Subgroups — #26115147

Order 12032.

486 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20244325	C₂	1	2	yes	maximal
2	C₂×#15688874	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
4	C₂×#15688844	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	#15688878	—	2	4	no	—
10	#15688874	—	2	2	no	—
11	Q₈⋊(C₄₇⋊C₈)	—	2	4	no	—
12	#15688844	—	2	2	no	—
13	C₂²×Q₁₆×C₄₇	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
15	C₄×Q₁₆×C₄₇	—	2	2	no	—
16	C₄×Q₁₆×C₄₇	—	2	2	no	—
17	C₂×#107×C₄₇	C₄	1	1	yes	characteristic
18	C₂×#103×C₄₇	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄×C₄₇⋊C₈	—	2	1	no	—
26	C₄×C₄₇⋊C₈	—	2	1	no	—
27	C₄×C₄₇⋊C₈	—	2	1	no	—
28	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
29	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₄₇	—	2	2	no	—
31	C₂×Q₁₆×C₄₇	—	2	2	no	—
32	C₂×Q₁₆×C₄₇	C₈	1	4	yes	complemented
33	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄×Q₈×C₄₇	—	2	2	no	—
36	C₄×Q₈×C₄₇	—	2	2	no	—
37	C₄×Q₈×C₄₇	—	2	2	no	—
38	C₄×Q₈×C₄₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	#107×C₄₇	—	2	2	no	—
44	#103×C₄₇	—	2	4	no	—
45	C₄×C₈×C₄₇	D₄	1	2	yes	—
46	C₄×C₈×C₄₇	D₄	1	2	yes	—
47	C₂×Q₈×C₄₇	—	2	2	no	—
48	C₂×Q₈×C₄₇	—	2	2	no	—
49	C₂×Q₈×C₄₇	—	2	2	no	—
50	C₂×Q₈×C₄₇	—	2	2	no	—
51	C₂×Q₈×C₄₇	—	2	2	no	—
52	C₂×Q₈×C₄₇	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
58	Q₁₆×C₄₇	—	2	8	no	—
59	C₂×C₈×C₄₇	—	2	1	no	—
60	C₂×C₈×C₄₇	—	2	1	no	—
61	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
65	C₄⋊C₄×C₄₇	—	2	2	no	—
66	C₄⋊C₄×C₄₇	—	2	2	no	—
67	C₄⋊C₄×C₄₇	—	4	2	no	—
68	C₄⋊C₄×C₄₇	—	4	2	no	—
69	C₄²×C₄₇	—	4	2	no	—
70	C₄²×C₄₇	—	4	2	no	—
71	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₇	#41	1	1	yes	characteristic
75	C₂×C₄₇⋊C₈	—	2	2	no	—
76	C₂×C₄₇⋊C₈	—	2	2	no	—
77	C₂×C₄₇⋊C₈	—	2	4	no	—
78	C₂×C₄₇⋊C₈	—	4	2	no	—
79	C₂×C₄₇⋊C₈	—	4	2	no	—
80	C₂×C₄₇⋊C₈	—	4	4	no	—
81	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
82	Q₈×C₄₇	—	2	4	no	—
83	Q₈×C₄₇	—	2	4	no	—
84	Q₈×C₄₇	—	4	2	no	—
85	Q₈×C₄₇	—	4	2	no	—
86	C₂×C₄×C₄₇	—	2	2	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	—	4	2	no	—
89	C₂×C₄×C₄₇	—	4	2	no	—
90	C₂×C₄×C₄₇	—	4	2	no	—
91	C₂×C₄×C₄₇	—	4	4	no	—
92	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
95	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
96	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#137	1	4	yes	—
103	C₈×C₄₇	—	2	2	no	—
104	C₈×C₄₇	C₂²⋊C₈	1	4	yes	—
105	C₄₇⋊C₈	—	4	2	no	—
106	C₄₇⋊C₈	—	4	2	no	—
107	C₄₇⋊C₈	—	8	2	no	—
108	#9658	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	#390	1	1	yes	characteristic
110	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
111	C₂²×C₄₇	#436	1	1	yes	characteristic
112	C₂²×C₄₇	#441	1	2	yes	—
113	C₂²×C₄₇	#444	1	2	yes	—
114	C₄×C₄₇	—	2	1	no	—
115	C₄×C₄₇	—	2	1	no	—
116	C₄×C₄₇	—	4	4	no	—
117	C₄×C₄₇	—	8	2	no	—
118	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₄×C₄₇	#408	1	1	yes	characteristic
120	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
121	C₄×C₄₇	#436	1	1	yes	characteristic
122	C₄×C₄₇	#470	1	2	yes	—
123	C₄×C₄₇	#497	1	2	yes	—
124	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic
125	#1984	—	47	2	no	—
126	C₂×#334	—	47	1	no	—
127	C₂×Q₈⋊C₈	—	47	2	no	—
128	C₂×#322	—	47	1	no	—
129	C₂×C₄₇	#1579	1	2	yes	—
130	C₂×C₄₇	#1593	1	2	yes	—
131	C₂×C₄₇	#1954	1	1	yes	characteristic
132	C₂×C₄₇	#1971	1	1	yes	characteristic
133	C₂×C₄₇	#1975	1	1	yes	characteristic
134	C₂²×Q₁₆	Dic₄₇	1	1	yes	characteristic
135	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
136	C₄×Q₁₆	—	2	2	no	—
137	C₄×Q₁₆	—	2	2	no	—
138	C₂×#107	Dic₄₇	1	1	yes	characteristic
