Subgroups — C₂×#20244653 #26244073

Order 12032.

1330 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689149	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689132	C₂	1	2	yes	maximal
3	C₂²×#13093943	C₂	1	2	yes	maximal
4	#20244653	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688888	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	1	yes	characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂×#13093983	C₂²	1	1	yes	characteristic
10	#15689149	—	2	2	no	—
11	#15689149	C₂²	1	4	yes	—
12	C₂×Dic₄₇⋊C₈	—	2	2	no	—
13	#15689132	—	2	4	no	—
14	#15689132	C₂²	1	8	yes	—
15	C₂×#13093948	—	2	4	no	—
16	C₂×#13093943	—	2	8	no	—
17	C₂×#13093943	C₂²	1	8	yes	—
18	#15688888	C₂²	1	4	yes	—
19	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
20	C₂²×Q₁₆×C₄₇	—	2	8	no	—
21	C₂²×Q₁₆×C₄₇	—	2	2	no	—
22	C₂²×Q₁₆×C₄₇	—	2	2	no	—
23	C₂²×Q₁₆×C₄₇	C₂²	1	4	yes	—
24	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	1	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	—	2	4	no	—
29	C₂×#10625329	—	2	4	no	—
30	C₂×#10625329	C₂³	1	2	yes	—
31	C₂×#10625329	C₂³	1	4	yes	—
32	C₂²×C₄₇⋊C₈	—	2	2	no	—
33	C₂²×C₄₇⋊C₈	—	2	1	no	—
34	#13093983	—	2	1	no	—
35	#13093983	—	2	1	no	—
36	#13093983	D₄	1	2	yes	—
37	#13093983	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₄₇	D₄	1	2	yes	—
39	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
40	C₂×C₄×Dic₄₇	—	2	1	no	—
41	Dic₄₇⋊C₈	—	2	8	no	—
42	#13093948	—	2	16	no	—
43	#13093943	—	2	16	no	—
44	C₂²×Q₈×C₄₇	—	2	4	no	—
45	C₂²×Q₈×C₄₇	—	2	2	no	—
46	C₂²×Q₈×C₄₇	—	2	2	no	—
47	C₂²×Q₈×C₄₇	—	2	2	no	—
48	C₂²×Q₈×C₄₇	—	2	4	no	—
49	C₂²×Q₈×C₄₇	—	2	2	no	—
50	C₂²×Q₈×C₄₇	—	2	2	no	—
51	C₂²×Q₈×C₄₇	—	2	4	no	—
52	C₂²×Q₈×C₄₇	D₄	1	8	yes	—
53	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
54	C₂³×C₄×C₄₇	—	2	2	no	—
55	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×Q₁₆×C₄₇	—	2	16	no	—
57	C₂×Q₁₆×C₄₇	—	2	16	no	—
58	C₂×Q₁₆×C₄₇	—	2	16	no	—
59	C₂×Q₁₆×C₄₇	—	2	8	no	—
60	C₂²×C₈×C₄₇	—	2	4	no	—
61	C₂²×C₈×C₄₇	D₄	1	2	yes	—
62	C₂²×C₈×C₄₇	D₄	1	2	yes	—
63	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
64	C₂⁴×C₄₇	#41	1	1	yes	characteristic
65	C₂×Q₈×C₄₇	—	2	8	no	—
66	C₂×Q₈×C₄₇	—	2	4	no	—
67	C₂×Q₈×C₄₇	—	2	8	no	—
68	C₂×Q₈×C₄₇	—	2	8	no	—
69	C₂×Q₈×C₄₇	—	2	8	no	—
70	C₂×Q₈×C₄₇	—	2	8	no	—
71	C₂×Q₈×C₄₇	—	2	4	no	—
72	C₂×Q₈×C₄₇	—	2	4	no	—
73	C₂×Q₈×C₄₇	—	2	4	no	—
74	C₂×Q₈×C₄₇	—	2	4	no	—
75	C₂×Q₈×C₄₇	—	4	2	no	—
76	C₂×Q₈×C₄₇	—	4	2	no	—
77	C₂×Q₈×C₄₇	—	4	4	no	—
78	C₂×Q₈×C₄₇	—	4	4	no	—
79	C₂×Q₈×C₄₇	—	4	4	no	—
80	C₂×Q₈×C₄₇	—	4	4	no	—
81	C₂×Q₈×C₄₇	—	4	4	no	—
82	C₂×Q₈×C₄₇	C₂×D₄	1	8	yes	—
