Subgroups — #26125198

Order 12032.

1254 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13094024	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688886	C₂	1	4	yes	maximal
3	#20241618	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₉₄	—	2	1	no	—
6	C₂×#13094024	—	2	8	no	—
7	C₂×#13094024	—	2	2	no	—
8	C₂×#13094024	C₄	1	4	yes	complemented
9	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
10	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×C₄₇⋊#95	—	2	1	no	—
13	C₂×C₄₇⋊#95	—	2	1	no	—
14	#15688886	—	2	8	no	—
15	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#336	—	2	2	no	—
18	C₂²×Dic₉₄	—	2	8	no	—
19	C₂²×Dic₉₄	—	2	4	no	—
20	C₂²×Dic₉₄	—	4	4	no	—
21	C₂²×Dic₉₄	—	4	2	no	—
22	C₂²×Dic₉₄	—	4	2	no	—
23	C₂³×Dic₄₇	—	2	2	no	—
24	C₂³×Dic₄₇	—	2	2	no	—
25	#13094024	—	2	16	no	—
26	#13094024	—	2	16	no	—
27	C₂×C₄⋊Dic₄₇	—	2	8	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	—	2	4	no	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄⋊Dic₄₇	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊Dic₄₇	D₄	1	8	yes	—
36	C₂×C₄⋊Dic₄₇	Q₈	1	4	yes	complemented
37	C₄₇⋊#95	—	4	2	no	—
38	C₄₇⋊#95	—	4	2	no	—
39	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	—	2	4	no	—
44	C₂²×C₈×C₄₇	—	2	2	no	—
45	C₂²×C₈×C₄₇	—	2	4	no	—
46	C₂²×C₈×C₄₇	Q₈	1	4	yes	complemented
47	C₂×C₄²×C₄₇	—	2	4	no	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₇	D₄	1	2	yes	—
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
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	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₇	D₈	1	2	yes	—
58	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
59	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	2	yes	—
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂²×Dic₄₇	—	2	8	no	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	4	2	no	—
72	C₂²×Dic₄₇	—	4	2	no	—
73	C₂²×Dic₄₇	—	4	4	no	—
74	C₂²×Dic₄₇	—	4	2	no	—
75	C₂²×Dic₄₇	—	4	2	no	—
76	C₂²×Dic₄₇	—	4	2	no	—
77	C₂²×Dic₄₇	—	4	2	no	—
78	C₂²×Dic₄₇	—	4	2	no	—
79	C₂²×Dic₄₇	—	4	2	no	—
80	C₂×Dic₉₄	—	2	16	no	—
81	C₂×Dic₉₄	—	4	8	no	—
82	C₂×Dic₉₄	—	4	16	no	—
83	C₂×Dic₉₄	—	4	16	no	—
84	C₂×Dic₉₄	—	4	8	no	—
85	C₂×C₈×C₄₇	—	2	8	no	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	—	4	2	no	—
88	C₂×C₈×C₄₇	—	4	4	no	—
89	C₂×C₈×C₄₇	—	4	2	no	—
90	C₄²×C₄₇	—	2	4	no	—
91	C₄²×C₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	2	16	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄⋊Dic₄₇	—	2	8	no	—
95	C₄⋊Dic₄₇	—	2	8	no	—
96	C₄⋊Dic₄₇	—	4	4	no	—
97	C₄⋊Dic₄₇	C₂²⋊C₄	1	8	yes	—
98	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂³×C₄₇	#120	1	2	yes	—
100	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₄₇	#124	1	1	yes	characteristic
102	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
