Subgroups — #26221224

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252024	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689182	C₂	1	1	yes	maximal, characteristic
3	#20251805	C₂	1	1	yes	maximal, characteristic
4	#20244162	C₂	1	2	yes	maximal, complemented
5	#20244160	C₂	1	2	yes	maximal, complemented
6	#20243594	C₂	1	2	yes	maximal
7	#20243591	C₂	1	2	yes	maximal
8	#20243295	C₂	1	1	yes	maximal, characteristic, complemented
9	#20243295	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13093960	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#108×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×D₄₇	C₂²	1	2	yes	—
13	C₂×#13093981	C₂²	1	1	yes	characteristic
14	#15689185	—	2	2	no	—
15	#15689182	—	2	2	no	—
16	#15689182	C₄	1	8	yes	—
17	C₂×#13093960	C₂²	1	2	yes	—
18	C₂×#13093959	C₂²	1	2	yes	—
19	C₂×#13093959	C₂²	1	2	yes	—
20	#15689146	C₂²	1	2	yes	—
21	C₂×Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
23	#15689086	C₂²	1	2	yes	—
24	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	1	yes	characteristic
26	#15689080	—	2	2	no	—
27	#15689077	—	2	2	no	—
28	C₄×#10625329	C₂²	1	1	yes	characteristic
29	C₄×#10625329	C₂²	1	1	yes	characteristic
30	M₁₆×Dic₄₇	C₂²	1	2	yes	—
31	C₂×#13093935	C₂²	1	1	yes	characteristic
32	#15688888	C₂²	1	2	yes	—
33	#15688887	C₂²	1	2	yes	—
34	#15688887	C₂²	1	2	yes	—
35	C₂×#108×C₄₇	C₂²	1	2	yes	—
36	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#341	C₂²	1	2	yes	—
40	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
41	C₂×#10625329	C₂³	1	2	yes	—
42	C₂×#10625329	C₂³	1	2	yes	—
43	C₂×#10625324	—	2	2	no	—
44	M₁₆×D₄₇	—	2	4	no	—
45	M₁₆×D₄₇	—	2	4	no	—
46	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₈×D₄₇	—	2	2	no	—
49	#13093981	C₂×C₄	1	4	yes	—
50	#13093981	D₄	1	2	yes	—
51	#13093981	D₄	1	2	yes	—
52	C₂×Dic₄₇⋊C₄	Q₈	1	2	yes	—
53	C₂×(C₂×D₄₇)⋊C₄	Q₈	1	2	yes	—
54	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
55	#13093960	—	2	4	no	—
56	#13093959	—	2	4	no	—
57	#13093959	—	2	4	no	—
58	Dic₄₇⋊C₈	—	2	4	no	—
59	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
60	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
61	(C₂×D₄₇)⋊C₈	—	2	4	no	—
62	(C₂×D₄₇)⋊C₈	C₂×C₄	1	4	yes	—
63	(C₂×D₄₇)⋊C₈	C₂×C₄	1	4	yes	—
64	#13093935	—	2	1	no	—
65	#13093935	—	2	1	no	—
66	#13093935	C₂×C₄	1	4	yes	—
67	#13093934	—	2	2	no	—
68	C₄×C₄₇⋊C₈	—	2	1	no	—
69	C₄×C₄₇⋊C₈	—	2	1	no	—
70	C₈×Dic₄₇	—	2	2	no	—
71	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
72	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
73	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
76	#108×C₄₇	—	2	4	no	—
77	#97×C₄₇	—	2	1	no	—
78	#97×C₄₇	—	2	1	no	—
79	C₄×C₈×C₄₇	—	2	1	no	—
80	C₄×C₈×C₄₇	—	2	1	no	—
81	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
82	C₂³×D₄₇	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
86	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
87	C₂×C₄×D₄₇	—	2	2	no	—
88	C₂×C₄×D₄₇	—	2	2	no	—
89	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
91	M₁₆×C₄₇	—	2	2	no	—
92	M₁₆×C₄₇	—	2	2	no	—
93	M₁₆×C₄₇	—	2	2	no	—
94	M₁₆×C₄₇	#41	1	4	yes	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
101	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
102	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
104	(C₂×D₄₇)⋊C₄	C₄⋊C₄	1	4	yes	—
105	(C₂×D₄₇)⋊C₄	C₄⋊C₄	1	4	yes	—
106	Dic₄₇⋊C₄	C₄⋊C₄	1	4	yes	—
