Subgroups — C₂×#20254044 #27086119

Order 12032.

1842 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689734	C₂	1	2	yes	maximal
2	#20254044	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689525	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15689491	C₂	1	2	yes	maximal, complemented
5	C₂×#15689378	C₂	1	2	yes	maximal
6	C₂×#15689361	C₂	1	2	yes	maximal
7	C₂×#15689230	C₂	1	2	yes	maximal, complemented
8	C₂×Q₁₆×Dic₄₇	C₂	1	2	yes	maximal, complemented
9	C₂×#15689077	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄₇×#494	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#10625324	C₂²	1	1	yes	characteristic
12	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
13	#15689734	—	2	8	no	—
14	#15689734	C₂²	1	16	yes	—
15	C₂×#13094019	—	2	4	no	—
16	C₂×#13094016	C₂²	1	2	yes	—
17	C₂×#13094014	—	2	4	no	—
18	C₂×#13094000	C₂²	1	4	yes	—
19	C₂×Q₈×Dic₄₇	C₂²	1	4	yes	—
20	C₂×#13093981	C₂²	1	1	yes	characteristic
21	#15689525	C₂²	1	8	yes	complemented
22	#15689491	C₂²	1	16	yes	complemented
23	#15689378	C₂²	1	16	yes	—
24	#15689361	C₂²	1	16	yes	—
25	#15689230	C₂²	1	16	yes	complemented
26	Q₁₆×Dic₄₇	C₂²	1	16	yes	complemented
27	C₂×#13093958	C₂²	1	2	yes	—
28	C₂×#13093956	C₂²	1	2	yes	—
29	C₂×#13093948	C₂²	1	4	yes	—
30	C₂×#13093948	C₂²	1	4	yes	—
31	#15689077	C₂²	1	8	yes	—
32	C₂×#13093935	C₂²	1	1	yes	characteristic
33	C₂×#13093934	C₂²	1	1	yes	characteristic
34	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
35	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
36	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
37	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
38	C₄₇×#494	C₂²	1	8	yes	complemented
39	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
41	C₂×#10625324	—	2	4	no	—
42	C₂×#10625324	C₂³	1	4	yes	—
43	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₈×D₄₇	—	2	4	no	—
46	C₂×C₈×D₄₇	C₂³	1	4	yes	—
47	#13094019	—	2	32	no	—
48	#13094016	—	2	4	no	—
49	#13094016	C₂³	1	8	yes	—
50	#13094014	—	2	32	no	—
51	#13094000	—	2	8	no	—
52	#13094000	C₂³	1	16	yes	—
53	Q₈×Dic₄₇	—	2	8	no	—
54	Q₈×Dic₄₇	C₂³	1	16	yes	—
55	#13093981	D₄	1	4	yes	—
56	#13093981	C₂³	1	4	yes	—
57	C₂×C₄⋊Dic₄₇	—	2	4	no	—
58	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
59	C₂×Dic₄₇⋊C₄	—	2	4	no	—
60	C₂×Dic₄₇⋊C₄	—	2	4	no	—
61	C₂×Dic₄₇⋊C₄	D₄	1	2	yes	—
62	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
63	C₂×(C₂×D₄₇)⋊C₄	D₄	1	2	yes	—
64	C₂×C₄×Dic₄₇	—	2	4	no	—
65	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
66	#13093958	C₂³	1	8	yes	—
67	#13093956	C₂³	1	8	yes	—
68	#13093948	C₂³	1	16	yes	—
69	#13093948	C₂³	1	16	yes	—
70	#13093935	C₂³	1	4	yes	—
71	#13093934	C₂³	1	4	yes	—
72	C₈×Dic₄₇	C₂³	1	4	yes	—
73	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
74	C₂×Q₁₆×C₄₇	—	2	8	no	—
75	C₂×Q₁₆×C₄₇	—	2	8	no	—
76	C₂×Q₁₆×C₄₇	C₂³	1	8	yes	—
77	C₂×Q₁₆×C₄₇	C₂³	1	8	yes	—
