Subgroups — C₂×#20245440 #26370511

Order 12032.

786 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093967	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13093965	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245440	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, complemented
8	C₂×C₄₇×#495	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₈₈	—	2	1	no	—
11	C₂×#13093967	—	2	4	no	—
12	C₂×#13093967	C₂²	1	4	yes	complemented
13	C₂×#13093965	—	2	4	no	—
14	C₂×#13093965	C₂²	1	4	yes	—
15	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₄×D₃₇₆	—	2	2	no	—
18	C₄×D₃₇₆	C₂²	1	4	yes	complemented
19	C₂×#13093958	C₂²	1	1	yes	characteristic
20	C₂×#13093946	—	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	complemented
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	complemented
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂²×D₁₈₈	—	2	1	no	—
32	C₂²×C₄×D₄₇	—	4	1	no	—
33	C₂×D₃₇₆	—	2	1	no	—
34	C₂×D₃₇₆	—	2	2	no	—
35	C₂×D₃₇₆	D₄	1	4	yes	—
36	C₂×D₃₇₆	C₂³	1	2	yes	—
37	C₄×D₁₈₈	—	2	4	no	—
38	C₄×D₁₈₈	—	4	2	no	—
39	C₂×C₄⋊Dic₄₇	—	2	1	no	—
40	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
41	#13093967	—	2	8	no	—
42	#13093965	—	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	#13093946	—	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	C₂³×D₄₇	—	4	1	no	—
73	C₂×Q₈×C₄₇	—	2	2	no	—
74	C₂×Q₈×C₄₇	—	4	1	no	—
75	C₂×Q₈×C₄₇	—	4	2	no	—
76	C₂×Q₈×C₄₇	—	4	2	no	—
77	C₂×D₄×C₄₇	—	2	2	no	—
78	C₂×D₄×C₄₇	—	4	2	no	—
79	C₂×D₄×C₄₇	—	4	1	no	—
80	C₂×D₄×C₄₇	—	4	2	no	—
81	C₂²×C₄×C₄₇	—	4	1	no	—
82	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×Dic₄₇	—	4	1	no	—
86	C₂×D₁₈₈	—	2	4	no	—
87	C₂×D₁₈₈	—	2	2	no	—
88	C₂×D₁₈₈	—	4	1	no	—
89	C₂×D₁₈₈	—	4	2	no	—
90	C₂×C₄×D₄₇	—	4	4	no	—
91	C₂×C₄×D₄₇	—	8	4	no	—
92	Q₁₆×C₄₇	—	2	2	no	—
93	Q₁₆×C₄₇	—	2	2	no	—
94	Q₁₆×C₄₇	—	4	2	no	—
95	SD₁₆×C₄₇	—	4	8	no	—
96	D₈×C₄₇	—	2	2	no	—
97	D₈×C₄₇	—	2	2	no	—
98	D₈×C₄₇	—	4	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₄₇	#41	1	2	yes	—
104	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
105	C₂²⋊C₄×C₄₇	—	4	4	no	—
106	C₂²⋊C₄×C₄₇	—	4	4	no	—
107	C₄²×C₄₇	D₈	1	2	yes	—
108	C₄²×C₄₇	C₂×D₄	1	2	yes	—
109	(C₂×D₄₇)⋊C₄	—	4	4	no	—
110	C₄⋊Dic₄₇	—	2	2	no	—
111	C₄⋊Dic₄₇	—	4	1	no	—
112	D₃₇₆	—	2	4	no	—
113	D₃₇₆	—	2	2	no	—
114	D₃₇₆	C₂×D₄	1	4	yes	—
115	C₄₇⋊C₁₆	—	2	2	no	—
116	C₄₇⋊C₁₆	—	2	2	no	—
117	C₂³×C₄₇	—	4	2	no	—
118	C₂³×C₄₇	—	8	2	no	—
119	C₂³×C₄₇	—	8	2	no	—
120	C₂³×C₄₇	—	8	2	no	—
121	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
122	C₂²×D₄₇	—	4	2	no	—
123	C₂²×D₄₇	—	4	4	no	—
124	C₂²×D₄₇	—	8	2	no	—
125	C₂²×D₄₇	—	8	2	no	—
126	Q₈×C₄₇	—	4	2	no	—
127	Q₈×C₄₇	—	4	2	no	—
128	Q₈×C₄₇	—	8	2	no	—
129	D₄×C₄₇	—	4	2	no	—
130	D₄×C₄₇	—	4	2	no	—
131	D₄×C₄₇	—	8	2	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	4	2	no	—
135	C₂×C₄×C₄₇	—	8	2	no	—
136	C₂×C₄×C₄₇	#121	1	2	yes	—
137	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	8	2	no	—
144	D₁₈₈	—	2	4	no	—
145	D₁₈₈	—	4	2	no	—
146	D₁₈₈	—	4	4	no	—
147	C₄×D₄₇	—	8	8	no	—
148	C₈×C₄₇	—	2	2	no	—
149	C₈×C₄₇	#121	1	2	yes	—
150	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
152	C₂×#2210	—	47	1	no	maximal, Sylow
153	C₂²×C₄₇	—	8	2	no	—
154	C₂²×C₄₇	—	8	2	no	—
155	C₂²×C₄₇	—	8	2	no	—
