Subgroups — C₂×#20254047 #27086122

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×#15689735	C₂	1	2	yes	maximal
2	#20254047	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689525	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15689503	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15689492	C₂	1	2	yes	maximal, complemented
6	C₂×#15689381	C₂	1	2	yes	maximal
7	C₂×#15689364	C₂	1	2	yes	maximal
8	C₂×#15689231	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Dic₁₈₈	C₂	1	2	yes	maximal, complemented
10	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
11	C₂²×Dic₁₈₈	C₂²	1	2	yes	—
12	C₂²×Dic₁₈₈	C₂²	1	2	yes	—
13	#15689735	—	2	8	no	—
14	#15689735	C₂²	1	16	yes	—
15	C₂×#13094024	C₂²	1	2	yes	—
16	C₂×#13094017	—	2	4	no	—
17	C₂×#13094016	C₂²	1	2	yes	—
18	C₂×#13094014	—	2	4	no	—
19	C₂×#13094003	C₂²	1	4	yes	—
20	C₂×C₄×Dic₉₄	C₂²	1	4	yes	—
21	#15689525	C₂²	1	8	yes	complemented
22	#15689503	C₂²	1	8	yes	complemented
23	#15689492	C₂²	1	16	yes	complemented
24	#15689381	C₂²	1	16	yes	—
25	#15689364	C₂²	1	16	yes	—
26	#15689231	C₂²	1	16	yes	complemented
27	C₄×Dic₁₈₈	C₂²	1	16	yes	complemented
28	C₂×#13093958	C₂²	1	2	yes	—
29	C₂×#13093956	C₂²	1	2	yes	—
30	C₂×#13093949	C₂²	1	4	yes	—
31	C₂×#13093949	C₂²	1	4	yes	—
32	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
33	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#405	C₂²	1	8	yes	—
36	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₂²×Dic₉₄	C₂³	1	4	yes	—
41	C₂×Dic₁₈₈	—	2	8	no	—
42	C₂×Dic₁₈₈	—	2	8	no	—
43	C₂×Dic₁₈₈	C₂³	1	8	yes	—
44	C₂×Dic₁₈₈	C₂³	1	8	yes	—
45	#13094024	—	2	4	no	—
46	#13094024	C₂³	1	8	yes	—
47	#13094017	—	2	32	no	—
48	#13094016	—	2	4	no	—
49	#13094016	C₂³	1	8	yes	—
50	#13094014	—	2	32	no	—
51	#13094003	—	2	8	no	—
52	#13094003	C₂³	1	16	yes	—
53	C₄×Dic₉₄	—	2	8	no	—
54	C₄×Dic₉₄	C₂³	1	16	yes	—
55	C₂×C₄⋊Dic₄₇	—	2	4	no	—
56	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
57	C₂×Dic₄₇⋊C₄	—	2	4	no	—
58	C₂×Dic₄₇⋊C₄	—	2	4	no	—
59	C₂×Dic₄₇⋊C₄	—	2	4	no	—
60	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
61	C₂×C₄×Dic₄₇	—	2	4	no	—
62	#13093958	C₂³	1	8	yes	—
63	#13093956	C₂³	1	8	yes	—
64	#13093949	C₂³	1	16	yes	—
65	#13093949	C₂³	1	16	yes	—
66	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₄₇	—	2	4	no	—
68	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
69	C₂²×C₈×C₄₇	—	2	4	no	—
70	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	#117×C₄₇	D₄	1	4	yes	—
76	#117×C₄₇	C₂³	1	4	yes	—
77	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
78	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
79	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
81	#97×C₄₇	C₂³	1	4	yes	—
82	#97×C₄₇	C₂³	1	4	yes	—
83	C₄×C₈×C₄₇	C₂³	1	4	yes	—
84	C₄×C₈×C₄₇	C₂³	1	4	yes	—
85	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	—	2	2	no	—
87	C₂²×C₄×C₄₇	—	2	2	no	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₄₇	C₂⁴	1	2	yes	—
93	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
94	C₂²×Dic₄₇	—	2	4	no	—
95	C₂²×Dic₄₇	—	2	4	no	—
96	C₂×Dic₉₄	—	2	8	no	—
97	C₂×Dic₉₄	—	2	8	no	—
98	C₂×Dic₉₄	—	2	4	no	—
99	C₂×Dic₉₄	C₂⁴	1	8	yes	—
100	M₁₆×C₄₇	—	2	8	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	—	2	8	no	—
104	C₂×C₈×C₄₇	#41	1	4	yes	—
105	C₂×C₈×C₄₇	#41	1	4	yes	—
106	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
107	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
108	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
109	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
110	C₄⋊C₄×C₄₇	C₂×D₄	1	8	yes	—