78	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₄⋊Q₈×C₄₇	—	2	4	no	—
81	C₄⋊Q₈×C₄₇	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₄₇	C₂³	1	4	yes	—
85	C₂³×D₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂×Q₈×C₄₇	—	2	4	no	—
87	C₂×Q₈×C₄₇	—	2	8	no	—
88	C₂×Q₈×C₄₇	—	2	8	no	—
89	C₂×Q₈×C₄₇	C₂⁴	1	8	yes	—
90	C₂²×C₄×C₄₇	—	2	4	no	—
91	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₂²×Dic₄₇	—	2	4	no	—
94	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
95	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₄₇	—	2	2	no	—
97	C₂×C₄×D₄₇	—	2	2	no	—
98	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
99	C₂×C₄×D₄₇	C₂⁴	1	2	yes	—
100	Q₁₆×C₄₇	—	2	16	no	—
101	Q₁₆×C₄₇	—	2	16	no	—
102	C₂×C₈×C₄₇	#41	1	4	yes	—
103	C₂×C₈×C₄₇	#41	1	4	yes	—
104	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
105	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
106	C₄⋊C₄×C₄₇	—	2	16	no	—
107	C₄²×C₄₇	C₂×D₄	1	2	yes	—
108	C₄²×C₄₇	C₂⁴	1	2	yes	—
109	(C₂×D₄₇)⋊C₄	—	2	16	no	—
110	(C₂×D₄₇)⋊C₄	C₂×D₄	1	8	yes	—
111	C₄⋊Dic₄₇	—	2	4	no	—
112	C₄⋊Dic₄₇	—	2	16	no	—
113	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
114	Dic₄₇⋊C₄	—	2	16	no	—
115	Dic₄₇⋊C₄	—	2	16	no	—
116	Dic₄₇⋊C₄	C₂×D₄	1	8	yes	—
117	C₄×Dic₄₇	—	2	16	no	—
118	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
119	C₄×Dic₄₇	C₂⁴	1	2	yes	—
120	C₂×C₄₇⋊C₈	—	2	2	no	—
121	C₂×C₄₇⋊C₈	—	2	2	no	—
122	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
123	C₂×C₄₇⋊C₈	C₂⁴	1	2	yes	—
124	#10625324	—	2	8	no	—
125	C₈×D₄₇	—	2	8	no	—
126	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
127	C₂²×D₄₇	—	2	2	no	—
128	C₂²×D₄₇	—	2	4	no	—
129	C₂²×D₄₇	C₂²×D₄	1	2	yes	—
130	Q₈×C₄₇	—	2	8	no	—
131	Q₈×C₄₇	—	2	8	no	—
132	Q₈×C₄₇	—	4	8	no	—
133	C₂×C₄×C₄₇	—	2	4	no	—
134	C₂×C₄×C₄₇	—	2	8	no	—
135	C₂×C₄×C₄₇	—	4	8	no	—
136	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
137	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
139	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
140	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
141	C₂×Dic₄₇	—	2	2	no	—
142	C₂×Dic₄₇	—	2	4	no	—
143	C₂×Dic₄₇	—	2	8	no	—
144	C₂×Dic₄₇	—	4	8	no	—
145	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
146	C₂×Dic₄₇	C₂²×D₄	1	2	yes	—
147	C₄×D₄₇	—	2	4	no	—
148	C₄×D₄₇	—	2	2	no	—
149	C₄×D₄₇	—	2	2	no	—
150	C₈×C₄₇	C₂×#41	1	4	yes	—
151	C₈×C₄₇	C₂×#41	1	4	yes	—
152	C₄₇⋊C₈	—	2	2	no	—
153	C₄₇⋊C₈	—	2	2	no	—
154	C₂×#3177	—	47	1	no	maximal, Sylow
155	C₂²×C₄₇	#532	1	2	yes	—
156	C₂²×C₄₇	#578	1	4	yes	—
157	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
158	C₂×D₄₇	—	2	2	no	—
159	C₂×D₄₇	—	2	4	no	—
160	C₂×D₄₇	—	4	4	no	—
161	C₄×C₄₇	—	2	4	no	—
162	C₄×C₄₇	—	4	8	no	—
163	C₄×C₄₇	#532	1	2	yes	—
164	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
165	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
166	Dic₄₇	—	2	2	no	—
167	Dic₄₇	—	2	4	no	—
168	Dic₄₇	—	4	8	no	—
169	C₂×#541	—	47	2	no	—
