Subgroups — #26124705

Order 12032.

622 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251978	C₂	1	1	yes	maximal, characteristic
2	#20243126	C₂	1	2	yes	maximal
3	C₂×#15688883	C₂	1	2	yes	maximal
4	C₂×C₄₇⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	2	yes	—
8	#15689165	—	2	4	no	—
9	#15689161	—	2	2	no	—
10	#15688891	—	2	4	no	—
11	#15688883	—	2	4	no	—
12	C₄₇⋊#338	C₄	1	4	yes	complemented
13	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	—	2	2	no	—
17	C₂×#10625329	—	2	2	no	—
18	C₂×#10625329	—	2	2	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	C₂×C₄	1	4	yes	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	#13093983	—	2	2	no	—
25	#13093983	—	2	2	no	—
26	#13093983	—	2	2	no	—
27	#13093983	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	D₄	1	4	yes	—
29	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	#13093958	—	2	4	no	—
32	#13093956	—	2	4	no	—
33	C₂³×C₄×C₄₇	—	2	1	no	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₄₇	—	2	2	no	—
36	C₂×M₁₆×C₄₇	—	2	2	no	—
37	C₂×M₁₆×C₄₇	—	2	1	no	—
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	—	2	2	no	—
40	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
42	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
43	C₂²×C₈×C₄₇	—	2	1	no	—
44	C₂²×C₈×C₄₇	D₄	1	2	yes	—
45	C₂×C₄²×C₄₇	—	2	2	no	—
46	C₂×C₄²×C₄₇	—	2	2	no	—
47	C₂×C₄²×C₄₇	—	2	1	no	—
48	C₂×C₄²×C₄₇	D₄	1	2	yes	—
49	C₂⁴×C₄₇	#41	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	4	4	no	—
52	C₂²×C₄×C₄₇	—	4	2	no	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
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	#41	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂²×Dic₄₇	—	2	4	no	—
64	M₁₆×C₄₇	—	2	4	no	—
65	M₁₆×C₄₇	—	2	4	no	—
66	M₁₆×C₄₇	—	2	4	no	—
67	M₁₆×C₄₇	—	4	2	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	4	1	no	—
72	C₂×C₈×C₄₇	—	4	1	no	—
73	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
74	C₄²×C₄₇	—	2	4	no	—
75	C₄²×C₄₇	—	4	1	no	—
76	C₄²×C₄₇	—	4	1	no	—
77	C₂²⋊Dic₄₇	—	4	2	no	—
78	C₂²⋊Dic₄₇	—	4	2	no	—
79	C₄⋊Dic₄₇	—	2	4	no	—
80	C₄⋊Dic₄₇	—	2	4	no	—
81	C₄×Dic₄₇	—	4	2	no	—
82	C₄×Dic₄₇	—	4	2	no	—
83	#10625329	—	2	4	no	—
84	#10625329	—	2	4	no	—
85	#10625329	—	4	2	no	—
86	#10625329	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂³×C₄₇	—	4	2	no	—
91	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₄₇	#121	1	1	yes	characteristic
93	C₂³×C₄₇	#123	1	2	yes	—
94	C₂³×C₄₇	#125	1	1	yes	characteristic
95	C₂³×C₄₇	#126	1	2	yes	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	—	4	8	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#117	1	2	yes	—
102	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
105	C₂×C₄×C₄₇	#121	1	2	yes	—
106	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#123	1	2	yes	—
111	C₂×C₄×C₄₇	#124	1	2	yes	—
112	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#125	1	2	yes	—
115	C₂×C₄×C₄₇	#125	1	2	yes	—
116	C₂×C₄×C₄₇	#126	1	2	yes	—
117	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×Dic₄₇	—	4	4	no	—
119	C₂×Dic₄₇	—	8	4	no	—
120	C₈×C₄₇	—	2	4	no	—
121	C₈×C₄₇	—	4	2	no	—
122	C₄₇⋊C₈	—	4	4	no	—
123	#11294	—	47	1	no	maximal, Sylow
124	C₂²×C₄₇	—	2	4	no	—
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	4	2	no	—
127	C₂²×C₄₇	—	4	1	no	—
128	C₂²×C₄₇	—	4	1	no	—
129	C₂²×C₄₇	#387	1	2	yes	—
130	C₂²×C₄₇	#389	1	1	yes	characteristic
131	C₂²×C₄₇	#394	1	1	yes	characteristic
132	C₂²×C₄₇	#397	1	1	yes	characteristic
133	C₂²×C₄₇	#400	1	2	yes	—
134	C₄×C₄₇	—	4	4	no	—
135	C₄×C₄₇	#382	1	1	yes	characteristic
136	C₄×C₄₇	#385	1	1	yes	characteristic
137	C₄×C₄₇	#388	1	2	yes	—
138	C₄×C₄₇	#393	1	1	yes	characteristic
139	C₄×C₄₇	#398	1	2	yes	—
140	C₄×C₄₇	#400	1	1	yes	characteristic
141	Dic₄₇	—	8	4	no	—
142	#2911	—	47	1	no	—
143	#1742	—	47	2	no	—
144	C₂×#339	—	47	2	no	—
145	C₂×#338	—	47	1	no	—
146	#1728	D₄₇	1	1	yes	characteristic
147	C₂×C₄₇	—	2	1	no	—
148	C₂×C₄₇	—	2	2	no	—
149	C₂×C₄₇	—	2	1	no	—
150	C₂×C₄₇	—	2	2	no	—
151	C₂×C₄₇	#1442	1	1	yes	characteristic
152	C₂×C₄₇	#2050	1	2	yes	—
153	C₂²×M₁₆	—	47	2	no	—
