Subgroups — #26225162

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253911	C₂	1	1	yes	maximal, characteristic
2	#20252320	C₂	1	2	yes	maximal
3	#20244367	C₂	1	2	yes	maximal
4	#20244137	C₂	1	1	yes	maximal, characteristic
5	#20244135	C₂	1	2	yes	maximal
6	#20244134	C₂	1	1	yes	maximal, characteristic
7	#20243561	C₂	1	1	yes	maximal, characteristic
8	#20243273	C₂	1	2	yes	maximal
9	#20243272	C₂	1	1	yes	maximal, characteristic
10	#20243076	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1773	C₂	1	1	yes	maximal, characteristic
12	#15689676	C₂²	1	2	yes	—
13	#15689531	C₄	1	4	yes	complemented
14	#15689530	C₄	1	4	yes	complemented
15	#15689251	—	2	4	no	—
16	C₈×Dic₉₄	—	2	4	no	—
17	C₂×#13093960	C₂²	1	1	yes	characteristic
18	C₂×#13093959	C₂²	1	2	yes	—
19	#15689161	C₂²	1	1	yes	characteristic
20	#15689159	C₂²	1	2	yes	—
21	#15689151	C₂²	1	2	yes	—
22	#15689150	C₂²	1	2	yes	—
23	#15689146	C₂²	1	2	yes	—
24	#15689078	C₂²	1	2	yes	—
25	C₂×#13093934	C₂²	1	1	yes	characteristic
26	C₂×#13093934	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
30	#15688969	C₂²	1	1	yes	characteristic
31	#15688884	C₂²	1	2	yes	—
32	#15688883	C₂²	1	2	yes	—
33	#15688882	C₂²	1	2	yes	—
34	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#427	C₂²	1	1	yes	characteristic
36	C₄₇×#405	C₂²	1	2	yes	—
37	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#339	C₂²	1	2	yes	—
39	C₂×#10625329	C₂³	1	2	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	#13094023	C₂×C₄	1	4	yes	—
43	#13094015	C₂×C₄	1	4	yes	—
44	#13094001	—	2	4	no	—
45	C₄×Dic₉₄	—	2	4	no	—
46	#13093983	C₂³	1	2	yes	—
47	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
49	#13093960	—	2	2	no	—
50	#13093959	—	2	4	no	—
51	#13093958	C₂×C₄	1	2	yes	—
52	#13093957	C₂×C₄	1	4	yes	—
53	#13093956	C₂×C₄	1	2	yes	—
54	#13093955	C₂×C₄	1	4	yes	—
55	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
56	Dic₄₇⋊C₈	—	2	4	no	—
57	Dic₄₇⋊C₈	—	2	4	no	—
58	#13093934	—	2	2	no	—
59	#13093934	C₂×C₄	1	2	yes	—
60	#13093934	C₂×C₄	1	2	yes	—
61	#13093934	Q₈	1	4	yes	complemented
62	C₄×C₄₇⋊C₈	—	2	2	no	—
63	C₈×Dic₄₇	—	2	2	no	—
64	C₈×Dic₄₇	—	2	2	no	—
65	C₈×Dic₄₇	D₄	1	4	yes	—
66	C₄₇⋊#95	—	2	2	no	—
67	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
70	#117×C₄₇	C₂³	1	2	yes	—
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	#108×C₄₇	—	2	2	no	—
73	#107×C₄₇	C₂×C₄	1	2	yes	—
74	#106×C₄₇	C₂×C₄	1	2	yes	—
75	#97×C₄₇	—	2	2	no	—
76	#97×C₄₇	Q₈	1	4	yes	complemented
77	C₄×C₈×C₄₇	—	2	2	no	—
78	C₂²×C₄×C₄₇	—	2	1	no	—
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₇	—	2	1	no	—
81	C₂²×Dic₄₇	#41	1	2	yes	—
82	C₂×Dic₉₄	—	2	4	no	—
83	M₁₆×C₄₇	—	2	2	no	—
84	M₁₆×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₄₇	#41	1	2	yes	—
93	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
94	C₂²⋊C₄×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	—	2	2	no	—
96	C₄²×C₄₇	—	2	1	no	—
97	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
98	C₂²⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	2	4	no	—
100	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
101	Dic₄₇⋊C₄	—	2	4	no	—
102	Dic₄₇⋊C₄	—	2	4	no	—
103	C₄×Dic₄₇	—	2	2	no	—
104	C₄×Dic₄₇	—	2	1	no	—
