Subgroups — #26116286

Order 12032.

566 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243540	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#20241229	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
9	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
13	C₄₇×#391	—	2	1	no	—
14	C₄₇×#389	—	2	1	no	—
15	C₄₇×#389	C₄	1	2	yes	—
16	C₄₇×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
18	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
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)	—	2	4	no	—
23	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
25	C₄×C₄₇⋊C₈	—	2	4	no	—
26	C₄×C₄₇⋊C₈	—	2	4	no	—
27	C₄×C₄₇⋊C₈	—	2	2	no	—
28	C₄×C₄₇⋊C₈	Q₈	1	4	yes	complemented
29	C₂²×Q₈×C₄₇	—	2	1	no	—
30	C₄⋊Q₈×C₄₇	—	2	2	no	—
31	C₄⋊Q₈×C₄₇	C₈	1	4	yes	complemented
32	C₄×Q₈×C₄₇	—	4	2	no	—
33	C₄×Q₈×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	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	#95×C₄₇	—	2	2	no	—
49	#95×C₄₇	—	2	2	no	—
50	C₂×Q₈×C₄₇	—	2	4	no	—
51	C₂×Q₈×C₄₇	—	4	2	no	—
52	C₂×Q₈×C₄₇	—	4	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₄₇	—	2	4	no	—
63	C₄⋊C₄×C₄₇	—	2	2	no	—
64	C₄⋊C₄×C₄₇	—	2	4	no	—
65	C₄⋊C₄×C₄₇	—	4	4	no	—
66	C₄⋊C₄×C₄₇	—	4	2	no	—
67	C₄⋊C₄×C₄₇	—	4	2	no	—
68	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
69	C₄²×C₄₇	—	2	2	no	—
70	C₄²×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	—	2	2	no	—
72	C₄²×C₄₇	—	4	2	no	—
73	C₄²×C₄₇	—	4	2	no	—
74	C₄²×C₄₇	C₂×C₈	1	2	yes	—
75	C₄²×C₄₇	M₁₆	1	2	yes	—
76	C₄²×C₄₇	D₈	1	2	yes	—
77	C₄²×C₄₇	SD₁₆	1	2	yes	—
78	C₄²×C₄₇	SD₁₆	1	2	yes	—
79	C₄²×C₄₇	Q₁₆	1	2	yes	—
80	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
81	C₄²×C₄₇	#41	1	2	yes	—
82	C₂×C₄₇⋊C₈	—	2	4	no	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	4	2	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	Q₈×C₄₇	—	4	8	no	—
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	1	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	—	4	2	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	—	4	2	no	—
102	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₄₇	#102	1	2	yes	—
105	C₂×C₄×C₄₇	#102	1	2	yes	—
106	C₂×C₄×C₄₇	#103	1	2	yes	—
107	C₂×C₄×C₄₇	#103	1	2	yes	—
108	C₂×C₄×C₄₇	#104	1	2	yes	—
109	C₂×C₄×C₄₇	#104	1	2	yes	—
110	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#131	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
121	C₄₇⋊C₈	—	2	8	no	—
122	C₄₇⋊C₈	—	8	2	no	—
123	#9870	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	#391	1	1	yes	characteristic
125	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₂²×C₄₇	#408	1	1	yes	characteristic
127	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
128	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
129	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#104	1	1	yes	characteristic
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	4	4	no	—
135	C₄×C₄₇	—	8	2	no	—
136	C₄×C₄₇	D₄⋊C₈	1	4	yes	—
137	C₄×C₄₇	Q₈⋊C₈	1	4	yes	—
138	C₄×C₄₇	C₈×Q₈	1	2	yes	—
139	C₄×C₄₇	#446	1	2	yes	—
140	C₄×C₄₇	#486	1	1	yes	characteristic
141	C₄×C₄₇	#487	1	1	yes	characteristic
142	C₄×C₄₇	#498	1	1	yes	characteristic
143	C₄×C₄₇	#500	1	1	yes	characteristic
144	C₄×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
145	#1842	—	47	1	no	—
146	C₄²×C₈	—	47	1	no	—
147	#1279	—	47	4	no	—
148	C₂×C₄₇	C₂×D₄⋊C₈	1	1	yes	characteristic
149	C₂×C₄₇	C₂×Q₈⋊C₈	1	1	yes	characteristic
150	C₂×C₄₇	#1492	1	1	yes	characteristic
151	C₂×C₄₇	#1493	1	1	yes	characteristic
152	C₂×C₄₇	#1984	1	1	yes	characteristic
153	C₂×C₄₇	#1990	1	1	yes	characteristic
154	C₂×C₄₇	#1994	1	1	yes	characteristic
155	C₂×C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