154	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
155	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
156	C₂×#117	—	47	2	no	—
157	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
158	#428	—	94	4	no	—
159	#427	—	94	2	no	—
160	#343	—	94	4	no	—
161	#339	—	94	4	no	—
162	#338	—	47	4	no	—
163	#336	—	2	2	no	—
164	C₄₇	#11294	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	—	2	1	no	—
166	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	47	4	no	—
173	C₂×M₁₆	—	94	2	no	—
174	C₂×M₁₆	—	94	2	no	—
175	C₂×M₁₆	—	94	2	no	—
176	C₂×M₁₆	—	94	2	no	—
177	C₂×M₁₆	—	94	2	no	—
178	C₂×M₁₆	Dic₉₄	1	2	yes	—
179	C₂×M₁₆	Dic₉₄	1	2	yes	—
180	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	94	2	no	—
183	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
184	#117	—	94	2	no	—
185	#117	—	94	2	no	—
186	#117	—	94	2	no	—
187	#117	—	94	2	no	—
188	C₂×C₄⋊C₄	—	47	4	no	—
189	C₂×C₂²⋊C₄	—	94	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	94	2	no	—
194	C₂×C₄²	D₁₈₈	1	2	yes	—
195	#107	—	94	4	no	—
196	#106	—	94	4	no	—
197	C₂⁴	#10625350	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	94	4	no	—
202	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
203	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
206	C₂²×C₄	#10625350	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₄₇	1	2	yes	—
208	C₂²×C₄	#10625352	1	2	yes	—
209	C₂²×C₄	#10625352	1	2	yes	—
210	C₂²×C₄	#10625354	1	2	yes	—
211	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	94	4	no	—
217	M₁₆	—	94	4	no	—
218	M₁₆	—	188	2	no	—
219	M₁₆	—	188	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	—	188	2	no	—
227	C₂×C₈	—	188	2	no	—
228	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
229	C₄⋊C₄	—	94	4	no	—
230	C₄⋊C₄	—	94	4	no	—
231	C₂²⋊C₄	—	188	2	no	—
232	C₂²⋊C₄	—	188	2	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	188	2	no	—
237	C₄²	—	188	2	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₄×Dic₉₄	1	1	yes	characteristic
240	C₂³	#13093998	1	1	yes	characteristic
241	C₂³	#13094009	1	2	yes	—
242	C₂³	#13094016	1	1	yes	characteristic
243	C₂³	#13094017	1	2	yes	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	376	4	no	—
249	C₂×C₄	—	188	4	no	—
250	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
251	C₂×C₄	#13093982	1	2	yes	—
252	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
253	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
254	C₂×C₄	#13093992	1	1	yes	characteristic
255	C₂×C₄	#13093999	1	2	yes	—
256	C₂×C₄	#13094001	1	2	yes	—
257	C₂×C₄	#13094001	1	2	yes	—
258	C₂×C₄	#13094003	1	1	yes	characteristic
259	C₂×C₄	#13094005	1	1	yes	characteristic
260	C₂×C₄	#13094008	1	2	yes	—
261	C₂×C₄	#13094010	1	2	yes	—
262	C₂×C₄	#13094011	1	1	yes	characteristic
263	C₂×C₄	#13094013	1	1	yes	characteristic
264	C₂×C₄	#13094015	1	2	yes	—
265	C₂×C₄	#13094015	1	2	yes	—
266	C₂×C₄	#13094019	1	2	yes	—
267	C₂×C₄	#13094024	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	4	2	no	—
270	C₈	—	188	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	#15689010	1	2	yes	—
277	C₂²	#15689021	1	1	yes	characteristic
278	C₂²	#15689040	1	1	yes	characteristic
279	C₂²	#15689048	1	1	yes	characteristic
280	C₂²	#15689065	1	2	yes	—
281	C₄	—	4	4	no	—
282	C₄	—	376	4	no	—
283	C₄	#15688988	1	1	yes	characteristic
284	C₄	#15688995	1	1	yes	characteristic
285	C₄	#15689015	1	2	yes	—
286	C₄	#15689035	1	1	yes	characteristic
287	C₄	#15689055	1	2	yes	—
288	C₄	#15689062	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	#20241644	1	1	yes	characteristic
294	C₂	#20244675	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13094013
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1728	C₂
socle	C₂²×C₄₇	#394
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1728	D₄₇
hypocenter	C₄₇	#11294
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
12. C₄₇⋊#338	C₄	semidirect
164. C₄₇	#11294	semidirect

Maximal subgroups

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

subgroup	index
#20251978	2
#20243126	2
C₂×#15688883	2
C₂×C₄₇⋊#338	2
C₄₇×#1728	2
#11294	47