111	C₂²⋊C₄×C₄₇	C₂×D₄	1	8	yes	—
112	C₄²×C₄₇	C₂×D₄	1	2	yes	—
113	C₄²×C₄₇	C₂×D₄	1	2	yes	—
114	C₄²×C₄₇	C₂⁴	1	2	yes	—
115	C₄²×C₄₇	C₂⁴	1	2	yes	—
116	C₂²⋊Dic₄₇	—	2	16	no	—
117	C₄⋊Dic₄₇	—	2	16	no	—
118	C₄⋊Dic₄₇	—	2	4	no	—
119	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
120	Dic₄₇⋊C₄	—	2	16	no	—
121	Dic₄₇⋊C₄	—	2	16	no	—
122	Dic₄₇⋊C₄	—	2	16	no	—
123	C₄×Dic₄₇	—	2	16	no	—
124	Dic₁₈₈	—	2	16	no	—
125	Dic₁₈₈	—	2	16	no	—
126	C₂³×C₄₇	—	2	2	no	—
127	C₂³×C₄₇	—	2	4	no	—
128	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
129	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	—	2	2	no	—
131	C₂×C₄×C₄₇	—	2	4	no	—
132	C₂×C₄×C₄₇	—	2	4	no	—
133	C₂×C₄×C₄₇	—	2	4	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
137	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₄₇	C₂²×D₄	1	4	yes	—
139	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
141	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
142	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
143	C₂×Dic₄₇	—	2	8	no	—
144	C₂×Dic₄₇	—	2	8	no	—
145	C₂×Dic₄₇	—	4	8	no	—
146	C₂×Dic₄₇	—	4	8	no	—
147	Dic₉₄	—	2	8	no	—
148	Dic₉₄	—	2	8	no	—
149	Dic₉₄	—	4	8	no	—
150	C₈×C₄₇	—	2	2	no	—
151	C₈×C₄₇	—	2	2	no	—
152	C₈×C₄₇	C₂×#41	1	4	yes	—
153	C₈×C₄₇	C₂×#41	1	4	yes	—
154	C₂×#3177	—	47	1	no	maximal, Sylow
155	C₂²×C₄₇	—	2	2	no	—
156	C₂²×C₄₇	—	2	4	no	—
157	C₂²×C₄₇	—	4	4	no	—
158	C₂²×C₄₇	#532	1	2	yes	—
159	C₂²×C₄₇	#578	1	4	yes	—
160	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	4	no	—
163	C₄×C₄₇	—	2	4	no	—
164	C₄×C₄₇	#532	1	2	yes	—
165	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
166	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
167	Dic₄₇	—	4	8	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	—	47	1	no	—
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	D₄₇	1	1	yes	characteristic
179	C₂×C₄₇	—	4	2	no	—
180	C₂×C₄₇	#3177	1	4	yes	complemented
181	C₂×C₄₇	C₂×#532	1	1	yes	characteristic
182	C₂×C₄₇	C₂×#578	1	2	yes	—
183	C₂²×Q₁₆	—	47	2	no	—
184	C₂²×Q₁₆	—	47	2	no	—
185	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
186	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
187	#541	—	47	16	no	—
188	#541	—	94	8	no	—
189	C₂×C₄⋊Q₈	—	47	2	no	—
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	C₂×D₄₇	1	1	yes	characteristic
196	#499	—	47	8	no	—
197	#494	—	47	8	no	—
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	C₂×D₄₇	1	8	yes	—
208	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
209	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
211	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
212	C₄₇	C₂×#3177	1	1	yes	characteristic, complemented, Sylow
213	C₂²×Q₈	—	47	4	no	—
214	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
215	C₂×Q₁₆	—	47	8	no	—
216	C₂×Q₁₆	—	47	8	no	—
217	C₂×Q₁₆	—	94	8	no	—
218	C₂×Q₁₆	—	94	8	no	—
219	C₂×M₁₆	—	2	4	no	—
220	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
221	C₂²×C₈	—	2	4	no	—
222	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
223	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
224	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
227	C₄⋊Q₈	—	47	8	no	—
228	C₄⋊Q₈	—	94	4	no	—
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	D₁₈₈	1	4	yes	—
238	#117	C₂²×D₄₇	1	4	yes	—
239	C₂×C₄⋊C₄	—	47	4	no	—
240	C₂×C₄⋊C₄	—	94	4	no	—
241	C₂×C₄⋊C₄	—	94	4	no	—
242	C₂×C₄⋊C₄	—	94	4	no	—
243	C₂×C₄⋊C₄	—	94	4	no	—
244	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
245	C₂×C₂²⋊C₄	—	94	4	no	—
246	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