107	Dic₄₇⋊C₄	C₄⋊C₄	1	4	yes	—
108	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
109	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
110	#10625329	—	2	2	no	—
111	#10625329	—	2	2	no	—
112	#10625329	—	2	2	no	—
113	#10625329	#41	1	4	yes	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	—	2	2	no	—
117	C₂×C₄₇⋊C₈	—	2	2	no	—
118	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
119	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
120	#10625324	—	2	8	no	—
121	C₈×D₄₇	—	2	8	no	—
122	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂²×D₄₇	—	2	2	no	—
124	C₂²×D₄₇	—	4	2	no	—
125	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
133	C₂×Dic₄₇	—	2	2	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
136	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
137	C₄×D₄₇	—	2	2	no	—
138	C₄×D₄₇	—	2	2	no	—
139	C₄×D₄₇	—	2	2	no	—
140	C₄×D₄₇	—	2	2	no	—
141	C₈×C₄₇	—	2	2	no	—
142	C₈×C₄₇	—	2	2	no	—
143	C₈×C₄₇	—	2	4	no	—
144	C₈×C₄₇	—	4	2	no	—
145	C₄₇⋊C₈	—	2	2	no	—
146	C₄₇⋊C₈	—	2	2	no	—
147	C₄₇⋊C₈	—	2	4	no	—
148	C₄₇⋊C₈	—	4	2	no	—
149	#21372	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	—	2	2	no	—
151	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₄₇	#533	1	2	yes	—
153	C₂×D₄₇	—	2	4	no	—
154	C₂×D₄₇	—	2	2	no	—
155	C₂×D₄₇	—	8	2	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
159	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
160	C₄×C₄₇	#525	1	2	yes	—
161	Dic₄₇	—	2	4	no	—
162	Dic₄₇	—	2	2	no	—
163	#2921	—	47	1	no	—
164	C₂×#433	—	47	1	no	—
165	#2873	—	47	1	no	—
166	#1950	—	47	2	no	—
167	#1950	—	47	2	no	—
168	#1855	—	47	2	no	—
169	#1855	—	47	2	no	—
170	#1778	—	47	1	no	—
171	#1778	—	47	1	no	—
172	C₄×#108	—	47	1	no	—
173	C₄×#108	D₄₇	1	1	yes	characteristic, complemented
174	C₂×C₄₇	—	2	2	no	—
175	C₂×C₄₇	#2296	1	1	yes	characteristic
176	C₂×C₄₇	#2916	1	2	yes	—
177	D₄₇	—	4	2	no	—
178	C₂²×M₁₆	—	47	2	no	—
179	C₂×#117	—	47	1	no	—
180	#433	—	47	8	no	—
181	#433	—	94	2	no	—
182	#433	—	94	2	no	—
183	C₂×#108	—	47	2	no	—
184	C₂×#108	—	47	2	no	—
185	C₂×#108	—	47	2	no	—
186	C₂×#108	C₂×D₄₇	1	2	yes	—
187	#423	—	47	2	no	—
188	C₂×C₄⋊C₈	—	47	1	no	—
189	C₂×C₄⋊C₈	—	47	1	no	—
190	#407	—	47	2	no	—
191	C₂×C₂²⋊C₈	—	47	1	no	—
192	C₂×C₂²⋊C₈	—	47	1	no	—
193	#405	—	94	2	no	—
194	#405	—	94	2	no	—
195	C₄×M₁₆	—	47	2	no	—
196	C₄×M₁₆	—	47	1	no	—
197	C₄×M₁₆	—	47	1	no	—
198	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
199	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
200	C₂×#97	—	47	1	no	—
201	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
202	#341	—	47	2	no	—
203	#341	—	47	2	no	—
204	#341	—	47	2	no	—
205	#341	C₂×D₄₇	1	2	yes	—
206	C₄₇	#21372	1	1	yes	characteristic, complemented, Sylow
207	C₂³×C₄	—	47	1	no	—
208	C₂×M₁₆	—	47	2	no	—
209	C₂×M₁₆	—	47	2	no	—
210	C₂×M₁₆	—	94	2	no	—
211	C₂×M₁₆	—	94	4	no	—
212	C₂×M₁₆	—	94	4	no	—
213	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
214	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
215	C₂²×C₈	—	47	1	no	—
216	C₂²×C₈	—	47	1	no	—
217	C₂²×C₈	—	94	2	no	—
218	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
219	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
220	#117	—	47	4	no	—
221	#117	—	47	2	no	—
222	#117	—	47	2	no	—
223	C₂×C₄⋊C₄	—	47	2	no	—
224	C₂×C₂²⋊C₄	—	47	2	no	—
225	C₂×C₄²	—	47	1	no	—
226	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