139	C₂×C₄⋊C₈	—	47	2	no	—
140	C₂×#103	Dic₄₇	1	2	yes	—
141	C₂×C₄×C₈	—	47	1	no	—
142	C₂×C₄×C₈	—	47	1	no	—
143	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
144	#336	—	94	4	no	—
145	#334	—	94	2	no	—
146	Q₈⋊C₈	—	94	4	no	—
147	#322	—	94	2	no	—
148	C₄₇	#9658	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	C₂×Dic₄₇	1	2	yes	—
150	C₂×Q₁₆	—	2	2	no	—
151	C₂×Q₁₆	—	2	2	no	—
152	C₂×Q₁₆	C₄₇⋊C₈	1	4	yes	complemented
153	C₂²×C₈	—	47	2	no	—
154	C₂²×C₈	—	94	1	no	—
155	C₂²×C₈	—	94	2	no	—
156	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
158	C₄×Q₈	—	2	2	no	—
159	C₄×Q₈	—	2	2	no	—
160	C₄×Q₈	—	2	2	no	—
161	C₄×Q₈	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
166	#107	—	2	2	no	—
167	C₄⋊C₈	—	94	2	no	—
168	C₄⋊C₈	—	94	2	no	—
169	#103	—	2	4	no	—
170	C₄×C₈	—	47	2	no	—
171	C₄×C₈	—	94	1	no	—
172	C₄×C₈	—	94	1	no	—
173	C₄×C₈	—	94	1	no	—
174	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
175	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	C₂×C₄₇⋊C₈	1	4	yes	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
185	C₂²×C₄	#10625352	1	1	yes	characteristic
186	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
187	Q₁₆	—	2	8	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	94	2	no	—
191	C₂×C₈	—	94	2	no	—
192	C₂×C₈	—	94	4	no	—
193	C₂×C₈	—	188	2	no	—
194	C₂×C₈	—	188	4	no	—
195	C₂×C₈	—	188	2	no	—
196	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
197	C₂×C₈	#10625329	1	2	yes	—
198	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
199	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄²	—	4	2	no	—
205	C₄²	—	4	2	no	—
206	C₄²	D₄×D₄₇	1	1	yes	characteristic
207	C₄²	#10625352	1	1	yes	characteristic
208	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
209	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
210	C₂³	D₄×Dic₄₇	1	1	yes	characteristic
211	Q₈	—	2	4	no	—
212	Q₈	—	2	4	no	—
213	Q₈	—	4	2	no	—
214	Q₈	—	4	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
222	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
223	C₂×C₄	#13093995	1	1	yes	characteristic
224	C₂×C₄	#13093995	1	1	yes	characteristic
225	C₂×C₄	#13094011	1	1	yes	characteristic
226	C₂×C₄	#13094020	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
228	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
229	C₂×C₄	#13094031	1	1	yes	characteristic
230	C₂×C₄	#13094031	1	1	yes	characteristic
231	C₂×C₄	#13094056	1	4	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	376	2	no	—
234	C₈	—	188	2	no	—
235	C₈	—	188	2	no	—
236	C₈	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
237	C₂²	#15689026	1	1	yes	characteristic
238	C₂²	D₄×C₄₇⋊C₈	1	1	yes	characteristic
239	C₂²	#15689201	1	1	yes	characteristic
240	C₂²	#15689230	1	2	yes	—
241	C₂²	#15689245	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	4	4	no	—
245	C₄	—	8	2	no	—
246	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
247	C₄	#15689090	1	1	yes	characteristic
248	C₄	D₄×C₄₇⋊C₈	1	1	yes	characteristic
249	C₄	#15689201	1	1	yes	characteristic
250	C₄	#15689378	1	2	yes	—
251	C₄	#15689516	1	2	yes	—
252	C₂	#20242207	1	2	yes	—
253	C₂	#20242297	1	2	yes	—
254	C₂	#20244183	1	1	yes	characteristic
255	C₂	#20244260	1	1	yes	characteristic
256	C₂	#20244282	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₄×D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	#9658
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
32. C₂×Q₁₆×C₄₇	C₈	semidirect
148. C₄₇	#9658	semidirect
152. C₂×Q₁₆	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20244325	2
C₂×#15688874	2
C₂×Q₈⋊(C₄₇⋊C₈)	2
C₂×#15688844	2
C₂×C₄×Q₁₆×C₄₇	2
#9658	47