Subgroups — #26122311

Order 12032.

762 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245745	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688892	C₂	1	1	yes	maximal, characteristic
3	#20243097	C₂	1	2	yes	maximal
4	C₂×#15688879	C₂	1	2	yes	maximal
5	C₄₇×#2317	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	2	yes	—
8	C₄⋊C₄×Dic₄₇	—	2	4	no	—
9	C₄×C₂²⋊Dic₄₇	—	2	2	no	—
10	#15688892	—	2	2	no	—
11	#15688882	—	2	4	no	—
12	#15688879	C₄	1	8	yes	complemented
13	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#389	—	2	2	no	—
17	C₄₇×#385	—	2	2	no	—
18	C₄₇×#384	C₄	1	2	yes	complemented
19	C₄₇×#382	C₄	1	2	yes	complemented
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	#13093983	—	2	2	no	—
28	#13093983	—	2	2	no	—
29	#13093983	C₂×C₄	1	4	yes	—
30	#13093983	Q₈	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	4	no	—
34	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
35	C₄₇⋊#95	—	2	4	no	—
36	C₂³×C₄×C₄₇	D₄	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	#117×C₄₇	—	2	2	no	—
39	#117×C₄₇	—	2	2	no	—
40	#117×C₄₇	—	2	2	no	—
41	#117×C₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
50	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
51	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
52	C₂×C₄²×C₄₇	—	2	1	no	—
53	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
54	#95×C₄₇	—	2	4	no	—
55	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	—	8	2	no	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	2	yes	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₄⋊C₄×C₄₇	—	2	4	no	—
73	C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	—	2	2	no	—
76	C₄⋊C₄×C₄₇	—	2	2	no	—
77	C₄⋊C₄×C₄₇	—	8	4	no	—
78	C₂²⋊C₄×C₄₇	—	2	4	no	—
79	C₂²⋊C₄×C₄₇	—	4	1	no	—
80	C₂²⋊C₄×C₄₇	—	4	1	no	—
81	C₄²×C₄₇	—	2	2	no	—
82	C₄²×C₄₇	—	4	2	no	—
83	C₄²×C₄₇	C₄²	1	4	yes	complemented
84	C₂²⋊Dic₄₇	—	2	4	no	—
85	C₂²⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	—	2	4	no	—
88	C₄×Dic₄₇	—	2	4	no	—
89	C₄×Dic₄₇	—	8	4	no	—
90	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	complemented
91	#10625329	—	2	4	no	—
92	#10625329	—	2	4	no	—
93	#10625329	—	4	2	no	—
94	#10625329	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂³×C₄₇	—	4	1	no	—
99	C₂³×C₄₇	—	4	1	no	—
100	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₄₇	C₄×D₄	1	2	yes	—
102	C₂³×C₄₇	#120	1	1	yes	characteristic
103	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
104	C₂³×C₄₇	#123	1	1	yes	characteristic
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	4	4	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	1	no	—
114	C₂×C₄×C₄₇	—	4	1	no	—
115	C₂×C₄×C₄₇	—	8	2	no	—
116	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	#117	1	2	yes	—
119	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₄₇	#121	1	2	yes	—
126	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	#123	1	2	yes	—
128	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#125	1	2	yes	—
130	C₂×C₄×C₄₇	#126	1	2	yes	—
131	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×Dic₄₇	—	2	8	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	—	4	4	no	—
135	C₄₇⋊C₈	—	4	4	no	—
136	#10888	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	4	2	no	—
144	C₂²×C₄₇	—	4	2	no	—
145	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄₇	#385	1	2	yes	—
147	C₂²×C₄₇	#386	1	1	yes	characteristic
148	C₂²×C₄₇	#387	1	2	yes	—
149	C₂²×C₄₇	#391	1	1	yes	characteristic
150	C₄×C₄₇	—	4	4	no	—
151	C₄×C₄₇	—	4	4	no	—
152	C₄×C₄₇	—	8	1	no	—
153	C₄×C₄₇	—	8	1	no	—
154	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
155	C₄×C₄₇	#387	1	2	yes	—
156	C₄×C₄₇	#388	1	2	yes	—
157	C₄×C₄₇	#389	1	2	yes	—
158	Dic₄₇	—	4	8	no	—
159	#2317	D₄₇	1	1	yes	characteristic
160	#2274	—	47	1	no	—
161	C₂×#343	—	47	1	no	—
162	#1733	—	47	2	no	—
163	C₂×H(3,ℤ/4)	—	47	2	no	—
164	C₂×C₄₇	—	2	2	no	—
165	C₂×C₄₇	—	2	1	no	—
166	C₂×C₄₇	—	2	2	no	—
167	C₂×C₄₇	—	2	1	no	—
168	C₂×C₄₇	#1436	1	1	yes	characteristic
169	C₂×C₄₇	#1911	1	1	yes	characteristic
170	C₂×C₄₇	#1912	1	1	yes	characteristic
171	C₂²×M₁₆	—	47	2	no	—
172	C₂×#117	—	47	2	no	—
173	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
174	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
175	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
176	#389	—	2	2	no	—
177	#385	—	2	2	no	—
178	#384	Dic₄₇	1	2	yes	complemented
179	#382	Dic₄₇	1	2	yes	complemented
180	C₄×C₄⋊C₄	—	94	4	no	—
181	C₄×C₂²⋊C₄	—	94	2	no	—
182	#343	—	94	2	no	—
183	#339	—	94	4	no	—
