Subgroups — #26104334

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243505	C₂	1	1	yes	maximal, characteristic, complemented
2	#20243134	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241209	C₂	1	4	yes	maximal, complemented
4	C₄×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
7	#15688878	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	C₂²	1	2	yes	—
10	#15688878	C₂²	1	2	yes	—
11	C₂×#127×C₄₇	C₄	1	1	yes	characteristic
12	C₂×C₄×D₄×C₄₇	—	2	1	no	—
13	C₂×C₄×D₄×C₄₇	—	2	1	no	—
14	C₂×C₄×D₄×C₄₇	C₄	1	2	yes	—
15	C₄₇×#390	—	2	1	no	—
16	C₄₇×#390	—	2	1	no	—
17	C₄₇×#390	C₂²	1	2	yes	—
18	C₄₇×#384	C₄	1	1	yes	characteristic
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
23	C₄×C₄₇⋊C₈	—	2	4	no	—
24	C₂²×D₄×C₄₇	—	2	1	no	—
25	C₂²×D₄×C₄₇	—	2	1	no	—
26	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₄₇	—	2	2	no	—
28	#127×C₄₇	—	2	2	no	—
29	#127×C₄₇	C₈	1	4	yes	complemented
30	C₄×D₄×C₄₇	—	4	4	no	—
31	C₄×D₄×C₄₇	—	4	4	no	—
32	C₄×D₄×C₄₇	—	4	2	no	—
33	C₄×D₄×C₄₇	—	4	2	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₂⁴×C₄₇	—	2	2	no	—
48	C₂×D₄×C₄₇	—	2	4	no	—
49	C₂×D₄×C₄₇	—	2	4	no	—
50	C₂×D₄×C₄₇	—	2	2	no	—
51	C₂×D₄×C₄₇	—	4	2	no	—
52	C₂×D₄×C₄₇	—	4	2	no	—
53	C₂×D₄×C₄₇	—	4	2	no	—
54	C₂×D₄×C₄₇	—	4	2	no	—
55	C₂×D₄×C₄₇	—	4	2	no	—
56	C₂×D₄×C₄₇	—	4	2	no	—
57	C₂×D₄×C₄₇	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	—	4	4	no	—
61	C₂²×C₄×C₄₇	—	4	4	no	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₄₇	—	4	2	no	—
70	C₄⋊C₄×C₄₇	—	4	2	no	—
71	C₄⋊C₄×C₄₇	—	4	2	no	—
72	C₂²⋊C₄×C₄₇	—	4	4	no	—
73	C₂²⋊C₄×C₄₇	—	4	2	no	—
74	C₂²⋊C₄×C₄₇	—	4	2	no	—
75	C₂²⋊C₄×C₄₇	—	4	4	no	—
76	C₄²×C₄₇	—	2	2	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₄²×C₄₇	—	2	2	no	—
79	C₄²×C₄₇	—	4	1	no	—
80	C₄²×C₄₇	—	4	1	no	—
81	C₄²×C₄₇	C₂×C₈	1	2	yes	—
82	C₄²×C₄₇	M₁₆	1	2	yes	—
83	C₄²×C₄₇	D₈	1	2	yes	—
84	C₄²×C₄₇	SD₁₆	1	2	yes	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂³×C₄₇	—	2	4	no	—
92	C₂³×C₄₇	—	4	2	no	—
93	C₂³×C₄₇	—	4	2	no	—
94	C₂³×C₄₇	—	4	2	no	—
95	C₂³×C₄₇	—	4	2	no	—
96	C₂³×C₄₇	—	4	2	no	—
97	C₂³×C₄₇	—	4	2	no	—
98	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	D₄×C₄₇	—	4	4	no	—
100	D₄×C₄₇	—	4	4	no	—
101	D₄×C₄₇	—	4	8	no	—
102	D₄×C₄₇	—	4	8	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	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	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	8	8	no	—
119	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₄₇	#100	1	2	yes	—
122	C₂×C₄×C₄₇	#102	1	2	yes	—
123	C₂×C₄×C₄₇	#102	1	2	yes	—
124	C₂×C₄×C₄₇	#104	1	2	yes	—
125	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
130	C₄₇⋊C₈	—	4	4	no	—
131	C₄₇⋊C₈	—	8	2	no	—
132	#7587	—	47	1	no	maximal, Sylow
133	C₂²×C₄₇	—	4	4	no	—
134	C₂²×C₄₇	—	4	4	no	—
135	C₂²×C₄₇	—	4	4	no	—
136	C₂²×C₄₇	—	8	2	no	—
137	C₂²×C₄₇	—	8	2	no	—
138	C₂²×C₄₇	—	8	2	no	—
139	C₂²×C₄₇	—	8	2	no	—
140	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₄₇	#410	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
144	C₂²×C₄₇	#418	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
146	C₂²×C₄₇	#421	1	1	yes	characteristic
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	2	2	no	—
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	8	2	no	—
156	C₄×C₄₇	D₄⋊C₈	1	4	yes	—
157	C₄×#127	D₄₇	1	1	yes	characteristic
158	#1831	—	47	1	no	—
159	#1744	—	47	1	no	—
160	#1270	—	47	4	no	—
161	C₂×C₄₇	—	8	4	no	—
162	C₂×C₄₇	#1460	1	1	yes	characteristic
163	C₂×C₄₇	C₂×D₄⋊C₈	1	1	yes	characteristic
164	C₂×C₄₇	#1478	1	1	yes	characteristic
165	C₂×C₄₇	#1505	1	1	yes	characteristic
166	C₂×C₄₇	#1517	1	1	yes	characteristic
167	C₂×C₄₇	#1530	1	1	yes	characteristic
168	C₂×C₄₇	#1545	1	1	yes	characteristic
