Subgroups — #26371682

Order 12032.

834 conjugacy classes of subgroups.

The subgroup lattice has 342 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	#20245647	C₂	1	4	yes	maximal, complemented
4	#20245645	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245643	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245640	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245637	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094046	—	2	1	no	—
11	C₂×#13094046	C₂²	1	2	yes	—
12	#15689673	—	2	1	no	—
13	#15689553	—	2	16	no	—
14	C₂×#13093970	C₂²	1	4	yes	—
15	C₂×#13093968	C₂²	1	4	yes	—
16	C₂×#13093967	C₂²	1	4	yes	—
17	C₂×#13093965	C₂²	1	4	yes	—
18	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
19	#15689507	—	2	1	no	—
20	#15689507	C₂²	1	2	yes	—
21	#15689503	C₂²	1	1	yes	characteristic
22	#15689500	C₂²	1	1	yes	characteristic
23	#15689496	—	2	1	no	—
24	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
27	C₂×#135×C₄₇	—	2	1	no	—
28	C₂×#135×C₄₇	C₂²	1	2	yes	—
29	C₄₇×#532	—	2	1	no	—
30	C₄₇×#495	—	2	1	no	—
31	C₄₇×#495	C₂²	1	2	yes	—
32	C₄₇×#494	C₂²	1	1	yes	characteristic
33	C₄₇×#493	C₂²	1	1	yes	characteristic
34	C₄₇×#492	—	2	1	no	—
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#10625350	—	2	2	no	—
37	#13094046	—	2	8	no	—
38	#13094046	—	4	4	no	—
39	C₂×Dic₁₈₈	—	2	1	no	—
40	C₂×Dic₁₈₈	C₂³	1	2	yes	—
41	C₂×#10625325	—	2	2	no	—
42	C₂×#10625325	—	2	2	no	—
43	C₂×D₃₇₆	—	2	1	no	—
44	C₂×D₃₇₆	C₂³	1	2	yes	—
45	#13094024	—	2	1	no	—
46	#13094021	—	2	1	no	—
47	#13094012	—	2	2	no	—
48	#13093998	—	4	2	no	—
49	C₄×D₁₈₈	—	4	2	no	—
50	#13093970	—	2	8	no	—
51	#13093968	—	2	8	no	—
52	#13093967	—	2	8	no	—
53	#13093965	—	2	8	no	—
54	C₂×C₄₇⋊C₁₆	D₄	1	8	yes	complemented
55	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
56	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
57	C₂×#41×C₄₇	—	2	2	no	—
58	#135×C₄₇	—	2	8	no	—
59	#135×C₄₇	—	4	4	no	—
60	C₂×Q₁₆×C₄₇	—	2	1	no	—
61	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₄₇	—	2	2	no	—
63	C₂×SD₁₆×C₄₇	—	2	2	no	—
64	C₂×D₈×C₄₇	—	2	1	no	—
65	C₂×D₈×C₄₇	C₂³	1	2	yes	—
66	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₄₇	—	2	1	no	—
69	#127×C₄₇	—	2	1	no	—
70	#124×C₄₇	—	2	2	no	—
71	#121×C₄₇	—	4	2	no	—
72	C₄×D₄×C₄₇	—	4	2	no	—
73	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
74	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
75	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
76	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	#41×C₄₇	—	4	4	no	—
79	#41×C₄₇	—	4	4	no	—
80	C₂×Q₈×C₄₇	—	2	2	no	—
81	C₂×D₄×C₄₇	—	2	2	no	—
82	C₂×D₄×C₄₇	—	4	2	no	—
83	C₂×D₄×C₄₇	—	4	2	no	—
84	C₂²×C₄×C₄₇	—	4	2	no	—
85	C₂²×C₄×C₄₇	D₈	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₂²⋊D₄₇	—	4	2	no	—
88	#10625350	—	4	4	no	—
89	#10625350	—	4	4	no	—
90	C₂×D₁₈₈	—	2	2	no	—
91	C₂×D₁₈₈	—	4	2	no	—
92	C₂×C₄×D₄₇	—	4	2	no	—
93	C₂×Dic₉₄	—	2	2	no	—
94	Q₁₆×C₄₇	—	2	4	no	—
95	Q₁₆×C₄₇	—	4	2	no	—
96	SD₁₆×C₄₇	—	4	4	no	—
97	SD₁₆×C₄₇	—	4	4	no	—
98	D₈×C₄₇	—	2	4	no	—
99	D₈×C₄₇	—	4	2	no	—
100	C₂×C₈×C₄₇	D₈	1	4	yes	—
