Subgroups — #26371700

Order 12032.

826 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689553	C₂	1	4	yes	maximal, complemented
2	#20253816	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245660	C₂	1	2	yes	maximal, complemented
4	#20245657	C₂	1	2	yes	maximal, complemented
5	#20245655	C₂	1	2	yes	maximal, complemented
6	#20245652	C₂	1	2	yes	maximal, complemented
7	C₂×#15688899	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	—	2	1	no	—
10	C₂×#13094046	C₂²	1	2	yes	—
11	#15689673	—	2	1	no	—
12	#15689553	—	2	16	no	—
13	C₂×#13093970	C₂²	1	4	yes	—
14	C₂×#13093968	C₂²	1	4	yes	—
15	C₂×#13093967	C₂²	1	4	yes	—
16	C₂×#13093965	C₂²	1	4	yes	—
17	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
18	#15689507	—	2	1	no	—
19	#15689507	C₂²	1	2	yes	—
20	#15689503	C₂²	1	1	yes	characteristic
21	#15689500	C₂²	1	1	yes	characteristic
22	#15689496	—	2	1	no	—
23	#15688899	C₂²	1	2	yes	—
24	#15688899	C₂²	1	2	yes	—
25	C₂×#135×C₄₇	—	2	1	no	—
26	C₂×#135×C₄₇	C₂²	1	2	yes	—
27	C₄₇×#532	—	2	1	no	—
28	C₄₇×#495	—	2	1	no	—
29	C₄₇×#495	C₂²	1	2	yes	—
30	C₄₇×#494	C₂²	1	1	yes	characteristic
31	C₄₇×#493	C₂²	1	1	yes	characteristic
32	C₄₇×#492	—	2	1	no	—
33	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#10625350	—	2	2	no	—
35	#13094046	—	2	8	no	—
36	#13094046	—	4	4	no	—
37	C₂×Dic₁₈₈	—	2	1	no	—
38	C₂×Dic₁₈₈	C₂³	1	2	yes	—
39	C₂×#10625325	—	2	2	no	—
40	C₂×#10625325	—	2	2	no	—
41	C₂×D₃₇₆	—	2	1	no	—
42	C₂×D₃₇₆	C₂³	1	2	yes	—
43	#13094024	—	2	1	no	—
44	#13094021	—	2	1	no	—
45	#13094012	—	2	2	no	—
46	#13093998	—	4	2	no	—
47	C₄×D₁₈₈	—	4	2	no	—
48	#13093970	—	2	8	no	—
49	#13093968	—	2	8	no	—
50	#13093967	—	2	8	no	—
51	#13093965	—	2	8	no	—
52	C₂×C₄₇⋊C₁₆	D₄	1	8	yes	complemented
53	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
54	C₂×#41×C₄₇	—	2	2	no	—
55	#135×C₄₇	—	2	8	no	—
56	#135×C₄₇	—	4	4	no	—
57	C₂×Q₁₆×C₄₇	—	2	1	no	—
58	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
59	C₂×SD₁₆×C₄₇	—	2	2	no	—
60	C₂×SD₁₆×C₄₇	—	2	2	no	—
61	C₂×D₈×C₄₇	—	2	1	no	—
62	C₂×D₈×C₄₇	C₂³	1	2	yes	—
63	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₄₇	—	2	1	no	—
66	#127×C₄₇	—	2	1	no	—
67	#124×C₄₇	—	2	2	no	—
68	#121×C₄₇	—	4	2	no	—
69	C₄×D₄×C₄₇	—	4	2	no	—
70	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
71	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
72	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
73	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	#41×C₄₇	—	4	4	no	—
76	#41×C₄₇	—	4	4	no	—
77	C₂×Q₈×C₄₇	—	2	2	no	—
78	C₂×D₄×C₄₇	—	2	2	no	—
79	C₂×D₄×C₄₇	—	4	2	no	—
80	C₂×D₄×C₄₇	—	4	2	no	—
81	C₂²×C₄×C₄₇	—	4	2	no	—
82	C₂²×C₄×C₄₇	D₈	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₄₇	—	4	2	no	—
85	#10625350	—	4	4	no	—
86	#10625350	—	4	4	no	—
87	C₂×D₁₈₈	—	2	2	no	—
88	C₂×D₁₈₈	—	4	2	no	—
89	C₂×C₄×D₄₇	—	4	2	no	—
90	C₂×Dic₉₄	—	2	2	no	—
91	Q₁₆×C₄₇	—	2	4	no	—
92	Q₁₆×C₄₇	—	4	2	no	—
93	SD₁₆×C₄₇	—	4	4	no	—
94	SD₁₆×C₄₇	—	4	4	no	—
95	D₈×C₄₇	—	2	4	no	—
96	D₈×C₄₇	—	4	2	no	—
97	C₂×C₈×C₄₇	D₈	1	4	yes	—
98	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₄₇	—	4	2	no	—
104	C₂²⋊C₄×C₄₇	—	4	4	no	—
105	C₄²×C₄₇	D₈	1	2	yes	—
