Subgroups — #26221001

Order 12032.

666 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245865	C₂	1	1	yes	maximal, characteristic
2	#20244327	C₂	1	2	yes	maximal
3	#20244109	C₂	1	2	yes	maximal
4	#20243943	C₂	1	4	yes	maximal
5	#20243246	C₂	1	2	yes	maximal
6	#20243139	C₂	1	2	yes	maximal
7	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
10	C₂×Dic₄₇⋊C₈	C₂²	1	4	yes	—
11	C₂×#13093935	C₂²	1	1	yes	characteristic
12	C₂×#13093935	C₂²	1	2	yes	—
13	C₂×#13093935	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
15	#15689029	C₄	1	2	yes	—
16	#15689021	C₄	1	2	yes	—
17	#15689007	C₄	1	4	yes	—
18	C₄×Dic₄₇⋊C₄	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
20	#15688969	C₂²	1	2	yes	—
21	#15688877	C₂²	1	4	yes	—
22	#15688877	C₂²	1	4	yes	—
23	#15688877	C₂²	1	4	yes	—
24	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
26	C₄³×C₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#336	C₂²	1	2	yes	—
28	C₂²×Dic₉₄	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	C₄×Dic₉₄	—	2	4	no	—
32	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
35	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
38	Dic₄₇⋊C₈	—	2	8	no	—
39	#13093935	—	2	4	no	—
40	#13093935	—	2	2	no	—
41	#13093935	Q₈	1	4	yes	complemented
42	C₄×C₄₇⋊C₈	—	2	4	no	—
43	C₄₇⋊#95	C₂×C₄	1	4	yes	—
44	C₄₇⋊#95	C₂×C₄	1	4	yes	—
45	C₄₇⋊#95	C₂×C₄	1	2	yes	—
46	C₄₇⋊#95	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₄₇	—	2	4	no	—
54	C₄×C₈×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
61	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
63	C₂×Dic₉₄	—	2	2	no	—
64	C₂×Dic₉₄	—	4	2	no	—
65	C₂×C₈×C₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	—	2	4	no	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₄²×C₄₇	—	2	4	no	—
69	C₄²×C₄₇	—	2	2	no	—
70	C₄²×C₄₇	—	4	2	no	—
71	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
72	C₄²×C₄₇	#41	1	4	yes	—
73	C₄²×C₄₇	#41	1	2	yes	—
74	C₄⋊Dic₄₇	—	2	2	no	—
75	C₄⋊Dic₄₇	—	4	2	no	—
76	Dic₄₇⋊C₄	—	2	4	no	—
77	Dic₄₇⋊C₄	—	2	8	no	—
78	C₄×Dic₄₇	—	2	4	no	—
79	C₄×Dic₄₇	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	2	2	no	—
81	C₂×C₄₇⋊C₈	—	2	2	no	—
82	C₂×C₄₇⋊C₈	—	2	2	no	—
83	C₂×C₄₇⋊C₈	—	2	2	no	—
84	C₂×C₄₇⋊C₈	C₂×Q₈	1	4	yes	—
85	C₂×C₄₇⋊C₈	#41	1	4	yes	—
86	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
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	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	#117	1	2	yes	—
94	C₂×C₄×C₄₇	#117	1	2	yes	—
95	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
96	C₂×C₄×C₄₇	#131	1	4	yes	—
97	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
99	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	—	2	4	no	—
104	C₂×Dic₄₇	—	2	4	no	—
105	C₂×Dic₄₇	—	2	4	no	—
106	C₂×Dic₄₇	—	2	4	no	—
107	C₂×Dic₄₇	—	2	4	no	—
108	Dic₉₄	—	4	4	no	—
109	C₈×C₄₇	—	4	4	no	—
110	C₄₇⋊C₈	—	2	4	no	—
111	C₄₇⋊C₈	—	2	4	no	—
112	#21344	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
115	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
116	C₂²×C₄₇	C₂×#131	1	2	yes	—
117	C₂²×C₄₇	#568	1	2	yes	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	2	2	no	—
120	C₄×C₄₇	—	2	4	no	—
121	C₄×C₄₇	—	2	4	no	—
122	C₄×C₄₇	#529	1	2	yes	—
123	C₄×C₄₇	#533	1	2	yes	—
124	Dic₄₇	—	4	2	no	—
125	Dic₄₇	—	4	2	no	—
126	Dic₄₇	—	4	4	no	—
127	#2295	—	47	1	no	—
128	#1985	—	47	2	no	—
129	#1940	—	47	2	no	—
130	#1916	—	47	4	no	—
131	#1768	—	47	2	no	—
132	C₄×C₄⋊C₈	D₄₇	1	1	yes	characteristic
133	#1746	—	47	2	no	—
134	C₄×#97	—	47	1	no	—
135	C₂×C₄₇	#2295	1	1	yes	characteristic
136	C₂×C₄₇	#2921	1	1	yes	characteristic
137	C₂×C₄₇	#2922	1	1	yes	characteristic
