Subgroups — #26369966

Order 12032.

532 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#10625352	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245290	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245220	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245218	C₂	1	2	yes	maximal, complemented
6	#20245217	C₂	1	2	yes	maximal, complemented
7	#20245185	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245184	C₂	1	2	yes	maximal, complemented
9	#20245143	C₂	1	2	yes	maximal, complemented
10	#20245142	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#10625352	C₄	1	1	yes	characteristic
13	C₂×#13093964	C₂²	1	2	yes	—
14	C₂×#13093963	C₂²	1	2	yes	—
15	C₈×Dic₉₄	C₂²	1	1	yes	characteristic
16	#15689247	C₄	1	1	yes	characteristic
17	D₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
18	C₈×C₂²⋊D₄₇	C₂²	1	2	yes	—
19	(C₂²⋊D₄₇)⋊C₈	C₄	1	2	yes	—
20	#15689175	C₄	1	1	yes	characteristic
21	#15689173	C₄	1	2	yes	—
22	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
23	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
24	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
25	Dic₄₇⋊C₁₆	C₂²	1	2	yes	—
26	Dic₄₇⋊C₁₆	C₂²	1	2	yes	—
27	Dic₄₇⋊C₁₆	C₂²	1	2	yes	—
28	Dic₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
29	Dic₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
31	(C₂×D₄₇)⋊C₁₆	C₂²	1	1	yes	characteristic
32	(C₂×D₄₇)⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₄₇	C₂²	1	2	yes	—
35	C₁₆×Dic₄₇	C₂²	1	2	yes	—
36	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
37	C₈×D₄×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₄₇×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×#10625352	C₈	1	1	yes	characteristic
42	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
45	C₄×Dic₉₄	C₂×C₄	1	1	yes	characteristic
46	#13093989	C₈	1	1	yes	characteristic
47	D₄×Dic₄₇	C₈	1	1	yes	characteristic
48	C₄×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
49	(C₂²⋊D₄₇)⋊C₄	C₈	1	2	yes	—
50	#13093982	C₈	1	1	yes	characteristic
51	#13093980	C₈	1	2	yes	—
52	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
53	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
54	#13093964	—	4	2	no	—
55	#13093963	—	4	2	no	—
56	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
57	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
60	Dic₄₇⋊C₈	C₂×C₄	1	2	yes	—
61	Dic₄₇⋊C₈	C₂×C₄	1	2	yes	—
62	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
63	(C₂×D₄₇)⋊C₈	C₂×C₄	1	2	yes	—
64	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
66	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
68	C₈×Dic₄₇	C₂³	1	2	yes	—
69	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
71	C₄×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
72	M₃₂×C₄₇	—	4	2	no	—
73	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
75	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂×C₂²⋊D₄₇	C₂×C₈	1	2	yes	—
82	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
83	C₂²×Dic₄₇	C₂×C₈	1	2	yes	—
84	#10625352	—	2	4	no	—
85	C₂×C₄×D₄₇	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₄₇	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×Dic₉₄	C₂×C₈	1	1	yes	characteristic
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
95	C₂²⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
96	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²⋊Dic₄₇	C₂×C₈	1	2	yes	—
98	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
100	Dic₄₇⋊C₄	C₂×C₈	1	2	yes	—
101	Dic₄₇⋊C₄	C₂×C₈	1	2	yes	—
102	C₄×Dic₄₇	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₄₇	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