83	C₂²×C₄×C₄₇	—	2	8	no	—
84	C₂²×C₄×C₄₇	—	2	4	no	—
85	C₂²×C₄×C₄₇	—	4	2	no	—
86	C₂²×C₄×C₄₇	—	4	2	no	—
87	C₂²×C₄×C₄₇	—	4	4	no	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
94	C₂²×C₄×C₄₇	#41	1	4	yes	—
95	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
96	C₂²×Dic₄₇	—	2	2	no	—
97	Q₁₆×C₄₇	—	2	32	no	—
98	C₂×C₈×C₄₇	—	2	4	no	—
99	C₂×C₈×C₄₇	—	2	4	no	—
100	C₂×C₈×C₄₇	—	2	4	no	—
101	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
102	C₂²⋊Dic₄₇	—	2	2	no	—
103	C₂²⋊Dic₄₇	—	2	2	no	—
104	C₄⋊Dic₄₇	—	2	2	no	—
105	C₄⋊Dic₄₇	C₂×D₄	1	4	yes	—
106	C₄×Dic₄₇	—	2	2	no	—
107	C₄×Dic₄₇	—	2	2	no	—
108	#10625329	—	2	4	no	—
109	#10625329	—	2	4	no	—
110	#10625329	—	2	4	no	—
111	#10625329	—	4	2	no	—
112	#10625329	—	4	4	no	—
113	C₂×C₄₇⋊C₈	—	2	4	no	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	4	2	no	—
116	C₂×C₄₇⋊C₈	—	4	4	no	—
117	C₂³×C₄₇	—	2	4	no	—
118	C₂³×C₄₇	#121	1	1	yes	characteristic
119	C₂³×C₄₇	#121	1	1	yes	characteristic
120	C₂³×C₄₇	#123	1	4	yes	—
121	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
122	Q₈×C₄₇	—	2	8	no	—
123	Q₈×C₄₇	—	2	8	no	—
124	Q₈×C₄₇	—	4	4	no	—
125	Q₈×C₄₇	—	4	4	no	—
126	Q₈×C₄₇	—	4	8	no	—
127	Q₈×C₄₇	—	4	8	no	—
128	C₂×C₄×C₄₇	—	2	8	no	—
129	C₂×C₄×C₄₇	—	4	4	no	—
130	C₂×C₄×C₄₇	—	4	4	no	—
131	C₂×C₄×C₄₇	—	4	8	no	—
132	C₂×C₄×C₄₇	—	4	8	no	—
133	C₂×C₄×C₄₇	#120	1	2	yes	—
134	C₂×C₄×C₄₇	#120	1	2	yes	—
135	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#121	1	4	yes	—
138	C₂×C₄×C₄₇	#121	1	4	yes	—
139	C₂×C₄×C₄₇	#123	1	4	yes	—
140	C₂×C₄×C₄₇	#124	1	2	yes	—
141	C₂×C₄×C₄₇	#124	1	2	yes	—
142	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
144	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
146	C₂×Dic₄₇	—	2	4	no	—
147	C₂×Dic₄₇	—	4	4	no	—
148	C₈×C₄₇	—	2	4	no	—
149	C₄₇⋊C₈	—	4	2	no	—
150	C₄₇⋊C₈	—	4	4	no	—
151	C₂×#2047	—	47	1	no	maximal, Sylow
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	—	2	8	no	—
155	C₂²×C₄₇	#394	1	2	yes	—
156	C₂²×C₄₇	#394	1	2	yes	—
157	C₂²×C₄₇	#473	1	4	yes	—
158	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
159	C₂²×C₄₇	C₂×#123	1	2	yes	—
160	C₄×C₄₇	—	4	8	no	—
161	C₄×C₄₇	#394	1	2	yes	—
162	C₄×C₄₇	#394	1	2	yes	—
163	C₄×C₄₇	C₂×#120	1	1	yes	characteristic
164	C₄×C₄₇	C₂×#121	1	2	yes	—
165	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
166	Dic₄₇	—	4	4	no	—
167	C₂³×Q₁₆	D₄₇	1	1	yes	characteristic
168	C₂×#424	—	47	1	no	—
169	C₂×#419	—	47	2	no	—
170	C₂²×#101	—	47	2	no	—
171	#2047	—	47	8	no	—
172	C₂×#341	—	47	1	no	—
173	C₂×C₄₇	—	2	2	no	—
174	C₂×C₄₇	—	2	2	no	—
175	C₂×C₄₇	#2047	1	4	yes	complemented
176	C₂×C₄₇	C₂×#394	1	1	yes	characteristic
