Subgroups — #26316143

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251751	C₂	1	1	yes	maximal, characteristic
2	#20244539	C₂	1	4	yes	maximal
3	#20244191	C₂	1	2	yes	maximal
4	#20243515	C₂	1	2	yes	maximal
5	#20243403	C₂	1	4	yes	maximal
6	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13094016	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×#13093948	C₂²	1	8	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
12	#15689021	—	2	2	no	—
13	#15689021	—	2	2	no	—
14	#15688993	C₂²	1	2	yes	—
15	#15688989	C₂²	1	2	yes	—
16	#15688885	C₂²	1	8	yes	—
17	#15688878	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
20	C₄₇×#391	C₂²	1	2	yes	—
21	C₄₇×#389	—	2	2	no	—
22	C₄₇×#384	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
24	C₄³×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
26	#13094016	—	4	4	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
30	C₂×Dic₄₇⋊C₄	—	2	4	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
32	#13093948	—	4	8	no	—
33	C₄×C₄₇⋊C₈	—	2	4	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
37	C₄⋊Q₈×C₄₇	—	2	2	no	—
38	C₄⋊Q₈×C₄₇	Q₈	1	4	yes	—
39	C₄×Q₈×C₄₇	—	2	8	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	D₄	1	2	yes	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	#95×C₄₇	—	2	4	no	—
51	C₂×Q₈×C₄₇	—	2	2	no	—
52	C₂×Q₈×C₄₇	—	2	4	no	—
53	C₂×Q₈×C₄₇	—	2	4	no	—
54	C₂×Q₈×C₄₇	C₂×Q₈	1	4	yes	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₄₇	—	2	4	no	—
63	C₄⋊C₄×C₄₇	—	2	8	no	—
64	C₄⋊C₄×C₄₇	—	2	4	no	—
65	C₄⋊C₄×C₄₇	—	4	4	no	—
66	C₄⋊C₄×C₄₇	—	4	4	no	—
67	C₄²×C₄₇	—	2	4	no	—
68	C₄²×C₄₇	—	2	4	no	—
69	C₄²×C₄₇	—	2	2	no	—
70	C₄²×C₄₇	—	4	4	no	—
71	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
72	C₄²×C₄₇	#41	1	2	yes	—
73	C₄²×C₄₇	#41	1	4	yes	—
74	C₄⋊Dic₄₇	—	2	8	no	—
75	C₄⋊Dic₄₇	—	4	4	no	—
76	C₄⋊Dic₄₇	—	4	4	no	—
77	Dic₄₇⋊C₄	—	4	8	no	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	2	4	no	—
81	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₄₇	—	2	8	no	—
83	Q₈×C₄₇	—	4	4	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	4	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	2	no	—
90	C₂×C₄×C₄₇	—	4	2	no	—
91	C₂×C₄×C₄₇	—	4	2	no	—
92	C₂×C₄×C₄₇	—	4	2	no	—
93	C₂×C₄×C₄₇	—	4	4	no	—
94	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
95	C₂×C₄×C₄₇	#124	1	2	yes	—
96	C₂×C₄×C₄₇	#124	1	2	yes	—
97	C₂×C₄×C₄₇	#135	1	4	yes	—
98	C₂×C₄×C₄₇	#137	1	4	yes	—
99	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
103	C₂×Dic₄₇	—	4	4	no	—
104	C₂×Dic₄₇	—	4	4	no	—
105	C₂×Dic₄₇	—	4	4	no	—
106	C₄₇⋊C₈	—	4	4	no	—
107	#29144	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
110	C₂²×C₄₇	#535	1	1	yes	characteristic
111	C₂²×C₄₇	C₂×#135	1	2	yes	—
112	C₂²×C₄₇	C₂×#137	1	2	yes	—
113	C₄×C₄₇	—	2	2	no	—
114	C₄×C₄₇	—	2	2	no	—
115	C₄×C₄₇	—	4	4	no	—
116	C₄×C₄₇	—	4	2	no	—
117	C₄×C₄₇	—	8	2	no	—
118	C₄×C₄₇	#479	1	8	yes	—
119	C₄×C₄₇	#554	1	2	yes	—
120	C₄×C₄₇	#555	1	2	yes	—
121	Dic₄₇	—	8	4	no	—
122	#2859	—	47	1	no	—
123	C₄×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
124	#2026	—	47	4	no	—
125	#1956	—	47	2	no	—
126	#1834	—	47	2	no	—
127	#1805	—	47	4	no	—
128	C₄×C₄⋊C₈	—	47	1	no	—
129	C₂×C₄₇	#2563	1	1	yes	characteristic
130	C₂×C₄₇	C₂×#479	1	2	yes	—
131	C₂×C₄₇	#3131	1	1	yes	characteristic
132	C₂×C₄₇	#3139	1	1	yes	characteristic
133	C₂×C₄₇	#3210	1	2	yes	—
