Subgroups — #26106642

Order 12032.

914 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	C₂×#15689087	C₂	1	2	yes	maximal
2	#20241296	C₂	1	4	yes	maximal, complemented
3	C₄₇×#2280	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	—	2	2	no	—
5	#15689087	—	2	2	no	—
6	#15689087	—	2	2	no	—
7	#15689087	C₄	1	4	yes	complemented
8	#15689087	C₂²	1	4	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
10	C₂×C₂²⋊(C₄₇⋊C₈)	C₄	1	4	yes	complemented
11	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₄₇	C₄	1	2	yes	—
13	C₄₇×#381	C₂²	1	2	yes	—
14	C₄₇×#380	—	2	2	no	—
15	C₄₇×#379	C₄	1	2	yes	—
16	C₄×C₂²⋊C₄×C₄₇	—	2	2	no	—
17	C₂×#10625329	—	2	4	no	—
18	C₂×#10625329	—	2	4	no	—
19	C₂×#10625329	—	4	8	no	—
20	C₂²×C₄₇⋊C₈	—	2	4	no	—
21	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
24	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
25	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₇	—	4	4	no	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	D₄	1	2	yes	—
31	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	D₄	1	2	yes	—
34	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	#117×C₄₇	—	4	4	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
40	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₄₇	—	2	2	no	—
42	#95×C₄₇	—	2	4	no	—
43	C₂⁴×C₄₇	—	4	2	no	—
44	C₂⁴×C₄₇	—	4	4	no	—
45	C₂⁴×C₄₇	C₄²	1	2	yes	—
46	C₂⁴×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	2	8	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	—	4	4	no	—
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
79	C₄⋊C₄×C₄₇	—	4	4	no	—
80	C₂²⋊C₄×C₄₇	—	4	8	no	—
81	C₂²⋊C₄×C₄₇	—	4	4	no	—
82	C₄²×C₄₇	—	4	4	no	—
83	#10625329	—	4	8	no	—
84	#10625329	—	4	8	no	—
85	C₂×C₄₇⋊C₈	—	2	8	no	—
86	C₂×C₄₇⋊C₈	—	4	8	no	—
87	C₂³×C₄₇	—	2	4	no	—
88	C₂³×C₄₇	—	2	4	no	—
89	C₂³×C₄₇	—	2	4	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	4	no	—
92	C₂³×C₄₇	—	4	2	no	—
93	C₂³×C₄₇	—	4	1	no	—
94	C₂³×C₄₇	—	4	1	no	—
95	C₂³×C₄₇	—	4	2	no	—
96	C₂³×C₄₇	—	4	2	no	—
97	C₂³×C₄₇	—	4	2	no	—
98	C₂³×C₄₇	—	4	2	no	—
99	C₂³×C₄₇	—	4	16	no	—
100	C₂³×C₄₇	#95	1	2	yes	—
101	C₂³×C₄₇	#95	1	2	yes	—
102	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
103	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	2	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	1	no	—
121	C₂×C₄×C₄₇	—	4	1	no	—
122	C₂×C₄×C₄₇	—	4	8	no	—
123	C₂×C₄×C₄₇	—	4	4	no	—
124	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#99	1	4	yes	—
129	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
131	C₄₇⋊C₈	—	4	8	no	—
132	#8112	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	4	8	no	—
144	C₂²×C₄₇	—	4	4	no	—
145	C₂²×C₄₇	—	4	4	no	—
146	C₂²×C₄₇	—	4	8	no	—
147	C₂²×C₄₇	—	4	4	no	—
148	C₂²×C₄₇	#344	1	4	yes	—
149	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#99	1	2	yes	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	8	4	no	—
156	C₄×C₄₇	#342	1	2	yes	—
157	C₄×C₄₇	#342	1	2	yes	—
158	C₂×#407	—	47	2	no	—
159	#2280	D₄₇	1	1	yes	characteristic
160	#1306	—	47	4	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	—	2	4	no	—
163	C₂×C₄₇	—	4	2	no	—
164	C₂×C₄₇	—	4	1	no	—
165	C₂×C₄₇	—	4	1	no	—
166	C₂×C₄₇	C₂×#342	1	1	yes	characteristic
167	C₂×C₄₇	C₂×#344	1	2	yes	—
168	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
169	C₂²×M₁₆	—	94	2	no	—
170	C₂×#117	Dic₄₇	1	2	yes	—
171	#407	—	47	4	no	—
172	#407	—	47	4	no	—
173	#407	—	94	2	no	—
174	#407	—	94	2	no	—
175	C₂×C₂²⋊C₈	—	47	4	no	—
176	C₂×C₂²⋊C₈	—	94	2	no	—
177	#381	C₂×D₄₇	1	2	yes	—
178	#380	—	2	2	no	—
179	#379	Dic₄₇	1	2	yes	—
180	C₄×C₂²⋊C₄	—	2	2	no	—
181	C₄₇	#8112	1	1	yes	characteristic, complemented, Sylow
182	C₂⁵	C₂×Dic₄₇	1	1	yes	characteristic
183	C₂³×C₄	—	4	4	no	—
184	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
185	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
186	C₂³×C₄	C₄×D₄₇	1	2	yes	—
187	C₂³×C₄	C₄×D₄₇	1	2	yes	—
