Subgroups — C₂×#20245482 #26370706

Order 12032.

754 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093968	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15689364	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245482	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688950	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688926	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15688924	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#466	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	C₂×#13094003	—	2	1	no	—
11	C₂×#13093968	—	2	4	no	—
12	C₂×#13093968	C₂²	1	4	yes	complemented
13	C₂²×C₄₇⋊C₁₆	—	2	4	no	—
14	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₁₆	C₂²	1	4	yes	complemented
17	#15689364	—	2	2	no	—
18	#15689364	C₂²	1	4	yes	complemented
19	C₂×#13093958	C₂²	1	1	yes	characteristic
20	C₂×#13093949	—	2	1	no	—
21	#15688950	C₂²	1	4	yes	complemented
22	#15688926	C₂²	1	4	yes	complemented
23	#15688924	C₂²	1	4	yes	complemented
24	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#121×C₄₇	—	2	1	no	—
27	C₄₇×#466	—	2	2	no	—
28	C₄₇×#466	C₂²	1	4	yes	—
29	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#102×C₄₇	—	2	1	no	—
31	C₂²×Dic₉₄	—	2	1	no	—
32	C₂×Dic₁₈₈	—	2	2	no	—
33	C₂×Dic₁₈₈	—	2	2	no	—
34	C₂×Dic₁₈₈	—	2	1	no	—
35	C₂×Dic₁₈₈	C₂³	1	2	yes	—
36	#13094003	—	2	4	no	—
37	#13094003	—	4	2	no	—
38	C₂×C₄⋊Dic₄₇	—	2	1	no	—
39	C₂×Dic₄₇⋊C₄	—	4	1	no	—
40	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
41	#13093968	—	2	8	no	—
42	C₂×C₄₇⋊C₁₆	—	2	2	no	—
43	C₂×C₄₇⋊C₁₆	—	2	8	no	—
44	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	complemented
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×D₄×C₄₇	—	2	1	no	—
51	C₂²×D₄×C₄₇	—	4	1	no	—
52	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
53	C₂×D₈×C₄₇	—	2	2	no	—
54	C₂×D₈×C₄₇	—	2	1	no	—
55	C₂×D₈×C₄₇	—	2	2	no	—
56	C₂×D₈×C₄₇	C₂³	1	2	yes	—
57	C₂²×C₈×C₄₇	—	2	2	no	—
58	C₂²×C₈×C₄₇	—	2	2	no	—
59	C₂²×C₈×C₄₇	D₄	1	2	yes	—
60	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	#121×C₄₇	—	2	4	no	—
64	#121×C₄₇	—	4	2	no	—
65	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
67	#107×C₄₇	—	2	1	no	—
68	#107×C₄₇	C₂³	1	2	yes	—
69	#102×C₄₇	—	2	4	no	—
70	C₂⁴×C₄₇	—	4	1	no	—
71	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
72	C₂×D₄×C₄₇	—	2	2	no	—
73	C₂×D₄×C₄₇	—	4	4	no	—
74	C₂×D₄×C₄₇	—	4	2	no	—
75	C₂×D₄×C₄₇	—	4	1	no	—
76	C₂×D₄×C₄₇	—	4	2	no	—
77	C₂×D₄×C₄₇	—	8	4	no	—
78	C₂²×C₄×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	4	1	no	—
81	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	D₈	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×Dic₄₇	—	4	1	no	—
86	C₂²×Dic₄₇	—	4	1	no	—
87	C₂×Dic₉₄	—	2	2	no	—
88	C₂×Dic₉₄	—	4	2	no	—
89	C₂×Dic₉₄	—	4	2	no	—
90	C₂×Dic₉₄	—	4	1	no	—
91	D₈×C₄₇	—	2	2	no	—
92	D₈×C₄₇	—	2	2	no	—
93	D₈×C₄₇	—	4	2	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	—	2	4	no	—
96	C₂×C₈×C₄₇	—	2	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	C₄⋊C₄×C₄₇	—	2	2	no	—
104	C₄⋊C₄×C₄₇	—	4	1	no	—
105	C₂²⋊C₄×C₄₇	—	4	4	no	—
106	C₂²⋊Dic₄₇	—	4	4	no	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	4	1	no	—
109	Dic₄₇⋊C₄	—	4	4	no	—
110	Dic₁₈₈	—	2	2	no	—
111	Dic₁₈₈	—	2	2	no	—
112	Dic₁₈₈	—	4	2	no	—
113	C₄₇⋊C₁₆	—	2	2	no	—
114	C₄₇⋊C₁₆	C₂×D₄	1	4	yes	complemented
115	C₂³×C₄₇	—	2	2	no	—
116	C₂³×C₄₇	—	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	SD₃₂	1	4	yes	complemented
122	C₂³×C₄₇	C₂×D₈	1	2	yes	—
123	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
124	D₄×C₄₇	—	4	2	no	—
125	D₄×C₄₇	—	4	2	no	—
126	D₄×C₄₇	—	8	8	no	—
127	D₄×C₄₇	—	8	2	no	—
