Subgroups — #26109626

Order 12032.

680 conjugacy classes of subgroups.

The subgroup lattice has 340 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	#20243368	C₂	1	1	yes	maximal, characteristic
3	#20241228	C₂	1	4	yes	maximal, complemented
4	C₄×C₂²⋊C₈×C₄₇	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×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
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×C₂²⋊C₈×C₄₇	—	2	1	no	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
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	2	no	—
33	C₂×C₄×Dic₄₇	—	4	2	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
35	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₄₇⋊#95	—	2	2	no	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	—	2	4	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²⋊C₈×C₄₇	—	2	4	no	—
48	C₄×C₈×C₄₇	—	2	4	no	—
49	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	4	1	no	—
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
62	C₂²×C₄×C₄₇	D₈	1	2	yes	—
63	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂²×Dic₄₇	—	2	2	no	—
70	C₂²×Dic₄₇	—	2	2	no	—
71	C₂²×Dic₄₇	—	4	2	no	—
72	C₂²×Dic₄₇	—	4	2	no	—
73	C₂²×Dic₄₇	—	4	2	no	—
74	C₂²×Dic₄₇	—	4	2	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	4	2	no	—
77	C₂×C₈×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	—	2	2	no	—
79	C₄²×C₄₇	—	2	2	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	#41	1	4	yes	—
82	C₄⋊Dic₄₇	—	2	4	no	—
83	C₄⋊Dic₄₇	—	2	4	no	—
84	C₄⋊Dic₄₇	—	2	2	no	—
85	C₄⋊Dic₄₇	—	2	2	no	—
86	C₄⋊Dic₄₇	C₂×C₈	1	4	yes	—
87	C₄⋊Dic₄₇	M₁₆	1	4	yes	—
88	C₄×Dic₄₇	—	4	4	no	—
89	C₄×Dic₄₇	—	4	2	no	—
90	C₄×Dic₄₇	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	2	1	no	—
96	C₂³×C₄₇	—	2	1	no	—
97	C₂³×C₄₇	—	2	1	no	—
98	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
99	C₂³×C₄₇	#102	1	2	yes	—
100	C₂³×C₄₇	#104	1	2	yes	—
101	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
117	C₂×C₄×C₄₇	#102	1	2	yes	—
118	C₂×C₄×C₄₇	#117	1	2	yes	—
119	C₂×C₄×C₄₇	#123	1	2	yes	—
120	C₂×C₄×C₄₇	#123	1	2	yes	—
121	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#137	1	2	yes	—
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	—	4	8	no	—
130	C₂×Dic₄₇	—	4	8	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	C₂×Dic₄₇	—	4	2	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₈×C₄₇	—	4	4	no	—
136	C₄₇⋊C₈	—	8	2	no	—
137	#8706	—	47	1	no	maximal, Sylow
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	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	D₄⋊C₈	1	4	yes	—
145	C₂²×C₄₇	#381	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₂²×C₄₇	#407	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
149	C₂²×C₄₇	#419	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
151	C₂²×C₄₇	#421	1	1	yes	characteristic
152	C₄×C₄₇	—	2	2	no	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	4	1	no	—
156	C₄×C₄₇	#431	1	2	yes	—
157	C₄×C₄₇	#432	1	2	yes	—
158	C₄×C₄₇	#480	1	2	yes	—
159	C₄×C₄₇	#483	1	2	yes	—
160	Dic₄₇	—	4	4	no	—
161	Dic₄₇	—	4	4	no	—
162	C₂×C₄×C₄⋊C₄	—	47	1	no	—
163	#1797	—	47	1	no	—
164	C₄×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
165	#1279	—	47	4	no	—
166	C₂×C₄₇	—	2	4	no	—
167	C₂×C₄₇	C₂×D₄⋊C₈	1	1	yes	characteristic
168	C₂×C₄₇	#1479	1	1	yes	characteristic
169	C₂×C₄₇	#1486	1	1	yes	characteristic
170	C₂×C₄₇	#1491	1	1	yes	characteristic
171	C₂×C₄₇	#1817	1	1	yes	characteristic
172	C₂×C₄₇	#1823	1	1	yes	characteristic
173	C₂×C₄₇	#1826	1	1	yes	characteristic
174	C₂²×C₄⋊C₄	—	47	1	no	—
175	C₂²×C₄⋊C₄	—	47	1	no	—
176	C₂²×C₄²	—	94	1	no	—
177	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
178	C₂×C₄⋊C₈	—	47	2	no	—
179	C₂×C₂²⋊C₈	—	2	1	no	—
180	C₂×C₂²⋊C₈	—	94	1	no	—
181	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
182	C₄×C₄⋊C₄	—	47	2	no	—
183	C₄×C₄⋊C₄	—	47	2	no	—
184	C₄×C₄⋊C₄	—	94	1	no	—
185	C₄×C₄⋊C₄	—	94	1	no	—
