Subgroups — #26216654

Order 12032.

608 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251814	C₂	1	1	yes	maximal, characteristic
2	#20243538	C₂	1	2	yes	maximal, complemented
3	#20243513	C₂	1	2	yes	maximal, complemented
4	#20243493	C₂	1	2	yes	maximal
5	#20243434	C₂	1	1	yes	maximal, characteristic
6	#20243275	C₂	1	2	yes	maximal, complemented
7	#20243191	C₂	1	2	yes	maximal
8	#20243144	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1813	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094031	C₂²	1	1	yes	characteristic
11	#41×Dic₄₇	C₂²	1	1	yes	characteristic
12	#15689161	C₂²	1	2	yes	—
13	#15689097	C₂²	1	2	yes	—
14	#15689081	C₄	1	2	yes	complemented
15	#15689081	C₂²	1	4	yes	—
16	#15689081	C₂²	1	1	yes	characteristic
17	#15689078	C₄	1	2	yes	complemented
18	#15689078	C₂²	1	1	yes	characteristic
19	M₁₆×Dic₄₇	C₄	1	2	yes	complemented
20	M₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
21	C₂×#13093934	C₄	1	2	yes	complemented
22	C₂×#13093934	C₂²	1	1	yes	characteristic
23	#15688895	C₂²	1	4	yes	—
24	#15688884	C₂²	1	4	yes	—
25	#15688880	C₂²	1	4	yes	—
26	#15688880	C₂²	1	2	yes	—
27	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#108×C₄₇	C₂²	1	2	yes	—
29	C₄₇×#413	C₂²	1	2	yes	—
30	C₄₇×#338	C₂²	1	2	yes	—
31	#13094031	—	2	4	no	—
32	C₂×#10625329	C₂×C₄	1	2	yes	—
33	C₂×#10625329	C₂³	1	1	yes	characteristic
34	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
36	Q₈×Dic₄₇	—	2	1	no	—
37	D₄×Dic₄₇	—	2	2	no	—
38	D₄×Dic₄₇	—	2	1	no	—
39	#13093983	C₂×C₄	1	2	yes	—
40	#13093983	C₂³	1	4	yes	—
41	#13093983	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
44	#13093958	—	2	2	no	—
45	#13093956	—	2	2	no	—
46	#13093939	—	2	4	no	—
47	#13093934	—	2	4	no	—
48	#13093934	C₂×C₄	1	1	yes	characteristic
49	#13093934	C₂×C₄	1	2	yes	—
50	#13093934	C₂×C₄	1	2	yes	—
51	#13093934	C₂×C₄	1	2	yes	—
52	#13093934	C₂×C₄	1	1	yes	characteristic
53	#13093934	C₂×C₄	1	1	yes	characteristic
54	#13093934	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₄₇	—	2	4	no	—
56	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
57	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
58	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
59	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
60	#131×C₄₇	—	2	2	no	—
61	#131×C₄₇	—	2	2	no	—
62	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
63	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
64	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	#108×C₄₇	—	2	2	no	—
68	#108×C₄₇	—	2	2	no	—
69	#101×C₄₇	—	2	2	no	—
70	#100×C₄₇	—	2	2	no	—
71	#41×C₄₇	—	2	2	no	—
72	#41×C₄₇	—	2	2	no	—
73	C₂×Q₈×C₄₇	#41	1	1	yes	characteristic
74	C₂×D₄×C₄₇	#41	1	2	yes	—
75	C₂×D₄×C₄₇	#41	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂²×Dic₄₇	—	2	1	no	—
80	M₁₆×C₄₇	—	2	2	no	—
81	M₁₆×C₄₇	—	2	4	no	—
82	M₁₆×C₄₇	—	2	4	no	—
83	M₁₆×C₄₇	—	2	4	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	—	2	2	no	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₄₇	#41	1	2	yes	—
91	C₂×C₈×C₄₇	#41	1	2	yes	—
92	C₂²⋊Dic₄₇	—	2	2	no	—
93	C₂²⋊Dic₄₇	—	2	1	no	—
94	C₂²⋊Dic₄₇	—	2	4	no	—
95	C₄⋊Dic₄₇	—	2	2	no	—
96	C₄⋊Dic₄₇	—	2	1	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄×Dic₄₇	—	2	2	no	—