103	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
104	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
106	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
107	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
108	C₂³×C₄₇	#137	1	2	yes	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	4	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	#102	1	4	yes	—
118	C₂×C₄×C₄₇	#103	1	4	yes	—
119	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₄₇	#120	1	2	yes	—
123	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
124	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
125	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₄₇	#124	1	4	yes	—
127	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
130	C₂×Dic₄₇	—	2	8	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	C₂×Dic₄₇	—	4	8	no	—
134	C₂×Dic₄₇	—	4	8	no	—
135	C₂×Dic₄₇	—	4	4	no	—
136	C₂×Dic₄₇	—	4	4	no	—
137	C₂×Dic₄₇	—	4	4	no	—
138	C₂×Dic₄₇	—	8	2	no	—
139	C₂×Dic₄₇	—	8	2	no	—
140	C₂×Dic₄₇	—	8	2	no	—
141	C₂×Dic₄₇	—	8	2	no	—
142	Dic₉₄	—	4	32	no	—
143	C₈×C₄₇	—	4	4	no	—
144	#11384	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	#379	1	1	yes	characteristic
146	C₂²×C₄₇	#391	1	1	yes	characteristic
147	C₂²×C₄₇	#391	1	1	yes	characteristic
148	C₂²×C₄₇	#393	1	2	yes	—
149	C₂²×C₄₇	#395	1	2	yes	—
150	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
152	C₂²×C₄₇	#418	1	1	yes	characteristic
153	C₂²×C₄₇	#419	1	1	yes	characteristic
154	C₂²×C₄₇	#447	1	2	yes	—
155	C₂²×C₄₇	#448	1	2	yes	—
156	C₂²×C₄₇	#450	1	2	yes	—
157	C₂²×C₄₇	#451	1	2	yes	—
158	C₂²×C₄₇	#474	1	2	yes	—
159	C₂²×C₄₇	#475	1	2	yes	—
160	C₂²×C₄₇	#476	1	2	yes	—
161	C₂²×C₄₇	#477	1	2	yes	—
162	C₂²×C₄₇	#486	1	1	yes	characteristic
163	C₂²×C₄₇	#487	1	1	yes	characteristic
164	C₂²×C₄₇	#489	1	2	yes	—
165	C₂²×C₄₇	#498	1	1	yes	characteristic
166	C₂²×C₄₇	#500	1	1	yes	characteristic
167	C₂²×C₄₇	#501	1	2	yes	—
168	C₄×C₄₇	—	2	4	no	—
169	C₄×C₄₇	—	2	4	no	—
170	C₄×C₄₇	#391	1	4	yes	—
171	C₄×C₄₇	#393	1	2	yes	—
172	C₄×C₄₇	#395	1	2	yes	—
173	Dic₄₇	—	4	8	no	—
174	Dic₄₇	—	8	4	no	—
175	C₂²×C₄⋊Q₈	—	47	1	no	—
176	C₂×#340	—	47	4	no	—
177	C₂×#336	D₄₇	1	1	yes	characteristic
178	C₄⋊#95	—	47	1	no	—
179	C₂×C₄₇	#1446	1	1	yes	characteristic
180	C₂×C₄₇	#1786	1	1	yes	characteristic
181	C₂×C₄₇	#1787	1	1	yes	characteristic
182	C₂×C₄₇	#1987	1	1	yes	characteristic
183	C₂×C₄₇	#1988	1	1	yes	characteristic
184	C₂×C₄₇	#1990	1	1	yes	characteristic
185	C₂×C₄₇	#1993	1	1	yes	characteristic
186	C₂×C₄₇	#2023	1	2	yes	—
187	C₂×C₄₇	#2024	1	2	yes	—
188	C₂×C₄₇	#2057	1	2	yes	—
189	C₂×C₄₇	#2058	1	2	yes	—
190	C₂³×Q₈	—	94	1	no	—
191	C₂³×C₈	C₂×D₄₇	1	2	yes	—
192	C₂×C₄⋊Q₈	—	47	4	no	—
193	C₂×C₄⋊Q₈	—	94	8	no	—
194	C₂×C₄⋊Q₈	—	94	2	no	—
195	C₂²×C₄⋊C₄	—	47	2	no	—
196	C₂²×C₄⋊C₄	—	47	2	no	—
197	C₂²×C₄⋊C₄	—	94	1	no	—
