Subgroups — C₂×#20245450 #26370534

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093970	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13093967	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689507	C₂	1	1	yes	maximal, characteristic
4	#20245450	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688926	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15688923	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	C₂²×#10625325	—	2	1	no	—
11	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
12	C₂×#13094012	—	2	1	no	—
13	C₂×#13093970	—	2	4	no	—
14	C₂×#13093970	C₂²	1	4	yes	complemented
15	C₂×#13093967	—	2	4	no	—
16	C₂×#13093967	C₂²	1	4	yes	—
17	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	#15689507	—	2	2	no	—
20	#15689507	C₂²	1	4	yes	—
21	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	complemented
22	#15688926	C₂²	1	4	yes	complemented
23	#15688923	C₂²	1	4	yes	—
24	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
26	C₄×Q₁₆×C₄₇	—	2	2	no	—
27	C₄×Q₁₆×C₄₇	C₂²	1	4	yes	complemented
28	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#103×C₄₇	—	2	1	no	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂²×Dic₉₄	—	2	1	no	—
32	C₂²×D₁₈₈	—	2	1	no	—
33	C₂×Dic₁₈₈	—	2	2	no	—
34	C₂×Dic₁₈₈	—	2	2	no	—
35	C₂×Dic₁₈₈	—	2	1	no	—
36	C₂×Dic₁₈₈	C₂³	1	2	yes	—
37	C₂×#10625325	—	2	4	no	—
38	C₂×#10625325	—	4	4	no	—
39	C₂×D₃₇₆	—	2	2	no	—
40	C₂×D₃₇₆	—	2	2	no	—
41	C₂×D₃₇₆	—	2	1	no	—
42	C₂×D₃₇₆	C₂³	1	2	yes	—
43	#13094012	—	2	4	no	—
44	#13094012	—	4	2	no	—
45	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
46	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
47	#13093970	—	2	8	no	—
48	#13093967	—	2	8	no	—
49	C₂×C₄₇⋊C₁₆	—	2	2	no	—
50	C₂×C₄₇⋊C₁₆	—	2	2	no	—
51	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
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	D₄	1	4	yes	—
57	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₄×Q₈×C₄₇	—	2	4	no	—
61	C₄×Q₈×C₄₇	—	4	2	no	—
62	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
64	C₂×C₄²×C₄₇	—	4	1	no	—
65	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
66	#107×C₄₇	—	2	1	no	—
67	#107×C₄₇	C₂³	1	2	yes	—
68	#103×C₄₇	—	2	4	no	—
69	C₄×C₈×C₄₇	D₄	1	2	yes	—
70	C₄×C₈×C₄₇	C₂³	1	2	yes	—
71	C₂³×D₄₇	—	4	1	no	—
72	C₂×Q₈×C₄₇	—	2	4	no	—
73	C₂×Q₈×C₄₇	—	2	2	no	—
74	C₂×Q₈×C₄₇	—	4	1	no	—
75	C₂×Q₈×C₄₇	—	4	2	no	—
76	C₂²×C₄×C₄₇	—	4	1	no	—
77	C₂²×C₄×C₄₇	—	4	1	no	—
78	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×Dic₄₇	—	4	1	no	—
82	C₂×D₁₈₈	—	2	2	no	—
83	C₂×D₁₈₈	—	4	2	no	—
84	C₂×D₁₈₈	—	4	2	no	—
85	C₂×D₁₈₈	—	4	1	no	—
86	C₂×Dic₉₄	—	2	2	no	—
87	C₂×Dic₉₄	—	4	2	no	—
88	C₂×Dic₉₄	—	4	2	no	—
89	C₂×Dic₉₄	—	4	1	no	—
90	Q₁₆×C₄₇	—	2	4	no	—
91	Q₁₆×C₄₇	—	2	2	no	—
92	Q₁₆×C₄₇	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₄₇	#41	1	2	yes	—
98	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₄₇	—	2	2	no	—
100	C₄⋊C₄×C₄₇	—	4	1	no	—
101	C₄⋊C₄×C₄₇	—	4	4	no	—
102	C₄²×C₄₇	—	4	4	no	—
103	C₄²×C₄₇	D₈	1	2	yes	—
104	C₄²×C₄₇	C₂×D₄	1	2	yes	—
105	(C₂×D₄₇)⋊C₄	—	4	4	no	—
106	(C₂×D₄₇)⋊C₄	—	4	4	no	—
107	Dic₁₈₈	—	2	2	no	—
108	Dic₁₈₈	—	2	2	no	—
109	Dic₁₈₈	—	4	2	no	—
110	D₃₇₆	—	2	2	no	—
111	D₃₇₆	—	2	2	no	—
112	D₃₇₆	—	4	2	no	—
113	#10625325	—	4	8	no	—
114	C₄₇⋊C₁₆	—	2	2	no	—
115	C₄₇⋊C₁₆	—	2	2	no	—
116	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
117	C₂²×D₄₇	—	4	2	no	—
