Subgroups — #26109319

Order 12032.

542 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	C₂×C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic
2	#20243385	C₂	1	2	yes	maximal
3	C₂×#15688874	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241271	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₄₇	—	2	1	no	—
7	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
8	C₈×D₁₈₈	—	2	2	no	—
9	C₈×D₁₈₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#13093944	—	2	2	no	—
13	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
14	#15688885	—	2	2	no	—
15	#15688874	—	2	2	no	—
16	C₂×#106×C₄₇	—	2	1	no	—
17	C₂×#106×C₄₇	—	2	1	no	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#384	C₂²	1	2	yes	—
20	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
21	C₂²×C₄×D₄₇	—	2	1	no	—
22	C₂²×C₄₇⋊C₈	—	2	1	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂×C₈×D₄₇	—	2	4	no	—
25	C₂×C₈×D₄₇	—	4	2	no	—
26	C₂×C₈×D₄₇	—	4	2	no	—
27	C₄×D₁₈₈	—	2	1	no	—
28	C₄×D₁₈₈	—	2	1	no	—
29	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
30	C₄×D₁₈₈	Q₈	1	2	yes	—
31	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
32	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
37	#13093944	—	4	4	no	—
38	(C₂×D₄₇)⋊C₈	—	2	4	no	—
39	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	#106×C₄₇	—	4	2	no	—
46	#106×C₄₇	—	4	2	no	—
47	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₄₇	D₄	1	2	yes	—
49	C₂³×D₄₇	—	2	1	no	—
50	C₂²×C₄×C₄₇	—	4	2	no	—
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
54	C₂²×Dic₄₇	—	2	1	no	—
55	C₂×D₁₈₈	—	2	1	no	—
56	C₂×D₁₈₈	—	2	2	no	—
57	C₂×D₁₈₈	C₄⋊C₄	1	2	yes	—
58	C₂×D₁₈₈	SD₁₆	1	4	yes	—
59	C₂×C₄×D₄₇	—	2	2	no	—
60	C₂×C₄×D₄₇	—	2	2	no	—
61	C₂×C₄×D₄₇	—	4	1	no	—
62	C₂×C₄×D₄₇	—	4	1	no	—
63	C₂×C₄×D₄₇	—	4	1	no	—
64	C₂×C₄×D₄₇	—	4	1	no	—
65	C₂×C₈×C₄₇	—	2	1	no	—
66	C₂×C₈×C₄₇	—	2	1	no	—
67	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₄₇	D₈	1	2	yes	—
70	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
71	C₄⋊C₄×C₄₇	—	4	2	no	—
72	C₄⋊C₄×C₄₇	—	4	2	no	—
73	C₄⋊C₄×C₄₇	—	4	2	no	—
74	C₄⋊C₄×C₄₇	—	4	2	no	—
75	C₄⋊C₄×C₄₇	—	8	2	no	—
76	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₄₇	#41	1	1	yes	characteristic
80	(C₂×D₄₇)⋊C₄	—	2	2	no	—
81	(C₂×D₄₇)⋊C₄	—	2	2	no	—
82	C₄⋊Dic₄₇	—	2	1	no	—
83	C₄⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
84	C₂×C₄₇⋊C₈	—	2	2	no	—
85	C₂×C₄₇⋊C₈	—	4	1	no	—
86	C₂×C₄₇⋊C₈	—	4	1	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₈×D₄₇	—	4	8	no	—
91	C₂³×C₄₇	#121	1	1	yes	characteristic
92	C₂²×D₄₇	—	2	2	no	—
93	C₂²×D₄₇	—	2	4	no	—
94	C₂²×D₄₇	—	4	1	no	—
95	C₂²×D₄₇	—	4	1	no	—
96	C₂²×D₄₇	—	4	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	8	2	no	—
99	C₂×C₄×C₄₇	—	8	2	no	—
100	C₂×C₄×C₄₇	—	8	2	no	—
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₄₇	#135	1	2	yes	—
111	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
113	C₂×Dic₄₇	—	2	2	no	—
114	C₂×Dic₄₇	—	4	1	no	—
115	C₂×Dic₄₇	—	4	1	no	—
116	D₁₈₈	—	2	2	no	—
117	D₁₈₈	—	2	4	no	—
118	D₁₈₈	#106	1	4	yes	complemented
119	C₄×D₄₇	—	4	2	no	—
