Subgroups — #26234762

Order 12032.

846 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689241	C₂	1	1	yes	maximal, characteristic
2	#20252265	C₂	1	2	yes	maximal
3	#20251980	C₂	1	2	yes	maximal, complemented
4	#20244173	C₂	1	2	yes	maximal, complemented
5	#20243622	C₂	1	4	yes	maximal, complemented
6	#20243552	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243444	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₂²	1	1	yes	characteristic
10	C₂×#13094030	C₂²	1	2	yes	—
11	#15689603	C₂²	1	4	yes	—
12	#15689241	C₄	1	8	yes	complemented
13	#15689225	—	2	4	no	—
14	C₄×Dic₁₈₈	—	2	2	no	—
15	C₄×D₃₇₆	—	2	2	no	—
16	C₂×#13093960	C₂²	1	1	yes	characteristic
17	#15689166	C₂²	1	4	yes	—
18	#15689164	—	2	4	no	—
19	#15689158	—	2	2	no	—
20	C₂×#13093958	C₂²	1	2	yes	—
21	C₂×#13093956	C₂²	1	2	yes	—
22	#106×D₄₇	—	2	2	no	—
23	C₂×#13093952	C₂²	1	2	yes	—
24	#15689125	C₂²	1	4	yes	—
25	#15688989	C₂²	1	2	yes	—
26	#15688891	C₂²	1	4	yes	—
27	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#427	C₂²	1	1	yes	characteristic
29	C₄₇×#405	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#339	C₂²	1	2	yes	—
32	C₂×#10625350	C₂³	1	2	yes	—
33	#13094046	C₂×C₄	1	4	yes	—
34	#13094046	Q₈	1	4	yes	—
35	C₂×Dic₁₈₈	D₄	1	1	yes	characteristic
36	C₂×#10625325	D₄	1	2	yes	—
37	C₂×D₃₇₆	D₄	1	1	yes	characteristic
38	#13094030	—	2	4	no	—
39	#13094030	C₂×C₄	1	8	yes	complemented
40	C₂×#10625329	C₂³	1	2	yes	—
41	C₂×#10625324	—	2	2	no	—
42	C₂×C₈×D₄₇	—	2	2	no	—
43	C₄×Dic₉₄	—	2	4	no	—
44	C₄×D₁₈₈	—	2	4	no	—
45	(C₂²⋊D₄₇)⋊C₄	—	4	4	no	—
46	#13093982	—	2	4	no	—
47	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
48	C₄⋊C₄×D₄₇	—	2	4	no	—
49	#13093960	C₂×C₄	1	4	yes	complemented
50	#13093958	—	2	2	no	—
51	#13093958	—	2	2	no	—
52	#13093957	—	2	4	no	—
53	#13093956	—	2	2	no	—
54	#13093956	—	2	2	no	—
55	#13093955	—	2	4	no	—
56	#13093952	C₂×C₄	1	8	yes	complemented
57	#13093949	—	2	4	no	—
58	#13093946	—	2	4	no	—
59	C₄₇⋊#95	—	4	2	no	—
60	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	#117×C₄₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	#108×C₄₇	—	2	2	no	—
66	#107×C₄₇	D₄	1	2	yes	—
67	#106×C₄₇	D₄	1	1	yes	characteristic
68	#106×C₄₇	D₄	1	1	yes	characteristic
69	#97×C₄₇	—	2	2	no	—
70	C₄×C₈×C₄₇	—	2	2	no	—
71	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
72	C₂²×C₄×C₄₇	—	2	1	no	—
73	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂×C₂²⋊D₄₇	—	2	2	no	—
75	C₂²×Dic₄₇	—	2	4	no	—
76	#10625350	—	2	2	no	—
77	#10625350	—	2	2	no	—
78	#10625350	C₂²×C₄	1	4	yes	—
79	#10625350	C₂×Q₈	1	4	yes	—
80	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
81	C₂×C₄×D₄₇	—	2	2	no	—
82	C₂×C₄×D₄₇	—	4	4	no	—
83	C₂×Dic₉₄	C₂×D₄	1	2	yes	—
84	M₁₆×C₄₇	—	2	2	no	—
85	M₁₆×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₄₇	#41	1	2	yes	—
95	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
96	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
97	C₂²⋊C₄×C₄₇	—	2	2	no	—
98	C₄²×C₄₇	—	2	2	no	—
99	C₄²×C₄₇	—	2	1	no	—
100	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
101	(C₂×D₄₇)⋊C₄	—	4	4	no	—
102	C₄⋊Dic₄₇	—	2	4	no	—
103	C₄⋊Dic₄₇	—	2	4	no	—
104	Dic₄₇⋊C₄	—	4	4	no	—
105	C₄×Dic₄₇	—	4	4	no	—
106	#10625329	—	2	2	no	—
107	#10625329	C₂²×C₄	1	4	yes	—
108	C₂×C₄₇⋊C₈	—	2	2	no	—
109	Dic₁₈₈	C₄⋊C₄	1	4	yes	—
110	D₃₇₆	C₄⋊C₄	1	4	yes	—
111	#10625325	C₄⋊C₄	1	8	yes	—
112	#10625324	—	2	8	no	—