118	C₂²×D₄₇	—	8	2	no	—
119	C₂²×D₄₇	—	8	2	no	—
120	C₂²×D₄₇	—	8	2	no	—
121	Q₈×C₄₇	—	2	4	no	—
122	Q₈×C₄₇	—	4	2	no	—
123	Q₈×C₄₇	—	4	4	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	4	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	8	2	no	—
130	C₂×C₄×C₄₇	#121	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₄₇	—	4	2	no	—
137	C₂×Dic₄₇	—	8	2	no	—
138	D₁₈₈	—	4	2	no	—
139	D₁₈₈	—	4	2	no	—
140	D₁₈₈	—	8	2	no	—
141	Dic₉₄	—	4	2	no	—
142	Dic₉₄	—	4	2	no	—
143	Dic₉₄	—	8	2	no	—
144	C₈×C₄₇	—	2	2	no	—
145	C₈×C₄₇	#121	1	2	yes	—
146	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂×#2211	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	#459	1	2	yes	—
150	C₂²×C₄₇	#508	1	2	yes	—
151	C₂²×C₄₇	#510	1	2	yes	—
152	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₄₇	—	8	2	no	—
154	C₂×D₄₇	—	8	2	no	—
155	C₂×D₄₇	—	8	2	no	—
156	C₂×D₄₇	—	16	4	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	4	4	no	—
160	C₄×C₄₇	—	8	2	no	—
161	C₄×C₄₇	#459	1	2	yes	—
162	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
163	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	Dic₄₇	—	8	2	no	—
165	C₂²×Q₃₂	—	47	1	no	—
166	C₂²×SD₃₂	—	47	1	no	—
167	C₂×#495	—	47	1	no	—
168	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic, complemented
169	#2211	—	47	8	no	—
170	C₂×C₄⋊C₁₆	—	47	1	no	—
171	C₂×#358	—	47	1	no	—
172	C₂×#357	—	47	1	no	—
173	C₂×C₄₇	#2211	1	4	yes	complemented
174	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
175	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
177	D₄₇	—	16	2	no	—
178	C₂²×Q₁₆	—	47	1	no	—
179	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
180	C₂²×SD₁₆	—	94	1	no	—
181	C₂²×D₈	—	47	1	no	—
182	C₂×#124	—	94	1	no	—
183	C₂×C₄×Q₈	—	2	1	no	—
184	C₂×Q₃₂	—	47	4	no	—
185	C₂×Q₃₂	—	94	4	no	—
186	C₂×SD₃₂	—	47	4	no	—
187	C₂×SD₃₂	—	94	4	no	—
188	C₂²×C₁₆	—	47	1	no	—
189	C₂²×C₁₆	—	47	1	no	—
190	#495	—	47	4	no	—
191	#495	—	94	2	no	—
192	C₄×Q₁₆	—	2	2	no	—
193	C₄×Q₁₆	C₂×D₄₇	1	4	yes	complemented
194	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
195	C₂×#103	—	2	1	no	—
196	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
197	C₄⋊C₁₆	—	47	4	no	—
198	#358	—	47	4	no	—
199	#357	—	47	4	no	—
200	C₄₇	C₂×#2211	1	1	yes	characteristic, complemented, Sylow
201	C₂²×Q₈	—	2	1	no	—
202	C₂²×Q₈	—	94	1	no	—
203	C₂²×D₄	—	94	1	no	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	—	47	2	no	—
207	C₂×Q₁₆	—	94	2	no	—
208	C₂×Q₁₆	—	94	2	no	—
209	C₂×Q₁₆	—	94	1	no	—
210	C₂×Q₁₆	D₁₈₈	1	4	yes	—
211	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
212	C₂×SD₁₆	—	94	4	no	—
213	C₂×SD₁₆	—	188	4	no	—
214	C₂×D₈	—	47	2	no	—
215	C₂×D₈	—	94	2	no	—
216	C₂×D₈	—	94	2	no	—
217	C₂×D₈	—	94	1	no	—
218	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
220	#124	—	94	4	no	—
221	#124	—	188	2	no	—
222	C₄×Q₈	—	2	4	no	—
223	C₄×Q₈	—	4	2	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	4	1	no	—
226	C₂×C₂²⋊C₄	—	188	1	no	—
227	C₂×C₂²⋊C₄	—	188	1	no	—
228	C₂×C₄²	—	4	1	no	—
229	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
230	Q₃₂	—	94	8	no	—
231	SD₃₂	—	94	8	no	—
232	C₂×C₁₆	—	47	2	no	—
233	C₂×C₁₆	—	47	2	no	—
234	C₂×C₁₆	—	94	2	no	—
235	C₂×C₁₆	—	94	2	no	—
236	#107	—	2	1	no	—
237	#107	C₂²×D₄₇	1	2	yes	—
238	#103	—	2	4	no	—
239	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
240	C₄×C₈	C₂²×D₄₇	1	2	yes	—
