Subgroups — #26109627

Order 12032.

680 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20241227	C₂	1	4	yes	maximal, complemented
4	C₄₇×#1797	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
6	C₂²×C₄⋊Dic₄₇	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₄₇	—	2	1	no	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
10	C₄×C₄⋊Dic₄₇	—	2	1	no	—
11	C₄×C₄⋊Dic₄₇	—	2	1	no	—
12	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
13	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
14	C₂×C₄₇⋊#95	—	2	1	no	—
15	#15688878	—	2	1	no	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
19	C₄₇×#336	—	2	1	no	—
20	C₂³×Dic₄₇	—	2	1	no	—
21	C₂³×Dic₄₇	—	2	1	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	—	2	1	no	—
24	C₂×C₄⋊Dic₄₇	—	2	1	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	4	no	—
27	C₂×C₄⋊Dic₄₇	C₈	1	4	yes	complemented
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	—	4	4	no	—
33	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
34	C₄×C₄₇⋊C₈	—	2	4	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	—	2	4	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₄₇	—	2	4	no	—
47	C₂²⋊C₈×C₄₇	—	4	2	no	—
48	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	4	1	no	—
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
61	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	4	no	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	2	2	no	—
70	C₂²×Dic₄₇	—	4	2	no	—
71	C₂²×Dic₄₇	—	4	2	no	—
72	C₂²×Dic₄₇	—	4	2	no	—
73	C₂²×Dic₄₇	—	4	2	no	—
74	C₂×C₈×C₄₇	—	4	2	no	—
75	C₂×C₈×C₄₇	—	4	2	no	—
76	C₂×C₈×C₄₇	—	4	2	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₄²×C₄₇	—	2	2	no	—
79	C₄²×C₄₇	—	2	2	no	—
80	C₄²×C₄₇	#41	1	4	yes	—
81	C₄⋊Dic₄₇	—	2	4	no	—
82	C₄⋊Dic₄₇	—	2	4	no	—
83	C₄⋊Dic₄₇	—	2	2	no	—
84	C₄⋊Dic₄₇	—	2	2	no	—
85	C₄⋊Dic₄₇	C₂×C₈	1	4	yes	—
86	C₄⋊Dic₄₇	M₁₆	1	4	yes	—
87	C₄×Dic₄₇	—	4	4	no	—
88	C₄×Dic₄₇	—	4	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂³×C₄₇	—	2	1	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	2	1	no	—
96	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
97	C₂³×C₄₇	#103	1	2	yes	—
98	C₂³×C₄₇	#104	1	2	yes	—
99	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
115	C₂×C₄×C₄₇	#103	1	2	yes	—
116	C₂×C₄×C₄₇	#117	1	2	yes	—
117	C₂×C₄×C₄₇	#123	1	2	yes	—
118	C₂×C₄×C₄₇	#123	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	—	2	4	no	—
127	C₂×Dic₄₇	—	4	8	no	—
128	C₂×Dic₄₇	—	4	8	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₈×C₄₇	—	8	2	no	—
134	C₄₇⋊C₈	—	4	4	no	—
135	#8707	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	2	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	Q₈⋊C₈	1	4	yes	—
143	C₂²×C₄₇	#381	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₂²×C₄₇	#407	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
147	C₂²×C₄₇	#418	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
149	C₂²×C₄₇	#421	1	1	yes	characteristic
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	4	1	no	—
154	C₄×C₄₇	#431	1	2	yes	—
155	C₄×C₄₇	#432	1	2	yes	—
156	C₄×C₄₇	#481	1	2	yes	—
157	C₄×C₄₇	#484	1	2	yes	—
158	Dic₄₇	—	4	4	no	—
159	Dic₄₇	—	4	4	no	—
160	C₂×C₄×C₄⋊C₄	—	47	1	no	—
161	#1797	D₄₇	1	1	yes	characteristic
162	C₄×C₂²⋊C₈	—	47	1	no	—
163	#1279	—	47	4	no	—
164	C₂×C₄₇	—	2	4	no	—
165	C₂×C₄₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
166	C₂×C₄₇	#1478	1	1	yes	characteristic
167	C₂×C₄₇	#1487	1	1	yes	characteristic
168	C₂×C₄₇	#1490	1	1	yes	characteristic
169	C₂×C₄₇	#1817	1	1	yes	characteristic
170	C₂×C₄₇	#1824	1	1	yes	characteristic
171	C₂×C₄₇	#1826	1	1	yes	characteristic
172	C₂²×C₄⋊C₄	—	47	1	no	—
173	C₂²×C₄⋊C₄	—	47	1	no	—
174	C₂²×C₄²	—	94	1	no	—
175	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
177	C₂×C₂²⋊C₈	—	2	1	no	—
178	C₂×C₂²⋊C₈	—	94	1	no	—
179	C₂×C₄×C₈	—	47	2	no	—
180	C₄×C₄⋊C₄	—	47	2	no	—
181	C₄×C₄⋊C₄	—	47	2	no	—
182	C₄×C₄⋊C₄	—	94	1	no	—
183	C₄×C₄⋊C₄	—	94	1	no	—
184	C₂×#95	—	94	1	no	—
185	#336	—	2	1	no	—
186	#336	—	94	1	no	—
