Subgroups — #26225152

Order 12032.

670 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253910	C₂	1	1	yes	maximal, characteristic
2	#20252329	C₂	1	1	yes	maximal, characteristic
3	#20252325	C₂	1	1	yes	maximal, characteristic
4	#20244365	C₂	1	1	yes	maximal, characteristic
5	#20244140	C₂	1	2	yes	maximal
6	#20244135	C₂	1	2	yes	maximal
7	#20243562	C₂	1	1	yes	maximal, characteristic
8	#20243271	C₂	1	2	yes	maximal
9	#20243266	C₂	1	2	yes	maximal
10	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1994	C₂	1	1	yes	maximal, characteristic
12	#15689680	C₂²	1	1	yes	characteristic
13	#15689533	C₄	1	4	yes	complemented
14	#15689529	C₄	1	4	yes	complemented
15	#15689254	—	2	4	no	—
16	#15689252	—	2	4	no	—
17	C₂×#13093959	C₂²	1	2	yes	—
18	C₂×#13093959	C₂²	1	2	yes	—
19	#15689159	C₂²	1	2	yes	—
20	#15689159	C₂²	1	2	yes	—
21	#15689151	C₂²	1	2	yes	—
22	#15689150	C₂²	1	2	yes	—
23	#15689147	C₂²	1	1	yes	characteristic
24	#15689081	C₂²	1	2	yes	—
25	#15689079	C₂²	1	2	yes	—
26	C₂×#13093935	C₂²	1	1	yes	characteristic
27	C₂×#13093935	C₂²	1	1	yes	characteristic
28	C₂×#13093935	C₂²	1	1	yes	characteristic
29	C₂×#13093935	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
32	#15688883	C₂²	1	2	yes	—
33	#15688883	C₂²	1	2	yes	—
34	C₄₇×#533	C₂²	1	1	yes	characteristic
35	C₄₇×#423	C₂²	1	1	yes	characteristic
36	C₄³×C₄₇	C₂²	1	1	yes	characteristic
37	C₄₇×#339	C₂²	1	2	yes	—
38	C₄₇×#339	C₂²	1	2	yes	—
39	C₂×#10625329	C₂³	1	2	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
44	#13094024	C₂×C₄	1	2	yes	—
45	#13094016	C₂×C₄	1	2	yes	—
46	#13094003	—	2	2	no	—
47	C₄×Dic₉₄	—	2	2	no	—
48	#13093983	C₂³	1	2	yes	—
49	#13093959	—	2	4	no	—
50	#13093959	—	2	4	no	—
51	#13093957	C₂×C₄	1	4	yes	—
52	#13093957	C₂×C₄	1	4	yes	—
53	#13093955	C₂×C₄	1	4	yes	—
54	#13093955	C₂×C₄	1	4	yes	—
55	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
56	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
57	Dic₄₇⋊C₈	—	2	4	no	—
58	#13093935	—	2	2	no	—
59	#13093935	—	2	2	no	—
60	#13093935	C₂×C₄	1	2	yes	—
61	#13093935	C₂×C₄	1	2	yes	—
62	#13093935	D₄	1	4	yes	—
63	#13093935	Q₈	1	4	yes	complemented
64	#13093934	—	2	2	no	—
65	C₄×C₄₇⋊C₈	—	2	2	no	—
66	C₄×C₄₇⋊C₈	—	2	2	no	—
67	C₄×C₄₇⋊C₈	Q₈	1	4	yes	complemented
68	C₈×Dic₄₇	—	2	2	no	—
69	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
70	C₄⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
71	#125×C₄₇	C₂×C₄	1	2	yes	—
72	C₂²⋊Q₈×C₄₇	—	2	2	no	—
73	C₄×Q₈×C₄₇	—	2	2	no	—
74	#117×C₄₇	C₂³	1	2	yes	—
75	C₂×C₄²×C₄₇	—	2	2	no	—
76	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₄₇	—	2	2	no	—
80	C₂×Q₈×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	—	2	1	no	—
82	C₂²×C₄×C₄₇	—	2	1	no	—
83	C₂²×C₄×C₄₇	#41	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂×Dic₉₄	—	2	2	no	—
86	M₁₆×C₄₇	—	2	2	no	—
87	M₁₆×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₄⋊C₄×C₄₇	—	2	4	no	—
90	C₄⋊C₄×C₄₇	—	2	2	no	—
91	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
92	C₂²⋊C₄×C₄₇	—	2	2	no	—
93	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	1	no	—
95	C₄²×C₄₇	—	2	1	no	—
96	C₄²×C₄₇	—	4	4	no	—
97	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₄₇	C₂×D₄	1	2	yes	—
100	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
101	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