120	C₄×D₄₇	—	4	2	no	—
121	C₄×D₄₇	—	4	2	no	—
122	C₄×D₄₇	—	4	2	no	—
123	C₈×C₄₇	—	2	2	no	—
124	C₈×C₄₇	#102	1	4	yes	complemented
125	C₄₇⋊C₈	—	4	2	no	—
126	C₄₇⋊C₈	—	8	2	no	—
127	#8642	—	47	1	no	maximal, Sylow
128	C₂²×C₄₇	#380	1	1	yes	characteristic
129	C₂²×C₄₇	#459	1	1	yes	characteristic
130	C₂²×C₄₇	#461	1	1	yes	characteristic
131	C₂²×C₄₇	#465	1	2	yes	—
132	C₂²×C₄₇	#471	1	2	yes	—
133	C₂×D₄₇	—	2	4	no	—
134	C₂×D₄₇	—	4	2	no	—
135	C₂×D₄₇	—	4	2	no	—
136	C₂×D₄₇	—	4	4	no	—
137	C₂×D₄₇	—	4	4	no	—
138	C₄×C₄₇	—	2	1	no	—
139	C₄×C₄₇	—	2	1	no	—
140	C₄×C₄₇	—	16	2	no	—
141	C₄×C₄₇	C₂×#102	1	1	yes	characteristic
142	C₄×C₄₇	#417	1	1	yes	characteristic
143	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
144	C₄×C₄₇	#427	1	1	yes	characteristic
145	C₄×C₄₇	#465	1	2	yes	—
146	C₄×C₄₇	#474	1	1	yes	characteristic
147	C₄×C₄₇	#476	1	1	yes	characteristic
148	Dic₄₇	—	4	2	no	—
149	C₂×C₈×D₄	—	47	1	no	—
150	#1844	D₄₇	1	1	yes	characteristic, complemented
151	#1802	—	47	2	no	—
152	C₂×#334	—	47	1	no	—
153	#1296	—	47	2	no	—
154	C₂×C₄₇	#1665	1	2	yes	—
155	C₂×C₄₇	#1671	1	2	yes	—
156	C₂×C₄₇	#1802	1	1	yes	characteristic
157	C₂×C₄₇	#1807	1	1	yes	characteristic
158	C₂×C₄₇	#1815	1	1	yes	characteristic
159	D₄₇	—	4	4	no	—
160	C₂³×C₈	—	94	1	no	—
161	C₂×C₄×D₄	—	47	1	no	—
162	C₈×D₄	—	47	4	no	—
163	C₈×D₄	—	94	2	no	—
164	C₂×#106	—	2	1	no	—
165	C₂×#106	—	2	1	no	—
166	C₂×C₄⋊C₈	—	47	1	no	—
167	C₂×C₄⋊C₈	—	47	2	no	—
168	C₂×#102	—	94	2	no	—
169	C₂×C₂²⋊C₈	—	94	1	no	—
170	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
171	#384	C₂×D₄₇	1	2	yes	—
172	#340	—	94	2	no	—
173	#334	—	94	2	no	—
174	C₄₇	#8642	1	1	yes	characteristic, complemented, Sylow
175	C₂²×D₄	—	47	1	no	—
176	C₂³×C₄	—	94	1	no	—
177	C₂²×C₈	—	94	1	no	—
178	C₂²×C₈	—	94	4	no	—
179	C₂²×C₈	—	94	2	no	—
180	C₂²×C₈	—	188	2	no	—
181	C₂²×C₈	—	188	2	no	—
182	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
183	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
184	C₄×D₄	—	47	2	no	—
185	C₄×D₄	—	47	2	no	—
186	C₄×D₄	—	94	1	no	—
187	C₄×D₄	—	94	1	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	47	1	no	—
192	C₂×C₂²⋊C₄	—	94	1	no	—
193	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
194	#106	—	4	2	no	—
195	#106	—	4	2	no	—
196	C₄⋊C₈	—	47	2	no	—
197	C₄⋊C₈	—	94	1	no	—
198	C₄⋊C₈	—	94	2	no	—
199	C₄⋊C₈	—	94	2	no	—
200	#102	—	188	4	no	—
201	C₂²⋊C₈	—	94	4	no	—
202	C₄×C₈	C₄×D₄₇	1	2	yes	—
203	C₄×C₈	D₁₈₈	1	2	yes	—
204	C₂⁴	—	94	1	no	—
205	C₂×D₄	—	47	2	no	—
206	C₂×D₄	—	47	4	no	—
207	C₂×D₄	—	94	1	no	—
208	C₂×D₄	—	94	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	94	1	no	—
211	C₂²×C₄	—	94	2	no	—
212	C₂²×C₄	—	94	2	no	—
213	C₂²×C₄	—	188	1	no	—
214	C₂²×C₄	—	188	1	no	—
215	C₂²×C₄	—	188	1	no	—
216	C₂²×C₄	—	188	1	no	—
217	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	#10625352	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	94	2	no	—
223	C₂×C₈	—	188	2	no	—
224	C₂×C₈	—	188	2	no	—
225	C₂×C₈	—	188	2	no	—
226	C₂×C₈	—	188	1	no	—
227	C₂×C₈	—	188	1	no	—
228	C₂×C₈	—	188	8	no	—