188	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
189	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
190	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
191	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
192	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
193	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
194	C₂×M₁₆	—	94	4	no	—
195	C₂×M₁₆	—	94	4	no	—
196	C₂×M₁₆	—	188	8	no	—
197	C₂²×C₈	—	94	4	no	—
198	#117	—	4	4	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	—	2	4	no	—
201	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂²⋊C₈	—	47	8	no	—
204	C₂²⋊C₈	—	94	4	no	—
205	C₂²⋊C₈	—	94	4	no	—
206	C₂²⋊C₈	—	94	4	no	—
207	#95	—	2	4	no	—
208	C₂⁴	—	4	2	no	—
209	C₂⁴	—	4	4	no	—
210	C₂⁴	C₄×Dic₄₇	1	2	yes	—
211	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
212	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
213	C₂⁴	C₂²⋊Dic₄₇	1	2	yes	—
214	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
228	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
229	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
230	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
231	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
232	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
234	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
235	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
236	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
237	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
238	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
242	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
244	M₁₆	—	188	8	no	—
245	M₁₆	—	188	8	no	—
246	C₂×C₈	—	94	8	no	—
247	C₂×C₈	—	188	8	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₂²⋊C₄	—	4	8	no	—
250	C₂²⋊C₄	—	4	4	no	—
251	C₄²	—	4	4	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	4	no	—
254	C₂³	—	2	4	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	2	4	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	16	no	—
265	C₂³	C₄₇⋊#95	1	2	yes	—
266	C₂³	C₄₇⋊#95	1	2	yes	—
267	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
268	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
269	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	8	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
290	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
291	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
292	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
293	C₂×C₄	#13093939	1	4	yes	—
294	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
295	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
296	C₈	—	188	8	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	8	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	8	no	—
311	C₂²	—	4	4	no	—
312	C₂²	#15688893	1	4	yes	—
313	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
314	C₂²	C₂×#13093939	1	2	yes	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	2	no	—
319	C₄	—	8	4	no	—
320	C₄	#15688890	1	2	yes	—
321	C₄	#15688890	1	2	yes	—
322	C₂	—	2	2	no	—
323	C₂	—	2	4	no	—
324	C₂	—	4	2	no	—
325	C₂	—	4	1	no	—
326	C₂	—	4	1	no	—
327	C₂	C₂×#15688890	1	1	yes	characteristic
328	C₂	C₂×#15688893	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄₇⋊#95
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2280	C₂
socle	C₂²×C₄₇	C₂×#95
prime radical	C₂⁵×C₄₇	C₂×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2280	D₄₇
hypocenter	C₄₇	#8112
LCS	G ▷ C₂³×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ #2280

Complements

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

N	H ≅ G/N	type
2. #20241296	C₂	semidirect
7. #15689087	C₄	semidirect
10. C₂×C₂²⋊(C₄₇⋊C₈)	C₄	semidirect
181. C₄₇	#8112	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689087	2
#20241296	2
C₄₇×#2280	2
#8112	47