156	C₂×C₄×Q₈	—	2	1	no	—
157	C₂×C₄⋊C₈	—	47	2	no	—
158	C₂×C₄⋊C₈	—	94	1	no	—
159	C₂×C₄⋊C₈	—	94	1	no	—
160	C₂×C₄×C₈	—	47	2	no	—
161	C₂×C₄×C₈	—	94	1	no	—
162	C₂×C₄×C₈	—	94	1	no	—
163	#391	—	2	1	no	—
164	#389	—	2	1	no	—
165	#389	Dic₄₇	1	2	yes	—
166	#384	Dic₄₇	1	1	yes	characteristic
167	C₄×C₄⋊C₄	—	2	1	no	—
168	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
169	C₄³	C₂×D₄₇	1	1	yes	characteristic
170	C₄₇	#9870	1	1	yes	characteristic, complemented, Sylow
171	C₂²×Q₈	—	2	1	no	—
172	C₂²×C₈	—	94	2	no	—
173	C₂²×C₈	—	94	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	C₄₇⋊C₈	1	4	yes	complemented
176	C₄×Q₈	—	4	2	no	—
177	C₄×Q₈	—	4	2	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	C₄⋊C₈	—	94	4	no	—
193	C₄⋊C₈	—	188	2	no	—
194	C₄⋊C₈	—	188	2	no	—
195	C₄×C₈	—	47	4	no	—
196	C₄×C₈	—	94	4	no	—
197	C₄×C₈	—	94	4	no	—
198	C₄×C₈	—	94	2	no	—
199	#95	—	2	2	no	—
200	#95	—	2	2	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	—	94	4	no	—
215	C₂×C₈	—	94	4	no	—
216	C₂×C₈	—	188	2	no	—
217	C₂×C₈	—	188	2	no	—
218	C₂×C₈	—	188	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	4	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
232	C₄²	#10625329	1	2	yes	—
233	C₄²	#10625334	1	2	yes	—
234	C₄²	#10625335	1	2	yes	—
235	C₄²	#10625336	1	2	yes	—
236	C₄²	#10625337	1	2	yes	—
237	C₄²	Q₈×D₄₇	1	2	yes	—
238	C₄²	#10625354	1	2	yes	—
239	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
240	Q₈	—	4	8	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
254	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
255	C₂×C₄	#13093945	1	2	yes	—
256	C₂×C₄	#13093945	1	2	yes	—
257	C₂×C₄	#13093948	1	2	yes	—
258	C₂×C₄	#13093948	1	2	yes	—
259	C₂×C₄	#13093951	1	2	yes	—
260	C₂×C₄	#13093951	1	2	yes	—
261	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
262	C₂×C₄	#13094000	1	2	yes	—
263	C₂×C₄	#13094013	1	1	yes	characteristic
264	C₂×C₄	#13094022	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
266	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
267	C₂×C₄	#13094031	1	2	yes	—
268	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
269	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
270	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
271	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
272	C₈	—	94	8	no	—
273	C₈	—	376	2	no	—
274	C₂²	#15689030	1	1	yes	characteristic
275	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
276	C₂²	#15689090	1	1	yes	characteristic
277	C₂²	C₂×#13093945	1	1	yes	characteristic
278	C₂²	C₂×#13093948	1	1	yes	characteristic
279	C₂²	C₂×#13093951	1	1	yes	characteristic
280	C₂²	C₂×#13093951	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	4	no	—
285	C₄	—	8	2	no	—
286	C₄	D₄⋊(C₄₇⋊C₈)	1	4	yes	—
287	C₄	Q₈⋊(C₄₇⋊C₈)	1	4	yes	—
288	C₄	Q₈×C₄₇⋊C₈	1	2	yes	—
289	C₄	#15689254	1	2	yes	—
290	C₄	#15689471	1	1	yes	characteristic
291	C₄	#15689475	1	1	yes	characteristic
292	C₄	#15689519	1	1	yes	characteristic
293	C₄	#15689526	1	1	yes	characteristic
294	C₂	C₂×D₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
295	C₂	C₂×Q₈⋊(C₄₇⋊C₈)	1	1	yes	characteristic
296	C₂	#20241839	1	1	yes	characteristic
297	C₂	#20241843	1	1	yes	characteristic
298	C₂	#20244325	1	1	yes	characteristic
299	C₂	#20244350	1	1	yes	characteristic
300	C₂	#20244366	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	C₄×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#9870
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
28. C₄×C₄₇⋊C₈	Q₈	semidirect
31. C₄⋊Q₈×C₄₇	C₈	semidirect
170. C₄₇	#9870	semidirect
175. C₄⋊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
#20243540	2
C₄²×C₄₇⋊C₈	2
#20241229	2
C₄×C₄⋊Q₈×C₄₇	2
#9870	47