187	C₄₇	#8707	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	94	1	no	—
189	C₂³×C₄	—	94	1	no	—
190	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
191	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
192	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	94	2	no	—
195	C₂×C₄⋊C₄	—	47	4	no	—
196	C₂×C₄⋊C₄	—	47	2	no	—
197	C₂×C₄⋊C₄	—	47	2	no	—
198	C₂×C₄⋊C₄	—	94	4	no	—
199	C₂×C₄⋊C₄	—	94	2	no	—
200	C₂×C₄⋊C₄	—	94	1	no	—
201	C₂×C₄⋊C₄	—	94	1	no	—
202	C₂×C₄²	—	2	4	no	—
203	C₂×C₄²	—	94	2	no	—
204	C₂×C₄²	—	94	2	no	—
205	C₂×C₄²	—	188	4	no	—
206	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
207	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
208	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
210	C₄⋊C₈	—	2	4	no	—
211	C₂²⋊C₈	—	4	2	no	—
212	C₂²⋊C₈	—	94	4	no	—
213	C₄×C₈	—	94	4	no	—
214	#95	—	94	2	no	—
215	#95	—	94	2	no	—
216	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	94	4	no	—
225	C₂²×C₄	—	94	4	no	—
226	C₂²×C₄	—	94	2	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	—	188	2	no	—
229	C₂²×C₄	—	188	2	no	—
230	C₂²×C₄	—	188	2	no	—
231	C₂²×C₄	—	188	2	no	—
232	C₂²×C₄	C₈×D₄₇	1	2	yes	—
233	C₂²×C₄	#10625324	1	2	yes	—
234	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
235	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
236	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
237	C₂²×C₄	#10625335	1	2	yes	—
238	C₂²×C₄	#10625337	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	94	4	no	—
246	C₂×C₈	—	188	2	no	—
247	C₂×C₈	—	188	2	no	—
248	C₄⋊C₄	—	47	4	no	—
249	C₄⋊C₄	—	47	4	no	—
250	C₄⋊C₄	—	94	4	no	—
251	C₄⋊C₄	—	94	4	no	—
252	C₄⋊C₄	—	94	2	no	—
253	C₄⋊C₄	—	94	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	188	4	no	—
258	C₄²	—	188	4	no	—
259	C₄²	#10625352	1	4	yes	—
260	C₂³	—	2	1	no	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	(C₂×D₄₇)⋊C₈	1	2	yes	—
265	C₂³	#13093947	1	2	yes	—
266	C₂³	#13093950	1	2	yes	—
267	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	94	4	no	—
283	C₂×C₄	—	94	4	no	—
284	C₂×C₄	—	188	8	no	—
285	C₂×C₄	—	188	8	no	—
286	C₂×C₄	—	188	2	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	—	188	2	no	—
290	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
291	C₂×C₄	#13093947	1	2	yes	—
292	C₂×C₄	#13093980	1	2	yes	—
293	C₂×C₄	#13094005	1	2	yes	—
294	C₂×C₄	#13094008	1	2	yes	—
295	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
296	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
297	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
298	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
299	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
300	C₂×C₄	#13094049	1	2	yes	—
301	C₈	—	8	2	no	—
302	C₈	—	188	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	#15688853	1	4	yes	—
310	C₂²	#15688984	1	1	yes	characteristic
311	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
312	C₂²	#15689086	1	1	yes	characteristic
313	C₂²	C₂×#13093947	1	1	yes	characteristic
314	C₂²	#15689127	1	1	yes	characteristic
315	C₂²	C₂×#13093950	1	1	yes	characteristic
316	C₂²	#15689139	1	1	yes	characteristic
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	1	no	—
321	C₄	—	188	4	no	—
322	C₄	—	188	4	no	—
323	C₄	#15689173	1	2	yes	—
324	C₄	#15689177	1	2	yes	—
325	C₄	#15689439	1	2	yes	—
326	C₄	#15689456	1	2	yes	—
327	C₂	—	2	4	no	—
328	C₂	C₂×#15688853	1	1	yes	characteristic
329	C₂	#20241778	1	1	yes	characteristic
330	C₂	#20241816	1	1	yes	characteristic
331	C₂	#20241830	1	1	yes	characteristic
332	C₂	#20243454	1	1	yes	characteristic
333	C₂	#20243476	1	1	yes	characteristic
334	C₂	#20243485	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1797	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1797	D₄₇
hypocenter	C₄₇	#8707
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. #20241227	C₂	semidirect
27. C₂×C₄⋊Dic₄₇	C₈	semidirect
187. C₄₇	#8707	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊Dic₄₇	2
C₄×C₂²⋊(C₄₇⋊C₈)	2
#20241227	2
C₄₇×#1797	2
#8707	47