99	C₄×Dic₄₇	—	2	1	no	—
100	C₄×Dic₄₇	—	2	4	no	—
101	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
102	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
103	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
104	#10625329	—	2	2	no	—
105	#10625329	—	2	4	no	—
106	C₂×C₄₇⋊C₈	—	2	4	no	—
107	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄₇⋊C₈	#41	1	4	yes	—
111	C₂³×C₄₇	#117	1	2	yes	—
112	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
113	Q₈×C₄₇	—	2	2	no	—
114	D₄×C₄₇	—	2	4	no	—
115	D₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	#117	1	2	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
121	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₄₇	—	2	1	no	—
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	2	1	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	8	1	no	—
128	C₂×Dic₄₇	—	8	2	no	—
129	C₈×C₄₇	—	2	2	no	—
130	C₈×C₄₇	—	4	4	no	—
131	C₈×C₄₇	#117	1	1	yes	characteristic
132	C₈×C₄₇	#117	1	1	yes	characteristic
133	C₈×C₄₇	#117	1	2	yes	—
134	C₄₇⋊C₈	—	2	2	no	—
135	C₄₇⋊C₈	#117	1	2	yes	—
136	C₄₇⋊C₈	#117	1	2	yes	—
137	#20788	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	4	1	no	—
140	C₂²×C₄₇	—	4	2	no	—
141	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
142	C₂²×C₄₇	#529	1	2	yes	—
143	C₄×C₄₇	—	2	2	no	—
144	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
145	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
146	C₄×C₄₇	#529	1	2	yes	—
147	Dic₄₇	—	4	2	no	—
148	Dic₄₇	—	4	1	no	—
149	Dic₄₇	—	4	1	no	—
150	Dic₄₇	—	8	4	no	—
151	#2875	—	47	1	no	—
152	#1841	—	47	2	no	—
153	#1833	—	47	2	no	—
154	#1827	—	47	2	no	—
155	#1813	D₄₇	1	1	yes	characteristic
156	#1812	—	47	1	no	—
157	#1773	—	47	2	no	—
158	#1757	—	47	2	no	—
159	#1747	—	47	2	no	—
160	C₂×C₄₇	—	2	1	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	—	4	2	no	—
163	C₂×C₄₇	#2283	1	1	yes	characteristic
164	C₂×#131	—	47	1	no	—
165	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
166	C₄×#41	—	47	1	no	—
167	C₂×#108	C₂×D₄₇	1	2	yes	—
168	#427	—	47	2	no	—
169	#413	C₂×D₄₇	1	2	yes	—
170	#410	—	47	2	no	—
171	#405	—	47	2	no	—
172	#405	—	47	1	no	—
173	#405	—	47	2	no	—
174	#405	—	47	1	no	—
175	#405	—	47	4	no	—
176	C₄×M₁₆	—	47	2	no	—
177	C₄×M₁₆	—	47	1	no	—
178	C₂×#97	—	47	2	no	—
179	C₂×#97	—	47	1	no	—
180	#344	—	47	4	no	—
181	#339	—	47	4	no	—
182	#338	C₂×D₄₇	1	2	yes	—
183	H(3,ℤ/4)	—	47	4	no	—
184	H(3,ℤ/4)	—	47	2	no	—
185	C₄₇	#20788	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	C₂²×D₄₇	1	1	yes	characteristic
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	#131	—	94	4	no	—
190	C₂×M₁₆	—	47	2	no	—
191	C₂×M₁₆	—	47	1	no	—
192	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
193	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
194	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	—	47	1	no	—
196	C₂²×C₈	—	47	2	no	—
197	C₂²×C₈	C₄×D₄₇	1	2	yes	—
198	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
199	C₄×Q₈	—	94	1	no	—
200	C₄×D₄	—	94	1	no	—
201	C₄×D₄	—	94	2	no	—
202	#117	—	47	2	no	—
203	#117	—	47	1	no	—
204	#117	—	47	4	no	—
205	C₂×C₄²	—	47	2	no	—
206	C₂×C₄²	—	47	1	no	—
207	#108	—	2	2	no	—
208	#108	—	2	2	no	—