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	—	2	2	no	—
131	C₂×C₄×C₄₇	—	2	4	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	—	8	2	no	—
134	C₂×C₄×C₄₇	#121	1	2	yes	—
135	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#135	1	2	yes	—
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	8	2	no	—
142	C₂×Dic₄₇	—	8	2	no	—
143	Dic₉₄	—	4	2	no	—
144	Dic₉₄	—	4	2	no	—
145	Dic₉₄	—	8	2	no	—
146	C₈×C₄₇	—	2	2	no	—
147	C₈×C₄₇	#121	1	2	yes	—
148	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
150	C₂×#2217	—	47	1	no	maximal, Sylow
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	—	2	8	no	—
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	—	8	2	no	—
155	C₂²×C₄₇	—	8	2	no	—
156	C₂²×C₄₇	—	8	2	no	—
157	C₂²×C₄₇	—	16	4	no	—
158	C₂²×C₄₇	#466	1	2	yes	—
159	C₂²×C₄₇	C₂×SD₃₂	1	2	yes	—
160	C₂²×C₄₇	C₂×SD₃₂	1	4	yes	complemented
161	C₂²×C₄₇	#508	1	2	yes	—
162	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₄₇	—	2	2	no	—
164	C₄×C₄₇	—	8	2	no	—
165	C₄×C₄₇	#466	1	2	yes	—
166	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
167	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
168	Dic₄₇	—	8	2	no	—
169	Dic₄₇	—	8	2	no	—
170	C₂²×SD₃₂	—	47	1	no	—
171	C₂³×C₁₆	—	47	1	no	—
172	C₂×#467	—	47	1	no	—
173	C₂×#466	D₄₇	1	1	yes	characteristic
174	#2217	—	47	8	no	—
175	C₂×#367	—	47	1	no	—
176	C₂×#358	—	47	1	no	—
177	C₂×#357	—	47	1	no	—
178	C₂×C₄₇	—	2	4	no	—
179	C₂×C₄₇	—	16	2	no	—
180	C₂×C₄₇	#2217	1	4	yes	complemented
181	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
182	C₂×C₄₇	C₂²×SD₃₂	1	1	yes	characteristic
183	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
184	C₂²×Q₁₆	—	47	1	no	—
185	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
186	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
187	C₂×C₂²⋊Q₈	—	94	1	no	—
188	C₂×#121	—	2	1	no	—
189	C₂×SD₃₂	—	47	4	no	—
190	C₂×SD₃₂	—	94	4	no	—
191	C₂²×C₁₆	—	47	1	no	—
192	C₂²×C₁₆	—	47	1	no	—
193	C₂²×C₁₆	—	47	4	no	—
194	C₂²×C₁₆	—	94	4	no	—
195	#467	—	47	4	no	—
196	#467	—	94	2	no	—
197	#466	—	2	2	no	—
198	#466	C₂×D₄₇	1	4	yes	—
199	C₂×#107	—	47	1	no	—
200	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
201	C₂×#103	—	94	1	no	—
202	C₂×#102	—	2	1	no	—
203	#367	—	47	4	no	—
204	#358	—	47	4	no	—
205	#357	—	47	4	no	—
206	C₄₇	C₂×#2217	1	1	yes	characteristic, complemented, Sylow
207	C₂²×Q₈	—	94	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	C₂²×D₄	—	4	1	no	—
210	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
211	C₂×Q₁₆	—	47	2	no	—
212	C₂×Q₁₆	—	94	2	no	—
213	C₂×Q₁₆	—	94	2	no	—
214	C₂×Q₁₆	—	94	1	no	—
215	C₂×D₈	—	2	2	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂×D₈	—	2	1	no	—
218	C₂×D₈	C₂²×D₄₇	1	2	yes	—
219	C₂²×C₈	—	2	2	no	—
220	C₂²×C₈	—	2	2	no	—
221	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
222	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
223	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
224	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
225	C₂²⋊Q₈	—	94	4	no	—
226	C₂²⋊Q₈	—	188	2	no	—
227	#121	—	2	4	no	—
228	#121	—	4	2	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	94	1	no	—
231	C₂×C₄⋊C₄	—	188	1	no	—
232	C₂×C₂²⋊C₄	—	4	1	no	—
233	C₂×C₂²⋊C₄	—	188	1	no	—
234	SD₃₂	—	94	8	no	—
235	C₂×C₁₆	—	47	4	no	—
236	C₂×C₁₆	—	47	2	no	—
237	C₂×C₁₆	—	47	2	no	—
238	C₂×C₁₆	—	94	8	no	—
239	C₂×C₁₆	—	94	2	no	—
240	#107	—	2	1	no	—
241	#107	—	47	2	no	—
242	#107	—	94	1	no	—
243	#107	C₂²×D₄₇	1	2	yes	—
244	#103	—	94	4	no	—
245	#102	—	2	4	no	—
246	C₂⁴	—	4	1	no	—
247	C₂⁴	#10625334	1	1	yes	characteristic
248	C₂×Q₈	—	94	2	no	—
249	C₂×Q₈	—	188	2	no	—