101	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
103	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₄₇	—	4	2	no	—
107	C₂²⋊C₄×C₄₇	—	4	4	no	—
108	C₄²×C₄₇	D₈	1	2	yes	—
109	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
111	(C₂×D₄₇)⋊C₄	—	4	4	no	—
112	C₄⋊Dic₄₇	—	4	2	no	—
113	Dic₁₈₈	—	2	4	no	—
114	Dic₁₈₈	—	4	2	no	—
115	D₃₇₆	—	2	4	no	—
116	D₃₇₆	—	4	2	no	—
117	#10625325	—	4	4	no	—
118	#10625325	—	4	4	no	—
119	C₄₇⋊C₁₆	C₂×D₄	1	4	yes	—
120	C₄₇⋊C₁₆	C₂×D₄	1	4	yes	—
121	C₂³×C₄₇	—	4	2	no	—
122	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₄₇	—	4	2	no	—
124	Q₈×C₄₇	—	4	2	no	—
125	Q₈×C₄₇	—	4	2	no	—
126	D₄×C₄₇	—	4	2	no	—
127	D₄×C₄₇	—	4	2	no	—
128	D₄×C₄₇	—	8	4	no	—
129	D₄×C₄₇	—	8	4	no	—
130	C₂×C₄×C₄₇	—	2	4	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	8	4	no	—
133	C₂×C₄×C₄₇	#127	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₄₇	—	8	4	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	D₁₈₈	—	4	2	no	—
142	D₁₈₈	—	4	2	no	—
143	D₁₈₈	—	8	4	no	—
144	C₄×D₄₇	—	8	4	no	—
145	Dic₉₄	—	4	2	no	—
146	Dic₉₄	—	4	2	no	—
147	C₈×C₄₇	#127	1	2	yes	—
148	C₈×C₄₇	C₂×D₈	1	2	yes	—
149	C₈×C₄₇	C₂×D₈	1	2	yes	—
150	C₈×C₄₇	C₂²×D₄	1	2	yes	—
151	#33713	—	47	1	no	maximal, Sylow
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	—	8	2	no	—
154	C₂²×C₄₇	—	8	4	no	—
155	C₂²×C₄₇	#493	1	2	yes	—
156	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₄₇	—	8	2	no	—
158	C₂×D₄₇	—	8	4	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	8	2	no	—
161	C₄×C₄₇	#493	1	2	yes	—
162	C₄×C₄₇	#508	1	4	yes	complemented
163	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
164	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
165	Dic₄₇	—	8	2	no	—
166	C₂×#508	—	47	4	no	—
167	#3147	—	47	1	no	—
168	#3147	D₄₇	1	1	yes	characteristic, complemented
169	#2249	—	47	4	no	—
170	#2248	—	47	1	no	—
171	#2248	—	47	1	no	—
172	#2247	—	47	1	no	—
173	#2246	—	47	1	no	—
174	C₂×C₄×C₁₆	—	47	1	no	—
175	C₂×C₄₇	—	2	2	no	—
176	C₂×C₄₇	—	16	2	no	—
177	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
178	C₂×C₄₇	C₂×#508	1	2	yes	—
179	D₄₇	—	16	2	no	—
180	C₂×#135	—	2	1	no	—
181	C₂×#135	—	47	2	no	—
182	C₂×#135	—	94	1	no	—
183	C₂×#135	C₂×D₄₇	1	2	yes	—
184	#532	—	2	1	no	—
185	#532	—	94	1	no	—
186	#508	—	94	16	no	—
187	C₂×Q₃₂	—	47	4	no	—
188	C₂×SD₃₂	—	47	4	no	—
189	C₂×SD₃₂	—	47	4	no	—
190	C₂×D₁₆	—	47	4	no	—
191	C₂²×C₁₆	—	47	2	no	—
192	#495	—	2	1	no	—
193	#495	—	47	2	no	—
194	#495	—	94	1	no	—
195	#495	C₂×D₄₇	1	2	yes	—
196	#494	—	47	1	no	—
197	#494	C₂×D₄₇	1	1	yes	characteristic
198	#493	—	47	1	no	—
199	#493	C₂×D₄₇	1	1	yes	characteristic
200	#492	—	2	1	no	—
201	#492	—	94	1	no	—
202	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
203	C₄×C₁₆	—	47	1	no	—
204	C₄×C₁₆	—	47	1	no	—
205	C₄×C₁₆	—	47	2	no	—
206	C₄₇	#33713	1	1	yes	characteristic, complemented, Sylow
207	C₂×#41	—	2	2	no	—
208	C₂×#41	—	94	2	no	—
209	#135	—	2	8	no	—
210	#135	—	4	4	no	—
211	#135	—	94	8	no	—
212	#135	—	188	4	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	47	2	no	—
215	C₂×Q₁₆	—	94	1	no	—