102	C₂²⋊Dic₄₇	—	2	2	no	—
103	C₄⋊Dic₄₇	—	2	2	no	—
104	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
105	Dic₄₇⋊C₄	—	2	4	no	—
106	C₄×Dic₄₇	—	2	2	no	—
107	#10625329	—	2	2	no	—
108	#10625329	—	2	2	no	—
109	C₂×C₄₇⋊C₈	—	2	1	no	—
110	C₂×C₄₇⋊C₈	—	2	1	no	—
111	C₂×C₄₇⋊C₈	—	2	2	no	—
112	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
113	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
115	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
116	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
117	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
118	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
119	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
120	C₂×C₄₇⋊C₈	#41	1	2	yes	—
121	C₂×C₄₇⋊C₈	#41	1	2	yes	—
122	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
123	Q₈×C₄₇	—	8	2	no	—
124	C₂×C₄×C₄₇	—	2	4	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	4	no	—
127	C₂×C₄×C₄₇	—	2	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₄₇	—	2	4	no	—
139	Dic₉₄	—	8	2	no	—
140	C₈×C₄₇	—	4	2	no	—
141	C₄₇⋊C₈	—	2	2	no	—
142	C₄₇⋊C₈	—	4	2	no	—
143	C₄₇⋊C₈	C₄×Q₈	1	4	yes	—
144	C₄₇⋊C₈	C₄×Q₈	1	4	yes	—
145	C₄₇⋊C₈	C₄⋊Q₈	1	4	yes	—
146	#21736	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂²×C₄₇	#533	1	1	yes	characteristic
150	C₂²×C₄₇	#533	1	1	yes	characteristic
151	C₄×C₄₇	—	2	4	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	4	2	no	—
155	C₄×C₄₇	—	4	4	no	—
156	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
157	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
159	C₄×C₄₇	#532	1	1	yes	characteristic
160	Dic₄₇	—	4	4	no	—
161	#3162	—	47	1	no	—
162	#2961	—	47	1	no	—
163	#2961	—	47	1	no	—
164	#1994	—	47	1	no	—
165	#1994	D₄₇	1	1	yes	characteristic
166	#1946	—	47	2	no	—
167	#1945	—	47	2	no	—
168	#1849	—	47	1	no	—
169	#1773	—	47	2	no	—
170	#1772	—	47	2	no	—
171	C₄×#97	—	47	1	no	—
172	C₂×C₄₇	—	2	1	no	—
173	C₂×C₄₇	—	2	1	no	—
174	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
175	C₂×C₄₇	#2956	1	2	yes	—
176	#533	—	47	1	no	—
177	#533	C₂×D₄₇	1	1	yes	characteristic
178	#501	—	47	4	no	—
179	#501	—	47	4	no	—
180	#446	—	94	4	no	—
181	#446	—	94	4	no	—
182	C₂×#108	—	47	2	no	—
183	C₂×#108	—	47	2	no	—
184	#427	—	47	2	no	—
185	#427	—	47	2	no	—
186	#424	—	47	2	no	—
187	#424	—	47	2	no	—
188	#423	—	47	1	no	—
189	#423	C₂×D₄₇	1	1	yes	characteristic
190	#405	—	47	2	no	—
191	#405	—	47	2	no	—
192	C₂×#97	—	47	1	no	—
193	C₂×#97	—	47	1	no	—
194	C₂×#97	—	47	1	no	—
195	C₂×#97	—	47	1	no	—
196	C₂×C₄×C₈	—	47	1	no	—
197	C₂×C₄×C₈	—	47	1	no	—
198	C₄³	C₂×D₄₇	1	1	yes	characteristic
199	#339	—	47	2	no	—
200	#339	—	47	2	no	—
201	#339	C₂×D₄₇	1	2	yes	—
202	#339	C₂×D₄₇	1	2	yes	—
203	C₄₇	#21736	1	1	yes	characteristic, complemented, Sylow
204	C₂×M₁₆	—	47	2	no	—
205	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
206	C₂²×C₈	—	47	1	no	—
207	C₂²×C₈	—	47	1	no	—
208	C₂²×C₈	—	47	1	no	—
209	C₂²×C₈	—	47	1	no	—
210	C₄⋊Q₈	—	47	2	no	—
211	C₄⋊Q₈	C₄×D₄₇	1	2	yes	—
212	#125	—	47	2	no	—
213	#125	C₄×D₄₇	1	2	yes	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	94	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	94	2	no	—
218	#117	—	47	2	no	—
219	#117	C₂²×D₄₇	1	2	yes	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
223	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
224	#108	—	94	4	no	—
