Subgroups — #26226761

Order 12032.

870 conjugacy classes of subgroups.

The subgroup lattice has 308 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	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20244111	C₂	1	4	yes	maximal, complemented
4	#20243570	C₂	1	4	yes	maximal
5	#20243539	C₂	1	2	yes	maximal, complemented
6	#20243504	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10625329	C₂²	1	2	yes	—
9	C₂×#13093983	C₄	1	2	yes	—
10	#15689147	—	2	4	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×Dic₄₇⋊C₈	C₂²	1	8	yes	—
14	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
16	C₄×C₂²⋊Dic₄₇	C₄	1	2	yes	—
17	C₄×C₂²⋊Dic₄₇	C₂²	1	2	yes	—
18	#15688969	C₄	1	1	yes	characteristic
19	#15688969	C₄	1	1	yes	characteristic
20	C₄²×Dic₄₇	C₂²	1	2	yes	—
21	#15688877	C₂²	1	8	yes	—
22	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#432	—	2	4	no	—
24	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
27	C₂×#97×C₄₇	C₂²	1	2	yes	—
28	C₂×#10625329	—	2	4	no	—
29	C₂×#10625329	—	4	4	no	—
30	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
31	#13093983	—	2	8	no	—
32	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
38	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
39	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
40	Dic₄₇⋊C₈	—	4	8	no	—
41	C₄₇⋊#95	C₂×C₄	1	2	yes	—
42	C₄₇⋊#95	C₂×C₄	1	2	yes	—
43	C₄₇⋊#95	C₂×C₄	1	2	yes	—
44	C₄₇⋊#95	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	D₄	1	4	yes	—
53	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₇	—	2	2	no	—
57	C₄⋊C₈×C₄₇	—	2	2	no	—
58	C₂²⋊C₈×C₄₇	—	2	4	no	—
59	#97×C₄₇	—	2	4	no	—
60	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	2	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
76	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	2	4	no	—
81	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
82	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
83	C₄²×C₄₇	C₂×D₄	1	2	yes	—
84	C₄²×C₄₇	C₂×D₄	1	2	yes	—
85	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
86	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
87	C₂²⋊Dic₄₇	—	2	8	no	—
88	C₄⋊Dic₄₇	—	2	8	no	—
89	C₄×Dic₄₇	—	2	4	no	—
90	C₄×Dic₄₇	—	2	4	no	—
91	C₄×Dic₄₇	—	2	4	no	—
92	C₄×Dic₄₇	—	4	4	no	—
93	C₄×Dic₄₇	M₁₆	1	8	yes	complemented
94	#10625329	—	4	8	no	—
95	C₂×C₄₇⋊C₈	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	8	no	—
97	C₂³×C₄₇	—	2	2	no	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	8	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
117	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
120	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
121	C₂×Dic₄₇	—	2	4	no	—
122	C₂×Dic₄₇	—	2	4	no	—
123	C₂×Dic₄₇	—	2	4	no	—
124	C₂×Dic₄₇	—	2	4	no	—
125	C₂×Dic₄₇	—	2	4	no	—
126	C₂×Dic₄₇	C₂×M₁₆	1	8	yes	—
127	C₈×C₄₇	—	4	2	no	—
128	C₈×C₄₇	—	4	2	no	—
129	C₄₇⋊C₈	—	4	4	no	—
130	#21978	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	2	no	—
134	C₂²×C₄₇	—	4	4	no	—
135	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₄₇	#520	1	2	yes	—
137	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
138	C₂²×C₄₇	#568	1	2	yes	—
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	2	no	—
143	C₄×C₄₇	#423	1	4	yes	—
144	C₄×C₄₇	#535	1	1	yes	characteristic
145	C₄×C₄₇	#536	1	1	yes	characteristic
146	C₄×C₄₇	#544	1	2	yes	—
147	Dic₄₇	—	2	8	no	—
148	Dic₄₇	—	4	4	no	—
149	C₂×#432	D₄₇	1	1	yes	characteristic
150	C₂×#423	—	47	1	no	—
151	C₄×#117	—	47	1	no	—
152	#1940	—	47	4	no	—
153	#1851	—	47	4	no	—
154	#1842	—	47	2	no	—
155	#1831	—	47	2	no	—
156	C₂×C₄₇	—	4	2	no	—
157	C₂×C₄₇	#2317	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#423	1	1	yes	characteristic
159	C₂×C₄₇	#2905	1	1	yes	characteristic
160	C₂×C₄₇	#2989	1	4	yes	—
161	C₂²×M₁₆	—	47	2	no	—
162	C₂×#117	—	47	2	no	—
163	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
164	#432	—	2	4	no	—
165	#423	—	94	4	no	—
166	C₂×C₄⋊C₈	—	47	2	no	—
167	C₂×C₄⋊C₈	—	47	2	no	—
168	C₂×C₄⋊C₈	—	47	8	no	—
169	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