170	#3177	—	47	16	no	—
171	C₂×#499	—	47	1	no	—
172	C₂×#494	D₄₇	1	1	yes	characteristic, complemented
173	C₂×#491	—	47	2	no	—
174	C₂×#470	—	47	2	no	—
175	C₂×#467	—	47	2	no	—
176	C₂×#441	—	47	2	no	—
177	C₂×C₄×Q₁₆	—	47	2	no	—
178	C₂×#405	—	47	1	no	—
179	C₂×C₄₇	#3177	1	4	yes	complemented
180	C₂×C₄₇	C₂×#532	1	1	yes	characteristic
181	C₂×C₄₇	C₂×#578	1	2	yes	—
182	D₄₇	—	4	2	no	—
183	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
184	C₂²×Q₁₆	C₂×D₄₇	1	2	yes	—
185	C₂²×M₁₆	—	47	1	no	—
186	C₂³×C₈	—	47	1	no	—
187	#541	—	47	16	no	—
188	#541	—	94	8	no	—
189	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
190	C₂×#126	—	94	4	no	—
191	C₂×#125	—	47	2	no	—
192	C₂×#125	—	94	4	no	—
193	C₂×C₂²⋊Q₈	—	47	4	no	—
194	C₂×C₄×Q₈	—	47	4	no	—
195	C₂×#117	—	47	1	no	—
196	#499	—	47	8	no	—
197	#494	C₂×D₄₇	1	8	yes	complemented
198	#491	—	47	16	no	—
199	#470	—	47	16	no	—
200	#467	—	47	16	no	—
201	#441	—	47	16	no	—
202	C₄×Q₁₆	—	47	16	no	—
203	C₂×#107	—	47	2	no	—
204	C₂×#106	—	47	2	no	—
205	C₂×#103	—	47	4	no	—
206	C₂×#103	—	47	4	no	—
207	#405	—	47	8	no	—
208	C₂×#97	—	47	1	no	—
209	C₂×#97	—	47	1	no	—
210	C₂×C₄×C₈	—	47	1	no	—
211	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
212	C₄₇	C₂×#3177	1	1	yes	characteristic, complemented, Sylow
213	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
214	C₂³×C₄	—	47	1	no	—
215	C₂×Q₁₆	—	2	8	no	—
216	C₂×Q₁₆	—	2	8	no	—
217	C₂×Q₁₆	C₂²×D₄₇	1	8	yes	—
218	C₂×Q₁₆	C₂²×D₄₇	1	8	yes	—
219	C₂×M₁₆	—	47	4	no	—
220	C₂×M₁₆	—	94	4	no	—
221	C₂²×C₈	—	47	1	no	—
222	C₂²×C₈	—	47	1	no	—
223	C₂²×C₈	—	47	4	no	—
224	C₂²×C₈	—	94	4	no	—
225	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
227	C₄⋊Q₈	—	2	4	no	—
228	C₄⋊Q₈	C₂²×D₄₇	1	8	yes	—
229	#126	—	94	32	no	—
230	#125	—	47	8	no	—
231	#125	—	94	32	no	—
232	#125	—	94	4	no	—
233	C₂²⋊Q₈	—	47	16	no	—
234	C₂²⋊Q₈	—	94	8	no	—
235	C₄×Q₈	—	47	16	no	—
236	C₄×Q₈	—	94	8	no	—
237	#117	—	47	4	no	—
238	#117	—	47	4	no	—
239	C₂×C₄⋊C₄	—	2	4	no	—
240	C₂×C₄⋊C₄	—	47	2	no	—
241	C₂×C₄⋊C₄	—	47	4	no	—
242	C₂×C₄⋊C₄	—	94	4	no	—
243	C₂×C₄⋊C₄	—	94	4	no	—
244	C₂×C₄⋊C₄	—	94	4	no	—
245	C₂×C₂²⋊C₄	—	47	2	no	—
246	C₂×C₂²⋊C₄	—	94	4	no	—
247	C₂×C₄²	—	47	1	no	—
248	C₂×C₄²	—	94	4	no	—
249	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
250	#107	—	47	8	no	—
251	#106	—	47	8	no	—
252	#103	—	47	16	no	—
253	#103	—	47	16	no	—
254	#97	—	47	4	no	—
255	#97	—	47	4	no	—
256	C₄×C₈	—	47	4	no	—
257	C₄×C₈	C₂²×D₄₇	1	4	yes	—
258	C₂⁴	—	47	1	no	—
259	C₂×Q₈	—	2	4	no	—
260	C₂×Q₈	—	2	8	no	—
261	C₂×Q₈	—	2	8	no	—
262	C₂×Q₈	C₂³×D₄₇	1	8	yes	—
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	47	2	no	—
265	C₂²×C₄	—	47	1	no	—
266	C₂²×C₄	—	47	2	no	—
267	C₂²×C₄	—	47	2	no	—
268	C₂²×C₄	—	94	2	no	—
269	C₂²×C₄	—	94	2	no	—
270	C₂²×C₄	—	94	4	no	—
271	C₂²×C₄	D₄×D₄₇	1	2	yes	—
272	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
273	Q₁₆	—	2	16	no	—
274	Q₁₆	—	2	16	no	—