229	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
230	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
231	C₂×C₈	#10625334	1	2	yes	—
232	C₂×C₈	#10625336	1	2	yes	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	8	2	no	—
238	C₄⋊C₄	—	47	2	no	—
239	C₄⋊C₄	—	94	1	no	—
240	C₂²⋊C₄	—	94	2	no	—
241	C₂²⋊C₄	—	94	2	no	—
242	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
243	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
244	C₄²	Q₈×D₄₇	1	1	yes	characteristic
245	C₄²	#10625354	1	1	yes	characteristic
246	C₂³	—	94	2	no	—
247	C₂³	—	94	4	no	—
248	C₂³	—	188	1	no	—
249	C₂³	—	188	1	no	—
250	C₂³	—	188	2	no	—
251	C₂³	#13093995	1	1	yes	characteristic
252	D₄	—	47	4	no	—
253	D₄	—	94	4	no	—
254	D₄	—	94	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	94	2	no	—
260	C₂×C₄	—	188	1	no	—
261	C₂×C₄	—	188	1	no	—
262	C₂×C₄	—	188	2	no	—
263	C₂×C₄	—	188	2	no	—
264	C₂×C₄	—	188	2	no	—
265	C₂×C₄	—	188	2	no	—
266	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
268	C₂×C₄	#13093982	1	1	yes	characteristic
269	C₂×C₄	#13093999	1	1	yes	characteristic
270	C₂×C₄	#13094000	1	1	yes	characteristic
271	C₂×C₄	#13094004	1	1	yes	characteristic
272	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
273	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
274	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
275	C₂×C₄	#13094045	1	2	yes	—
276	C₂×C₄	#13094050	1	1	yes	characteristic
277	C₂×C₄	#13094055	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	376	2	no	—
280	C₈	—	188	2	no	—
281	C₈	#13093944	1	4	yes	complemented
282	C₂²	—	94	4	no	—
283	C₂²	—	188	4	no	—
284	C₂²	—	188	4	no	—
285	C₂²	—	188	2	no	—
286	C₂²	—	188	2	no	—
287	C₂²	#15688980	1	1	yes	characteristic
288	C₂²	#15689315	1	1	yes	characteristic
289	C₂²	#15689325	1	1	yes	characteristic
290	C₂²	#15689349	1	2	yes	—
291	C₂²	#15689383	1	2	yes	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	16	2	no	—
295	C₄	—	188	2	no	—
296	C₄	C₂×#13093944	1	1	yes	characteristic
297	C₄	#15689122	1	1	yes	characteristic
298	C₄	#106×D₄₇	1	1	yes	characteristic
299	C₄	#15689158	1	1	yes	characteristic
300	C₄	#15689353	1	2	yes	—
301	C₄	#15689400	1	1	yes	characteristic
302	C₄	#15689410	1	1	yes	characteristic
303	C₂	—	188	4	no	—
304	C₂	#20242726	1	2	yes	—
305	C₂	#20242760	1	2	yes	—
306	C₂	#20243385	1	1	yes	characteristic
307	C₂	#20243410	1	1	yes	characteristic
308	C₂	#20243448	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093995
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1844	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂²×D₁₈₈	D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#8642
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
3. C₂×#15688874	C₂	semidirect
4. #20241271	C₂	semidirect
5. C₄₇×#1844	C₂	semidirect
9. C₈×D₁₈₈	C₄	semidirect
118. D₁₈₈	#106	semidirect
124. C₈×C₄₇	#102	semidirect
150. #1844	D₄₇	semidirect
174. C₄₇	#8642	semidirect
281. C₈	#13093944	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₁₈₈	2
#20243385	2
C₂×#15688874	2
#20241271	2
C₄₇×#1844	2
#8642	47