105	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
106	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
107	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
108	#10625329	—	2	2	no	—
109	#10625329	—	2	2	no	—
110	C₂×C₄₇⋊C₈	—	2	1	no	—
111	C₂×C₄₇⋊C₈	—	2	1	no	—
112	C₂×C₄₇⋊C₈	—	2	2	no	—
113	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
115	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
116	C₂×C₄₇⋊C₈	#41	1	2	yes	—
117	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	C₂³×C₄	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	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	2	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	C₄×D₄	1	4	yes	—
132	Dic₉₄	—	4	8	no	—
133	C₈×C₄₇	—	4	2	no	—
134	C₈×C₄₇	C₄×Q₈	1	2	yes	—
135	C₈×C₄₇	C₄×Q₈	1	2	yes	—
136	C₈×C₄₇	C₄⋊Q₈	1	4	yes	—
137	C₄₇⋊C₈	—	2	2	no	—
138	C₄₇⋊C₈	—	4	2	no	—
139	C₄₇⋊C₈	C₄×Q₈	1	4	yes	—
140	#21737	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
143	C₂²×C₄₇	#533	1	1	yes	characteristic
144	C₂²×C₄₇	#533	1	1	yes	characteristic
145	C₄×C₄₇	—	4	2	no	—
146	C₄×C₄₇	—	4	4	no	—
147	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
148	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
150	C₄×C₄₇	#532	1	1	yes	characteristic
151	Dic₄₇	—	2	4	no	—
152	Dic₄₇	—	4	2	no	—
153	Dic₄₇	—	4	4	no	—
154	#3162	—	47	1	no	—
155	#2960	—	47	2	no	—
156	#1995	—	47	2	no	—
157	#1945	—	47	1	no	—
158	#1945	—	47	1	no	—
159	#1945	—	47	2	no	—
160	#1849	—	47	1	no	—
161	#1773	—	47	2	no	—
162	#1773	—	47	1	no	—
163	#1773	D₄₇	1	1	yes	characteristic
164	#1726	—	47	1	no	—
165	C₂×C₄₇	—	2	1	no	—
166	C₂×C₄₇	—	2	1	no	—
167	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
168	C₂×C₄₇	#2955	1	2	yes	—
169	#533	—	47	2	no	—
170	#501	—	47	4	no	—
171	#501	—	47	4	no	—
172	#446	—	94	4	no	—
173	C₈×Q₈	—	94	4	no	—
174	C₂×#108	—	47	2	no	—
175	C₂×#108	—	47	1	no	—
176	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
177	#427	—	47	1	no	—
178	#427	—	47	2	no	—
179	#427	C₂×D₄₇	1	1	yes	characteristic
180	#424	—	47	2	no	—
181	#424	—	47	2	no	—
182	#423	—	47	2	no	—
183	#405	—	47	2	no	—
184	#405	C₂×D₄₇	1	2	yes	—
185	C₂×#97	—	47	1	no	—
186	C₂×#97	—	47	1	no	—
187	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
188	C₂×C₄×C₈	—	47	1	no	—
189	C₂×C₄×C₈	—	47	1	no	—
190	C₂×C₄×C₈	—	47	1	no	—
191	#376	—	47	1	no	—
192	#339	—	47	2	no	—
193	#339	—	47	2	no	—
194	#339	—	47	2	no	—
195	#339	C₂×D₄₇	1	2	yes	—
196	C₄₇	#21737	1	1	yes	characteristic, complemented, Sylow
197	C₂×M₁₆	—	47	2	no	—
198	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
199	C₂²×C₈	—	47	1	no	—
200	C₂²×C₈	—	47	1	no	—
201	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
203	C₄⋊Q₈	—	47	4	no	—
204	#125	—	47	4	no	—
205	C₂²⋊Q₈	—	94	4	no	—
206	C₄×Q₈	—	94	4	no	—
207	#117	—	47	2	no	—
208	#117	C₂²×D₄₇	1	2	yes	—
209	C₂×C₄²	—	47	1	no	—
210	C₂×C₄²	—	47	1	no	—
211	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
212	#108	—	2	2	no	—
213	#108	—	94	4	no	—
214	#108	—	94	2	no	—
215	#107	—	47	2	no	—
216	#107	—	47	4	no	—
217	#107	C₄×D₄₇	1	2	yes	—
218	#106	—	47	2	no	—
219	#106	—	47	4	no	—
220	#106	C₄×D₄₇	1	2	yes	—
221	C₄⋊C₈	—	94	4	no	—
222	C₄⋊C₈	—	94	4	no	—
223	C₄⋊C₈	—	94	4	no	—
