Subgroups — #26214187

Order 12032.

1150 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689147	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×Dic₄₇	C₂	1	2	yes	maximal
3	C₂×#15688969	C₂	1	1	yes	maximal, characteristic
4	#20243247	C₂	1	8	yes	maximal, complemented
5	#20243081	C₂	1	2	yes	maximal
6	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	#15689147	C₄	1	8	yes	complemented
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
12	M₁₆×Dic₄₇	—	2	8	no	—
13	C₂×#13093934	C₂²	1	4	yes	—
14	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
15	#15688969	C₄	1	4	yes	—
16	#15688969	C₂²	1	4	yes	—
17	C₂×C₄₇⋊#95	C₄	1	4	yes	—
18	#15688877	C₂²	1	8	yes	—
19	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#423	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
23	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
24	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
25	C₂×#10625329	—	2	4	no	—
26	C₂×#10625329	—	2	4	no	—
27	C₂×#10625329	C₂×C₄	1	8	yes	—
28	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
34	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
35	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
36	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
37	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	16	yes	complemented
38	#13093934	—	2	8	no	—
39	C₈×Dic₄₇	—	2	8	no	—
40	C₄₇⋊#95	C₂×C₄	1	16	yes	—
41	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₄₇	—	2	4	no	—
44	C₂×M₁₆×C₄₇	—	2	4	no	—
45	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
46	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
51	C₄⋊C₈×C₄₇	C₂×C₄	1	16	yes	complemented
52	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
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	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₄₇	—	2	4	no	—
65	C₂²×Dic₄₇	—	4	4	no	—
66	C₂²×Dic₄₇	—	4	4	no	—
67	C₂²×Dic₄₇	C₄⋊C₄	1	8	yes	—
68	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
69	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
70	M₁₆×C₄₇	—	2	16	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
74	C₄²×C₄₇	C₄²	1	8	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
77	C₄×Dic₄₇	—	2	4	no	—
78	C₄×Dic₄₇	—	2	2	no	—
79	C₄×Dic₄₇	—	2	2	no	—
80	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
81	C₄×Dic₄₇	C₂×D₄	1	4	yes	—
82	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
83	#10625329	—	2	16	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
87	C₂³×C₄₇	—	2	4	no	—
88	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
89	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
90	C₂³×C₄₇	#117	1	2	yes	—
91	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	—	2	8	no	—
93	C₂×C₄×C₄₇	—	4	8	no	—
94	C₂×C₄×C₄₇	C₂×C₄²	1	8	yes	—
95	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₄₇	#117	1	2	yes	—
98	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
99	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
101	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
108	C₂×Dic₄₇	—	2	8	no	—
109	C₂×Dic₄₇	—	2	4	no	—
110	C₂×Dic₄₇	—	2	8	no	—
111	C₂×Dic₄₇	—	4	8	no	—
112	C₂×Dic₄₇	—	4	8	no	—
113	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
114	C₈×C₄₇	—	2	8	no	—
115	C₄₇⋊C₈	—	2	8	no	—
116	#20429	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	2	4	no	—
118	C₂²×C₄₇	—	2	4	no	—
119	C₂²×C₄₇	—	2	4	no	—
120	C₂²×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
121	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
122	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
124	C₂²×C₄₇	#568	1	4	yes	—
125	C₄×C₄₇	—	2	8	no	—
126	C₄×C₄₇	C₄×C₄⋊C₄	1	4	yes	—
127	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
130	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
131	Dic₄₇	—	2	8	no	—
132	Dic₄₇	—	4	4	no	—
133	C₂×#423	—	47	1	no	—
134	C₂×#423	D₄₇	1	1	yes	characteristic
135	C₂×C₄×M₁₆	—	47	2	no	—
136	C₂×#376	—	47	1	no	—
137	#1768	—	47	8	no	—
138	#1728	—	47	2	no	—
139	C₂×C₄₇	—	2	4	no	—
140	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄₇	#2873	1	2	yes	—
142	C₂×C₄₇	#2905	1	2	yes	—
143	C₂×C₄₇	#2921	1	2	yes	—
144	C₂²×M₁₆	—	47	2	no	—
145	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
146	C₂²×C₄²	—	47	1	no	—
147	C₂²×C₄²	—	47	1	no	—
148	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
149	#423	—	47	8	no	—
150	#423	Dic₄₇	1	8	yes	complemented
151	C₂×C₄⋊C₈	—	47	4	no	—
152	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