106	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
108	(C₂×D₄₇)⋊C₄	—	4	4	no	—
109	C₄⋊Dic₄₇	—	4	2	no	—
110	Dic₁₈₈	—	2	4	no	—
111	Dic₁₈₈	—	4	2	no	—
112	D₃₇₆	—	2	4	no	—
113	D₃₇₆	—	4	2	no	—
114	#10625325	—	4	4	no	—
115	#10625325	—	4	4	no	—
116	C₄₇⋊C₁₆	C₂×D₄	1	8	yes	—
117	C₂³×C₄₇	—	4	2	no	—
118	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
119	C₂²×D₄₇	—	4	2	no	—
120	Q₈×C₄₇	—	4	2	no	—
121	Q₈×C₄₇	—	4	2	no	—
122	D₄×C₄₇	—	4	2	no	—
123	D₄×C₄₇	—	4	2	no	—
124	D₄×C₄₇	—	8	4	no	—
125	D₄×C₄₇	—	8	4	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	8	4	no	—
129	C₂×C₄×C₄₇	#127	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₄₇	—	8	4	no	—
136	C₂×Dic₄₇	—	4	2	no	—
137	D₁₈₈	—	4	2	no	—
138	D₁₈₈	—	4	2	no	—
139	D₁₈₈	—	8	4	no	—
140	C₄×D₄₇	—	8	4	no	—
141	Dic₉₄	—	4	2	no	—
142	Dic₉₄	—	4	2	no	—
143	C₈×C₄₇	#127	1	2	yes	—
144	C₈×C₄₇	C₂×D₈	1	2	yes	—
145	C₈×C₄₇	C₂×D₈	1	2	yes	—
146	C₈×C₄₇	C₂²×D₄	1	2	yes	—
147	#33716	—	47	1	no	maximal, Sylow
148	C₂²×C₄₇	—	2	2	no	—
149	C₂²×C₄₇	—	8	2	no	—
150	C₂²×C₄₇	—	8	4	no	—
151	C₂²×C₄₇	#493	1	2	yes	—
152	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₄₇	—	8	2	no	—
154	C₂×D₄₇	—	8	4	no	—
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	8	2	no	—
158	C₄×C₄₇	#493	1	2	yes	—
159	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
160	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
161	Dic₄₇	—	8	2	no	—
162	C₂×#508	—	47	4	no	—
163	#3147	—	47	1	no	—
164	#3147	D₄₇	1	1	yes	characteristic, complemented
165	#2251	—	47	2	no	—
166	#2251	—	47	2	no	—
167	#2250	—	47	2	no	—
168	#2250	—	47	2	no	—
169	C₂×#346	—	47	1	no	—
170	C₂×C₄₇	—	2	2	no	—
171	C₂×C₄₇	—	16	2	no	—
172	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
173	C₂×C₄₇	#3414	1	2	yes	—
174	D₄₇	—	16	2	no	—
175	C₂×#135	—	2	1	no	—
176	C₂×#135	—	47	2	no	—
177	C₂×#135	—	94	1	no	—
178	C₂×#135	C₂×D₄₇	1	2	yes	—
179	#532	—	2	1	no	—
180	#532	—	94	1	no	—
181	#508	—	94	16	no	—
182	C₂×Q₃₂	—	47	4	no	—
183	C₂×SD₃₂	—	47	4	no	—
184	C₂×SD₃₂	—	47	4	no	—
185	C₂×D₁₆	—	47	4	no	—
186	C₂²×C₁₆	—	47	2	no	—
187	#495	—	2	1	no	—
188	#495	—	47	2	no	—
189	#495	—	94	1	no	—
190	#495	C₂×D₄₇	1	2	yes	—
191	#494	—	47	1	no	—
192	#494	C₂×D₄₇	1	1	yes	characteristic
193	#493	—	47	1	no	—
194	#493	C₂×D₄₇	1	1	yes	characteristic
195	#492	—	2	1	no	—
196	#492	—	94	1	no	—
197	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
198	#346	—	47	2	no	—
199	#346	—	47	2	no	—
200	C₄₇	#33716	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	2	2	no	—
202	C₂×#41	—	94	2	no	—
203	#135	—	2	8	no	—
204	#135	—	4	4	no	—
205	#135	—	94	8	no	—
206	#135	—	188	4	no	—
207	C₂×Q₁₆	—	2	1	no	—
208	C₂×Q₁₆	—	47	2	no	—
209	C₂×Q₁₆	—	94	1	no	—
210	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×SD₁₆	—	94	2	no	—
214	C₂×SD₁₆	—	94	2	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	47	2	no	—
217	C₂×D₈	—	94	1	no	—
218	C₂×D₈	C₂²×D₄₇	1	2	yes	—
219	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	94	1	no	—
223	#127	—	2	1	no	—
224	#127	—	94	1	no	—
225	#124	—	2	2	no	—
226	#124	—	94	2	no	—
227	#121	—	4	2	no	—
228	#121	—	188	2	no	—
229	C₄×D₄	—	4	2	no	—
230	C₄×D₄	—	188	2	no	—
231	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
