Subgroups — C₂×#20243561 #26216736

Order 12032.

1006 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093960	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093959	C₂	1	2	yes	maximal
3	C₂×#15689146	C₂	1	2	yes	maximal
4	C₂²×#13093934	C₂	1	1	yes	maximal, characteristic
5	#20243561	C₂	1	8	yes	maximal, direct product factor
6	C₂²×#108×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂×#13093960	—	2	4	no	—
11	C₂×#13093960	C₂²	1	4	yes	—
12	C₂×#13093959	—	2	8	no	—
13	C₂×#13093959	C₂²	1	8	yes	—
14	#15689146	—	2	4	no	—
15	#15689146	C₂²	1	8	yes	—
16	C₂×Dic₄₇⋊C₈	—	2	4	no	—
17	C₂×#13093934	C₄	1	4	yes	—
18	C₂×#13093934	C₄	1	4	yes	—
19	C₂×#13093934	C₂²	1	4	yes	—
20	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
21	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#108×C₄₇	—	2	4	no	—
23	C₂×#108×C₄₇	C₂²	1	4	yes	—
24	C₂³×Dic₄₇	Q₈	1	2	yes	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	4	no	—
27	C₂×#10625329	—	2	4	no	—
28	C₂×#10625329	C₂³	1	4	yes	—
29	C₂²×C₄₇⋊C₈	—	2	2	no	—
30	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
33	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
34	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
41	#13093960	—	2	8	no	—
42	#13093959	—	2	16	no	—
43	Dic₄₇⋊C₈	—	2	16	no	—
44	#13093934	C₂×C₄	1	16	yes	—
45	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₄₇	—	2	2	no	—
47	C₂×M₁₆×C₄₇	—	2	4	no	—
48	C₂×M₁₆×C₄₇	—	2	4	no	—
49	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
50	C₂²×C₈×C₄₇	—	2	2	no	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₄₇	D₄	1	4	yes	—
54	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
56	#108×C₄₇	—	2	8	no	—
57	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	4	4	no	—
69	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
70	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
71	C₂²×Dic₄₇	C₂×Q₈	1	4	yes	—
72	M₁₆×C₄₇	—	2	4	no	—
73	M₁₆×C₄₇	—	2	4	no	—
74	M₁₆×C₄₇	—	4	4	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	4	4	no	—
77	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
78	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
81	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
82	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
83	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
84	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
85	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
86	#10625329	—	2	4	no	—
87	#10625329	—	2	4	no	—
88	#10625329	—	4	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	4	4	no	—
91	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
92	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
93	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
94	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
95	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
96	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
100	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
101	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₄₇	#127	1	2	yes	—
109	C₂×C₄×C₄₇	#127	1	2	yes	—
110	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
113	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂×Dic₄₇	—	2	8	no	—
121	C₂×Dic₄₇	—	4	4	no	—
122	C₂×Dic₄₇	—	4	4	no	—
123	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
124	C₈×C₄₇	—	4	4	no	—
125	C₈×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
126	C₄₇⋊C₈	—	4	4	no	—
127	C₄₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	—
128	C₂×#1849	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	#384	1	2	yes	—
134	C₂²×C₄₇	#384	1	2	yes	—
135	C₂²×C₄₇	#430	1	4	yes	—
136	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
139	C₄×C₄₇	#384	1	2	yes	—
140	C₄×C₄₇	#384	1	2	yes	—
141	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
142	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
144	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	Dic₄₇	—	2	8	no	—
146	C₂²×#108	—	47	2	no	—
147	C₂²×#108	—	47	1	no	—
148	C₂²×#108	D₄₇	1	1	yes	characteristic
149	C₂×#423	—	47	2	no	—
150	C₂²×#97	—	47	1	no	—
151	#1849	—	47	8	no	—
152	C₂×C₄₇	—	2	1	no	—
153	C₂×C₄₇	—	2	1	no	—
154	C₂×C₄₇	—	2	2	no	—
155	C₂×C₄₇	#1849	1	4	yes	complemented
156	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
157	C₂×C₄₇	C₂×#430	1	2	yes	—
158	C₂²×M₁₆	—	47	2	no	—
159	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
160	C₂³×C₈	—	47	1	no	—
161	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
162	C₂²×C₄²	—	47	1	no	—
163	C₂×#108	—	2	4	no	—
164	C₂×#108	—	47	8	no	—
165	C₂×#108	—	47	4	no	—
166	C₂×#108	—	94	8	no	—
167	C₂×#108	—	94	4	no	—