105	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
106	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
107	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
108	C₂×C₄₇⋊C₈	—	4	2	no	—
109	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
112	C₈×D₄₇	—	2	1	no	—
113	C₈×D₄₇	—	2	1	no	—
114	C₈×D₄₇	—	4	1	no	—
115	C₁₆×C₄₇	—	2	1	no	—
116	C₁₆×C₄₇	—	2	2	no	—
117	C₁₆×C₄₇	—	2	1	no	—
118	C₄₇⋊C₁₆	—	2	2	no	—
119	C₄₇⋊C₁₆	—	2	1	no	—
120	C₄₇⋊C₁₆	—	2	1	no	—
121	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
122	C₂²×D₄₇	C₂²×C₈	1	1	yes	characteristic
123	D₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
130	C₂²⋊D₄₇	—	2	4	no	—
131	C₂×Dic₄₇	—	2	4	no	—
132	C₂×Dic₄₇	—	2	4	no	—
133	C₂×Dic₄₇	C₂²×C₈	1	1	yes	characteristic
134	C₂×Dic₄₇	C₂²×C₈	1	1	yes	characteristic
135	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
136	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
137	C₂×Dic₄₇	C₂²×C₈	1	1	yes	characteristic
138	C₄×D₄₇	—	2	1	no	—
139	C₄×D₄₇	—	2	1	no	—
140	C₄×D₄₇	—	2	2	no	—
141	C₄×D₄₇	—	2	2	no	—
142	Dic₉₄	M₃₂	1	4	yes	complemented
143	C₈×C₄₇	—	2	2	no	—
144	C₈×C₄₇	—	2	1	no	—
145	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
146	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
147	C₄₇⋊C₈	—	2	1	no	—
148	C₄₇⋊C₈	—	2	1	no	—
149	C₄₇⋊C₈	—	2	2	no	—
150	#33505	—	47	1	no	maximal, Sylow
151	C₂²×C₄₇	—	2	2	no	—
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
155	C₂×D₄₇	—	2	1	no	—
156	C₂×D₄₇	—	2	1	no	—
157	C₂×D₄₇	—	2	1	no	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	C₂×M₃₂	1	2	yes	—
161	C₄×C₄₇	#520	1	1	yes	characteristic
162	C₄×C₄₇	#568	1	1	yes	characteristic
163	Dic₄₇	—	2	1	no	—
164	Dic₄₇	—	2	2	no	—
165	Dic₄₇	—	2	1	no	—
166	Dic₄₇	C₂×M₃₂	1	4	yes	—
167	Dic₄₇	#504	1	2	yes	—
168	C₈×#41	—	47	1	no	—
169	Q₈×C₁₆	—	47	1	no	—
170	Q₈×C₁₆	—	47	1	no	—
171	#2169	—	47	2	no	—
172	#2169	—	47	1	no	—
173	#2169	—	47	2	no	—
174	#2169	D₄₇	1	1	yes	characteristic, complemented
175	#2162	—	47	2	no	—
176	#2162	—	47	1	no	—
177	#2150	—	47	2	no	—
178	#2150	—	47	1	no	—
179	C₂×C₄₇	—	4	2	no	—
180	C₂×C₄₇	#2988	1	1	yes	characteristic
181	C₂×C₄₇	C₂²×M₃₂	1	1	yes	characteristic
182	C₂×C₄₇	C₂×#504	1	1	yes	characteristic
183	D₄₇	—	4	1	no	—
184	C₄×#41	—	47	1	no	—
185	C₂×M₃₂	—	47	2	no	—
186	C₂×M₃₂	—	47	2	no	—
187	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
188	C₈×Q₈	—	47	1	no	—
189	C₈×Q₈	—	47	1	no	—
190	C₈×D₄	—	47	2	no	—
191	C₈×D₄	—	47	1	no	—
192	C₈×D₄	—	47	2	no	—
193	C₈×D₄	C₂×D₄₇	1	1	yes	characteristic
194	#431	—	47	1	no	—
195	#431	—	47	2	no	—
196	C₂×C₄×C₈	—	47	2	no	—
197	C₂×C₄×C₈	—	47	1	no	—
198	C₄⋊C₁₆	—	47	2	no	—
199	C₄⋊C₁₆	—	47	1	no	—
200	C₄⋊C₁₆	—	47	2	no	—
201	C₄⋊C₁₆	—	47	2	no	—
202	C₄⋊C₁₆	—	47	1	no	—
203	C₄⋊C₁₆	—	47	1	no	—
204	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
205	C₂²⋊C₁₆	—	47	2	no	—
206	C₂²⋊C₁₆	—	47	1	no	—
207	C₂²⋊C₁₆	—	47	1	no	—
208	C₂²⋊C₁₆	C₂×D₄₇	1	2	yes	—
209	C₄×C₁₆	—	47	2	no	—
210	C₄×C₁₆	—	47	2	no	—
211	C₄×C₁₆	—	47	1	no	—
212	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
213	C₄₇	#33505	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	—	47	1	no	—
215	C₂²×C₈	—	47	2	no	—
216	C₂²×C₈	—	47	1	no	—
217	C₂²×C₈	—	47	1	no	—
218	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
219	C₄×Q₈	—	47	1	no	—
220	C₄×Q₈	—	47	1	no	—
221	C₄×D₄	—	47	1	no	—
222	C₄×D₄	—	47	2	no	—
223	C₄×D₄	—	47	2	no	—
224	C₄×D₄	C₄×D₄₇	1	1	yes	characteristic
225	#117	—	47	1	no	—
226	#117	—	47	2	no	—
227	C₂×C₄²	—	47	2	no	—
228	C₂×C₄²	—	47	1	no	—
229	M₃₂	—	4	2	no	—
230	M₃₂	—	188	2	no	—
231	M₃₂	—	188	2	no	—
232	C₂×C₁₆	—	47	1	no	—
233	C₂×C₁₆	—	47	2	no	—
234	C₂×C₁₆	—	47	1	no	—
235	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