232	Q₃₂	—	94	8	no	—
233	SD₃₂	—	94	8	no	—
234	SD₃₂	—	94	8	no	—
235	D₁₆	—	94	8	no	—
236	C₂×C₁₆	—	47	8	no	—
237	C₂×C₁₆	—	47	4	no	—
238	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
239	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
242	#41	—	4	4	no	—
243	#41	—	4	4	no	—
244	#41	—	188	4	no	—
245	#41	—	188	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	94	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	94	2	no	—
252	C₂×D₄	—	188	2	no	—
253	C₂×D₄	—	188	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	188	2	no	—
256	C₂²×C₄	#10625334	1	2	yes	—
257	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
258	Q₁₆	—	2	4	no	—
259	Q₁₆	—	4	2	no	—
260	Q₁₆	—	94	4	no	—
261	Q₁₆	—	188	2	no	—
262	SD₁₆	—	4	4	no	—
263	SD₁₆	—	4	4	no	—
264	SD₁₆	—	188	4	no	—
265	SD₁₆	—	188	4	no	—
266	D₈	—	2	4	no	—
267	D₈	—	4	2	no	—
268	D₈	—	94	4	no	—
269	D₈	—	188	2	no	—
270	C₂×C₈	#10625334	1	4	yes	—
271	C₂×C₈	D₄×D₄₇	1	4	yes	—
272	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
273	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
274	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	188	2	no	—
278	C₂²⋊C₄	—	4	4	no	—
279	C₂²⋊C₄	—	188	4	no	—
280	C₄²	#10625334	1	2	yes	—
281	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
283	C₁₆	—	47	8	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	188	2	no	—
286	C₂³	C₂×#10625334	1	1	yes	characteristic
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	188	2	no	—
290	Q₈	—	188	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	376	4	no	—
296	D₄	—	376	4	no	—
297	D₄	—	188	2	no	—
298	D₄	—	188	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	—	376	4	no	—
303	C₂×C₄	—	188	2	no	—
304	C₂×C₄	#13094020	1	2	yes	—
305	C₂×C₄	C₂×#10625334	1	2	yes	—
306	C₂×C₄	C₂×#10625334	1	2	yes	—
307	C₂×C₄	C₂×#10625334	1	2	yes	—
308	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
310	C₈	#13094020	1	2	yes	—
311	C₈	C₂×#10625334	1	2	yes	—
312	C₈	C₂×#10625334	1	2	yes	—
313	C₈	C₂×D₄×D₄₇	1	2	yes	—
314	C₂²	—	2	2	no	—
315	C₂²	—	8	4	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	376	4	no	—
318	C₂²	—	376	2	no	—
319	C₂²	#15689499	1	2	yes	—
320	C₂²	C₂²×#10625334	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	376	2	no	—
325	C₄	#15689499	1	2	yes	—
326	C₄	C₂×#13094020	1	1	yes	characteristic
327	C₄	C₂²×#10625334	1	1	yes	characteristic
328	C₂	—	2	2	no	—
329	C₂	—	16	2	no	—
330	C₂	—	752	2	no	—
331	C₂	C₂×#15689499	1	1	yes	characteristic
332	C₂	#20256522	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3147	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3147	D₄₇
hypocenter	C₄₇	#33716
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#15689553	C₂	semidirect
2. #20253816	C₂	semidirect
3. #20245660	C₂	semidirect
4. #20245657	C₂	semidirect
5. #20245655	C₂	semidirect
6. #20245652	C₂	semidirect
7. C₂×#15688899	C₂	semidirect
8. C₄₇×#3147	C₂	semidirect
52. C₂×C₄₇⋊C₁₆	D₄	semidirect
164. #3147	D₄₇	semidirect
200. C₄₇	#33716	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689553	2
#20253816	2
#20245660	2
#20245657	2
#20245655	2
#20245652	2
C₂×#15688899	2
C₄₇×#3147	2
#33716	47