227	#108	—	2	4	no	—
228	#108	—	94	4	no	—
229	#108	—	94	4	no	—
230	#108	—	94	4	no	—
231	C₄⋊C₈	—	47	4	no	—
232	C₄⋊C₈	—	47	4	no	—
233	C₄⋊C₈	—	94	4	no	—
234	C₂²⋊C₈	—	47	4	no	—
235	C₂²⋊C₈	—	47	4	no	—
236	C₂²⋊C₈	—	94	4	no	—
237	#97	—	2	1	no	—
238	#97	—	2	1	no	—
239	#97	—	47	4	no	—
240	#97	—	94	2	no	—
241	#97	—	94	1	no	—
242	#97	—	94	1	no	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	—	2	1	no	—
245	C₄×C₈	—	94	2	no	—
246	C₄×C₈	—	94	1	no	—
247	C₄×C₈	—	94	1	no	—
248	C₄×C₈	C₄×D₄₇	1	4	yes	—
249	C₂⁴	—	47	1	no	—
250	C₂²×C₄	—	47	1	no	—
251	C₂²×C₄	—	47	2	no	—
252	C₂²×C₄	—	47	2	no	—
253	C₂²×C₄	—	47	2	no	—
254	C₂²×C₄	—	94	2	no	—
255	C₂²×C₄	—	94	2	no	—
256	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
257	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
258	M₁₆	—	2	2	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	47	4	no	—
262	M₁₆	—	94	8	no	—
263	M₁₆	—	94	2	no	—
264	M₁₆	—	94	2	no	—
265	M₁₆	—	94	2	no	—
266	M₁₆	#10625350	1	4	yes	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	47	2	no	—
272	C₂×C₈	—	47	2	no	—
273	C₂×C₈	—	94	8	no	—
274	C₂×C₈	—	94	2	no	—
275	C₂×C₈	—	94	2	no	—
276	C₂×C₈	—	94	2	no	—
277	C₂×C₈	—	94	2	no	—
278	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
279	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
280	C₄⋊C₄	—	47	4	no	—
281	C₄⋊C₄	—	47	4	no	—
282	C₂²⋊C₄	—	47	4	no	—
283	C₂²⋊C₄	—	47	4	no	—
284	C₄²	—	47	2	no	—
285	C₄²	—	47	2	no	—
286	C₄²	C₂×C₄×D₄₇	1	2	yes	—
287	C₄²	D₄×D₄₇	1	1	yes	characteristic
288	C₄²	D₄×D₄₇	1	1	yes	characteristic
289	C₂³	—	47	2	no	—
290	C₂³	—	94	2	no	—
291	C₂³	—	188	2	no	—
292	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	47	4	no	—
295	C₂×C₄	—	47	2	no	—
296	C₂×C₄	—	94	2	no	—
297	C₂×C₄	—	94	2	no	—
298	C₂×C₄	—	94	2	no	—
299	C₂×C₄	—	94	2	no	—
300	C₂×C₄	—	94	2	no	—
301	C₂×C₄	—	188	2	no	—
302	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
303	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
304	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
306	C₂×C₄	#41×D₄₇	1	2	yes	—
307	C₂×C₄	C₂×#10625350	1	2	yes	—
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	2	4	no	—
311	C₈	—	4	2	no	—
312	C₈	—	94	2	no	—
313	C₈	—	94	2	no	—
314	C₈	—	94	4	no	—
315	C₈	—	188	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	94	4	no	—
318	C₂²	—	94	2	no	—
319	C₂²	—	376	2	no	—
320	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
321	C₂²	#15689677	1	2	yes	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	94	4	no	—
325	C₄	—	94	2	no	—
326	C₄	#15689591	1	1	yes	characteristic
327	C₄	C₄×#10625350	1	1	yes	characteristic
328	C₄	#15689632	1	2	yes	—
329	C₂	—	2	2	no	—
330	C₂	—	188	2	no	—
331	C₂	#20245868	1	1	yes	characteristic
332	C₂	#20251996	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#108×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#108	D₄₇
hypocenter	C₄₇	#21372
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
4. #20244162	C₂	semidirect
5. #20244160	C₂	semidirect
8. #20243295	C₂	semidirect
9. #20243295	C₂	semidirect
10. C₄×#13093960	C₂	semidirect
11. C₄×#108×C₄₇	C₂	semidirect
173. C₄×#108	D₄₇	semidirect
206. C₄₇	#21372	semidirect

Maximal subgroups

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

subgroup	index
#20252024	2
C₂×#15689182	2
#20251805	2
#20244162	2
#20244160	2
#20243594	2
#20243591	2
#20243295	2
C₄×#13093960	2
C₄×#108×C₄₇	2
#21372	47