94	1	no	—
195	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
196	#364	C₂×D₄₇	1	4	yes	complemented
197	#361	—	47	8	no	—
198	C₄₇	C₂×#2213	1	1	yes	characteristic, complemented, Sylow
199	C₂×#41	—	94	1	no	—
200	C₂²×D₄	—	94	2	no	—
201	C₂²×D₄	—	188	2	no	—
202	#135	—	47	2	no	—
203	#135	—	47	2	no	—
204	#135	—	94	1	no	—
205	#135	—	94	1	no	—
206	C₂×Q₁₆	—	47	1	no	—
207	C₂×SD₁₆	—	94	1	no	—
208	C₂×D₈	—	47	1	no	—
209	C₂×D₈	—	47	4	no	—
210	C₂×D₈	—	94	2	no	—
211	C₂×D₈	—	94	4	no	—
212	C₂×D₈	—	94	2	no	—
213	C₂×D₈	—	94	2	no	—
214	C₂×D₈	—	94	2	no	—
215	C₂×D₈	—	94	2	no	—
216	C₂×D₈	—	188	1	no	—
217	C₂×D₈	—	188	1	no	—
218	C₂×D₈	—	188	2	no	—
219	#131	—	94	4	no	—
220	#131	—	188	2	no	—
221	C₂×M₁₆	—	94	1	no	—
222	C₂×M₁₆	—	188	1	no	—
223	C₂²×C₈	—	188	1	no	—
224	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
226	#127	—	94	4	no	—
227	#127	—	188	2	no	—
228	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
229	SD₃₂	—	94	8	no	—
230	D₁₆	—	94	8	no	—
231	M₃₂	—	2	2	no	—
232	M₃₂	C₂²×D₄₇	1	4	yes	—
233	C₂×C₁₆	—	2	2	no	—
234	#108	—	47	2	no	—
235	#108	—	94	1	no	—
236	#104	—	94	4	no	—
237	C₄×C₈	D₁₈₈	1	2	yes	—
238	C₄×C₈	C₂²×D₄₇	1	2	yes	—
239	C₂⁴	—	188	2	no	—
240	#41	—	94	2	no	—
241	#41	—	94	2	no	—
242	#41	—	188	1	no	—
243	#41	—	188	1	no	—
244	C₂×Q₈	—	94	1	no	—
245	C₂×D₄	—	94	1	no	—
246	C₂×D₄	—	94	4	no	—
247	C₂×D₄	—	376	8	no	—
248	C₂×D₄	—	188	8	no	—
249	C₂×D₄	—	188	1	no	—
250	C₂×D₄	—	188	2	no	—
251	C₂×D₄	—	188	2	no	—
252	C₂×D₄	—	188	2	no	—
253	C₂×D₄	—	188	2	no	—
254	C₂×D₄	—	188	2	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	188	1	no	—
257	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
258	Q₁₆	—	47	2	no	—
259	Q₁₆	—	94	1	no	—
260	SD₁₆	—	94	2	no	—
261	SD₁₆	—	94	2	no	—
262	D₈	—	47	2	no	—
263	D₈	—	94	1	no	—
264	D₈	—	94	4	no	—
265	D₈	—	94	4	no	—
266	D₈	—	188	2	no	—
267	D₈	—	188	2	no	—
268	D₈	—	188	4	no	—
269	D₈	—	188	4	no	—
270	M₁₆	—	94	2	no	—
271	M₁₆	—	188	1	no	—
272	M₁₆	—	188	4	no	—
273	C₂×C₈	—	188	1	no	—
274	C₂×C₈	—	188	4	no	—
275	C₂×C₈	D₃₇₆	1	1	yes	characteristic
276	C₂×C₈	D₃₇₆	1	1	yes	characteristic
277	C₂×C₈	D₃₇₆	1	2	yes	—
278	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
279	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
280	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
281	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
282	C₂×C₈	#10625354	1	1	yes	characteristic
283	C₂×C₈	#10625354	1	1	yes	characteristic
284	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
285	C₄²	—	2	1	no	—
286	C₄²	C₂×D₁₈₈	1	2	yes	—
287	C₁₆	—	2	4	no	—
288	C₂³	—	376	8	no	—
289	C₂³	—	376	2	no	—
290	C₂³	—	376	2	no	—
291	C₂³	—	188	4	no	—
292	C₂³	—	188	1	no	—
