Subgroups — #26108026

Order 12032.

706 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243384	C₂	1	1	yes	maximal, characteristic
2	#20242069	C₂	1	2	yes	maximal
3	#20241817	C₂	1	2	yes	maximal
4	#20241712	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#557	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	#15689147	C₂²	1	2	yes	—
8	C₂×#13093943	—	2	1	no	—
9	#15688892	—	2	1	no	—
10	#15688872	—	2	4	no	—
11	Q₈⋊(C₄₇⋊C₈)	—	2	4	no	—
12	#15688849	—	2	2	no	—
13	C₄₇×#557	—	2	4	no	—
14	C₄₇×#557	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₄₇	C₄	1	2	yes	—
16	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#125×C₄₇	—	2	2	no	—
18	C₂×#125×C₄₇	—	2	1	no	—
19	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
20	C₂×C₄×Q₈×C₄₇	C₄	1	2	yes	—
21	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#10625329	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
27	#13093943	—	4	2	no	—
28	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
29	C₄×C₄₇⋊C₈	—	2	2	no	—
30	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₄₇	—	2	4	no	—
34	C₄⋊Q₈×C₄₇	—	2	4	no	—
35	C₄⋊Q₈×C₄₇	—	2	2	no	—
36	C₄⋊Q₈×C₄₇	D₄	1	4	yes	—
37	#125×C₄₇	—	2	16	no	—
38	#125×C₄₇	—	2	8	no	—
39	C₄×Q₈×C₄₇	—	2	8	no	—
40	C₄×Q₈×C₄₇	—	2	4	no	—
41	C₄×Q₈×C₄₇	—	2	4	no	—
42	C₄×Q₈×C₄₇	—	2	2	no	—
43	C₄×Q₈×C₄₇	—	2	2	no	—
44	C₄×Q₈×C₄₇	—	2	2	no	—
45	C₄×Q₈×C₄₇	D₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
50	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
51	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	—	2	2	no	—
56	C₂×C₄²×C₄₇	—	2	1	no	—
57	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×Q₈×C₄₇	—	2	1	no	—
60	C₂×Q₈×C₄₇	—	2	2	no	—
61	C₂×Q₈×C₄₇	—	2	2	no	—
62	C₂×Q₈×C₄₇	—	2	2	no	—
63	C₂×Q₈×C₄₇	—	4	2	no	—
64	C₂×Q₈×C₄₇	—	4	4	no	—
65	C₂×Q₈×C₄₇	C₂²⋊C₄	1	2	yes	—
66	C₂×Q₈×C₄₇	C₂²⋊C₄	1	4	yes	—
67	C₂×Q₈×C₄₇	SD₁₆	1	2	yes	—
68	C₂×Q₈×C₄₇	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₄₇	—	2	1	no	—
70	C₂²×C₄×C₄₇	—	2	2	no	—
71	C₂²×C₄×C₄₇	—	2	1	no	—
72	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₄₇	—	2	8	no	—
78	C₄⋊C₄×C₄₇	—	2	8	no	—
79	C₄⋊C₄×C₄₇	—	2	8	no	—
80	C₄⋊C₄×C₄₇	—	2	1	no	—
81	C₄⋊C₄×C₄₇	—	2	4	no	—
82	C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₄⋊C₄×C₄₇	—	2	4	no	—
84	C₄⋊C₄×C₄₇	—	2	4	no	—
85	C₄⋊C₄×C₄₇	—	2	2	no	—
86	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₄₇	—	2	8	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	2	2	no	—
90	C₄²×C₄₇	—	2	4	no	—
91	C₄²×C₄₇	C₂×D₄	1	2	yes	—
92	C₄²×C₄₇	C₂×D₄	1	2	yes	—
93	#10625329	—	4	2	no	—
94	#10625329	—	4	2	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	Q₈×C₄₇	—	2	4	no	—
99	Q₈×C₄₇	—	2	4	no	—
100	Q₈×C₄₇	—	2	4	no	—
101	Q₈×C₄₇	—	4	4	no	—
102	Q₈×C₄₇	—	4	2	no	—
103	Q₈×C₄₇	—	4	4	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	4	1	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	#103	1	4	yes	—
117	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₇	#120	1	2	yes	—
120	C₂×C₄×C₄₇	#120	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#137	1	2	yes	—
124	C₄₇⋊C₈	—	8	2	no	—
125	C₄₇⋊C₈	—	8	2	no	—
126	#8392	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	1	no	—
128	C₂²×C₄₇	—	2	1	no	—
129	C₂²×C₄₇	#379	1	1	yes	characteristic
130	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
131	C₂²×C₄₇	#419	1	1	yes	characteristic
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	1	no	—
134	C₄×C₄₇	—	2	4	no	—
135	C₄×C₄₇	—	4	2	no	—
136	C₄×C₄₇	—	4	2	no	—
137	C₄×C₄₇	—	4	2	no	—
138	C₄×C₄₇	—	4	4	no	—
139	C₄×C₄₇	#413	1	2	yes	—
140	C₄×C₄₇	#448	1	2	yes	—
141	C₄×C₄₇	#451	1	2	yes	—
142	C₂×#557	D₄₇	1	1	yes	characteristic
143	#1801	—	47	1	no	—
144	#1548	—	47	2	no	—
145	#1487	—	47	2	no	—
146	#1461	—	47	1	no	—
147	C₂×C₄₇	—	2	1	no	—
148	C₂×C₄₇	—	2	1	no	—
149	C₂×C₄₇	#1785	1	1	yes	characteristic
150	C₂×C₄₇	#1787	1	1	yes	characteristic
