Subgroups — #26104276

Order 12032.

668 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243368	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20241214	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₇×#391	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
9	#15688878	C₂²	1	2	yes	—
10	#15688878	C₂²	1	2	yes	—
11	C₂³×Q₈×C₄₇	C₄	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#391	—	2	1	no	—
15	C₄₇×#391	—	2	1	no	—
16	C₄₇×#391	C₄	1	2	yes	—
17	C₄₇×#391	C₂²	1	2	yes	—
18	C₂×#95×C₄₇	—	2	1	no	—
19	C₂×#95×C₄₇	—	2	1	no	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
25	C₄×C₄₇⋊C₈	—	4	2	no	—
26	C₂²×Q₈×C₄₇	—	2	1	no	—
27	C₂²×Q₈×C₄₇	—	2	2	no	—
28	C₂²×Q₈×C₄₇	—	4	2	no	—
29	C₂²×Q₈×C₄₇	—	4	2	no	—
30	C₂²×Q₈×C₄₇	C₈	1	4	yes	complemented
31	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₄₇	—	2	1	no	—
33	C₂³×C₄×C₄₇	—	2	1	no	—
34	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₇	—	4	2	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	#95×C₄₇	—	2	2	no	—
46	#95×C₄₇	—	2	2	no	—
47	#95×C₄₇	—	2	2	no	—
48	#95×C₄₇	—	2	2	no	—
49	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×Q₈×C₄₇	—	2	4	no	—
51	C₂×Q₈×C₄₇	—	2	2	no	—
52	C₂×Q₈×C₄₇	—	4	8	no	—
53	C₂×Q₈×C₄₇	—	4	8	no	—
54	C₂×Q₈×C₄₇	—	4	2	no	—
55	C₂×Q₈×C₄₇	—	4	2	no	—
56	C₂×Q₈×C₄₇	—	4	2	no	—
57	C₂×Q₈×C₄₇	—	4	2	no	—
58	C₂×Q₈×C₄₇	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	4	1	no	—
67	C₂²×C₄×C₄₇	—	4	1	no	—
68	C₂²×C₄×C₄₇	—	4	1	no	—
69	C₂²×C₄×C₄₇	—	4	2	no	—
70	C₂²×C₄×C₄₇	—	4	2	no	—
71	C₂²×C₄×C₄₇	—	4	2	no	—
72	C₂²×C₄×C₄₇	—	4	2	no	—
73	C₂²×C₄×C₄₇	—	4	2	no	—
74	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
78	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
79	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₄⋊C₄×C₄₇	—	4	2	no	—
85	C₄⋊C₄×C₄₇	—	4	2	no	—
86	C₄²×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	4	1	no	—
89	C₄²×C₄₇	—	4	1	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂×C₄₇⋊C₈	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂³×C₄₇	—	2	1	no	—
97	C₂³×C₄₇	—	2	1	no	—
98	C₂³×C₄₇	—	2	1	no	—
99	C₂³×C₄₇	—	2	1	no	—
100	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
101	C₂³×C₄₇	#103	1	2	yes	—
102	C₂³×C₄₇	#104	1	2	yes	—
103	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	Q₈×C₄₇	—	4	4	no	—
105	Q₈×C₄₇	—	4	4	no	—
106	Q₈×C₄₇	—	4	8	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	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	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	4	no	—
124	C₂×C₄×C₄₇	—	4	8	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	8	4	no	—
131	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
132	C₂×C₄×C₄₇	#99	1	2	yes	—
133	C₂×C₄×C₄₇	#100	1	2	yes	—
134	C₂×C₄×C₄₇	#103	1	2	yes	—
135	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#137	1	2	yes	—
140	C₄₇⋊C₈	—	4	4	no	—
141	C₄₇⋊C₈	—	8	2	no	—
142	#7575	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	Q₈⋊C₈	1	4	yes	—
150	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
153	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
154	C₂²×C₄₇	#419	1	1	yes	characteristic
155	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
156	C₂²×C₄₇	#421	1	1	yes	characteristic
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	2	1	no	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	1	no	—
163	C₄×C₄₇	—	4	4	no	—
164	C₄×C₄₇	—	4	1	no	—
165	C₄×C₄₇	—	8	2	no	—
166	C₄×C₄₇	#323	1	4	yes	—
167	C₂×#391	D₄₇	1	1	yes	characteristic
168	#1797	—	47	1	no	—
169	C₄×C₂²⋊C₈	—	47	1	no	—
170	#1272	—	47	4	no	—
171	C₂×C₄₇	—	2	4	no	—
172	C₂×C₄₇	C₂×#323	1	1	yes	characteristic
173	C₂×C₄₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
174	C₂×C₄₇	#1479	1	1	yes	characteristic
175	C₂×C₄₇	#1508	1	1	yes	characteristic
176	C₂×C₄₇	#1520	1	1	yes	characteristic
177	C₂×C₄₇	#1527	1	1	yes	characteristic
178	C₂×C₄₇	#1548	1	1	yes	characteristic
179	C₂³×Q₈	Dic₄₇	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
181	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
