Subgroups — #26215915

Order 12032.

1322 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689172	C₂	1	2	yes	maximal, complemented
2	#20251833	C₂	1	2	yes	maximal
3	#20243430	C₂	1	4	yes	maximal, complemented
4	#20243426	C₂	1	4	yes	maximal, complemented
5	C₂×#15688888	C₂	1	2	yes	maximal, complemented
6	C₂²×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	4	yes	—
8	#15689172	—	2	8	no	—
9	C₂×#13093960	C₂²	1	8	yes	—
10	#15689090	—	2	8	no	—
11	#15689087	C₂²	1	8	yes	—
12	#15688888	C₂²	1	4	yes	—
13	#15688888	C₂²	1	4	yes	—
14	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#131×C₄₇	—	2	8	no	—
16	C₂×#131×C₄₇	C₄	1	8	yes	—
17	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
21	C₂×#10625329	—	2	4	no	—
22	C₂×#10625329	—	2	8	no	—
23	C₂×#10625329	—	2	8	no	—
24	C₂×#10625329	C₂³	1	8	yes	—
25	C₂²×C₄₇⋊C₈	—	2	4	no	—
26	#13093960	—	2	16	no	—
27	C₂²⋊(C₄₇⋊C₈)	—	2	16	no	—
28	C₂×#41×C₄₇	—	2	4	no	—
29	C₂×#41×C₄₇	—	2	4	no	—
30	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×#41×C₄₇	D₄	1	4	yes	—
32	C₂²×Q₈×C₄₇	Q₈	1	1	yes	characteristic
33	C₂²×D₄×C₄₇	Q₈	1	2	yes	—
34	C₂²×D₄×C₄₇	Q₈	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	#131×C₄₇	—	2	32	no	—
38	C₂×M₁₆×C₄₇	—	2	8	no	—
39	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
40	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×M₁₆×C₄₇	D₄	1	8	yes	—
42	C₂²×C₈×C₄₇	—	2	4	no	—
43	C₂²×C₈×C₄₇	—	2	8	no	—
44	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	—
46	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂⁴×C₄₇	C₂×Q₈	1	2	yes	—
50	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	#41×C₄₇	—	2	8	no	—
52	#41×C₄₇	—	2	8	no	—
53	#41×C₄₇	—	2	16	no	—
54	C₂×Q₈×C₄₇	—	2	4	no	—
55	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
56	C₂×D₄×C₄₇	—	2	8	no	—
57	C₂×D₄×C₄₇	—	2	4	no	—
58	C₂×D₄×C₄₇	C₄⋊C₄	1	8	yes	—
59	C₂×D₄×C₄₇	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	#41	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	M₁₆×C₄₇	—	2	16	no	—
72	M₁₆×C₄₇	C₂²⋊C₄	1	16	yes	—
73	C₂×C₈×C₄₇	—	2	8	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	16	no	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₄₇	C₂×D₄	1	8	yes	—
81	#10625329	—	2	8	no	—
82	#10625329	—	2	8	no	—
83	#10625329	—	4	8	no	—
84	C₂×C₄₇⋊C₈	—	2	8	no	—
85	C₂×C₄₇⋊C₈	—	4	8	no	—
86	C₂³×C₄₇	—	2	4	no	—
87	C₂³×C₄₇	—	4	2	no	—
88	C₂³×C₄₇	—	4	4	no	—
89	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
90	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₄₇	C₂²⋊Q₈	1	4	yes	—
92	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
93	Q₈×C₄₇	—	2	8	no	—
94	D₄×C₄₇	—	2	16	no	—
95	D₄×C₄₇	—	2	8	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	8	no	—
100	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	#117	1	4	yes	—
104	C₂×C₄×C₄₇	#121	1	8	yes	—
105	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
110	C₈×C₄₇	—	2	2	no	—
111	C₈×C₄₇	—	2	2	no	—
112	C₈×C₄₇	C₂×C₂²⋊C₄	1	8	yes	—
113	C₄₇⋊C₈	—	4	8	no	—
114	#20669	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	4	no	—
117	C₂²×C₄₇	—	2	8	no	—
118	C₂²×C₄₇	—	2	4	no	—
119	C₂²×C₄₇	—	4	4	no	—
120	C₂²×C₄₇	—	4	8	no	—
121	C₂²×C₄₇	#380	1	4	yes	—
122	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	2	yes	—
124	C₄×C₄₇	—	2	4	no	—
125	C₄×C₄₇	#380	1	4	yes	—
126	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
128	C₄×C₄₇	C₂×#121	1	2	yes	—
129	C₂²×#131	D₄₇	1	1	yes	characteristic
130	C₂×#430	—	47	2	no	—
131	#2879	—	47	2	no	—
132	#1811	—	47	4	no	—
133	#1810	—	47	4	no	—
134	C₂×#341	—	47	2	no	—
135	C₂×C₄₇	—	2	2	no	—
136	C₂×C₄₇	—	2	4	no	—
137	C₂×C₄₇	—	4	4	no	—
138	C₂×C₄₇	C₂×#380	1	1	yes	characteristic
139	C₂×C₄₇	#2916	1	2	yes	—
140	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
141	C₂×#131	—	2	8	no	—
142	C₂×#131	Dic₄₇	1	8	yes	—
143	C₂²×M₁₆	—	47	4	no	—
144	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
145	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
146	C₂³×C₈	C₂×D₄₇	1	2	yes	—
147	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
148	#430	—	94	8	no	—
149	C₂×#108	—	47	8	no	—
150	#408	—	94	8	no	—
151	#407	—	47	8	no	—
152	#341	—	47	4	no	—
