Subgroups — #26215369

Order 12032.

798 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689254	C₂	1	2	yes	maximal
2	#20243263	C₂	1	2	yes	maximal
3	#20243263	C₂	1	2	yes	maximal
4	C₄⋊C₄×C₄₇⋊C₈	C₂	1	2	yes	maximal
5	#20243245	C₂	1	2	yes	maximal
6	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#20243140	C₂	1	2	yes	maximal
8	#20243137	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#2295	C₂	1	1	yes	maximal, characteristic
10	#15689254	C₄	1	16	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×#13093935	C₂²	1	2	yes	—
15	C₂×#13093935	C₂²	1	2	yes	—
16	C₂×#13093935	C₂²	1	2	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
21	#15688878	C₂²	1	2	yes	—
22	#15688878	C₂²	1	1	yes	characteristic
23	#15688878	C₂²	1	2	yes	—
24	#15688878	C₂²	1	2	yes	—
25	#15688878	C₂²	1	2	yes	—
26	#15688878	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#391	C₄	1	2	yes	—
29	C₄₇×#389	C₄	1	2	yes	—
30	C₄₇×#387	C₄	1	4	yes	—
31	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×#376	C₂²	1	2	yes	—
34	C₄³×C₄₇	C₂²	1	1	yes	characteristic
35	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
39	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
40	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
41	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	8	yes	—
42	#13093935	—	4	2	no	—
43	#13093935	C₂×C₄	1	8	yes	—
44	#13093935	C₂×C₄	1	8	yes	—
45	C₄×C₄₇⋊C₈	—	4	1	no	—
46	C₄×C₄₇⋊C₈	—	4	2	no	—
47	C₄×C₄₇⋊C₈	—	4	1	no	—
48	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
49	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
50	C₄×Q₈×C₄₇	C₂×C₄	1	8	yes	—
51	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	#95×C₄₇	C₂×C₄	1	4	yes	—
63	#95×C₄₇	C₂×C₄	1	4	yes	—
64	#95×C₄₇	C₂×C₄	1	2	yes	—
65	#95×C₄₇	C₂×C₄	1	4	yes	—
66	C₂×Q₈×C₄₇	—	4	2	no	—
67	C₂×Q₈×C₄₇	C₄²	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄×C₄₇	—	2	8	no	—
77	C₄⋊C₄×C₄₇	—	4	2	no	—
78	C₄⋊C₄×C₄₇	C₄²	1	8	yes	—
79	C₄⋊C₄×C₄₇	C₄²	1	4	yes	—
80	C₄²×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	—	2	1	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	1	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	4	2	no	—
87	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	—	2	2	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
99	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
100	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
101	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
102	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
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	4	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
117	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
119	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₄₇	#131	1	4	yes	—
121	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
125	C₄₇⋊C₈	—	2	4	no	—
126	C₄₇⋊C₈	—	2	4	no	—
127	C₄₇⋊C₈	—	2	4	no	—
128	C₄₇⋊C₈	—	2	4	no	—
129	#20575	—	47	1	no	maximal, Sylow
130	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
131	C₂²×C₄₇	#519	1	1	yes	characteristic
132	C₂²×C₄₇	#520	1	1	yes	characteristic
133	C₂²×C₄₇	C₂×#131	1	2	yes	—
134	C₂²×C₄₇	#568	1	2	yes	—
135	C₄×C₄₇	—	2	2	no	—
136	C₄×C₄₇	—	2	4	no	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	—	2	4	no	—
140	C₄×C₄₇	—	4	2	no	—
141	C₄×C₄₇	—	4	4	no	—
142	C₄×C₄₇	#519	1	2	yes	—
143	C₄×C₄₇	#520	1	2	yes	—
144	C₂×#446	—	47	2	no	—
145	#2295	D₄₇	1	1	yes	characteristic
146	#1771	—	47	2	no	—
147	#1771	—	47	2	no	—
148	C₈×C₄⋊C₄	—	47	2	no	—
149	#1768	—	47	2	no	—
150	C₄×C₄⋊C₈	—	47	1	no	—
151	#1746	—	47	2	no	—
152	#1745	—	47	1	no	—
153	C₂×C₄₇	#2276	1	1	yes	characteristic
154	C₂×C₄₇	#2873	1	1	yes	characteristic
155	C₂×C₄₇	C₄×#131	1	1	yes	characteristic
156	C₂×C₄₇	#2985	1	2	yes	—
157	C₂×C₄₇	#2992	1	2	yes	—
158	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
159	#446	—	47	16	no	—
160	C₂×C₄⋊C₈	—	47	2	no	—
161	C₂×C₄⋊C₈	—	47	2	no	—
162	C₂×C₄⋊C₈	—	47	2	no	—
163	C₂×#97	—	47	2	no	—
164	C₂×#97	—	47	2	no	—
165	C₂×#97	—	47	2	no	—
166	C₂×C₄×C₈	—	47	2	no	—
167	C₂×C₄×C₈	—	47	1	no	—