186	C₂×#95	—	94	1	no	—
187	#336	—	2	1	no	—
188	#336	—	94	1	no	—
189	C₄₇	#8706	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₄	—	94	1	no	—
191	C₂³×C₄	—	94	1	no	—
192	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
193	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
194	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	94	2	no	—
197	C₂×C₄⋊C₄	—	47	4	no	—
198	C₂×C₄⋊C₄	—	47	2	no	—
199	C₂×C₄⋊C₄	—	47	2	no	—
200	C₂×C₄⋊C₄	—	94	4	no	—
201	C₂×C₄⋊C₄	—	94	2	no	—
202	C₂×C₄⋊C₄	—	94	1	no	—
203	C₂×C₄⋊C₄	—	94	1	no	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	94	2	no	—
206	C₂×C₄²	—	94	2	no	—
207	C₂×C₄²	—	188	2	no	—
208	C₂×C₄²	—	188	2	no	—
209	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
210	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
213	C₄⋊C₈	—	94	4	no	—
214	C₂²⋊C₈	—	2	4	no	—
215	C₂²⋊C₈	—	188	2	no	—
216	C₄×C₈	—	2	4	no	—
217	#95	—	94	2	no	—
218	#95	—	94	2	no	—
219	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	94	4	no	—
228	C₂²×C₄	—	94	4	no	—
229	C₂²×C₄	—	94	2	no	—
230	C₂²×C₄	—	94	2	no	—
231	C₂²×C₄	—	188	2	no	—
232	C₂²×C₄	—	188	2	no	—
233	C₂²×C₄	—	188	2	no	—
234	C₂²×C₄	—	188	2	no	—
235	C₂²×C₄	C₈×D₄₇	1	2	yes	—
236	C₂²×C₄	#10625324	1	2	yes	—
237	C₂²×C₄	#10625325	1	2	yes	—
238	C₂²×C₄	D₃₇₆	1	2	yes	—
239	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
240	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
241	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	188	2	no	—
249	C₂×C₈	—	188	2	no	—
250	C₂×C₈	—	188	2	no	—
251	C₄⋊C₄	—	47	4	no	—
252	C₄⋊C₄	—	47	4	no	—
253	C₄⋊C₄	—	94	4	no	—
254	C₄⋊C₄	—	94	4	no	—
255	C₄⋊C₄	—	94	2	no	—
256	C₄⋊C₄	—	94	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	188	4	no	—
261	C₄²	—	188	2	no	—
262	C₄²	—	188	2	no	—
263	C₄²	#10625352	1	4	yes	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	(C₂×D₄₇)⋊C₈	1	2	yes	—
269	C₂³	#13093946	1	2	yes	—
270	C₂³	#13093952	1	2	yes	—
271	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	94	4	no	—
287	C₂×C₄	—	94	4	no	—
288	C₂×C₄	—	188	8	no	—
289	C₂×C₄	—	188	8	no	—
290	C₂×C₄	—	188	2	no	—
291	C₂×C₄	—	188	2	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
295	C₂×C₄	#13093946	1	2	yes	—
296	C₂×C₄	#13093980	1	2	yes	—
297	C₂×C₄	#13094005	1	2	yes	—
298	C₂×C₄	#13094008	1	2	yes	—
299	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
300	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
301	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
302	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
303	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
304	C₂×C₄	#13094057	1	2	yes	—
305	C₈	—	4	4	no	—
306	C₈	—	376	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	#15688852	1	4	yes	—
314	C₂²	#15688984	1	1	yes	characteristic
315	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
316	C₂²	#15689086	1	1	yes	characteristic
317	C₂²	C₂×#13093946	1	1	yes	characteristic
318	C₂²	#15689133	1	1	yes	characteristic
319	C₂²	C₂×#13093952	1	1	yes	characteristic
320	C₂²	#15689141	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	188	4	no	—
326	C₄	—	188	4	no	—
327	C₄	#15689173	1	2	yes	—
328	C₄	#15689177	1	2	yes	—
329	C₄	#15689437	1	2	yes	—
330	C₄	#15689454	1	2	yes	—
331	C₂	—	2	4	no	—
332	C₂	C₂×#15688852	1	1	yes	characteristic
333	C₂	#20241784	1	1	yes	characteristic
334	C₂	#20241814	1	1	yes	characteristic
335	C₂	#20241836	1	1	yes	characteristic
336	C₂	#20243454	1	1	yes	characteristic
337	C₂	#20243475	1	1	yes	characteristic
338	C₂	#20243488	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×C₂²⋊C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	#8706
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. #20241228	C₂	semidirect
27. C₂×C₄⋊Dic₄₇	C₈	semidirect
189. C₄₇	#8706	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
#20243368	2
#20241228	2
C₄×C₂²⋊C₈×C₄₇	2
#8706	47