241	C₂⁴	—	188	1	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	94	2	no	—
247	C₂×Q₈	—	188	2	no	—
248	C₂×Q₈	—	188	2	no	—
249	C₂×Q₈	—	188	1	no	—
250	C₂×D₄	—	94	2	no	—
251	C₂×D₄	—	188	2	no	—
252	C₂×D₄	—	188	1	no	—
253	C₂×D₄	—	188	2	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	188	1	no	—
257	C₂²×C₄	#10625334	1	1	yes	characteristic
258	C₂²×C₄	#10625334	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
260	Q₁₆	—	2	4	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	94	2	no	—
263	Q₁₆	—	94	2	no	—
264	Q₁₆	—	188	2	no	—
265	Q₁₆	C₂×D₁₈₈	1	4	yes	—
266	SD₁₆	—	188	8	no	—
267	D₈	—	94	2	no	—
268	D₈	—	94	2	no	—
269	D₈	—	188	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
272	C₂×C₈	#10625350	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
274	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₂²⋊C₄	—	188	4	no	—
280	C₂²⋊C₄	—	188	4	no	—
281	C₄²	—	4	4	no	—
282	C₄²	#10625334	1	2	yes	—
283	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
284	C₁₆	—	94	2	no	—
285	C₁₆	—	94	2	no	—
286	C₂³	—	376	2	no	—
287	C₂³	—	376	2	no	—
288	C₂³	—	376	2	no	—
289	C₂³	—	188	2	no	—
290	C₂³	C₂×#10625334	1	1	yes	characteristic
291	Q₈	—	2	4	no	—
292	Q₈	—	4	4	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	376	2	no	—
295	Q₈	—	188	2	no	—
296	Q₈	—	188	2	no	—
297	D₄	—	376	2	no	—
298	D₄	—	188	2	no	—
299	D₄	—	188	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	376	2	no	—
307	C₂×C₄	—	188	2	no	—
308	C₂×C₄	#13093998	1	2	yes	—
309	C₂×C₄	C₂×#10625334	1	2	yes	—
310	C₂×C₄	C₂×#10625334	1	2	yes	—
311	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
312	C₂×C₄	#13094049	1	2	yes	—
313	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	#13093998	1	2	yes	—
316	C₈	C₂²×D₁₈₈	1	1	yes	characteristic
317	C₈	C₂×#10625350	1	1	yes	characteristic
318	C₂²	—	376	2	no	—
319	C₂²	—	376	2	no	—
320	C₂²	—	376	2	no	—
321	C₂²	—	752	4	no	—
322	C₂²	#15689318	1	2	yes	—
323	C₂²	#15689553	1	2	yes	—
324	C₂²	#15689564	1	2	yes	—
325	C₂²	C₂²×#10625334	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	4	4	no	—
329	C₄	—	8	2	no	—
330	C₄	—	376	2	no	—
331	C₄	#15689318	1	2	yes	—
332	C₄	C₂×#13093998	1	1	yes	characteristic
333	C₄	C₂×#13094049	1	1	yes	characteristic
334	C₂	—	752	2	no	—
335	C₂	#20245450	1	4	yes	direct product factor
336	C₂	C₂×#15689318	1	1	yes	characteristic
337	C₂	C₂×#15689553	1	1	yes	characteristic
338	C₂	C₂×#15689564	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂²×D₃₇₆	C₂²
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#2211
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093970	C₂	semidirect
4. #20245450	C₂	direct (N × H)
5. C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
6. C₂×#15688926	C₂	semidirect
8. C₂×C₄×Q₁₆×C₄₇	C₂	semidirect
14. C₂×#13093970	C₂²	semidirect
21. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
22. #15688926	C₂²	semidirect
27. C₄×Q₁₆×C₄₇	C₂²	semidirect
168. C₂×C₄×Q₁₆	D₄₇	semidirect
173. C₂×C₄₇	#2211	semidirect
193. C₄×Q₁₆	C₂×D₄₇	semidirect
200. C₄₇	C₂×#2211	semidirect
335. C₂	#20245450	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×#13093970	2
C₂²×#13093967	2
C₂×#15689507	2
#20245450	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
C₂×#15688926	2
C₂×#15688923	2
C₂×C₄×Q₁₆×C₄₇	2
C₂×#2211	47

Subgroups — C2×#20245450 #26370534

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245450 #26370534