198	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
199	C₂×#95	—	94	1	no	—
200	C₂×#95	—	94	1	no	—
201	#340	—	94	8	no	—
202	#336	—	2	2	no	—
203	C₄₇	#11384	1	1	yes	characteristic, complemented, Sylow
204	C₂²×Q₈	—	94	8	no	—
205	C₂²×Q₈	—	94	4	no	—
206	C₂²×Q₈	—	188	4	no	—
207	C₂²×Q₈	—	188	2	no	—
208	C₂²×Q₈	—	188	2	no	—
209	C₂³×C₄	—	94	2	no	—
210	C₂³×C₄	—	94	2	no	—
211	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
212	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
213	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
214	C₂²×C₈	—	2	2	no	—
215	C₂²×C₈	—	2	4	no	—
216	C₂²×C₈	—	2	4	no	—
217	C₂²×C₈	—	2	2	no	—
218	C₂²×C₈	Dic₉₄	1	4	yes	complemented
219	C₄⋊Q₈	—	94	16	no	—
220	C₄⋊Q₈	—	94	16	no	—
221	C₂×C₄⋊C₄	—	47	4	no	—
222	C₂×C₄⋊C₄	—	47	8	no	—
223	C₂×C₄⋊C₄	—	47	4	no	—
224	C₂×C₄⋊C₄	—	94	8	no	—
225	C₂×C₄⋊C₄	—	94	4	no	—
226	C₂×C₄⋊C₄	—	94	4	no	—
227	C₂×C₄⋊C₄	—	94	2	no	—
228	C₂×C₄⋊C₄	—	94	4	no	—
229	C₂×C₄⋊C₄	—	94	4	no	—
230	C₂×C₄⋊C₄	—	94	2	no	—
231	C₂×C₄²	—	2	4	no	—
232	C₂×C₄²	C₄×D₄₇	1	2	yes	—
233	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
234	#95	—	188	2	no	—
235	#95	—	188	2	no	—
236	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
237	C₂×Q₈	—	94	16	no	—
238	C₂×Q₈	—	188	16	no	—
239	C₂×Q₈	—	188	16	no	—
240	C₂×Q₈	—	188	8	no	—
241	C₂×Q₈	—	188	8	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	94	4	no	—
247	C₂²×C₄	—	94	8	no	—
248	C₂²×C₄	—	94	4	no	—
249	C₂²×C₄	—	188	2	no	—
250	C₂²×C₄	—	188	2	no	—
251	C₂²×C₄	—	188	2	no	—
252	C₂²×C₄	—	188	2	no	—
253	C₂²×C₄	—	188	2	no	—
254	C₂²×C₄	—	188	2	no	—
255	C₂²×C₄	—	188	2	no	—
256	C₂²×C₄	—	188	2	no	—
257	C₂²×C₄	—	188	4	no	—
258	C₂²×C₄	#10625325	1	2	yes	—
259	C₂²×C₄	#10625325	1	2	yes	—
260	C₂²×C₄	D₃₇₆	1	2	yes	—
261	C₂²×C₄	Dic₁₈₈	1	2	yes	—
262	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
263	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
264	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
265	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
266	C₂²×C₄	#10625350	1	2	yes	—
267	C₂²×C₄	D₄×D₄₇	1	4	yes	—
268	C₂²×C₄	#10625352	1	2	yes	—
269	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
270	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
271	C₂×C₈	—	2	8	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	—	4	2	no	—
276	C₄⋊C₄	—	47	8	no	—
277	C₄⋊C₄	—	94	16	no	—
278	C₄⋊C₄	—	94	4	no	—
279	C₄⋊C₄	—	94	8	no	—
280	C₄⋊C₄	—	94	8	no	—
281	C₄⋊C₄	—	188	4	no	—
282	C₄²	—	2	4	no	—
283	C₄²	—	2	4	no	—
284	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
285	C₂³	#13093994	1	2	yes	—
286	C₂³	#13094004	1	2	yes	—
287	C₂³	#13094012	1	1	yes	characteristic
288	C₂³	#13094024	1	1	yes	characteristic
289	C₂³	C₂×D₃₇₆	1	1	yes	characteristic
290	C₂³	C₂×#10625325	1	1	yes	characteristic