113	C₈×D₄₇	—	2	8	no	—
114	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂²×D₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
126	C₂²⋊D₄₇	—	2	4	no	—
127	C₂²⋊D₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	2	2	no	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	8	4	no	—
131	D₁₈₈	—	2	2	no	—
132	D₁₈₈	C₂×C₄⋊C₄	1	4	yes	—
133	C₄×D₄₇	—	2	4	no	—
134	C₄×D₄₇	—	2	4	no	—
135	C₄×D₄₇	—	16	4	no	—
136	Dic₉₄	—	2	2	no	—
137	Dic₉₄	C₂×C₄⋊C₄	1	4	yes	—
138	C₈×C₄₇	—	4	2	no	—
139	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
141	C₈×C₄₇	C₄×D₄	1	2	yes	—
142	C₈×C₄₇	C₄×D₄	1	2	yes	—
143	C₄₇⋊C₈	—	2	4	no	—
144	#22690	—	47	1	no	maximal, Sylow
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	#520	1	1	yes	characteristic
147	C₂²×C₄₇	D₄²	1	1	yes	characteristic
148	C₂²×C₄₇	#554	1	1	yes	characteristic
149	C₂×D₄₇	—	2	4	no	—
150	C₂×D₄₇	—	4	4	no	—
151	C₄×C₄₇	—	4	2	no	—
152	C₄×C₄₇	—	4	2	no	—
153	C₄×C₄₇	—	4	2	no	—
154	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
156	C₄×C₄₇	#547	1	1	yes	characteristic
157	C₄×C₄₇	D₄×Q₈	1	1	yes	characteristic
158	Dic₄₇	—	2	4	no	—
159	Dic₄₇	—	8	4	no	—
160	C₂×#443	—	47	1	no	—
161	#2951	—	47	2	no	—
162	#2912	—	47	2	no	—
163	#1952	—	47	2	no	—
164	#1860	—	47	4	no	—
165	#1846	—	47	1	no	—
166	#1815	—	47	2	no	—
167	#1772	D₄₇	1	1	yes	characteristic, complemented
168	C₂×C₄₇	—	2	1	no	—
169	C₂×C₄₇	—	2	1	no	—
170	C₂×C₄₇	D₄×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄₇	#3290	1	2	yes	—
172	D₄₇	—	4	4	no	—
173	C₂×#135	—	47	1	no	—
174	C₂×#131	—	47	2	no	—
175	#520	—	47	4	no	—
176	#443	—	47	8	no	—
177	#440	—	94	4	no	—
178	C₄×Q₁₆	—	94	2	no	—
179	C₄×D₈	—	94	2	no	—
180	C₂×#108	—	47	1	no	—
181	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
182	#428	—	47	4	no	—
183	#428	—	94	4	no	—
184	#427	—	94	2	no	—
185	#427	C₂×D₄₇	1	1	yes	characteristic
186	C₂×#107	—	47	2	no	—
187	C₂×#106	—	47	2	no	—
188	C₂×#106	—	94	2	no	—
189	C₂×#104	—	47	2	no	—
190	#417	—	47	4	no	—
191	#405	C₂×D₄₇	1	2	yes	—
192	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
193	#382	—	47	2	no	—
194	#343	—	47	4	no	—
195	#339	C₂×D₄₇	1	2	yes	—
196	C₄₇	#22690	1	1	yes	characteristic, complemented, Sylow
197	C₂×#41	—	47	2	no	—
198	#135	—	47	4	no	—
199	#135	—	47	4	no	—
200	C₂×Q₁₆	—	47	1	no	—
201	C₂×SD₁₆	—	47	2	no	—
202	C₂×D₈	—	47	1	no	—
203	#131	—	47	8	no	—
204	#131	—	94	4	no	—
205	C₂×M₁₆	—	47	2	no	—
206	C₂×M₁₆	—	94	2	no	—
207	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
208	C₂²×C₈	—	94	2	no	—
209	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
211	C₄×Q₈	—	94	4	no	—
212	C₄×D₄	—	94	4	no	—
213	C₄×D₄	—	188	4	no	—
214	#117	—	94	4	no	—
215	#117	C₂²×D₄₇	1	2	yes	—
216	C₂×C₄⋊C₄	—	47	4	no	—
217	C₂×C₄⋊C₄	—	94	4	no	—
218	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
219	#108	—	2	2	no	—
220	#108	—	47	4	no	—
221	#107	—	94	2	no	—
222	#107	—	94	2	no	—
223	#107	—	94	4	no	—
224	#107	D₁₈₈	1	2	yes	—
225	#106	—	94	2	no	—
226	#106	—	94	2	no	—
227	#106	—	94	4	no	—
228	#106	D₁₈₈	1	1	yes	characteristic
229	#106	D₁₈₈	1	1	yes	characteristic
230	#104	—	47	8	no	—
231	#103	—	94	4	no	—
232	#102	—	94	4	no	—
233	#97	—	2	2	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	C₄×D₄₇	1	4	yes	complemented
236	#95	—	188	2	no	—
237	#41	—	47	4	no	—
238	#41	—	47	4	no	—
239	#41	—	94	2	no	—
240	#41	—	94	2	no	—