237	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
238	C₄⋊C₈	—	47	2	no	—
239	C₄⋊C₈	—	47	2	no	—
240	C₄⋊C₈	—	47	1	no	—
241	C₄⋊C₈	C₄×D₄₇	1	1	yes	characteristic
242	C₂²⋊C₈	—	47	2	no	—
243	C₂²⋊C₈	—	47	2	no	—
244	C₂²⋊C₈	C₄×D₄₇	1	2	yes	—
245	C₄×C₈	—	47	2	no	—
246	C₄×C₈	—	47	1	no	—
247	C₄×C₈	—	47	2	no	—
248	C₄×C₈	—	47	2	no	—
249	C₄×C₈	—	47	1	no	—
250	C₄×C₈	—	47	1	no	—
251	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
252	#41	—	94	4	no	—
253	C₂×Q₈	—	47	1	no	—
254	C₂×D₄	—	47	2	no	—
255	C₂×D₄	C₈×D₄₇	1	1	yes	characteristic
256	C₂²×C₄	—	47	2	no	—
257	C₂²×C₄	—	47	2	no	—
258	C₂²×C₄	—	47	1	no	—
259	C₂²×C₄	—	47	1	no	—
260	C₂²×C₄	—	47	1	no	—
261	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	47	1	no	—
265	C₂×C₈	—	47	1	no	—
266	C₂×C₈	—	47	2	no	—
267	C₂×C₈	—	94	1	no	—
268	C₂×C₈	—	94	1	no	—
269	C₂×C₈	—	188	1	no	—
270	C₂×C₈	—	188	2	no	—
271	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
272	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
273	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
274	C₄⋊C₄	—	47	2	no	—
275	C₄⋊C₄	—	47	2	no	—
276	C₄⋊C₄	—	47	1	no	—
277	C₄⋊C₄	C₈×D₄₇	1	1	yes	characteristic
278	C₂²⋊C₄	—	47	2	no	—
279	C₂²⋊C₄	—	47	2	no	—
280	C₂²⋊C₄	C₈×D₄₇	1	2	yes	—
281	C₄²	—	47	2	no	—
282	C₄²	—	47	1	no	—
283	C₄²	—	47	2	no	—
284	C₄²	—	47	1	no	—
285	C₄²	—	47	1	no	—
286	C₄²	—	47	2	no	—
287	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
288	C₁₆	—	2	1	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	2	1	no	—
291	C₁₆	—	94	2	no	—
292	C₁₆	—	94	1	no	—
293	C₁₆	—	94	1	no	—
294	C₂³	—	47	1	no	—
295	C₂³	C₂×C₈×D₄₇	1	2	yes	—
296	Q₈	—	47	4	no	—
297	D₄	—	2	2	no	—
298	D₄	—	94	4	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	47	1	no	—
302	C₂×C₄	—	47	2	no	—
303	C₂×C₄	—	47	1	no	—
304	C₂×C₄	—	47	1	no	—
305	C₂×C₄	—	47	2	no	—
306	C₂×C₄	—	94	2	no	—
307	C₂×C₄	—	94	4	no	—
308	C₂×C₄	—	94	4	no	—
309	C₂×C₄	—	94	2	no	—
310	C₂×C₄	—	94	1	no	—
311	C₂×C₄	—	94	1	no	—
312	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
313	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	2	1	no	—
318	C₈	—	94	1	no	—
319	C₈	—	94	2	no	—
320	C₈	—	94	1	no	—
321	C₈	#13094069	1	1	yes	characteristic
322	C₈	#13094071	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	94	1	no	—
327	C₂²	—	94	1	no	—
328	C₂²	—	94	1	no	—
329	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	47	4	no	—
333	C₄	—	47	2	no	—
334	C₄	—	94	1	no	—
335	C₄	—	94	1	no	—
336	C₄	—	94	2	no	—
337	C₄	M₃₂×D₄₇	1	2	yes	—
338	C₄	#15689602	1	1	yes	characteristic
339	C₄	#15689888	1	1	yes	characteristic
340	C₂	—	4	2	no	—
341	C₂	—	188	1	no	—
342	C₂	#20252495	1	1	yes	characteristic
343	C₂	C₂×M₃₂×D₄₇	1	1	yes	characteristic
344	C₂	#504×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₄₇×#2169	C₂
prime radical	C₂×#10625352	C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2169	D₄₇
hypocenter	C₄₇	#33505
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ #2169

Complements

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

N	H ≅ G/N	type
2. C₁₆×Dic₉₄	C₂	semidirect
3. #20245290	C₂	semidirect
4. #20245220	C₂	semidirect
5. #20245218	C₂	semidirect
6. #20245217	C₂	semidirect
7. #20245185	C₂	semidirect
8. #20245184	C₂	semidirect
9. #20245143	C₂	semidirect
10. #20245142	C₂	semidirect
11. C₄₇×#2169	C₂	semidirect
142. Dic₉₄	M₃₂	semidirect
174. #2169	D₄₇	semidirect
213. C₄₇	#33505	semidirect

Maximal subgroups

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

subgroup	index
C₈×#10625352	2
C₁₆×Dic₉₄	2
#20245290	2
#20245220	2
#20245218	2
#20245217	2
#20245185	2
#20245184	2
#20245143	2
#20245142	2
C₄₇×#2169	2
#33505	47