Subgroups — #26125005

Order 12032.

698 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	#20251952	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689149	C₂	1	2	yes	maximal
3	C₂×#15688895	C₂	1	2	yes	maximal
4	#20243121	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	2	yes	—
8	#15689150	—	2	4	no	—
9	#15689149	—	2	4	no	—
10	#15689149	—	2	4	no	—
11	#15689147	—	2	2	no	—
12	C₂×Dic₄₇⋊C₈	—	2	4	no	—
13	#15688895	—	2	4	no	—
14	#15688890	C₄	1	4	yes	—
15	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#336	—	2	2	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
26	#13093983	—	2	2	no	—
27	#13093983	—	2	2	no	—
28	#13093983	D₄	1	4	yes	—
29	#13093983	Q₈	1	4	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
34	Dic₄₇⋊C₈	—	2	16	no	—
35	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₄₇	—	2	1	no	—
37	C₂×M₁₆×C₄₇	—	2	2	no	—
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
40	C₂²×C₈×C₄₇	—	2	1	no	—
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	—	2	1	no	—
44	C₂²×C₈×C₄₇	—	2	1	no	—
45	C₂²×C₈×C₄₇	D₄	1	2	yes	—
46	#117×C₄₇	—	2	2	no	—
47	#117×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	Q₈	1	2	yes	—
49	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	D₄	1	2	yes	—
51	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	C₂²×Dic₄₇	—	2	4	no	—
64	M₁₆×C₄₇	—	2	4	no	—
65	M₁₆×C₄₇	—	4	2	no	—
66	C₂×C₈×C₄₇	—	2	4	no	—
67	C₂×C₈×C₄₇	—	2	2	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	4	1	no	—
70	C₂×C₈×C₄₇	—	4	2	no	—
71	C₂×C₈×C₄₇	—	4	1	no	—
72	C₂×C₈×C₄₇	—	4	1	no	—
73	C₂×C₈×C₄₇	—	4	1	no	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₂²⋊C₄×C₄₇	—	2	4	no	—
76	C₄²×C₄₇	—	2	2	no	—
77	C₄²×C₄₇	—	2	2	no	—
78	C₂²⋊Dic₄₇	—	2	4	no	—
79	C₂²⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
80	C₄⋊Dic₄₇	—	2	4	no	—
81	C₄⋊Dic₄₇	—	2	4	no	—
82	C₄×Dic₄₇	—	2	4	no	—
83	C₄×Dic₄₇	—	2	4	no	—
84	#10625329	—	4	2	no	—
85	#10625329	—	4	2	no	—
86	#10625329	—	4	2	no	—
87	#10625329	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	4	no	—
91	C₂³×C₄₇	—	4	2	no	—
92	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₄₇	C₄×D₄	1	2	yes	—
95	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
96	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	4	4	no	—
99	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
100	C₂×C₄×C₄₇	#120	1	2	yes	—
101	C₂×C₄×C₄₇	#121	1	2	yes	—
102	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#123	1	2	yes	—
110	C₂×C₄×C₄₇	#125	1	2	yes	—
111	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×Dic₄₇	—	2	8	no	—
118	C₂×Dic₄₇	—	4	8	no	—
119	C₈×C₄₇	—	4	2	no	—
120	C₈×C₄₇	—	4	2	no	—
121	C₄₇⋊C₈	—	4	4	no	—
122	#11345	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	4	no	—
124	C₂²×C₄₇	—	2	2	no	—
125	C₂²×C₄₇	—	4	2	no	—
126	C₂²×C₄₇	—	4	1	no	—
127	C₂²×C₄₇	—	4	1	no	—
128	C₂²×C₄₇	#384	1	1	yes	characteristic
129	C₂²×C₄₇	#385	1	1	yes	characteristic
130	C₂²×C₄₇	#385	1	2	yes	—
131	C₂²×C₄₇	#389	1	2	yes	—
132	C₂²×C₄₇	#389	1	1	yes	characteristic
133	C₄×C₄₇	—	4	4	no	—
134	C₄×C₄₇	#392	1	1	yes	characteristic
135	C₄×C₄₇	#395	1	1	yes	characteristic
136	C₄×C₄₇	#395	1	2	yes	—
137	C₄×C₄₇	#398	1	1	yes	characteristic
138	C₄×C₄₇	#398	1	2	yes	—
139	C₄×C₄₇	#399	1	1	yes	characteristic
140	Dic₄₇	—	4	8	no	—
141	#2905	—	47	1	no	—
142	C₂×#424	—	47	2	no	—
143	C₂×#344	—	47	2	no	—
144	#1740	—	47	1	no	—
145	#1729	D₄₇	1	1	yes	characteristic
146	C₂×C₄₇	—	2	2	no	—
147	C₂×C₄₇	—	2	1	no	—
148	C₂×C₄₇	—	2	2	no	—
149	C₂×C₄₇	—	2	1	no	—
150	C₂×C₄₇	#1444	1	1	yes	characteristic
151	C₂×C₄₇	#1868	1	2	yes	—
152	C₂²×M₁₆	—	47	2	no	—
153	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
154	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
155	C₂×#117	—	47	2	no	—
156	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
157	#424	—	94	4	no	—
158	#424	—	94	4	no	—
159	#424	—	94	4	no	—
160	#423	—	94	2	no	—
161	C₂×C₄⋊C₈	—	94	4	no	—