151	C₂×C₄₇	#1796	1	1	yes	characteristic
152	#557	—	2	4	no	—
153	#557	—	2	4	no	—
154	C₂×C₄⋊Q₈	Dic₄₇	1	2	yes	—
155	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
156	C₂×#125	—	2	2	no	—
157	C₂×#125	—	2	1	no	—
158	C₂×C₄×Q₈	—	2	1	no	—
159	C₂×C₄×Q₈	Dic₄₇	1	2	yes	—
160	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
161	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
162	#431	—	47	2	no	—
163	#423	—	47	2	no	—
164	C₂×#101	—	94	1	no	—
165	#343	—	94	1	no	—
166	#333	—	94	4	no	—
167	Q₈⋊C₈	—	94	4	no	—
168	#324	—	94	2	no	—
169	C₄₇	#8392	1	1	yes	characteristic, complemented, Sylow
170	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
171	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
172	C₂²×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
173	C₂×M₁₆	—	94	2	no	—
174	C₄⋊Q₈	—	2	4	no	—
175	C₄⋊Q₈	—	2	4	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	C₂²⋊D₄₇	1	4	yes	—
178	#125	—	2	16	no	—
179	#125	—	2	8	no	—
180	C₄×Q₈	—	2	8	no	—
181	C₄×Q₈	—	2	4	no	—
182	C₄×Q₈	—	2	4	no	—
183	C₄×Q₈	—	2	2	no	—
184	C₄×Q₈	—	2	2	no	—
185	C₄×Q₈	—	2	2	no	—
186	C₄×Q₈	C₂²⋊D₄₇	1	4	yes	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
194	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
195	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
199	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
200	C₄⋊C₈	—	94	2	no	—
201	C₄⋊C₈	—	94	2	no	—
202	C₄⋊C₈	—	94	2	no	—
203	#101	—	188	2	no	—
204	C₂²⋊C₈	—	94	2	no	—
205	C₄×C₈	—	94	2	no	—
206	C₂×Q₈	—	2	1	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×Q₈	#10625336	1	2	yes	—
213	C₂×Q₈	#10625337	1	2	yes	—
214	C₂×Q₈	C₂²⋊Dic₄₇	1	2	yes	—
215	C₂×Q₈	C₂²⋊Dic₄₇	1	4	yes	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
220	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
221	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
224	M₁₆	—	188	2	no	—
225	M₁₆	—	188	2	no	—
226	C₂×C₈	—	188	2	no	—
227	C₂×C₈	—	188	2	no	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	C₂²⋊Dic₄₇	1	2	yes	—
238	C₄²	—	2	8	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	4	no	—
242	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
243	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
244	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
245	Q₈	—	2	4	no	—
246	Q₈	—	2	4	no	—
247	Q₈	—	2	4	no	—
248	Q₈	—	4	4	no	—
249	Q₈	—	4	2	no	—
250	Q₈	—	4	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	#13093948	1	4	yes	—
264	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
266	C₂×C₄	#13093993	1	2	yes	—
267	C₂×C₄	#13093993	1	2	yes	—
268	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
269	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
270	C₂×C₄	#13094055	1	2	yes	—
271	C₈	—	376	2	no	—
272	C₈	—	376	2	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	#15688979	1	1	yes	characteristic
276	C₂²	C₂×#13093948	1	1	yes	characteristic
277	C₂²	#15689132	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	4	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	2	no	—
285	C₄	#15689107	1	2	yes	—
286	C₄	#15689262	1	2	yes	—
287	C₄	#15689279	1	2	yes	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	#20243316	1	1	yes	characteristic
291	C₂	#20243323	1	1	yes	characteristic
292	C₂	#20243363	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15688979
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#557	C₂
socle	C₂²×C₄₇	#379
prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#557	D₄₇
hypocenter	C₄₇	#8392
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#557

Complements

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

N	H ≅ G/N	type
169. C₄₇	#8392	semidirect

Maximal subgroups

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

subgroup	index
#20243384	2
#20242069	2
#20241817	2
#20241712	2
C₂×C₄₇×#557	2
#8392	47