224	#97	—	2	2	no	—
225	#97	—	47	2	no	—
226	#97	—	47	2	no	—
227	#97	—	47	4	no	—
228	#97	—	94	2	no	—
229	#97	Dic₉₄	1	4	yes	complemented
230	C₄×C₈	—	2	2	no	—
231	C₄×C₈	—	47	4	no	—
232	C₄×C₈	—	94	2	no	—
233	C₄×C₈	—	94	2	no	—
234	C₄×C₈	—	94	2	no	—
235	#95	—	94	2	no	—
236	C₂×Q₈	—	94	4	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	47	2	no	—
239	C₂²×C₄	—	94	1	no	—
240	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
241	M₁₆	—	2	2	no	—
242	M₁₆	—	2	2	no	—
243	M₁₆	—	94	2	no	—
244	M₁₆	—	94	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	47	2	no	—
247	C₂×C₈	—	47	2	no	—
248	C₂×C₈	—	47	2	no	—
249	C₂×C₈	—	47	2	no	—
250	C₂×C₈	—	94	1	no	—
251	C₂×C₈	—	94	1	no	—
252	C₂×C₈	—	94	2	no	—
253	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
254	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
257	C₂×C₈	#10625350	1	2	yes	—
258	C₂×C₈	D₄×D₄₇	1	2	yes	—
259	C₂×C₈	Q₈×D₄₇	1	2	yes	—
260	C₄⋊C₄	—	47	4	no	—
261	C₄⋊C₄	—	94	4	no	—
262	C₄⋊C₄	—	94	4	no	—
263	C₄⋊C₄	—	94	4	no	—
264	C₄⋊C₄	C₂×C₄×D₄₇	1	4	yes	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	94	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	47	2	no	—
270	C₄²	—	47	2	no	—
271	C₄²	—	47	2	no	—
272	C₄²	—	94	2	no	—
273	C₄²	—	94	1	no	—
274	C₄²	C₂×Dic₉₄	1	2	yes	—
275	C₂³	#41×D₄₇	1	1	yes	characteristic
276	Q₈	—	188	8	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	47	4	no	—
281	C₂×C₄	—	94	4	no	—
282	C₂×C₄	—	94	2	no	—
283	C₂×C₄	—	188	2	no	—
284	C₂×C₄	—	188	2	no	—
285	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
286	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
287	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
288	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
289	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
290	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
291	C₈	—	4	2	no	—
292	C₈	—	47	4	no	—
293	C₈	—	94	2	no	—
294	C₈	—	188	2	no	—
295	C₈	C₄×Dic₉₄	1	2	yes	—
296	C₈	C₄×Dic₉₄	1	2	yes	—
297	C₈	#13094023	1	4	yes	—
298	C₂²	—	2	2	no	—
299	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
300	C₂²	#15689676	1	1	yes	characteristic
301	C₂²	#15689677	1	1	yes	characteristic
302	C₄	—	4	2	no	—
303	C₄	—	4	4	no	—
304	C₄	—	94	4	no	—
305	C₄	—	188	2	no	—
306	C₄	—	188	4	no	—
307	C₄	C₄×Q₈×D₄₇	1	1	yes	characteristic
308	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
309	C₄	C₂×#13094023	1	1	yes	characteristic
310	C₄	#15689670	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	C₄×#13094023	1	1	yes	characteristic
314	C₂	#20252288	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1773	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1773	D₄₇
hypocenter	C₄₇	#21737
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1773

Complements

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

N	H ≅ G/N	type
13. #15689531	C₄	semidirect
14. #15689530	C₄	semidirect
61. #13093934	Q₈	semidirect
76. #97×C₄₇	Q₈	semidirect
196. C₄₇	#21737	semidirect
229. #97	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20253911	2
#20252320	2
#20244367	2
#20244137	2
#20244135	2
#20244134	2
#20243561	2
#20243273	2
#20243272	2
#20243076	2
C₄₇×#1773	2
#21737	47