168	C₂×C₄×C₈	—	47	1	no	—
169	C₂×C₄×C₈	—	47	2	no	—
170	#391	Dic₄₇	1	2	yes	—
171	#389	Dic₄₇	1	2	yes	—
172	#387	Dic₄₇	1	4	yes	—
173	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
174	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
175	#376	C₂×D₄₇	1	2	yes	—
176	C₄³	C₂×D₄₇	1	1	yes	characteristic
177	#336	—	47	1	no	—
178	#336	—	47	1	no	—
179	#336	—	47	2	no	—
180	#336	—	47	2	no	—
181	#336	—	47	2	no	—
182	#336	—	47	2	no	—
183	C₄₇	#20575	1	1	yes	characteristic, complemented, Sylow
184	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
185	C₂²×C₈	—	47	2	no	—
186	C₂²×C₈	—	47	2	no	—
187	C₂²×C₈	—	47	2	no	—
188	C₂²×C₈	—	47	2	no	—
189	C₄×Q₈	C₄×D₄₇	1	8	yes	—
190	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
191	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
192	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
193	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
194	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
195	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
196	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
197	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
199	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
201	C₄⋊C₈	—	47	8	no	—
202	C₄⋊C₈	—	47	8	no	—
203	C₄⋊C₈	—	47	8	no	—
204	#97	—	47	8	no	—
205	#97	—	47	8	no	—
206	#97	—	188	2	no	—
207	C₄×C₈	—	47	8	no	—
208	C₄×C₈	—	188	1	no	—
209	C₄×C₈	—	188	1	no	—
210	C₄×C₈	—	188	2	no	—
211	#95	C₂×Dic₄₇	1	4	yes	—
212	#95	C₂×Dic₄₇	1	4	yes	—
213	#95	C₂×Dic₄₇	1	2	yes	—
214	#95	C₂×Dic₄₇	1	4	yes	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	C₄×Dic₄₇	1	4	yes	—
217	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
219	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
222	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
224	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
225	C₂×C₈	—	47	4	no	—
226	C₂×C₈	—	47	4	no	—
227	C₂×C₈	—	47	4	no	—
228	C₂×C₈	—	47	4	no	—
229	C₂×C₈	—	94	2	no	—
230	C₂×C₈	—	94	2	no	—
231	C₂×C₈	—	94	2	no	—
232	C₂×C₈	—	94	2	no	—
233	C₂×C₈	—	94	2	no	—
234	C₂×C₈	—	94	2	no	—
235	C₂×C₈	—	94	2	no	—
236	C₂×C₈	—	94	2	no	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	C₄×Dic₄₇	1	8	yes	—
240	C₄⋊C₄	C₄×Dic₄₇	1	4	yes	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₂×C₄×D₄₇	1	4	yes	—
249	C₄²	C₂×C₄×D₄₇	1	4	yes	—
250	C₄²	C₂×C₄×D₄₇	1	4	yes	—
251	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
252	Q₈	—	4	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
266	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
267	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
268	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
269	C₂×C₄	#13094031	1	4	yes	—
270	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
271	C₂×C₄	#13094070	1	1	yes	characteristic
272	C₂×C₄	#13094071	1	1	yes	characteristic
273	C₂×C₄	#13094072	1	2	yes	—
274	C₈	—	94	4	no	—
275	C₈	—	94	4	no	—
276	C₈	—	94	4	no	—
277	C₈	—	94	4	no	—
278	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
279	C₂²	#15689601	1	1	yes	characteristic
280	C₂²	#15689606	1	1	yes	characteristic
281	C₂²	C₂×#13094031	1	2	yes	—
282	C₂²	#15689890	1	2	yes	—
283	C₄	—	2	4	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	4	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	4	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	4	no	—
290	C₄	#15689599	1	2	yes	—
291	C₄	#15689605	1	2	yes	—
292	C₂	#20245753	1	1	yes	characteristic
293	C₂	#20251807	1	1	yes	characteristic
294	C₂	C₄×#13094031	1	1	yes	characteristic
295	C₂	#20252483	1	2	yes	—
296	C₂	#20252527	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#2295	C₂
socle, prime radical	C₂³×C₄₇	C₂³×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#2295	D₄₇
hypocenter	C₄₇	#20575
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
10. #15689254	C₄	semidirect
183. C₄₇	#20575	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689254	2
#20243263	2
C₄⋊C₄×C₄₇⋊C₈	2
#20243245	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243140	2
#20243137	2
C₄₇×#2295	2
#20575	47