170	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
171	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
172	C₂×#97	C₂×D₄₇	1	2	yes	—
173	C₄×C₄⋊C₄	—	47	2	no	—
174	C₄×C₄⋊C₄	—	47	2	no	—
175	C₄×C₂²⋊C₄	—	47	2	no	—
176	C₄×C₂²⋊C₄	—	47	2	no	—
177	#376	—	47	1	no	—
178	#376	—	47	1	no	—
179	C₄³	—	47	2	no	—
180	#336	—	47	8	no	—
181	C₄₇	#21978	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
183	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
184	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
185	C₂×M₁₆	—	94	4	no	—
186	C₂×M₁₆	—	188	4	no	—
187	C₂²×C₈	—	47	4	no	—
188	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
189	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
190	#117	—	94	8	no	—
191	C₂×C₄⋊C₄	—	47	4	no	—
192	C₂×C₂²⋊C₄	—	47	4	no	—
193	C₂×C₄²	—	47	2	no	—
194	C₂×C₄²	—	47	2	no	—
195	C₂×C₄²	—	47	4	no	—
196	C₂×C₄²	—	47	4	no	—
197	C₂×C₄²	Dic₉₄	1	4	yes	—
198	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
199	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
201	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
202	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	—	2	2	no	—
205	C₄⋊C₈	—	94	4	no	—
206	C₄⋊C₈	—	94	4	no	—
207	C₄⋊C₈	—	188	8	no	—
208	C₂²⋊C₈	—	2	4	no	—
209	#97	—	2	4	no	—
210	#95	—	47	2	no	—
211	#95	—	47	2	no	—
212	#95	—	47	2	no	—
213	#95	—	47	2	no	—
214	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	47	4	no	—
220	C₂²×C₄	—	47	4	no	—
221	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
222	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
223	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
224	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
230	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
231	M₁₆	—	188	8	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	94	4	no	—
237	C₂×C₈	—	94	8	no	—
238	C₄⋊C₄	—	94	8	no	—
239	C₂²⋊C₄	—	94	8	no	—
240	C₄²	—	47	8	no	—
241	C₄²	—	94	4	no	—
242	C₄²	—	94	4	no	—
243	C₄²	—	94	4	no	—
244	C₄²	—	188	4	no	—
245	C₄²	Dic₄₇⋊C₄	1	4	yes	—
246	C₄²	Dic₄₇⋊C₄	1	4	yes	—
247	C₄²	C₂×Dic₉₄	1	2	yes	—
248	C₄²	C₂×Dic₉₄	1	2	yes	—
249	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
250	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
255	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	47	8	no	—
266	C₂×C₄	—	94	4	no	—
267	C₂×C₄	—	94	4	no	—
268	C₂×C₄	—	94	4	no	—
269	C₂×C₄	—	94	4	no	—
270	C₂×C₄	—	94	4	no	—
271	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
272	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
273	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
274	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
275	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
276	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
277	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
278	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
279	C₂×C₄	#13094070	1	2	yes	—
280	C₂×C₄	#13094071	1	2	yes	—
281	C₈	—	4	2	no	—
282	C₈	—	4	2	no	—
283	C₈	—	188	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
289	C₂²	#15689603	1	2	yes	—
290	C₂²	C₂×M₁₆×D₄₇	1	2	yes	—
291	C₂²	#15689889	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	94	8	no	—
297	C₄	—	188	4	no	—
298	C₄	#15689146	1	4	yes	—
299	C₄	#15689687	1	1	yes	characteristic
300	C₄	#15689694	1	1	yes	characteristic
301	C₄	#15689751	1	2	yes	—
302	C₂	—	4	2	no	—
303	C₂	#20246008	1	1	yes	characteristic
304	C₂	C₂×#15689146	1	1	yes	characteristic
305	C₂	#20251951	1	1	yes	characteristic
306	C₂	#20252507	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₂×C₄₇×#432	C₂
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#21978
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20244111	C₂	semidirect
5. #20243539	C₂	semidirect
6. #20243504	C₂	semidirect
93. C₄×Dic₄₇	M₁₆	semidirect
181. C₄₇	#21978	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
C₄×#13093983	2
#20244111	2
#20243570	2
#20243539	2
#20243504	2
C₂×C₄₇×#432	2
#21978	47