177	C₂×C₄₇	C₂×#473	1	2	yes	—
178	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
179	C₂²×Q₁₆	—	2	8	no	—
180	C₂²×Q₁₆	—	2	2	no	—
181	C₂²×Q₁₆	—	2	2	no	—
182	C₂²×Q₁₆	C₂×D₄₇	1	4	yes	—
183	C₂²×M₁₆	—	47	1	no	—
184	C₂²×M₁₆	—	47	2	no	—
185	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
186	C₂×#117	—	47	1	no	—
187	#424	—	47	4	no	—
188	#424	—	94	2	no	—
189	C₂×C₄⋊C₈	—	94	2	no	—
190	#419	—	47	8	no	—
191	#419	—	94	4	no	—
192	C₂×#103	—	94	4	no	—
193	C₂×#101	—	47	8	no	—
194	C₂×#101	—	94	8	no	—
195	#341	—	47	4	no	—
196	C₄₇	C₂×#2047	1	1	yes	characteristic, complemented, Sylow
197	C₂²×Q₈	—	2	4	no	—
198	C₂²×Q₈	—	2	2	no	—
199	C₂²×Q₈	—	2	2	no	—
200	C₂²×Q₈	—	2	2	no	—
201	C₂²×Q₈	—	2	4	no	—
202	C₂²×Q₈	—	2	2	no	—
203	C₂²×Q₈	—	2	2	no	—
204	C₂²×Q₈	—	2	4	no	—
205	C₂²×Q₈	C₂²⋊D₄₇	1	8	yes	—
206	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
207	C₂³×C₄	—	2	2	no	—
208	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
209	C₂×Q₁₆	—	2	16	no	—
210	C₂×Q₁₆	—	2	16	no	—
211	C₂×Q₁₆	—	2	16	no	—
212	C₂×Q₁₆	—	2	8	no	—
213	C₂×M₁₆	—	47	4	no	—
214	C₂×M₁₆	—	47	2	no	—
215	C₂×M₁₆	—	94	2	no	—
216	C₂×M₁₆	—	94	1	no	—
217	C₂×M₁₆	—	94	4	no	—
218	C₂×M₁₆	—	94	4	no	—
219	C₂×M₁₆	—	94	4	no	—
220	C₂²×C₈	—	2	4	no	—
221	C₂²×C₈	—	94	2	no	—
222	C₂²×C₈	—	94	1	no	—
223	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
224	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
225	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
226	#117	—	47	2	no	—
227	#117	—	47	2	no	—
228	#117	—	94	1	no	—
229	#117	—	94	1	no	—
230	C₂×C₄⋊C₄	—	47	2	no	—
231	C₂×C₂²⋊C₄	—	94	1	no	—
232	C₂×C₄²	—	94	1	no	—
233	C₄⋊C₈	—	94	8	no	—
234	#103	—	94	16	no	—
235	#101	—	94	16	no	—
236	C₂⁴	#10625352	1	1	yes	characteristic
237	C₂×Q₈	—	2	8	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	2	8	no	—
240	C₂×Q₈	—	2	8	no	—
241	C₂×Q₈	—	2	8	no	—
242	C₂×Q₈	—	2	8	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	—	4	2	no	—
249	C₂×Q₈	—	4	4	no	—
250	C₂×Q₈	—	4	4	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	4	4	no	—
253	C₂×Q₈	—	4	4	no	—
254	C₂×Q₈	C₂×C₂²⋊D₄₇	1	8	yes	—
255	C₂²×C₄	—	2	8	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	4	no	—
260	C₂²×C₄	—	94	2	no	—
261	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
262	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
263	C₂²×C₄	#10625352	1	1	yes	characteristic
264	C₂²×C₄	#10625352	1	4	yes	—
265	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
268	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
269	Q₁₆	—	2	32	no	—
270	M₁₆	—	94	4	no	—
271	M₁₆	—	94	4	no	—