156	C₂²×C₄₇	—	16	4	no	—
157	C₂²×C₄₇	#459	1	2	yes	—
158	C₂²×C₄₇	#508	1	2	yes	—
159	C₂²×C₄₇	#509	1	2	yes	—
160	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
161	C₂×D₄₇	—	4	4	no	—
162	C₂×D₄₇	—	8	2	no	—
163	C₂×D₄₇	—	8	2	no	—
164	C₂×D₄₇	—	8	8	no	—
165	C₄×C₄₇	—	2	2	no	—
166	C₄×C₄₇	—	2	2	no	—
167	C₄×C₄₇	—	8	2	no	—
168	C₄×C₄₇	#459	1	2	yes	—
169	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
170	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
171	Dic₄₇	—	8	2	no	—
172	C₂²×SD₃₂	—	47	1	no	—
173	C₂²×D₁₆	—	47	1	no	—
174	C₂×#495	D₄₇	1	1	yes	characteristic, complemented
175	C₂×C₄×D₈	—	47	1	no	—
176	#2210	—	47	8	no	—
177	C₂×C₄⋊C₁₆	—	47	1	no	—
178	C₂×#358	—	47	1	no	—
179	C₂×#357	—	47	1	no	—
180	C₂×C₄₇	—	16	2	no	—
181	C₂×C₄₇	#2210	1	4	yes	complemented
182	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
183	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
184	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
185	D₄₇	—	8	4	no	—
186	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
187	C₂²×SD₁₆	—	2	1	no	—
188	C₂²×D₈	—	47	1	no	—
189	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
190	C₂×#124	—	2	1	no	—
191	C₂×C₄×D₄	—	94	1	no	—
192	C₂×SD₃₂	—	47	4	no	—
193	C₂×SD₃₂	—	94	4	no	—
194	C₂×D₁₆	—	47	4	no	—
195	C₂×D₁₆	—	94	4	no	—
196	C₂²×C₁₆	—	47	1	no	—
197	C₂²×C₁₆	—	47	1	no	—
198	#495	—	2	2	no	—
199	#495	C₂×D₄₇	1	4	yes	complemented
200	C₄×D₈	—	47	4	no	—
201	C₄×D₈	—	94	2	no	—
202	C₂×#107	—	47	1	no	—
203	C₂×#102	—	94	1	no	—
204	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
205	C₄⋊C₁₆	—	47	4	no	—
206	#358	—	47	4	no	—
207	#357	—	47	4	no	—
208	C₄₇	C₂×#2210	1	1	yes	characteristic, complemented, Sylow
209	C₂²×Q₈	—	2	1	no	—
210	C₂²×D₄	—	2	1	no	—
211	C₂²×D₄	—	94	1	no	—
212	C₂³×C₄	—	188	1	no	—
213	C₂×Q₁₆	—	2	2	no	—
214	C₂×Q₁₆	—	2	2	no	—
215	C₂×Q₁₆	—	2	1	no	—
216	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
217	C₂×SD₁₆	—	2	4	no	—
218	C₂×SD₁₆	—	4	4	no	—
219	C₂×D₈	—	2	2	no	—
220	C₂×D₈	—	2	2	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	—	47	4	no	—
223	C₂×D₈	—	47	2	no	—
224	C₂×D₈	—	94	2	no	—
225	C₂×D₈	—	94	1	no	—
226	C₂×D₈	C₂²×D₄₇	1	2	yes	—
227	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
229	#124	—	2	4	no	—
230	#124	—	4	2	no	—
231	C₄×D₄	—	94	4	no	—
232	C₄×D₄	—	188	2	no	—
233	C₂×C₄⋊C₄	—	94	1	no	—
234	C₂×C₂²⋊C₄	—	4	1	no	—
235	C₂×C₂²⋊C₄	—	4	1	no	—
236	C₂×C₂²⋊C₄	—	188	1	no	—
237	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
238	SD₃₂	—	94	8	no	—
239	D₁₆	—	94	8	no	—
240	C₂×C₁₆	—	47	2	no	—
241	C₂×C₁₆	—	47	2	no	—
242	C₂×C₁₆	—	94	2	no	—
243	C₂×C₁₆	—	94	2	no	—
244	#107	—	47	2	no	—
245	#107	—	94	1	no	—
246	#102	—	94	4	no	—
247	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
248	C₄×C₈	C₂²×D₄₇	1	2	yes	—
249	C₂⁴	—	4	1	no	—
250	C₂⁴	—	188	1	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	94	4	no	—
260	C₂×D₄	—	94	2	no	—
261	C₂×D₄	—	188	1	no	—
262	C₂×D₄	—	188	2	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	376	4	no	—
265	C₂²×C₄	—	188	1	no	—
266	C₂²×C₄	—	188	4	no	—
267	C₂²×C₄	#10625334	1	1	yes	characteristic
268	C₂²×C₄	#10625334	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	2	2	no	—
272	Q₁₆	—	4	2	no	—
273	SD₁₆	—	4	8	no	—
274	D₈	—	2	2	no	—
275	D₈	—	2	2	no	—