168	C₂×#108	C₂×D₄₇	1	4	yes	—
169	#423	—	47	8	no	—
170	#423	—	94	4	no	—
171	C₂×C₄⋊C₈	—	94	4	no	—
172	C₂×#97	—	47	4	no	—
173	C₂×#97	—	47	4	no	—
174	C₂×#97	—	47	4	no	—
175	C₄₇	C₂×#1849	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	—	47	2	no	—
177	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	47	4	no	—
182	C₂×M₁₆	—	94	2	no	—
183	C₂×M₁₆	—	94	4	no	—
184	C₂×M₁₆	—	94	4	no	—
185	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	47	2	no	—
188	C₂²×C₈	—	47	2	no	—
189	C₂²×C₈	—	47	4	no	—
190	C₂²×C₈	—	47	4	no	—
191	C₂²×C₈	—	47	2	no	—
192	C₂²×C₈	—	94	2	no	—
193	C₂²×C₈	Dic₉₄	1	4	yes	—
194	C₂²×C₈	C₄×D₄₇	1	2	yes	—
195	C₂²×C₈	C₄×D₄₇	1	2	yes	—
196	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
197	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
198	C₂×C₄²	—	47	2	no	—
199	C₂×C₄²	—	47	2	no	—
200	C₂×C₄²	—	47	2	no	—
201	C₂×C₄²	—	47	2	no	—
202	C₂×C₄²	—	47	2	no	—
203	C₂×C₄²	—	47	2	no	—
204	#108	—	2	8	no	—
205	#108	—	94	16	no	—
206	#108	—	94	8	no	—
207	C₄⋊C₈	—	94	16	no	—
208	#97	—	47	16	no	—
209	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
210	C₂²×C₄	—	47	4	no	—
211	C₂²×C₄	—	47	4	no	—
212	C₂²×C₄	—	47	4	no	—
213	C₂²×C₄	—	188	4	no	—
214	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
215	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
217	C₂²×C₄	D₄×D₄₇	1	2	yes	—
218	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
220	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
221	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
223	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	4	4	no	—
227	M₁₆	—	94	4	no	—
228	M₁₆	—	94	4	no	—
229	M₁₆	—	188	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	4	no	—
232	C₂×C₈	—	47	8	no	—
233	C₂×C₈	—	47	8	no	—
234	C₂×C₈	—	47	4	no	—
235	C₂×C₈	—	47	4	no	—
236	C₂×C₈	—	47	4	no	—
237	C₂×C₈	—	94	4	no	—
238	C₂×C₈	—	188	4	no	—
239	C₂×C₈	Dic₄₇⋊C₄	1	8	yes	—
240	C₂×C₈	Dic₄₇⋊C₄	1	8	yes	—
241	C₂×C₈	C₂×Dic₉₄	1	4	yes	—
242	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
243	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
244	C₄²	—	47	4	no	—
245	C₄²	—	47	4	no	—
246	C₄²	—	47	4	no	—
247	C₄²	—	47	4	no	—
248	C₂³	—	2	4	no	—
249	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
250	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
251	C₂³	#13094022	1	2	yes	—
252	C₂³	#13094023	1	2	yes	—
253	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
254	C₂×C₄	—	47	8	no	—
255	C₂×C₄	—	94	8	no	—
256	C₂×C₄	—	188	4	no	—
257	C₂×C₄	—	188	4	no	—
258	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
259	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
260	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
261	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
262	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
263	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
264	C₂×C₄	#13094020	1	2	yes	—
265	C₂×C₄	#13094020	1	2	yes	—
266	C₂×C₄	#13094022	1	2	yes	—
267	C₂×C₄	#13094023	1	2	yes	—
268	C₂×C₄	#13094023	1	2	yes	—
269	C₂×C₄	#13094023	1	2	yes	—
270	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
271	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
272	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
276	C₈	—	4	4	no	—
277	C₈	—	47	8	no	—
278	C₈	—	188	4	no	—
279	C₈	C₂×Dic₄₇⋊C₄	1	8	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	#15688992	1	2	yes	—
285	C₂²	#15688992	1	2	yes	—
286	C₂²	#15689170	1	4	yes	—
287	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
288	C₂²	C₂×#13094022	1	1	yes	characteristic
289	C₂²	C₂×#13094023	1	1	yes	characteristic
290	C₄	—	94	8	no	—
291	C₄	#15688992	1	2	yes	—
292	C₄	#15688992	1	2	yes	—
293	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
294	C₄	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
295	C₄	C₂×#13094020	1	1	yes	characteristic
296	C₄	C₂×#13094023	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	#20243561	1	4	yes	direct product factor
301	C₂	C₂×#15688992	1	1	yes	characteristic
302	C₂	C₂×#15689170	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#108×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#108	D₄₇
hypocenter	C₄₇	C₂×#1849
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
5. #20243561	C₂	direct (N × H)
155. C₂×C₄₇	#1849	semidirect
175. C₄₇	C₂×#1849	semidirect
300. C₂	#20243561	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×#13093960	2
C₂²×#13093959	2
C₂×#15689146	2
C₂²×#13093934	2
#20243561	2
C₂²×#108×C₄₇	2
C₂×#1849	47

Subgroups — C2×#20243561 #26216736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243561 #26216736