Subgroups — #26221133

Order 12032.

870 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245786	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093983	C₂	1	1	yes	maximal, characteristic
3	#20244158	C₂	1	4	yes	maximal, complemented
4	#20243589	C₂	1	4	yes	maximal
5	C₄×#13093958	C₂	1	2	yes	maximal, complemented
6	C₄×#13093956	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093983	C₂²	1	2	yes	—
9	C₂×#13093958	C₂²	1	4	yes	—
10	C₂×#13093956	C₂²	1	4	yes	—
11	#15688991	—	2	1	no	—
12	#15688989	C₂²	1	2	yes	—
13	#15688986	—	2	1	no	—
14	#15688983	C₂²	1	2	yes	—
15	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
16	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
18	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
19	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
20	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
21	#15688969	—	2	1	no	—
22	#15688969	—	2	1	no	—
23	C₄²×Dic₄₇	—	2	1	no	—
24	#15688886	C₂²	1	4	yes	—
25	#15688886	C₂²	1	4	yes	—
26	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#431	C₄	1	8	yes	complemented
28	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	#13093983	—	2	8	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
36	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	—	2	2	no	—
38	C₂×C₄×Dic₄₇	—	2	2	no	—
39	C₂×C₄×Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	—	2	2	no	—
41	#13093958	—	2	8	no	—
42	#13093956	—	2	8	no	—
43	C₄₇⋊#95	—	2	2	no	—
44	C₄₇⋊#95	—	2	2	no	—
45	C₄₇⋊#95	—	2	2	no	—
46	C₄₇⋊#95	—	2	2	no	—
47	C₄₇⋊#95	—	2	2	no	—
48	C₄₇⋊#95	—	2	2	no	—
49	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
61	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
62	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
63	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
64	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	4	2	no	—
68	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	4	no	—
79	C₂²×Dic₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	2	4	no	—
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
84	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
85	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
86	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
89	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
90	C₂²⋊Dic₄₇	—	2	8	no	—
91	C₂²⋊Dic₄₇	—	4	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	2	4	no	—
94	C₄⋊Dic₄₇	—	2	4	no	—
95	C₄⋊Dic₄₇	#41	1	8	yes	—
96	C₄×Dic₄₇	—	2	8	no	—
97	C₄×Dic₄₇	—	4	4	no	—
98	C₄×Dic₄₇	—	4	4	no	—
99	C₄×Dic₄₇	—	4	4	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	—	2	2	no	—
102	C₂³×C₄₇	—	2	2	no	—
103	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
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	2	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	8	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
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×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₄₇	#135	1	4	yes	—
119	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
122	C₂×Dic₄₇	—	2	8	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	4	4	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₈×C₄₇	—	2	4	no	—
129	C₈×C₄₇	—	2	4	no	—
130	#21360	—	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	#525	1	2	yes	—
137	C₂²×C₄₇	C₂×#135	1	2	yes	—
138	C₂²×C₄₇	#575	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	#427	1	4	yes	—
144	C₄×C₄₇	C₄×#41	1	2	yes	—
145	C₄×C₄₇	#533	1	2	yes	—
146	Dic₄₇	—	4	8	no	—
147	Dic₄₇	—	8	4	no	—
148	C₂×#431	D₄₇	1	1	yes	characteristic
149	#2283	—	47	1	no	—
150	C₄×#117	—	47	1	no	—
151	#1949	—	47	4	no	—
152	#1854	—	47	4	no	—
153	C₄×#107	—	47	2	no	—
154	C₄×#106	—	47	2	no	—
155	C₂×C₄₇	—	4	2	no	—
156	C₂×C₄₇	#2296	1	1	yes	characteristic
157	C₂×C₄₇	C₂×#427	1	1	yes	characteristic
158	C₂×C₄₇	#2911	1	1	yes	characteristic
159	C₂×C₄₇	#3102	1	4	yes	—
160	C₂×#117	—	47	2	no	—
161	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
162	#431	Dic₄₇	1	8	yes	complemented
163	C₂×#107	—	47	4	no	—
164	C₂×#106	—	47	4	no	—