184	H(3,ℤ/4)	—	47	8	no	—
185	C₄₇	#10888	1	1	yes	characteristic, complemented, Sylow
186	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
187	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
188	C₂×M₁₆	—	47	4	no	—
189	C₂×M₁₆	—	94	2	no	—
190	C₂×M₁₆	—	94	2	no	—
191	C₂×M₁₆	—	94	2	no	—
192	C₂×M₁₆	—	94	2	no	—
193	C₂×M₁₆	—	94	2	no	—
194	C₂²×C₈	—	94	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	47	4	no	—
199	#117	—	47	4	no	—
200	#117	—	94	2	no	—
201	#117	—	94	2	no	—
202	#117	C₄×D₄₇	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	94	2	no	—
208	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
209	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
210	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
211	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
212	C₂×C₂²⋊C₄	—	2	1	no	—
213	C₂×C₂²⋊C₄	—	2	1	no	—
214	C₂×C₂²⋊C₄	—	94	2	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	—	47	4	no	—
217	C₂×C₄²	—	94	4	no	—
218	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
219	#95	—	2	4	no	—
220	#95	—	94	4	no	—
221	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	8	2	no	—
228	C₂²×C₄	—	94	4	no	—
229	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
230	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
231	C₂²×C₄	#10625350	1	2	yes	—
232	C₂²×C₄	D₄×D₄₇	1	2	yes	—
233	C₂²×C₄	#10625352	1	2	yes	—
234	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
235	C₂²×C₄	#10625354	1	2	yes	—
236	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
238	M₁₆	—	94	4	no	—
239	M₁₆	—	94	4	no	—
240	M₁₆	—	188	2	no	—
241	M₁₆	—	188	2	no	—
242	C₂×C₈	—	94	4	no	—
243	C₂×C₈	—	188	2	no	—
244	C₂×C₈	—	188	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	8	4	no	—
251	C₄⋊C₄	—	94	4	no	—
252	C₄⋊C₄	—	94	4	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	4	1	no	—
255	C₂²⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	94	4	no	—
257	C₂²⋊C₄	—	94	4	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	—	47	8	no	—
261	C₄²	—	94	4	no	—
262	C₄²	—	376	4	no	—
263	C₄²	C₄×Dic₄₇	1	4	yes	complemented
264	C₂³	—	4	1	no	—
265	C₂³	—	4	1	no	—
266	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
267	C₂³	C₄×C₂²⋊D₄₇	1	2	yes	—
268	C₂³	#13093992	1	1	yes	characteristic
269	C₂³	#13094000	1	2	yes	—
270	C₂³	#13094005	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	8	2	no	—
282	C₂×C₄	—	94	8	no	—
283	C₂×C₄	—	188	4	no	—
284	C₂×C₄	—	188	4	no	—
285	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
286	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
287	C₂×C₄	#13093982	1	2	yes	—
288	C₂×C₄	#13093983	1	1	yes	characteristic
289	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
290	C₂×C₄	#13093986	1	2	yes	—
291	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
292	C₂×C₄	#13093989	1	2	yes	—
293	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
294	C₂×C₄	#13093996	1	2	yes	—
295	C₂×C₄	#13094002	1	2	yes	—
296	C₂×C₄	#13094009	1	2	yes	—
297	C₂×C₄	#13094013	1	1	yes	characteristic
298	C₂×C₄	#13094014	1	2	yes	—
299	C₂×C₄	#13094019	1	2	yes	—
300	C₂×C₄	#13094022	1	1	yes	characteristic
301	C₈	—	188	4	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
311	C₂²	#15688994	1	2	yes	—
312	C₂²	#15689003	1	1	yes	characteristic
313	C₂²	#15689004	1	2	yes	—
314	C₂²	#15689030	1	1	yes	characteristic
315	C₄	—	4	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	8	1	no	—
318	C₄	—	8	1	no	—
319	C₄	—	188	8	no	—
320	C₄	C₄⋊C₄×Dic₄₇	1	2	yes	—
321	C₄	#15689009	1	2	yes	—
322	C₄	#15689013	1	2	yes	—
323	C₄	#15689018	1	2	yes	—
324	C₂	—	2	1	no	—
325	C₂	—	2	2	no	—
326	C₂	—	2	2	no	—
327	C₂	—	2	1	no	—
328	C₂	#20241620	1	1	yes	characteristic
329	C₂	#20243921	1	1	yes	characteristic
330	C₂	#20243926	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689003
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2317	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2317	D₄₇
hypocenter	C₄₇	#10888
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2317

Complements

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

N	H ≅ G/N	type
12. #15688879	C₄	semidirect
18. C₄₇×#384	C₄	semidirect
19. C₄₇×#382	C₄	semidirect
83. C₄²×C₄₇	C₄²	semidirect
90. C₄×Dic₄₇	C₄⋊C₄	semidirect
178. #384	Dic₄₇	semidirect
179. #382	Dic₄₇	semidirect
185. C₄₇	#10888	semidirect
263. C₄²	C₄×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20245745	2
C₂×#15688892	2
#20243097	2
C₂×#15688879	2
C₄₇×#2317	2
#10888	47