209	#107	—	94	2	no	—
210	#106	—	94	2	no	—
211	#101	—	2	2	no	—
212	#100	—	2	2	no	—
213	#99	—	94	4	no	—
214	#97	—	47	1	no	—
215	#97	—	47	1	no	—
216	#97	—	47	2	no	—
217	#97	—	47	1	no	—
218	#97	—	47	2	no	—
219	#97	—	47	1	no	—
220	#97	—	47	2	no	—
221	#97	—	94	4	no	—
222	C₄×C₈	—	47	2	no	—
223	C₄×C₈	—	47	2	no	—
224	C₄×C₈	—	47	2	no	—
225	C₄×C₈	—	94	4	no	—
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	#10625352	1	1	yes	characteristic
229	C₂×D₄	#10625352	1	1	yes	characteristic
230	C₂×D₄	#10625354	1	2	yes	—
231	C₂²×C₄	—	94	2	no	—
232	C₂²×C₄	—	94	1	no	—
233	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
234	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	94	4	no	—
240	M₁₆	—	94	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	47	1	no	—
245	C₂×C₈	—	47	1	no	—
246	C₂×C₈	—	47	4	no	—
247	C₂×C₈	—	47	2	no	—
248	C₂×C₈	—	94	4	no	—
249	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
251	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
252	C₂×C₈	#10625352	1	2	yes	—
253	C₂×C₈	#10625352	1	2	yes	—
254	C₄⋊C₄	—	94	2	no	—
255	C₄⋊C₄	—	94	1	no	—
256	C₄⋊C₄	—	94	4	no	—
257	C₂²⋊C₄	—	94	2	no	—
258	C₂²⋊C₄	—	94	1	no	—
259	C₂²⋊C₄	—	94	4	no	—
260	C₄²	—	47	2	no	—
261	C₄²	—	47	1	no	—
262	C₄²	—	47	1	no	—
263	C₄²	—	94	2	no	—
264	C₄²	—	94	1	no	—
265	C₄²	—	94	4	no	—
266	C₂³	#13093982	1	2	yes	—
267	C₂³	C₂×#10625354	1	1	yes	characteristic
268	Q₈	—	2	2	no	—
269	D₄	—	2	4	no	—
270	D₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	94	1	no	—
274	C₂×C₄	—	94	2	no	—
275	C₂×C₄	—	94	1	no	—
276	C₂×C₄	—	376	1	no	—
277	C₂×C₄	—	376	2	no	—
278	C₂×C₄	—	188	4	no	—
279	C₂×C₄	#13093982	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
281	C₂×C₄	#41×D₄₇	1	2	yes	—
282	C₂×C₄	C₂×#10625352	1	2	yes	—
283	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	4	4	no	—
286	C₈	—	47	2	no	—
287	C₈	—	47	2	no	—
288	C₈	—	94	2	no	—
289	C₈	#13093980	1	1	yes	characteristic
290	C₈	#13093980	1	1	yes	characteristic
291	C₈	#13093980	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₂×#13093982	1	1	yes	characteristic
296	C₂²	#15689659	1	2	yes	—
297	C₄	—	2	2	no	—
298	C₄	—	376	4	no	—
299	C₄	—	188	1	no	—
300	C₄	—	188	2	no	—
301	C₄	—	188	1	no	—
302	C₄	#117×D₄₇	1	1	yes	characteristic
303	C₄	C₂×#13093980	1	1	yes	characteristic
304	C₄	#15689654	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	4	2	no	—
308	C₂	#20245783	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1813	C₂
socle	C₂×C₄₇	#2283
prime radical	C₂×#41×C₄₇	C₂³
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1813	D₄₇
hypocenter	C₄₇	#20788
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #1813

Complements

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

N	H ≅ G/N	type
2. #20243538	C₂	semidirect
3. #20243513	C₂	semidirect
6. #20243275	C₂	semidirect
8. #20243144	C₂	semidirect
14. #15689081	C₄	semidirect
17. #15689078	C₄	semidirect
19. M₁₆×Dic₄₇	C₄	semidirect
21. C₂×#13093934	C₄	semidirect
185. C₄₇	#20788	semidirect

Maximal subgroups

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

subgroup	index
#20251814	2
#20243538	2
#20243513	2
#20243493	2
#20243434	2
#20243275	2
#20243191	2
#20243144	2
C₄₇×#1813	2
#20788	47