138	C₂×C₄₇	#2962	1	2	yes	—
139	C₂×C₄₇	#2968	1	2	yes	—
140	C₂×C₄×Q₈	—	47	1	no	—
141	C₂×C₄⋊C₈	—	47	4	no	—
142	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
143	C₂×#97	—	47	1	no	—
144	C₂×#97	—	47	2	no	—
145	C₂×#97	—	47	1	no	—
146	C₂×C₄×C₈	—	47	2	no	—
147	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
148	#391	—	47	2	no	—
149	#389	—	47	2	no	—
150	#387	—	47	4	no	—
151	C₄×C₄⋊C₄	—	47	1	no	—
152	C₄×C₄⋊C₄	—	47	2	no	—
153	#376	—	47	2	no	—
154	C₄³	C₂×D₄₇	1	1	yes	characteristic
155	#336	—	47	4	no	—
156	#336	—	47	4	no	—
157	#336	—	47	4	no	—
158	#336	C₂×D₄₇	1	2	yes	—
159	C₄₇	#21344	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	47	1	no	—
161	C₂²×C₈	—	47	2	no	—
162	C₂²×C₈	—	47	2	no	—
163	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
164	C₄×Q₈	—	94	4	no	—
165	C₂×C₄⋊C₄	—	47	2	no	—
166	C₂×C₄⋊C₄	—	47	1	no	—
167	C₂×C₄⋊C₄	—	47	4	no	—
168	C₂×C₄⋊C₄	—	47	2	no	—
169	C₂×C₄²	—	47	2	no	—
170	C₂×C₄²	—	47	2	no	—
171	C₂×C₄²	C₄×D₄₇	1	2	yes	—
172	C₂×C₄²	C₄×D₄₇	1	2	yes	—
173	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	94	8	no	—
178	#97	—	47	4	no	—
179	#97	—	94	4	no	—
180	#97	—	94	2	no	—
181	C₄×C₈	—	2	4	no	—
182	C₄×C₈	—	94	4	no	—
183	#95	—	47	4	no	—
184	#95	—	47	4	no	—
185	#95	—	47	2	no	—
186	#95	—	47	4	no	—
187	C₂×Q₈	—	94	2	no	—
188	C₂×Q₈	—	188	2	no	—
189	C₂²×C₄	—	47	4	no	—
190	C₂²×C₄	—	47	2	no	—
191	C₂²×C₄	—	47	2	no	—
192	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
195	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
196	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	47	4	no	—
201	C₂×C₈	—	47	4	no	—
202	C₂×C₈	—	94	2	no	—
203	C₂×C₈	—	94	2	no	—
204	C₂×C₈	—	94	2	no	—
205	C₂×C₈	—	94	2	no	—
206	C₄⋊C₄	—	94	4	no	—
207	C₄⋊C₄	—	94	2	no	—
208	C₄⋊C₄	—	94	8	no	—
209	C₄⋊C₄	—	188	2	no	—
210	C₄²	—	2	4	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	—	94	4	no	—
214	C₄²	—	94	4	no	—
215	C₄²	#10625350	1	4	yes	—
216	C₄²	#10625352	1	2	yes	—
217	C₄²	Q₈×D₄₇	1	2	yes	—
218	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
219	Q₈	—	188	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	94	2	no	—
227	C₂×C₄	—	94	2	no	—
228	C₂×C₄	—	94	4	no	—
229	C₂×C₄	—	94	4	no	—
230	C₂×C₄	—	94	4	no	—
231	C₂×C₄	—	94	4	no	—
232	C₂×C₄	—	94	4	no	—
233	C₂×C₄	#13093981	1	2	yes	—
234	C₂×C₄	#13093981	1	2	yes	—
235	C₂×C₄	#13093989	1	4	yes	—
236	C₂×C₄	#13094030	1	4	yes	—
237	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
238	C₂×C₄	#41×D₄₇	1	2	yes	—
239	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
240	C₂×C₄	C₂×#10625350	1	2	yes	—
241	C₈	—	4	4	no	—
242	C₈	—	94	4	no	—
243	C₈	—	94	4	no	—
244	C₂²	C₂×#13093981	1	1	yes	characteristic
245	C₂²	C₂×#13093989	1	1	yes	characteristic
246	C₂²	#15689591	1	1	yes	characteristic
247	C₂²	C₂×#13094030	1	2	yes	—
248	C₂²	#15689889	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	—	188	2	no	—
254	C₄	—	188	2	no	—
255	C₄	—	188	4	no	—
256	C₄	#15689655	1	2	yes	—
257	C₄	#15689677	1	2	yes	—
258	C₂	#20245862	1	1	yes	characteristic
259	C₂	#20252024	1	1	yes	characteristic
260	C₂	#20252030	1	1	yes	characteristic
261	C₂	#20252331	1	2	yes	—
262	C₂	#20252377	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂³×C₄
Fitting	C₄×C₄⋊C₈×C₄₇	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#21344
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
41. #13093935	Q₈	semidirect
159. C₄₇	#21344	semidirect

Maximal subgroups

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

subgroup	index
#20245865	2
#20244327	2
#20244109	2
#20243943	2
#20243246	2
#20243139	2
C₄×#13093935	2
C₄×C₄⋊C₈×C₄₇	2
#21344	47