182	C₂×C₄⋊C₈	—	94	1	no	—
183	C₂×C₂²⋊C₈	—	94	1	no	—
184	C₂×C₂²⋊C₈	—	94	1	no	—
185	C₂×C₄×C₈	—	94	1	no	—
186	#391	—	2	1	no	—
187	#391	—	2	1	no	—
188	#391	Dic₄₇	1	2	yes	—
189	#391	C₂×D₄₇	1	2	yes	—
190	C₂×#95	—	2	1	no	—
191	C₂×#95	—	2	1	no	—
192	#336	—	47	2	no	—
193	#336	—	47	2	no	—
194	C₄₇	#7575	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	—	2	2	no	—
196	C₂²×Q₈	—	2	1	no	—
197	C₂²×Q₈	—	4	2	no	—
198	C₂²×Q₈	—	4	2	no	—
199	C₂²×Q₈	C₄₇⋊C₈	1	4	yes	complemented
200	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
201	C₂³×C₄	—	2	1	no	—
202	C₂³×C₄	—	2	1	no	—
203	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
204	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂²×C₈	—	94	2	no	—
207	C₂²×C₈	—	94	2	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	4	2	no	—
210	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
211	C₂×C₄²	—	4	2	no	—
212	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
216	C₄⋊C₈	—	188	2	no	—
217	C₂²⋊C₈	—	94	4	no	—
218	C₂²⋊C₈	—	188	2	no	—
219	C₄×C₈	—	188	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	C₂⁴	C₂²⋊Dic₄₇	1	1	yes	characteristic
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	8	no	—
228	C₂×Q₈	—	4	8	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×Q₈	—	4	2	no	—
232	C₂×Q₈	—	4	2	no	—
233	C₂×Q₈	C₂×C₄₇⋊C₈	1	4	yes	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
250	C₂²×C₄	#10625329	1	2	yes	—
251	C₂²×C₄	#10625336	1	2	yes	—
252	C₂²×C₄	#10625337	1	2	yes	—
253	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
255	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
256	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
259	C₂×C₈	—	94	4	no	—
260	C₂×C₈	—	188	2	no	—
261	C₂×C₈	—	188	2	no	—
262	C₂×C₈	—	188	2	no	—
263	C₂×C₈	—	188	2	no	—
264	C₂×C₈	—	188	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	4	1	no	—
270	C₄²	—	4	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
276	C₂³	#13093948	1	2	yes	—
277	C₂³	#13093951	1	2	yes	—
278	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
279	Q₈	—	4	4	no	—
280	Q₈	—	4	4	no	—
281	Q₈	—	4	8	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	8	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	4	no	—
306	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
307	C₂×C₄	#13093939	1	2	yes	—
308	C₂×C₄	#13093941	1	2	yes	—
309	C₂×C₄	#13093948	1	2	yes	—
310	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
311	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
312	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
313	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
314	C₂×C₄	#13094055	1	2	yes	—
315	C₈	—	376	2	no	—
316	C₈	—	188	4	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	Q₈⋊(C₄₇⋊C₈)	1	4	yes	—
324	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
325	C₂²	C₂×#13093939	1	1	yes	characteristic
326	C₂²	C₂×#13093941	1	1	yes	characteristic
327	C₂²	C₂×#13093948	1	1	yes	characteristic
328	C₂²	#15689132	1	1	yes	characteristic
329	C₂²	C₂×#13093951	1	1	yes	characteristic
330	C₂²	#15689140	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	2	no	—
337	C₄	—	4	4	no	—
338	C₄	—	4	1	no	—
339	C₄	—	8	2	no	—
340	C₄	#15688847	1	4	yes	—
341	C₂	—	2	4	no	—
342	C₂	C₂×#15688847	1	1	yes	characteristic
343	C₂	C₂×Q₈⋊(C₄₇⋊C₈)	1	1	yes	characteristic
344	C₂	#20241783	1	1	yes	characteristic
345	C₂	#20241908	1	1	yes	characteristic
346	C₂	#20241960	1	1	yes	characteristic
347	C₂	#20241986	1	1	yes	characteristic
348	C₂	#20242069	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#391	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#391	D₄₇
hypocenter	C₄₇	#7575
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#391

Complements

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

N	H ≅ G/N	type
3. #20241214	C₂	semidirect
30. C₂²×Q₈×C₄₇	C₈	semidirect
194. C₄₇	#7575	semidirect
199. C₂²×Q₈	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20243368	2
C₄×C₂²⋊(C₄₇⋊C₈)	2
#20241214	2
C₂×C₄₇×#391	2
#7575	47