153	C₄×M₁₆	—	94	8	no	—
154	C₂×#97	—	47	4	no	—
155	C₂×C₄×C₈	—	47	4	no	—
156	#376	—	47	4	no	—
157	#376	—	47	4	no	—
158	C₂×#95	—	47	4	no	—
159	#336	—	47	8	no	—
160	C₄₇	#20429	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	—	47	2	no	—
162	C₂³×C₄	—	47	2	no	—
163	C₂³×C₄	C₄×D₄₇	1	2	yes	—
164	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	—	47	8	no	—
168	C₂×M₁₆	—	94	4	no	—
169	C₂×M₁₆	—	94	4	no	—
170	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
171	C₂²×C₈	—	47	4	no	—
172	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
173	C₂×C₄²	—	47	2	no	—
174	C₂×C₄²	—	47	4	no	—
175	C₂×C₄²	—	47	4	no	—
176	C₂×C₄²	—	47	2	no	—
177	C₂×C₄²	—	47	2	no	—
178	C₂×C₄²	—	47	2	no	—
179	C₂×C₄²	—	94	2	no	—
180	C₂×C₄²	—	94	2	no	—
181	C₂×C₄²	C₄×D₄₇	1	2	yes	—
182	C₂×C₄²	C₄×D₄₇	1	4	yes	—
183	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
184	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
185	C₄⋊C₈	—	47	16	no	—
186	C₄⋊C₈	C₂×Dic₄₇	1	16	yes	complemented
187	#97	—	94	8	no	—
188	C₄×C₈	—	94	8	no	—
189	#95	—	47	16	no	—
190	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	—	47	4	no	—
193	C₂²×C₄	—	47	8	no	—
194	C₂²×C₄	—	47	4	no	—
195	C₂²×C₄	—	94	4	no	—
196	C₂²×C₄	—	188	4	no	—
197	C₂²×C₄	—	188	4	no	—
198	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
199	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
200	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
202	C₂²×C₄	D₄×D₄₇	1	2	yes	—
203	C₂²×C₄	#10625352	1	2	yes	—
204	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
205	C₂²×C₄	#10625354	1	2	yes	—
206	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
207	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
208	M₁₆	—	2	16	no	—
209	M₁₆	—	94	16	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	47	8	no	—
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	—	94	4	no	—
215	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
216	C₄²	—	47	8	no	—
217	C₄²	—	47	4	no	—
218	C₄²	—	47	4	no	—
219	C₄²	—	94	4	no	—
220	C₄²	—	94	2	no	—
221	C₄²	—	94	2	no	—
222	C₄²	C₄×Dic₄₇	1	8	yes	—
223	C₄²	C₂×C₄×D₄₇	1	4	yes	—
224	C₄²	C₂²×Dic₄₇	1	4	yes	—
225	C₂³	—	2	4	no	—
226	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
227	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
228	C₂³	#13093982	1	2	yes	—
229	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	4	8	no	—
232	C₂×C₄	—	47	8	no	—
233	C₂×C₄	—	94	8	no	—
234	C₂×C₄	—	94	4	no	—
235	C₂×C₄	—	94	8	no	—
236	C₂×C₄	—	188	8	no	—
237	C₂×C₄	—	188	8	no	—
238	C₂×C₄	C₂×C₄×Dic₄₇	1	8	yes	—
239	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
240	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
241	C₂×C₄	#13093982	1	2	yes	—
242	C₂×C₄	#13093986	1	4	yes	—
243	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
244	C₂×C₄	#13093989	1	4	yes	—
245	C₂×C₄	Q₈×Dic₄₇	1	4	yes	—
246	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
247	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
249	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
250	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
251	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
252	C₈	—	2	8	no	—
253	C₈	—	94	8	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	2	4	no	—
257	C₂²	C₄⋊C₄×Dic₄₇	1	4	yes	—
258	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
259	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
260	C₂²	C₂×#13093982	1	1	yes	characteristic
261	C₂²	#15689889	1	4	yes	—
262	C₄	—	2	8	no	—
263	C₄	—	94	8	no	—
264	C₄	—	188	4	no	—
265	C₄	C₄⋊C₄×Dic₄₇	1	4	yes	—
266	C₄	C₂×#13093986	1	1	yes	characteristic
267	C₄	C₂×D₄×Dic₄₇	1	1	yes	characteristic
268	C₄	C₂×#13093989	1	1	yes	characteristic
269	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
270	C₂	—	2	4	no	—
271	C₂	C₂×C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
272	C₂	#20251806	1	2	yes	—
273	C₂	#20251950	1	2	yes	—
274	C₂	#20252025	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#20429
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #20243247	C₂	semidirect
10. #15689147	C₄	semidirect
21. C₄₇×#423	C₄	semidirect
37. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
51. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
150. #423	Dic₄₇	semidirect
160. C₄₇	#20429	semidirect
186. C₄⋊C₈	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689147	2
C₂×M₁₆×Dic₄₇	2
C₂×#15688969	2
#20243247	2
#20243081	2
C₂×C₄₇×#423	2
#20429	47