162	#344	—	94	4	no	—
163	#342	—	47	4	no	—
164	#336	—	2	2	no	—
165	C₄₇	#11345	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	94	2	no	—
171	C₂×M₁₆	—	94	2	no	—
172	C₂×M₁₆	—	94	2	no	—
173	C₂×M₁₆	—	94	2	no	—
174	C₂×M₁₆	—	94	2	no	—
175	C₂×M₁₆	—	94	2	no	—
176	C₂×M₁₆	Dic₉₄	1	2	yes	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	47	4	no	—
183	C₂²×C₈	D₁₈₈	1	2	yes	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	47	4	no	—
187	#117	—	47	4	no	—
188	#117	—	94	2	no	—
189	#117	—	94	2	no	—
190	C₂×C₄⋊C₄	—	94	2	no	—
191	C₂×C₄⋊C₄	Dic₉₄	1	2	yes	—
192	C₂×C₂²⋊C₄	—	47	4	no	—
193	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
194	C₂×C₄²	—	94	2	no	—
195	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
196	C₄⋊C₈	—	94	8	no	—
197	C₄⋊C₈	—	94	16	no	—
198	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
199	C₂²×C₄	—	94	4	no	—
200	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
201	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
202	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
204	C₂²×C₄	#10625350	1	1	yes	characteristic
205	C₂²×C₄	#10625350	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₄₇	1	2	yes	—
207	C₂²×C₄	D₄×D₄₇	1	2	yes	—
208	C₂²×C₄	#10625352	1	2	yes	—
209	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
210	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	188	2	no	—
214	M₁₆	—	188	2	no	—
215	M₁₆	—	188	2	no	—
216	M₁₆	—	188	2	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	—	94	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	94	4	no	—
230	C₄⋊C₄	—	94	4	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	47	8	no	—
233	C₂²⋊C₄	—	94	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	94	4	no	—
237	C₄²	—	94	4	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
240	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
241	C₂³	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
242	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
243	C₂³	C₄×Dic₉₄	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	94	8	no	—
247	C₂×C₄	—	188	8	no	—
248	C₂×C₄	#13093992	1	1	yes	characteristic
249	C₂×C₄	#13093994	1	2	yes	—
250	C₂×C₄	#13093997	1	2	yes	—
251	C₂×C₄	#13093998	1	1	yes	characteristic
252	C₂×C₄	#13094001	1	2	yes	—
253	C₂×C₄	#13094001	1	2	yes	—
254	C₂×C₄	#13094002	1	2	yes	—
255	C₂×C₄	#13094003	1	1	yes	characteristic
256	C₂×C₄	#13094004	1	2	yes	—
257	C₂×C₄	#13094005	1	1	yes	characteristic
258	C₂×C₄	#13094006	1	2	yes	—
259	C₂×C₄	#13094014	1	2	yes	—
260	C₂×C₄	#13094016	1	1	yes	characteristic
261	C₂×C₄	#13094020	1	1	yes	characteristic
262	C₂×C₄	#13094022	1	1	yes	characteristic
263	C₂×C₄	#13094023	1	2	yes	—
264	C₂×C₄	#13094023	1	2	yes	—
265	C₂×C₄	#13094024	1	1	yes	characteristic
266	C₈	—	4	2	no	—
267	C₈	—	4	2	no	—
268	C₈	—	188	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	#15688992	1	1	yes	characteristic
275	C₂²	#15688995	1	1	yes	characteristic
276	C₂²	#15688997	1	2	yes	—
277	C₂²	#15689020	1	2	yes	—
278	C₂²	#15689021	1	1	yes	characteristic
279	C₄	—	4	4	no	—
280	C₄	—	188	8	no	—
281	C₄	#15689032	1	1	yes	characteristic
282	C₄	#15689043	1	1	yes	characteristic
283	C₄	#15689044	1	2	yes	—
284	C₄	#15689053	1	1	yes	characteristic
285	C₄	#15689057	1	2	yes	—
286	C₄	#15689059	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	1	no	—
291	C₂	#20241652	1	1	yes	characteristic
292	C₂	#20243673	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093992
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1729	C₂
socle	C₂²×C₄₇	#385
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1729	D₄₇
hypocenter	C₄₇	#11345
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
165. C₄₇	#11345	semidirect

Maximal subgroups

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

subgroup	index
#20251952	2
C₂×#15689149	2
C₂×#15688895	2
#20243121	2
C₄₇×#1729	2
#11345	47