Subgroups — C₂×#20245447 #26370531

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×#13093968	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245447	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688927	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15688924	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#495	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₉₄	—	2	1	no	—
11	C₂×#13093970	—	2	4	no	—
12	C₂×#13093970	C₂²	1	4	yes	complemented
13	C₂×#13093968	—	2	4	no	—
14	C₂×#13093968	C₂²	1	4	yes	—
15	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₈₈	—	2	2	no	—
18	C₄×Dic₁₈₈	C₂²	1	4	yes	complemented
19	C₂×#13093958	C₂²	1	1	yes	characteristic
20	C₂×#13093949	—	2	1	no	—
21	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	complemented
22	#15688927	C₂²	1	4	yes	complemented
23	#15688924	C₂²	1	4	yes	—
24	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×SD₁₆×C₄₇	—	2	1	no	—
26	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#124×C₄₇	—	2	1	no	—
28	C₄₇×#495	—	2	2	no	—
29	C₄₇×#495	C₂²	1	4	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂²×Dic₉₄	—	2	1	no	—
32	C₂×Dic₁₈₈	—	2	1	no	—
33	C₂×Dic₁₈₈	—	2	2	no	—
34	C₂×Dic₁₈₈	D₄	1	4	yes	—
35	C₂×Dic₁₈₈	C₂³	1	2	yes	—
36	C₄×Dic₉₄	—	2	4	no	—
37	C₄×Dic₉₄	—	4	2	no	—
38	C₂×C₄⋊Dic₄₇	—	2	1	no	—
39	C₂×Dic₄₇⋊C₄	—	4	1	no	—
40	C₂×C₄×Dic₄₇	—	4	1	no	—
41	#13093970	—	2	8	no	—
42	#13093968	—	2	8	no	—
43	C₂×C₄₇⋊C₁₆	—	2	2	no	—
44	C₂×C₄₇⋊C₁₆	—	2	2	no	—
45	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
46	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
47	#13093958	—	2	1	no	—
48	#13093958	C₂³	1	2	yes	—
49	#13093949	—	2	4	no	—
50	C₂²×Q₈×C₄₇	—	2	1	no	—
51	C₂²×D₄×C₄₇	—	2	1	no	—
52	C₂×Q₁₆×C₄₇	—	2	2	no	—
53	C₂×Q₁₆×C₄₇	—	2	1	no	—
54	C₂×Q₁₆×C₄₇	—	2	2	no	—
55	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
56	C₂×SD₁₆×C₄₇	—	2	4	no	—
57	C₂×SD₁₆×C₄₇	—	4	4	no	—
58	C₂×D₈×C₄₇	—	2	2	no	—
59	C₂×D₈×C₄₇	—	2	1	no	—
60	C₂×D₈×C₄₇	—	2	2	no	—
61	C₂×D₈×C₄₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	#124×C₄₇	—	2	4	no	—
65	#124×C₄₇	—	4	2	no	—
66	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
67	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
68	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
69	C₄×C₈×C₄₇	D₄	1	2	yes	—
70	C₄×C₈×C₄₇	C₂³	1	2	yes	—
71	C₂⁴×C₄₇	—	4	1	no	—
72	C₂×Q₈×C₄₇	—	2	2	no	—
73	C₂×Q₈×C₄₇	—	4	1	no	—
74	C₂×Q₈×C₄₇	—	4	2	no	—
75	C₂×Q₈×C₄₇	—	4	2	no	—
76	C₂×D₄×C₄₇	—	2	2	no	—
77	C₂×D₄×C₄₇	—	4	2	no	—
78	C₂×D₄×C₄₇	—	4	1	no	—
79	C₂×D₄×C₄₇	—	4	2	no	—
80	C₂²×C₄×C₄₇	—	4	1	no	—
81	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×Dic₄₇	—	4	1	no	—
85	C₂²×Dic₄₇	—	4	1	no	—
86	C₂×Dic₉₄	—	2	4	no	—
87	C₂×Dic₉₄	—	2	2	no	—
88	C₂×Dic₉₄	—	4	1	no	—
89	C₂×Dic₉₄	—	4	2	no	—
90	Q₁₆×C₄₇	—	2	2	no	—
91	Q₁₆×C₄₇	—	2	2	no	—
92	Q₁₆×C₄₇	—	4	2	no	—
93	SD₁₆×C₄₇	—	4	8	no	—
94	D₈×C₄₇	—	2	2	no	—
95	D₈×C₄₇	—	2	2	no	—
96	D₈×C₄₇	—	4	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₄₇	#41	1	2	yes	—
102	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
103	C₂²⋊C₄×C₄₇	—	4	4	no	—