291	C₂³	C₂×#10625325	1	1	yes	characteristic
292	C₂³	C₂×Dic₁₈₈	1	1	yes	characteristic
293	C₂³	#13094052	1	2	yes	—
294	C₂³	#13094057	1	2	yes	—
295	Q₈	—	188	32	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	94	8	no	—
305	C₂×C₄	—	376	2	no	—
306	C₂×C₄	—	376	2	no	—
307	C₂×C₄	—	376	2	no	—
308	C₂×C₄	—	376	2	no	—
309	C₂×C₄	—	188	4	no	—
310	C₂×C₄	—	188	4	no	—
311	C₂×C₄	—	188	4	no	—
312	C₂×C₄	—	188	8	no	—
313	C₂×C₄	—	188	8	no	—
314	C₂×C₄	—	188	4	no	—
315	C₂×C₄	—	188	4	no	—
316	C₂×C₄	#13093946	1	4	yes	—
317	C₂×C₄	#13093949	1	4	yes	—
318	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
319	C₂×C₄	#13093989	1	2	yes	—
320	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
321	C₂×C₄	#13093992	1	2	yes	—
322	C₂×C₄	#13094001	1	4	yes	—
323	C₂×C₄	#13094002	1	4	yes	—
324	C₂×C₄	#13094003	1	2	yes	—
325	C₂×C₄	#13094010	1	4	yes	—
326	C₂×C₄	#13094011	1	1	yes	characteristic
327	C₂×C₄	#13094022	1	1	yes	characteristic
328	C₂×C₄	#13094023	1	4	yes	—
329	C₈	—	4	4	no	—
330	C₂²	#15688978	1	1	yes	characteristic
331	C₂²	#15689028	1	1	yes	characteristic
332	C₂²	#15689029	1	1	yes	characteristic
333	C₂²	#15689036	1	2	yes	—
334	C₂²	#15689044	1	2	yes	—
335	C₂²	C₂×#13093946	1	1	yes	characteristic
336	C₂²	C₂×#13093949	1	1	yes	characteristic
337	C₂²	#15689129	1	1	yes	characteristic
338	C₂²	#15689133	1	1	yes	characteristic
339	C₂²	#15689259	1	2	yes	—
340	C₂²	#15689264	1	2	yes	—
341	C₂²	#15689276	1	2	yes	—
342	C₂²	#15689281	1	2	yes	—
343	C₂²	#15689404	1	2	yes	—
344	C₂²	#15689409	1	2	yes	—
345	C₂²	#15689414	1	2	yes	—
346	C₂²	#15689419	1	2	yes	—
347	C₂²	#15689470	1	1	yes	characteristic
348	C₂²	#15689474	1	1	yes	characteristic
349	C₂²	#15689483	1	2	yes	—
350	C₂²	#15689521	1	1	yes	characteristic
351	C₂²	#15689528	1	1	yes	characteristic
352	C₂²	#15689532	1	2	yes	—
353	C₄	—	2	4	no	—
354	C₄	—	2	4	no	—
355	C₄	—	376	4	no	—
356	C₄	—	188	8	no	—
357	C₄	#15689027	1	4	yes	—
358	C₄	#15689035	1	2	yes	—
359	C₄	#15689043	1	2	yes	—
360	C₂	#20241658	1	1	yes	characteristic
361	C₂	#20243320	1	1	yes	characteristic
362	C₂	#20243324	1	1	yes	characteristic
363	C₂	#20244335	1	1	yes	characteristic
364	C₂	#20244339	1	1	yes	characteristic
365	C₂	#20244349	1	1	yes	characteristic
366	C₂	#20244361	1	1	yes	characteristic
367	C₂	#20244525	1	2	yes	—
368	C₂	#20244530	1	2	yes	—
369	C₂	#20244717	1	2	yes	—
370	C₂	#20244722	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#336	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#336	D₄₇
hypocenter	C₄₇	#11384
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
8. C₂×#13094024	C₄	semidirect
36. C₂×C₄⋊Dic₄₇	Q₈	semidirect
46. C₂²×C₈×C₄₇	Q₈	semidirect
203. C₄₇	#11384	semidirect
218. C₂²×C₈	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13094024	2
C₂×#15688886	2
#20241618	2
C₂×C₄₇×#336	2
#11384	47