275	M₁₆	—	94	8	no	—
276	C₂×C₈	—	47	2	no	—
277	C₂×C₈	—	47	2	no	—
278	C₂×C₈	—	94	2	no	—
279	C₂×C₈	—	94	2	no	—
280	C₂×C₈	—	94	8	no	—
281	C₂×C₈	#10625352	1	4	yes	—
282	C₂×C₈	#10625352	1	4	yes	—
283	C₂×C₈	C₂³×D₄₇	1	2	yes	—
284	C₂×C₈	C₂³×D₄₇	1	2	yes	—
285	C₄⋊C₄	—	2	16	no	—
286	C₄⋊C₄	—	47	8	no	—
287	C₄⋊C₄	—	47	8	no	—
288	C₄⋊C₄	—	94	16	no	—
289	C₄⋊C₄	—	94	16	no	—
290	C₄⋊C₄	—	94	16	no	—
291	C₄⋊C₄	—	94	4	no	—
292	C₂²⋊C₄	—	47	8	no	—
293	C₂²⋊C₄	—	94	16	no	—
294	C₄²	—	47	2	no	—
295	C₄²	—	47	2	no	—
296	C₄²	—	94	16	no	—
297	C₄²	D₄×D₄₇	1	2	yes	—
298	C₄²	C₂³×D₄₇	1	2	yes	—
299	C₂³	—	47	2	no	—
300	C₂³	—	94	2	no	—
301	C₂³	—	94	4	no	—
302	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
303	Q₈	—	2	8	no	—
304	Q₈	—	2	8	no	—
305	Q₈	—	4	8	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	8	no	—
308	C₂×C₄	—	4	8	no	—
309	C₂×C₄	—	47	4	no	—
310	C₂×C₄	—	47	2	no	—
311	C₂×C₄	—	94	2	no	—
312	C₂×C₄	—	94	4	no	—
313	C₂×C₄	—	94	4	no	—
314	C₂×C₄	—	94	2	no	—
315	C₂×C₄	—	94	2	no	—
316	C₂×C₄	—	94	8	no	—
317	C₂×C₄	—	188	8	no	—
318	C₂×C₄	C₂×D₄×D₄₇	1	4	yes	—
319	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
320	C₂×C₄	C₂×#10625352	1	2	yes	—
321	C₂×C₄	C₂×#10625352	1	2	yes	—
322	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
323	C₈	—	94	2	no	—
324	C₈	—	94	2	no	—
325	C₈	C₂×#10625352	1	4	yes	—
326	C₈	C₂×#10625352	1	4	yes	—
327	C₂²	—	94	4	no	—
328	C₂²	—	94	2	no	—
329	C₂²	—	188	4	no	—
330	C₂²	#15689671	1	2	yes	—
331	C₂²	#15689936	1	4	yes	—
332	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	4	8	no	—
335	C₄	—	94	4	no	—
336	C₄	—	94	2	no	—
337	C₄	—	188	8	no	—
338	C₄	#15689671	1	2	yes	—
339	C₄	C₂²×#10625352	1	1	yes	characteristic
340	C₄	C₂²×#10625352	1	1	yes	characteristic
341	C₂	—	188	2	no	—
342	C₂	#20254044	1	4	yes	direct product factor
343	C₂	C₂×#15689671	1	1	yes	characteristic
344	C₂	C₂×#15689936	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#494	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂³×D₄₇	C₂×D₄
hypercenter	C₂×#494	D₄₇
hypocenter	C₄₇	C₂×#3177
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
2. #20254044	C₂	direct (N × H)
3. C₂×#15689525	C₂	semidirect
4. C₂×#15689491	C₂	semidirect
7. C₂×#15689230	C₂	semidirect
8. C₂×Q₁₆×Dic₄₇	C₂	semidirect
10. C₂×C₄₇×#494	C₂	semidirect
21. #15689525	C₂²	semidirect
22. #15689491	C₂²	semidirect
25. #15689230	C₂²	semidirect
26. Q₁₆×Dic₄₇	C₂²	semidirect
38. C₄₇×#494	C₂²	semidirect
172. C₂×#494	D₄₇	semidirect
179. C₂×C₄₇	#3177	semidirect
197. #494	C₂×D₄₇	semidirect
212. C₄₇	C₂×#3177	semidirect
342. C₂	#20254044	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×#15689734	2
#20254044	2
C₂×#15689525	2
C₂×#15689491	2
C₂×#15689378	2
C₂×#15689361	2
C₂×#15689230	2
C₂×Q₁₆×Dic₄₇	2
C₂×#15689077	2
C₂×C₄₇×#494	2
C₂×#3177	47

Subgroups — C2×#20254044 #27086119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20254044 #27086119