293	C₂³	#13094052	1	1	yes	characteristic
294	Q₈	—	94	2	no	—
295	Q₈	—	188	1	no	—
296	D₄	—	94	2	no	—
297	D₄	—	376	16	no	—
298	D₄	—	376	2	no	—
299	D₄	—	376	2	no	—
300	D₄	—	188	2	no	—
301	D₄	—	188	2	no	—
302	D₄	—	188	1	no	—
303	D₄	—	188	4	no	—
304	D₄	—	188	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	188	1	no	—
308	C₂×C₄	—	188	2	no	—
309	C₂×C₄	—	188	2	no	—
310	C₂×C₄	#13093999	1	1	yes	characteristic
311	C₂×C₄	#13093999	1	1	yes	characteristic
312	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
313	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
314	C₂×C₄	#13094052	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
316	C₈	—	188	2	no	—
317	C₈	#13093999	1	1	yes	characteristic
318	C₈	#13093999	1	1	yes	characteristic
319	C₈	C₂×D₃₇₆	1	2	yes	—
320	C₈	C₂×D₃₇₆	1	2	yes	—
321	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
322	C₈	C₂×#10625354	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	376	8	no	—
325	C₂²	—	376	2	no	—
326	C₂²	—	376	4	no	—
327	C₂²	—	188	2	no	—
328	C₂²	—	752	4	no	—
329	C₂²	—	752	4	no	—
330	C₂²	#15689319	1	1	yes	characteristic
331	C₂²	#15689319	1	1	yes	characteristic
332	C₂²	C₂×#13094052	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	188	2	no	—
335	C₄	#15689319	1	1	yes	characteristic
336	C₄	#15689319	1	1	yes	characteristic
337	C₄	C₂×#13093999	1	1	yes	characteristic
338	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	376	2	no	—
342	C₂	—	752	4	no	—
343	C₂	#20245465	1	2	yes	direct product factor
344	C₂	C₂×#15689319	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689319
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#364	C₂
socle	C₂²×C₄₇	#459
hypercenter	C₂×#364	D₄₇
hypocenter	C₄₇	C₂×#2213
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
1. C₂×#15689561	C₂	semidirect
2. C₂×#15689500	C₂	semidirect
3. C₂×#15689241	C₂	semidirect
4. #20245465	C₂	direct (N × H)
5. C₂×#15688937	C₂	semidirect
6. C₂×C₄₇×#364	C₂	semidirect
13. #15689561	C₂²	semidirect
17. #15689500	C₂²	semidirect
19. #15689241	C₂²	semidirect
22. #15688937	C₂²	semidirect
25. C₄₇×#364	C₂²	semidirect
31. #13094046	D₄	semidirect
33. C₂×Dic₁₈₈	D₄	semidirect
44. C₂×D₃₇₆	D₄	semidirect
101. Dic₁₈₈	C₂×D₄	semidirect
109. D₃₇₆	C₂×D₄	semidirect
171. C₂×#364	D₄₇	semidirect
175. C₂×C₄₇	#2213	semidirect
196. #364	C₂×D₄₇	semidirect
198. C₄₇	C₂×#2213	semidirect
343. C₂	#20245465	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×#15689561	2
C₂×#15689500	2
C₂×#15689241	2
#20245465	2
C₂×#15688937	2
C₂×C₄₇×#364	2
C₂×#2213	47

Subgroups — C2×#20245465 #26370659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245465 #26370659