165	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
166	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
167	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
169	#383	—	94	1	no	—
170	#382	—	47	2	no	—
171	#381	—	94	1	no	—
172	#380	—	47	2	no	—
173	C₄×C₄⋊C₄	—	47	2	no	—
174	C₄×C₄⋊C₄	—	47	2	no	—
175	C₄×C₄⋊C₄	—	47	2	no	—
176	C₄×C₂²⋊C₄	—	94	1	no	—
177	C₄×C₂²⋊C₄	—	94	1	no	—
178	C₄×C₂²⋊C₄	—	94	1	no	—
179	#376	—	94	1	no	—
180	#376	—	94	1	no	—
181	C₄³	—	94	1	no	—
182	#340	—	47	4	no	—
183	#340	—	47	4	no	—
184	C₄₇	#21360	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
186	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
187	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
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×C₄⋊C₄	—	47	4	no	—
193	C₂×C₂²⋊C₄	—	94	2	no	—
194	C₂×C₂²⋊C₄	—	94	2	no	—
195	C₂×C₄²	—	94	2	no	—
196	C₂×C₄²	—	94	2	no	—
197	C₂×C₄²	—	94	2	no	—
198	C₂×C₄²	—	94	2	no	—
199	C₂×C₄²	Dic₉₄	1	4	yes	—
200	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
201	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
202	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
203	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
204	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
205	#107	—	94	8	no	—
206	#106	—	94	8	no	—
207	C₄⋊C₈	C₂×Dic₄₇	1	8	yes	—
208	C₂²⋊C₈	C₂×Dic₄₇	1	8	yes	—
209	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
210	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
211	#95	—	94	2	no	—
212	#95	—	94	2	no	—
213	#95	—	94	2	no	—
214	#95	—	94	2	no	—
215	#95	—	94	2	no	—
216	#95	—	94	2	no	—
217	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	94	4	no	—
222	C₂²×C₄	—	94	4	no	—
223	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
224	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
225	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
226	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
231	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
232	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
236	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
237	C₄⋊C₄	—	47	8	no	—
238	C₄⋊C₄	—	94	4	no	—
239	C₄⋊C₄	—	94	4	no	—
240	C₄⋊C₄	—	94	4	no	—
241	C₂²⋊C₄	—	94	8	no	—
242	C₂²⋊C₄	—	188	4	no	—
243	C₄²	—	94	8	no	—
244	C₄²	—	188	4	no	—
245	C₄²	—	188	4	no	—
246	C₄²	—	188	4	no	—
247	C₄²	C₄⋊Dic₄₇	1	4	yes	—
248	C₄²	C₄⋊Dic₄₇	1	4	yes	—
249	C₄²	C₂×Dic₉₄	1	2	yes	—
250	C₄²	C₂×Dic₉₄	1	2	yes	—
251	C₄²	C₂²×Dic₄₇	1	2	yes	—
252	C₄²	C₂²×Dic₄₇	1	2	yes	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
257	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	94	8	no	—
267	C₂×C₄	—	188	4	no	—
268	C₂×C₄	—	188	4	no	—
269	C₂×C₄	—	188	4	no	—
270	C₂×C₄	—	188	4	no	—
271	C₂×C₄	—	188	4	no	—
272	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
276	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
277	C₂×C₄	#13094046	1	4	yes	—
278	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
279	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
280	C₂×C₄	#41×D₄₇	1	4	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	4	4	no	—
287	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
288	C₂²	#15689635	1	2	yes	—
289	C₂²	C₂×#13094046	1	2	yes	—
290	C₂²	#15689923	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	376	4	no	—
296	C₄	—	188	8	no	—
297	C₄	#15689161	1	4	yes	—
298	C₄	#41×Dic₄₇	1	2	yes	—
299	C₄	#15689676	1	2	yes	—
300	C₂	—	4	2	no	—
301	C₂	#20245872	1	1	yes	characteristic
302	C₂	C₂×#15689161	1	1	yes	characteristic
303	C₂	#20251978	1	1	yes	characteristic
304	C₂	#20253408	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#21360
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20244158	C₂	semidirect
5. C₄×#13093958	C₂	semidirect
6. C₄×#13093956	C₂	semidirect
27. C₄₇×#431	C₄	semidirect
162. #431	Dic₄₇	semidirect
184. C₄₇	#21360	semidirect

Maximal subgroups

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

subgroup	index
#20245786	2
C₄×#13093983	2
#20244158	2
#20243589	2
C₄×#13093958	2
C₄×#13093956	2
C₂×C₄₇×#431	2
#21360	47