134	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
135	C₂×#125	—	47	2	no	—
136	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
137	C₂×C₄⋊C₈	—	47	2	no	—
138	C₂×#103	—	47	8	no	—
139	C₂×C₄×C₈	—	47	2	no	—
140	#391	C₂×D₄₇	1	2	yes	—
141	#389	—	2	2	no	—
142	#389	—	94	2	no	—
143	#389	—	94	2	no	—
144	#384	—	47	2	no	—
145	#384	C₂×D₄₇	1	1	yes	characteristic
146	#382	—	47	2	no	—
147	C₄×C₄⋊C₄	—	2	2	no	—
148	C₄³	C₂×D₄₇	1	1	yes	characteristic
149	#340	—	47	8	no	—
150	#336	—	47	2	no	—
151	C₄₇	#29144	1	1	yes	characteristic, complemented, Sylow
152	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
153	C₂²×C₈	—	47	4	no	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	Dic₉₄	1	4	yes	—
156	#125	—	188	4	no	—
157	C₄×Q₈	—	2	8	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	47	4	no	—
162	C₂×C₄⋊C₄	—	94	4	no	—
163	C₂×C₄⋊C₄	—	94	2	no	—
164	C₂×C₄⋊C₄	—	94	2	no	—
165	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
168	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
172	C₄⋊C₈	—	94	4	no	—
173	#103	—	188	8	no	—
174	C₄×C₈	—	94	4	no	—
175	#95	—	2	4	no	—
176	#95	—	94	2	no	—
177	#95	—	94	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	C₂×Dic₉₄	1	4	yes	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	94	4	no	—
185	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
186	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
189	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
190	C₂×C₈	—	94	4	no	—
191	C₂×C₈	—	94	4	no	—
192	C₂×C₈	—	94	4	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	94	8	no	—
198	C₄⋊C₄	—	188	8	no	—
199	C₄⋊C₄	—	188	4	no	—
200	C₄⋊C₄	—	188	4	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	4	4	no	—
205	C₄²	C₂×Dic₉₄	1	2	yes	—
206	C₄²	#10625350	1	2	yes	—
207	C₄²	#10625352	1	4	yes	—
208	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
209	Q₈	—	2	8	no	—
210	Q₈	—	4	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	188	4	no	—
222	C₂×C₄	—	188	4	no	—
223	C₂×C₄	—	188	4	no	—
224	C₂×C₄	#13094003	1	4	yes	—
225	C₂×C₄	#13094011	1	2	yes	—
226	C₂×C₄	#13094011	1	2	yes	—
227	C₂×C₄	#13094044	1	4	yes	—
228	C₂×C₄	#13094055	1	4	yes	—
229	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
230	C₂×C₄	C₂×#10625352	1	2	yes	—
231	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
232	C₂×C₄	#13094070	1	2	yes	—
233	C₈	—	188	4	no	—
234	C₂²	C₂×#13094003	1	1	yes	characteristic
235	C₂²	C₂×#13094011	1	1	yes	characteristic
236	C₂²	#15689687	1	1	yes	characteristic
237	C₂²	C₂×#13094044	1	2	yes	—
238	C₂²	C₂×#13094055	1	2	yes	—
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	4	no	—
243	C₄	—	8	2	no	—
244	C₄	—	376	4	no	—
245	C₄	#15689430	1	8	yes	—
246	C₄	#15689830	1	2	yes	—
247	C₄	#15689838	1	2	yes	—
248	C₂	#20248561	1	1	yes	characteristic
249	C₂	C₂×#15689430	1	2	yes	—
250	C₂	#20253710	1	1	yes	characteristic
251	C₂	#20253768	1	1	yes	characteristic
252	C₂	#20254340	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₄×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#29144
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
151. C₄₇	#29144	semidirect

Maximal subgroups

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

subgroup	index
#20251751	2
#20244539	2
#20244191	2
#20243515	2
#20243403	2
C₄×C₄⋊(C₄₇⋊C₈)	2
C₄×C₄⋊Q₈×C₄₇	2
#29144	47