272	M₁₆	—	94	4	no	—
273	M₁₆	—	188	2	no	—
274	M₁₆	—	188	4	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	94	4	no	—
279	C₂×C₈	—	94	2	no	—
280	C₂×C₈	—	188	2	no	—
281	C₂×C₈	—	188	4	no	—
282	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
283	C₄⋊C₄	—	47	4	no	—
284	C₄⋊C₄	—	94	2	no	—
285	C₂²⋊C₄	—	94	2	no	—
286	C₂²⋊C₄	—	94	2	no	—
287	C₄²	—	94	2	no	—
288	C₄²	—	94	2	no	—
289	C₂³	—	2	4	no	—
290	C₂³	#13093995	1	1	yes	characteristic
291	C₂³	#13093995	1	1	yes	characteristic
292	C₂³	#13094005	1	4	yes	—
293	C₂³	C₂×#10625352	1	1	yes	characteristic
294	Q₈	—	2	8	no	—
295	Q₈	—	2	8	no	—
296	Q₈	—	4	8	no	—
297	Q₈	—	4	8	no	—
298	Q₈	—	4	4	no	—
299	Q₈	—	4	4	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	4	8	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	94	4	no	—
306	C₂×C₄	—	188	4	no	—
307	C₂×C₄	#13093993	1	2	yes	—
308	C₂×C₄	#13093993	1	2	yes	—
309	C₂×C₄	#13093995	1	1	yes	characteristic
310	C₂×C₄	#13093995	1	1	yes	characteristic
311	C₂×C₄	#13093996	1	4	yes	—
312	C₂×C₄	#13093996	1	4	yes	—
313	C₂×C₄	#13094005	1	4	yes	—
314	C₂×C₄	#13094011	1	2	yes	—
315	C₂×C₄	#13094011	1	2	yes	—
316	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
318	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
319	C₂×C₄	C₂×#10625352	1	2	yes	—
320	C₈	—	2	4	no	—
321	C₈	—	188	2	no	—
322	C₈	—	188	4	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	8	no	—
326	C₂²	#15689037	1	2	yes	—
327	C₂²	#15689037	1	2	yes	—
328	C₂²	#15689395	1	4	yes	—
329	C₂²	C₂×#13093995	1	1	yes	characteristic
330	C₂²	C₂×#13094005	1	2	yes	—
331	C₄	—	4	8	no	—
332	C₄	—	188	4	no	—
333	C₄	#15689037	1	2	yes	—
334	C₄	#15689037	1	2	yes	—
335	C₄	C₂×#13093993	1	1	yes	characteristic
336	C₄	C₂×#13093996	1	2	yes	—
337	C₄	C₂×#13094011	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	2	2	no	—
340	C₂	#20244653	1	4	yes	direct product factor
341	C₂	C₂×#15689037	1	1	yes	characteristic
342	C₂	C₂×#15689395	1	2	yes	—

	subgroup	G/H
center	C₂³	#13093995
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂³×Q₁₆×C₄₇	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂⁴×C₄₇	#41
hypercenter	C₂³×Q₁₆	D₄₇
hypocenter	C₄₇	C₂×#2047
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
4. #20244653	C₂	direct (N × H)
175. C₂×C₄₇	#2047	semidirect
196. C₄₇	C₂×#2047	semidirect
340. C₂	#20244653	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×#15689149	2
C₂×#15689132	2
C₂²×#13093943	2
#20244653	2
C₂×#15688888	2
C₂³×Q₁₆×C₄₇	2
C₂×#2047	47

Subgroups — C2×#20244653 #26244073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20244653 #26244073