104	C₂²⋊C₄×C₄₇	—	4	4	no	—
105	C₄²×C₄₇	D₈	1	2	yes	—
106	C₄²×C₄₇	C₂×D₄	1	2	yes	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	4	1	no	—
109	Dic₄₇⋊C₄	—	4	4	no	—
110	C₄×Dic₄₇	—	4	4	no	—
111	Dic₁₈₈	—	2	4	no	—
112	Dic₁₈₈	—	2	2	no	—
113	Dic₁₈₈	C₂×D₄	1	4	yes	—
114	C₄₇⋊C₁₆	—	2	2	no	—
115	C₄₇⋊C₁₆	—	2	2	no	—
116	C₂³×C₄₇	—	4	2	no	—
117	C₂³×C₄₇	—	8	2	no	—
118	C₂³×C₄₇	—	8	2	no	—
119	C₂³×C₄₇	—	8	2	no	—
120	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₄₇	—	4	2	no	—
122	Q₈×C₄₇	—	4	2	no	—
123	Q₈×C₄₇	—	8	2	no	—
124	D₄×C₄₇	—	4	2	no	—
125	D₄×C₄₇	—	4	2	no	—
126	D₄×C₄₇	—	8	2	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	8	2	no	—
131	C₂×C₄×C₄₇	#121	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
136	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	4	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	8	2	no	—
141	Dic₉₄	—	2	4	no	—
142	Dic₉₄	—	4	2	no	—
143	Dic₉₄	—	4	4	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	—	8	2	no	—
150	C₂²×C₄₇	—	8	2	no	—
151	C₂²×C₄₇	—	8	2	no	—
152	C₂²×C₄₇	—	16	4	no	—
153	C₂²×C₄₇	#459	1	2	yes	—
154	C₂²×C₄₇	#508	1	2	yes	—
155	C₂²×C₄₇	#510	1	2	yes	—
156	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	8	2	no	—
160	C₄×C₄₇	#459	1	2	yes	—
161	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
162	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	Dic₄₇	—	4	4	no	—
164	Dic₄₇	—	8	2	no	—
165	C₂²×Q₃₂	—	47	1	no	—
166	C₂²×SD₃₂	—	47	1	no	—
167	C₂×#495	D₄₇	1	1	yes	characteristic
168	C₂×C₄×Q₁₆	—	47	1	no	—
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	—	16	2	no	—
174	C₂×C₄₇	#2211	1	4	yes	complemented
175	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
176	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
177	C₂×C₄₇	C₂×#510	1	1	yes	characteristic
178	C₂²×Q₁₆	—	47	1	no	—
179	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
180	C₂²×SD₁₆	—	2	1	no	—
181	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
182	C₂×#124	—	2	1	no	—
183	C₂×C₄×Q₈	—	94	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	—	2	2	no	—
191	#495	C₂×D₄₇	1	4	yes	—
192	C₄×Q₁₆	—	47	4	no	—
193	C₄×Q₁₆	—	94	2	no	—
194	C₂×#107	—	47	1	no	—
195	C₂×#103	—	94	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	—	2	1	no	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	—	47	4	no	—
208	C₂×Q₁₆	—	47	2	no	—
209	C₂×Q₁₆	—	94	2	no	—
210	C₂×Q₁₆	—	94	1	no	—
211	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
212	C₂×SD₁₆	—	2	4	no	—
213	C₂×SD₁₆	—	4	4	no	—
214	C₂×D₈	—	2	2	no	—
215	C₂×D₈	—	2	2	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	C₂²×D₄₇	1	2	yes	—
218	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
220	#124	—	2	4	no	—
221	#124	—	4	2	no	—
222	C₄×Q₈	—	94	4	no	—
223	C₄×Q₈	—	188	2	no	—
224	C₂×C₄⋊C₄	—	94	1	no	—
225	C₂×C₄⋊C₄	—	188	1	no	—
226	C₂×C₂²⋊C₄	—	4	1	no	—
227	C₂×C₂²⋊C₄	—	4	1	no	—
228	C₂×C₄²	—	188	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	—	47	2	no	—
237	#107	—	94	1	no	—
238	#103	—	94	4	no	—
239	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
240	C₄×C₈	C₂²×D₄₇	1	2	yes	—