276	D₈	—	4	2	no	—
277	D₈	—	47	4	no	—
278	D₈	—	94	4	no	—
279	D₈	—	94	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	D₄×D₄₇	1	2	yes	—
282	C₂×C₈	#10625352	1	2	yes	—
283	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
284	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
285	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
286	C₄⋊C₄	—	94	2	no	—
287	C₄⋊C₄	—	188	1	no	—
288	C₂²⋊C₄	—	4	4	no	—
289	C₂²⋊C₄	—	4	4	no	—
290	C₂²⋊C₄	—	188	4	no	—
291	C₄²	#10625334	1	2	yes	—
292	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
293	C₁₆	—	94	2	no	—
294	C₁₆	—	94	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	8	2	no	—
297	C₂³	—	8	2	no	—
298	C₂³	—	8	2	no	—
299	C₂³	—	376	2	no	—
300	C₂³	—	376	2	no	—
301	C₂³	—	188	4	no	—
302	C₂³	—	188	2	no	—
303	C₂³	C₂×#10625334	1	1	yes	characteristic
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	8	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	8	2	no	—
310	D₄	—	94	4	no	—
311	D₄	—	188	4	no	—
312	D₄	—	188	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	—	376	8	no	—
318	C₂×C₄	—	376	2	no	—
319	C₂×C₄	—	188	2	no	—
320	C₂×C₄	#13093995	1	2	yes	—
321	C₂×C₄	C₂×#10625334	1	2	yes	—
322	C₂×C₄	C₂×#10625334	1	2	yes	—
323	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
324	C₂×C₄	#13094050	1	2	yes	—
325	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	#13093995	1	2	yes	—
328	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
329	C₈	C₂×#10625352	1	1	yes	characteristic
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	16	4	no	—
334	C₂²	—	376	8	no	—
335	C₂²	—	376	2	no	—
336	C₂²	—	376	2	no	—
337	C₂²	—	188	4	no	—
338	C₂²	#15689315	1	2	yes	—
339	C₂²	#15689553	1	2	yes	—
340	C₂²	#15689558	1	2	yes	—
341	C₂²	C₂²×#10625334	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	8	2	no	—
345	C₄	—	376	2	no	—
346	C₄	#15689315	1	2	yes	—
347	C₄	C₂×#13093995	1	1	yes	characteristic
348	C₄	C₂×#13094050	1	1	yes	characteristic
349	C₂	—	16	2	no	—
350	C₂	—	376	4	no	—
351	C₂	#20245440	1	4	yes	direct product factor
352	C₂	C₂×#15689315	1	1	yes	characteristic
353	C₂	C₂×#15689553	1	1	yes	characteristic
354	C₂	C₂×#15689558	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×#13093965	C₂
hypercenter	C₂×#495	D₄₇
hypocenter	C₄₇	C₂×#2210
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×#13093967	C₂	semidirect
3. C₂×C₄×D₃₇₆	C₂	semidirect
4. #20245440	C₂	direct (N × H)
5. C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
6. C₂×#15688927	C₂	semidirect
7. C₂×#15688924	C₂	semidirect
8. C₂×C₄₇×#495	C₂	semidirect
12. C₂×#13093967	C₂²	semidirect
18. C₄×D₃₇₆	C₂²	semidirect
21. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
22. #15688927	C₂²	semidirect
23. #15688924	C₂²	semidirect
29. C₄₇×#495	C₂²	semidirect
174. C₂×#495	D₄₇	semidirect
181. C₂×C₄₇	#2210	semidirect
199. #495	C₂×D₄₇	semidirect
208. C₄₇	C₂×#2210	semidirect
351. C₂	#20245440	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×#13093967	2
C₂²×#13093965	2
C₂×C₄×D₃₇₆	2
#20245440	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
C₂×#15688927	2
C₂×#15688924	2
C₂×C₄₇×#495	2
C₂×#2210	47

Subgroups — C2×#20245440 #26370511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245440 #26370511