225	#108	—	94	4	no	—
226	#107	—	47	4	no	—
227	#107	—	47	4	no	—
228	#106	—	47	4	no	—
229	#106	—	47	4	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	94	4	no	—
232	C₄⋊C₈	—	94	4	no	—
233	C₄⋊C₈	—	94	2	no	—
234	#97	—	47	2	no	—
235	#97	—	47	2	no	—
236	#97	—	47	4	no	—
237	#97	—	47	4	no	—
238	#97	—	94	2	no	—
239	#97	—	94	2	no	—
240	#97	—	94	2	no	—
241	C₄×C₈	—	47	4	no	—
242	C₄×C₈	—	94	2	no	—
243	C₄×C₈	—	94	2	no	—
244	C₄×C₈	—	94	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	94	2	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	#10625352	1	2	yes	—
250	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
251	M₁₆	—	2	2	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	94	2	no	—
254	M₁₆	—	94	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	47	2	no	—
257	C₂×C₈	—	47	2	no	—
258	C₂×C₈	—	47	2	no	—
259	C₂×C₈	—	47	2	no	—
260	C₂×C₈	—	47	2	no	—
261	C₂×C₈	—	47	2	no	—
262	C₂×C₈	—	47	2	no	—
263	C₂×C₈	—	47	2	no	—
264	C₂×C₈	—	47	2	no	—
265	C₂×C₈	—	47	2	no	—
266	C₂×C₈	—	94	1	no	—
267	C₂×C₈	—	94	1	no	—
268	C₂×C₈	—	94	2	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	47	4	no	—
272	C₄⋊C₄	—	94	4	no	—
273	C₄⋊C₄	—	94	2	no	—
274	C₄⋊C₄	C₂×C₄×D₄₇	1	4	yes	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	94	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	4	4	no	—
281	C₄²	—	94	2	no	—
282	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
283	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
284	C₄²	D₄×D₄₇	1	2	yes	—
285	C₄²	Q₈×D₄₇	1	2	yes	—
286	C₄²	Q₈×D₄₇	1	2	yes	—
287	C₂³	C₂×#10625352	1	1	yes	characteristic
288	Q₈	—	8	2	no	—
289	Q₈	—	376	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	94	4	no	—
297	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
298	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
299	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
301	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
302	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
303	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
304	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	47	4	no	—
307	C₈	—	47	4	no	—
308	C₈	—	47	4	no	—
309	C₈	—	94	2	no	—
310	C₈	—	188	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
313	C₂²	#15689675	1	1	yes	characteristic
314	C₂²	#15689675	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	4	no	—
320	C₄	—	188	4	no	—
321	C₄	C₄×Q₈×D₄₇	1	1	yes	characteristic
322	C₄	C₄×Q₈×D₄₇	1	1	yes	characteristic
323	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
324	C₄	#15689672	1	1	yes	characteristic
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	#20245957	1	1	yes	characteristic
328	C₂	#20252294	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1994	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1994	D₄₇
hypocenter	C₄₇	#21736
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1994

Complements

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

N	H ≅ G/N	type
13. #15689533	C₄	semidirect
14. #15689529	C₄	semidirect
63. #13093935	Q₈	semidirect
67. C₄×C₄₇⋊C₈	Q₈	semidirect
203. C₄₇	#21736	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20253910	2
#20252329	2
#20252325	2
#20244365	2
#20244140	2
#20244135	2
#20243562	2
#20243271	2
#20243266	2
C₄×#13093935	2
C₄₇×#1994	2
#21736	47