241	C₂⁴	—	4	1	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	94	4	no	—
247	C₂×Q₈	—	94	2	no	—
248	C₂×Q₈	—	188	1	no	—
249	C₂×Q₈	—	188	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	188	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	2	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	4	2	no	—
263	Q₁₆	—	47	4	no	—
264	Q₁₆	—	94	4	no	—
265	Q₁₆	—	94	2	no	—
266	SD₁₆	—	4	8	no	—
267	D₈	—	2	2	no	—
268	D₈	—	2	2	no	—
269	D₈	—	4	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	D₄×D₄₇	1	2	yes	—
272	C₂×C₈	#10625352	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	—	94	2	no	—
277	C₄⋊C₄	—	188	4	no	—
278	C₄⋊C₄	—	188	1	no	—
279	C₂²⋊C₄	—	4	4	no	—
280	C₂²⋊C₄	—	4	4	no	—
281	C₄²	—	188	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	—	4	2	no	—
287	C₂³	—	8	2	no	—
288	C₂³	—	8	2	no	—
289	C₂³	—	8	2	no	—
290	C₂³	C₂×#10625334	1	1	yes	characteristic
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	8	2	no	—
294	Q₈	—	94	4	no	—
295	Q₈	—	188	4	no	—
296	Q₈	—	188	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	8	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	376	2	no	—
305	C₂×C₄	—	188	2	no	—
306	C₂×C₄	—	188	4	no	—
307	C₂×C₄	—	188	2	no	—
308	C₂×C₄	#13093995	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	#13094050	1	2	yes	—
313	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	#13093995	1	2	yes	—
316	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
317	C₈	C₂×#10625352	1	1	yes	characteristic
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	16	4	no	—
322	C₂²	#15689315	1	2	yes	—
323	C₂²	#15689553	1	2	yes	—
324	C₂²	#15689563	1	2	yes	—
325	C₂²	C₂²×#10625334	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	8	2	no	—
329	C₄	—	376	2	no	—
330	C₄	—	188	4	no	—
331	C₄	#15689315	1	2	yes	—
332	C₄	C₂×#13093995	1	1	yes	characteristic
333	C₄	C₂×#13094050	1	1	yes	characteristic
334	C₂	—	16	2	no	—
335	C₂	#20245447	1	4	yes	direct product factor
336	C₂	C₂×#15689315	1	1	yes	characteristic
337	C₂	C₂×#15689553	1	1	yes	characteristic
338	C₂	C₂×#15689563	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#495	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂²×D₈×C₄₇	C₂²
hypercenter	C₂×#495	D₄₇
hypocenter	C₄₇	C₂×#2211
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#495

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093970	C₂	semidirect
3. C₂×C₄×Dic₁₈₈	C₂	semidirect
4. #20245447	C₂	direct (N × H)
5. C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
6. C₂×#15688927	C₂	semidirect
12. C₂×#13093970	C₂²	semidirect
18. C₄×Dic₁₈₈	C₂²	semidirect
21. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
22. #15688927	C₂²	semidirect
174. C₂×C₄₇	#2211	semidirect
200. C₄₇	C₂×#2211	semidirect
335. C₂	#20245447	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×#13093968	2
C₂×C₄×Dic₁₈₈	2
#20245447	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
C₂×#15688927	2
C₂×#15688924	2
C₂×C₄₇×#495	2
C₂×#2211	47

Subgroups — C2×#20245447 #26370531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245447 #26370531