153	#341	—	47	4	no	—
154	C₄₇	#20669	1	1	yes	characteristic, complemented, Sylow
155	C₂×#41	—	2	4	no	—
156	C₂×#41	—	2	4	no	—
157	C₂×#41	D₁₈₈	1	4	yes	—
158	C₂×#41	C₂×Dic₄₇	1	4	yes	—
159	C₂²×Q₈	Dic₉₄	1	1	yes	characteristic
160	C₂²×D₄	Dic₉₄	1	2	yes	—
161	C₂²×D₄	Dic₉₄	1	1	yes	characteristic
162	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
163	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
164	#131	—	2	32	no	—
165	C₂×M₁₆	—	2	8	no	—
166	C₂×M₁₆	—	47	8	no	—
167	C₂×M₁₆	—	94	4	no	—
168	C₂×M₁₆	—	94	8	no	—
169	C₂×M₁₆	—	94	8	no	—
170	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
171	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
172	C₂×M₁₆	C₂²⋊D₄₇	1	8	yes	—
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	2	8	no	—
175	C₂²×C₈	—	94	4	no	—
176	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
177	C₂²×C₈	C₂×Dic₄₇	1	8	yes	—
178	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
181	#108	—	94	16	no	—
182	C₂²⋊C₈	—	94	16	no	—
183	C₂⁴	C₂×Dic₉₄	1	2	yes	—
184	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
185	#41	—	2	8	no	—
186	#41	—	2	8	no	—
187	#41	—	2	16	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	C₄⋊Dic₄₇	1	4	yes	—
190	C₂×D₄	—	2	8	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	C₄⋊Dic₄₇	1	8	yes	—
193	C₂×D₄	C₄⋊Dic₄₇	1	4	yes	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
197	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
199	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
200	C₂²×C₄	#10625350	1	4	yes	—
201	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
202	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
203	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
204	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
205	M₁₆	—	2	16	no	—
206	M₁₆	—	94	8	no	—
207	M₁₆	—	94	8	no	—
208	M₁₆	—	188	8	no	—
209	M₁₆	C₂²⋊Dic₄₇	1	16	yes	—
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	16	no	—
214	C₂×C₈	—	94	8	no	—
215	C₂×C₈	—	188	8	no	—
216	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
217	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
218	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
219	C₂×C₈	C₂×C₂²⋊D₄₇	1	8	yes	—
220	C₂³	—	2	4	no	—
221	C₂³	—	4	2	no	—
222	C₂³	—	4	4	no	—
223	C₂³	C₂×C₄⋊Dic₄₇	1	4	yes	—
224	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
225	C₂³	#13094003	1	4	yes	—
226	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
227	Q₈	—	2	8	no	—
228	D₄	—	2	16	no	—
229	D₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
235	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
236	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
237	C₂×C₄	#13093983	1	4	yes	—
238	C₂×C₄	#13093998	1	8	yes	—
239	C₂×C₄	#13094003	1	4	yes	—
240	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
242	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
243	C₂×C₄	C₂×#10625350	1	2	yes	—
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₈	—	188	8	no	—
247	C₈	C₂×C₂²⋊Dic₄₇	1	8	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	8	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	8	no	—
254	C₂²	#15688983	1	4	yes	—
255	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
256	C₂²	C₂×#13094003	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	#15688983	1	4	yes	—
259	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
260	C₄	C₂×#13093983	1	1	yes	characteristic
261	C₄	C₂×#13093998	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	2	4	no	—
264	C₂	—	4	4	no	—
265	C₂	C₂×#15688983	1	1	yes	characteristic
266	C₂	#20251999	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×#131×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂²×#41×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×#131	D₄₇
hypocenter	C₄₇	#20669
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#15689172	C₂	semidirect
3. #20243430	C₂	semidirect
4. #20243426	C₂	semidirect
5. C₂×#15688888	C₂	semidirect
154. C₄₇	#20669	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689172	2
#20251833	2
#20243430	2
#20243426	2
C₂×#15688888	2
C₂²×#131×C₄₇	2
#20669	47