169	C₂×#127	Dic₄₇	1	1	yes	characteristic
170	C₂×C₄×D₄	—	2	1	no	—
171	C₂×C₄×D₄	—	2	1	no	—
172	C₂×C₄×D₄	Dic₄₇	1	2	yes	—
173	C₂×C₄⋊C₈	—	94	1	no	—
174	C₂×C₄×C₈	—	94	1	no	—
175	#390	—	2	1	no	—
176	#390	—	2	1	no	—
177	#390	C₂×D₄₇	1	2	yes	—
178	#384	Dic₄₇	1	1	yes	characteristic
179	C₄³	C₂×D₄₇	1	1	yes	characteristic
180	#336	—	47	2	no	—
181	#336	—	47	2	no	—
182	#336	—	94	1	no	—
183	#336	—	94	1	no	—
184	C₄₇	#7587	1	1	yes	characteristic, complemented, Sylow
185	C₂²×D₄	—	2	1	no	—
186	C₂²×D₄	—	2	1	no	—
187	C₂²×D₄	C₂×Dic₄₇	1	2	yes	—
188	C₂³×C₄	—	2	2	no	—
189	C₂²×C₈	—	94	2	no	—
190	C₂²×C₈	—	94	2	no	—
191	#127	—	2	2	no	—
192	#127	C₄₇⋊C₈	1	4	yes	complemented
193	C₄×D₄	—	4	4	no	—
194	C₄×D₄	—	4	4	no	—
195	C₄×D₄	—	4	2	no	—
196	C₄×D₄	—	4	2	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	—	2	2	no	—
202	C₂×C₂²⋊C₄	—	2	2	no	—
203	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
207	C₂×C₄²	C₂²⋊D₄₇	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	—	188	2	no	—
211	C₄×C₈	—	94	4	no	—
212	C₂⁴	—	2	2	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	C₂×C₄₇⋊C₈	1	4	yes	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
231	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
234	C₂×C₈	—	94	4	no	—
235	C₂×C₈	—	188	2	no	—
236	C₂×C₈	—	188	2	no	—
237	C₂×C₈	—	188	2	no	—
238	C₂×C₈	—	188	2	no	—
239	C₂×C₈	—	188	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	4	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	4	1	no	—
251	C₄²	—	4	1	no	—
252	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
253	C₄²	#10625329	1	2	yes	—
254	C₄²	#10625334	1	2	yes	—
255	C₄²	#10625335	1	2	yes	—
256	C₂³	—	2	4	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	2	no	—
263	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
264	D₄	—	4	4	no	—
265	D₄	—	4	4	no	—
266	D₄	—	4	8	no	—
267	D₄	—	4	8	no	—
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	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	8	no	—
284	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
285	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
286	C₂×C₄	#13093941	1	2	yes	—
287	C₂×C₄	#13093945	1	2	yes	—
288	C₂×C₄	#13093945	1	2	yes	—
289	C₂×C₄	#13093951	1	2	yes	—
290	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
291	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
292	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
293	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
294	C₂×C₄	#13094050	1	2	yes	—
295	C₈	—	376	2	no	—
296	C₈	—	188	4	no	—
297	C₂²	—	4	4	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
305	C₂²	#15689097	1	1	yes	characteristic
306	C₂²	C₂×#13093941	1	1	yes	characteristic
307	C₂²	C₂×#13093945	1	1	yes	characteristic
308	C₂²	#15689128	1	1	yes	characteristic
309	C₂²	C₂×#13093951	1	1	yes	characteristic
310	C₂²	#15689140	1	1	yes	characteristic
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	8	2	no	—
320	C₄	D₄⋊(C₄₇⋊C₈)	1	4	yes	—
321	C₂	—	8	4	no	—
322	C₂	#20241709	1	1	yes	characteristic
323	C₂	C₂×D₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
324	C₂	#20241779	1	1	yes	characteristic
325	C₂	#20241896	1	1	yes	characteristic
326	C₂	#20241948	1	1	yes	characteristic
327	C₂	#20241999	1	1	yes	characteristic
328	C₂	#20242056	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₄₇	C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#127	D₄₇
hypocenter	C₄₇	#7587
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. #20243505	C₂	semidirect
2. #20243134	C₂	semidirect
3. #20241209	C₂	semidirect
29. #127×C₄₇	C₈	semidirect
184. C₄₇	#7587	semidirect
192. #127	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20243505	2
#20243134	2
#20241209	2
C₄×#127×C₄₇	2
#7587	47