247	C₂×C₄²	—	94	4	no	—
248	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
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	C₂²×D₄₇	1	4	yes	—
255	#97	C₂²×D₄₇	1	4	yes	—
256	C₄×C₈	C₂²×D₄₇	1	4	yes	—
257	C₄×C₈	C₂²×D₄₇	1	4	yes	—
258	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
259	C₂×Q₈	—	47	8	no	—
260	C₂×Q₈	—	94	4	no	—
261	C₂×Q₈	—	94	8	no	—
262	C₂×Q₈	—	94	8	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	94	4	no	—
266	C₂²×C₄	—	94	4	no	—
267	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
268	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
269	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
270	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
271	C₂²×C₄	C₂³×D₄₇	1	2	yes	—
272	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
273	Q₁₆	—	94	16	no	—
274	Q₁₆	—	94	16	no	—
275	M₁₆	—	2	8	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	8	no	—
279	C₂×C₈	#10625350	1	4	yes	—
280	C₂×C₈	#10625350	1	4	yes	—
281	C₂×C₈	C₂³×D₄₇	1	2	yes	—
282	C₂×C₈	C₂³×D₄₇	1	2	yes	—
283	C₂×C₈	C₂³×D₄₇	1	2	yes	—
284	C₂×C₈	C₂³×D₄₇	1	2	yes	—
285	C₄⋊C₄	—	47	8	no	—
286	C₄⋊C₄	—	94	16	no	—
287	C₄⋊C₄	—	94	16	no	—
288	C₄⋊C₄	—	94	16	no	—
289	C₄⋊C₄	—	94	4	no	—
290	C₄⋊C₄	—	94	16	no	—
291	C₄⋊C₄	C₂×D₁₈₈	1	8	yes	—
292	C₂²⋊C₄	—	94	16	no	—
293	C₂²⋊C₄	C₂×D₁₈₈	1	8	yes	—
294	C₄²	—	94	16	no	—
295	C₄²	C₂×D₁₈₈	1	2	yes	—
296	C₄²	C₂×D₁₈₈	1	2	yes	—
297	C₄²	C₂³×D₄₇	1	2	yes	—
298	C₄²	C₂³×D₄₇	1	2	yes	—
299	C₂³	—	2	2	no	—
300	C₂³	—	2	4	no	—
301	C₂³	C₂²×D₁₈₈	1	2	yes	—
302	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
303	Q₈	—	94	8	no	—
304	Q₈	—	94	8	no	—
305	Q₈	—	188	8	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	94	8	no	—
313	C₂×C₄	—	94	8	no	—
314	C₂×C₄	—	188	8	no	—
315	C₂×C₄	—	188	8	no	—
316	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
317	C₂×C₄	C₂²×D₁₈₈	1	4	yes	—
318	C₂×C₄	C₂²×D₁₈₈	1	2	yes	—
319	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
320	C₂×C₄	C₂×#10625350	1	2	yes	—
321	C₂×C₄	C₂×#10625350	1	2	yes	—
322	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
323	C₈	—	2	2	no	—
324	C₈	—	2	2	no	—
325	C₈	C₂×#10625350	1	4	yes	—
326	C₈	C₂×#10625350	1	4	yes	—
327	C₂²	—	2	4	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	4	no	—
330	C₂²	#15689673	1	2	yes	—
331	C₂²	#15689940	1	4	yes	—
332	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	—	2	2	no	—
336	C₄	—	188	8	no	—
337	C₄	—	188	8	no	—
338	C₄	#15689673	1	2	yes	—
339	C₄	C₂²×#10625350	1	1	yes	characteristic
340	C₄	C₂²×#10625350	1	1	yes	characteristic
341	C₂	—	4	2	no	—
342	C₂	#20254047	1	4	yes	direct product factor
343	C₂	C₂×#15689673	1	1	yes	characteristic
344	C₂	C₂×#15689940	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	C₂×#3177
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20254047	C₂	direct (N × H)
3. C₂×#15689525	C₂	semidirect
4. C₂×#15689503	C₂	semidirect
5. C₂×#15689492	C₂	semidirect
8. C₂×#15689231	C₂	semidirect
9. C₂×C₄×Dic₁₈₈	C₂	semidirect
21. #15689525	C₂²	semidirect
22. #15689503	C₂²	semidirect
23. #15689492	C₂²	semidirect
26. #15689231	C₂²	semidirect
27. C₄×Dic₁₈₈	C₂²	semidirect
180. C₂×C₄₇	#3177	semidirect
212. C₄₇	C₂×#3177	semidirect
342. C₂	#20254047	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×#15689735	2
#20254047	2
C₂×#15689525	2
C₂×#15689503	2
C₂×#15689492	2
C₂×#15689381	2
C₂×#15689364	2
C₂×#15689231	2
C₂×C₄×Dic₁₈₈	2
C₂×C₄₇×#405	2
C₂×#3177	47

Subgroups — C2×#20254047 #27086122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20254047 #27086122