241	C₂×Q₈	—	47	2	no	—
242	C₂×D₄	—	47	2	no	—
243	C₂×D₄	—	94	2	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	94	2	no	—
246	C₂²×C₄	—	94	4	no	—
247	C₂²×C₄	—	188	4	no	—
248	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
249	Q₁₆	—	47	4	no	—
250	SD₁₆	—	47	8	no	—
251	D₈	—	47	4	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	47	4	no	—
255	M₁₆	—	94	2	no	—
256	M₁₆	—	94	8	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	94	2	no	—
259	C₂×C₈	—	94	8	no	—
260	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
261	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
262	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
263	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
264	C₂×C₈	#10625350	1	2	yes	—
265	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
266	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
267	C₂×C₈	Q₈×D₄₇	1	2	yes	—
268	C₄⋊C₄	—	94	4	no	—
269	C₄⋊C₄	—	94	4	no	—
270	C₄⋊C₄	—	188	4	no	—
271	C₄⋊C₄	C₂×D₁₈₈	1	2	yes	—
272	C₄⋊C₄	C₂×D₁₈₈	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	188	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	188	4	no	—
278	C₄²	C₂×C₄×D₄₇	1	2	yes	—
279	C₂³	—	94	2	no	—
280	C₂³	#13094070	1	1	yes	characteristic
281	Q₈	—	47	4	no	—
282	Q₈	—	94	2	no	—
283	D₄	—	47	4	no	—
284	D₄	—	94	4	no	—
285	D₄	—	94	4	no	—
286	D₄	—	94	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	94	2	no	—
292	C₂×C₄	—	94	4	no	—
293	C₂×C₄	—	94	4	no	—
294	C₂×C₄	—	376	4	no	—
295	C₂×C₄	—	188	4	no	—
296	C₂×C₄	—	752	4	no	—
297	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
300	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
301	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
302	C₂×C₄	#13094071	1	1	yes	characteristic
303	C₈	—	4	2	no	—
304	C₈	—	94	4	no	—
305	C₈	C₄⋊C₄×D₄₇	1	2	yes	—
306	C₈	C₄⋊C₄×D₄₇	1	2	yes	—
307	C₈	C₄×D₁₈₈	1	2	yes	—
308	C₈	C₄×D₁₈₈	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	94	4	no	—
311	C₂²	—	188	4	no	—
312	C₂²	#15689603	1	1	yes	characteristic
313	C₂²	D₄×D₁₈₈	1	1	yes	characteristic
314	C₂²	#15689833	1	1	yes	characteristic
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	94	4	no	—
319	C₄	—	376	4	no	—
320	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
321	C₄	C₂×C₄×D₁₈₈	1	1	yes	characteristic
322	C₄	#15689778	1	1	yes	characteristic
323	C₄	Q₈×D₁₈₈	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	188	4	no	—
327	C₂	C₄⋊C₄×D₁₈₈	1	1	yes	characteristic
328	C₂	#20255175	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1772	C₂
socle	C₂²×C₄₇	D₄²
prime radical	C₂×#13094046	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1772	D₄₇
hypocenter	C₄₇	#22690
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772

Complements

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

N	H ≅ G/N	type
3. #20251980	C₂	semidirect
4. #20244173	C₂	semidirect
5. #20243622	C₂	semidirect
6. #20243552	C₂	semidirect
7. #20243444	C₂	semidirect
8. C₄₇×#1772	C₂	semidirect
12. #15689241	C₄	semidirect
39. #13094030	C₂×C₄	semidirect
49. #13093960	C₂×C₄	semidirect
56. #13093952	C₂×C₄	semidirect
71. C₄×C₈×C₄₇	C₂×C₄	semidirect
167. #1772	D₄₇	semidirect
196. C₄₇	#22690	semidirect
235. C₄×C₈	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689241	2
#20252265	2
#20251980	2
#20244173	2
#20243622	2
#20243552	2
#20243444	2
C₄₇×#1772	2
#22690	47