250	C₂×Q₈	—	188	2	no	—
251	C₂×Q₈	—	188	1	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	4	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	8	4	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	188	1	no	—
262	C₂²×C₄	—	188	1	no	—
263	C₂²×C₄	#10625334	1	2	yes	—
264	C₂²×C₄	#10625334	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
266	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
267	Q₁₆	—	94	2	no	—
268	Q₁₆	—	94	2	no	—
269	Q₁₆	—	188	2	no	—
270	D₈	—	2	2	no	—
271	D₈	—	2	2	no	—
272	D₈	—	4	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	D₄×D₄₇	1	2	yes	—
278	C₂×C₈	#10625352	1	2	yes	—
279	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
280	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
281	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	94	2	no	—
285	C₄⋊C₄	—	188	4	no	—
286	C₄⋊C₄	—	188	1	no	—
287	C₂²⋊C₄	—	4	4	no	—
288	C₂²⋊C₄	—	188	4	no	—
289	C₁₆	—	47	4	no	—
290	C₁₆	—	94	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	8	2	no	—
295	C₂³	—	8	2	no	—
296	C₂³	—	8	2	no	—
297	C₂³	#13093968	1	4	yes	complemented
298	C₂³	C₂×#10625334	1	2	yes	—
299	C₂³	C₂×#10625334	1	1	yes	characteristic
300	Q₈	—	376	2	no	—
301	Q₈	—	188	2	no	—
302	Q₈	—	188	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	8	no	—
306	D₄	—	8	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	376	2	no	—
314	C₂×C₄	—	376	2	no	—
315	C₂×C₄	—	188	2	no	—
316	C₂×C₄	—	188	2	no	—
317	C₂×C₄	#13093996	1	2	yes	—
318	C₂×C₄	C₂×#10625334	1	2	yes	—
319	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
320	C₂×C₄	#13094044	1	2	yes	—
321	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	#13093996	1	2	yes	—
324	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
325	C₈	C₂×#10625352	1	1	yes	characteristic
326	C₂²	—	2	2	no	—
327	C₂²	—	2	8	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	16	4	no	—
333	C₂²	#15689357	1	2	yes	—
334	C₂²	C₂×#13093968	1	2	yes	—
335	C₂²	C₂×#13093968	1	4	yes	complemented
336	C₂²	#15689553	1	2	yes	—
337	C₂²	C₂²×#10625334	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	8	2	no	—
340	C₄	—	376	2	no	—
341	C₄	—	376	2	no	—
342	C₄	#15689357	1	2	yes	—
343	C₄	C₂×#13093996	1	1	yes	characteristic
344	C₄	C₂×#13094044	1	1	yes	characteristic
345	C₂	—	2	4	no	—
346	C₂	—	16	2	no	—
347	C₂	#20245482	1	4	yes	direct product factor
348	C₂	C₂×#15689357	1	1	yes	characteristic
349	C₂	C₂²×#13093968	1	1	yes	characteristic
350	C₂	C₂×#15689553	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₂×C₄₇×#466	C₂
socle	C₂³×C₄₇	C₂×D₈
hypercenter	C₂×#466	D₄₇
hypocenter	C₄₇	C₂×#2217
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093968	C₂	semidirect
2. C₂³×C₄₇⋊C₁₆	C₂	semidirect
3. C₂×#15689364	C₂	semidirect
4. #20245482	C₂	direct (N × H)
5. C₂×#15688950	C₂	semidirect
6. C₂×#15688926	C₂	semidirect
7. C₂×#15688924	C₂	semidirect
12. C₂×#13093968	C₂²	semidirect
16. C₂²×C₄₇⋊C₁₆	C₂²	semidirect
18. #15689364	C₂²	semidirect
21. #15688950	C₂²	semidirect
22. #15688926	C₂²	semidirect
23. #15688924	C₂²	semidirect
44. C₂×C₄₇⋊C₁₆	D₄	semidirect
114. C₄₇⋊C₁₆	C₂×D₄	semidirect
121. C₂³×C₄₇	SD₃₂	semidirect
160. C₂²×C₄₇	C₂×SD₃₂	semidirect
180. C₂×C₄₇	#2217	semidirect
206. C₄₇	C₂×#2217	semidirect
297. C₂³	#13093968	semidirect
335. C₂²	C₂×#13093968	semidirect
347. C₂	#20245482	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×#13093968	2
C₂³×C₄₇⋊C₁₆	2
C₂×#15689364	2
#20245482	2
C₂×#15688950	2
C₂×#15688926	2
C₂×#15688924	2
C₂×C₄₇×#466	2
C₂×#2217	47

Subgroups — C2×#20245482 #26370706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245482 #26370706