216	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×SD₁₆	—	2	2	no	—
219	C₂×SD₁₆	—	94	2	no	—
220	C₂×SD₁₆	—	94	2	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	—	47	2	no	—
223	C₂×D₈	—	94	1	no	—
224	C₂×D₈	C₂²×D₄₇	1	2	yes	—
225	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	94	1	no	—
229	#127	—	2	1	no	—
230	#127	—	94	1	no	—
231	#124	—	2	2	no	—
232	#124	—	94	2	no	—
233	#121	—	4	2	no	—
234	#121	—	188	2	no	—
235	C₄×D₄	—	4	2	no	—
236	C₄×D₄	—	188	2	no	—
237	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
238	Q₃₂	—	94	8	no	—
239	SD₃₂	—	94	8	no	—
240	SD₃₂	—	94	8	no	—
241	D₁₆	—	94	8	no	—
242	C₂×C₁₆	—	47	8	no	—
243	C₂×C₁₆	—	47	2	no	—
244	C₂×C₁₆	—	47	2	no	—
245	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
246	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
249	#41	—	4	4	no	—
250	#41	—	4	4	no	—
251	#41	—	188	4	no	—
252	#41	—	188	4	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	94	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	94	2	no	—
259	C₂×D₄	—	188	2	no	—
260	C₂×D₄	—	188	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	188	2	no	—
263	C₂²×C₄	#10625334	1	2	yes	—
264	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
265	Q₁₆	—	2	4	no	—
266	Q₁₆	—	4	2	no	—
267	Q₁₆	—	94	4	no	—
268	Q₁₆	—	188	2	no	—
269	SD₁₆	—	4	4	no	—
270	SD₁₆	—	4	4	no	—
271	SD₁₆	—	188	4	no	—
272	SD₁₆	—	188	4	no	—
273	D₈	—	2	4	no	—
274	D₈	—	4	2	no	—
275	D₈	—	94	4	no	—
276	D₈	—	188	2	no	—
277	C₂×C₈	#10625334	1	4	yes	—
278	C₂×C₈	D₄×D₄₇	1	4	yes	—
279	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
280	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
281	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
282	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	188	2	no	—
285	C₂²⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	—	188	4	no	—
287	C₄²	#10625334	1	2	yes	—
288	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
289	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
290	C₁₆	—	47	4	no	—
291	C₁₆	—	47	4	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	188	2	no	—
294	C₂³	C₂×#10625334	1	1	yes	characteristic
295	Q₈	—	4	2	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	188	2	no	—
298	Q₈	—	188	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	4	no	—
302	D₄	—	8	4	no	—
303	D₄	—	376	4	no	—
304	D₄	—	376	4	no	—
305	D₄	—	188	2	no	—
306	D₄	—	188	2	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	—	376	4	no	—
311	C₂×C₄	—	188	2	no	—
312	C₂×C₄	#13094020	1	2	yes	—
313	C₂×C₄	C₂×#10625334	1	2	yes	—
314	C₂×C₄	C₂×#10625334	1	2	yes	—
315	C₂×C₄	C₂×#10625334	1	2	yes	—
316	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
318	C₈	#13094020	1	2	yes	—
319	C₈	C₂×#10625334	1	2	yes	—
320	C₈	C₂×#10625334	1	2	yes	—
321	C₈	C₂×D₄×D₄₇	1	2	yes	—
322	C₂²	—	2	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	4	no	—
325	C₂²	—	376	2	no	—
326	C₂²	—	376	4	no	—
327	C₂²	#15689499	1	2	yes	—
328	C₂²	C₂²×#10625334	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	376	2	no	—
332	C₄	#15689499	1	2	yes	—
333	C₄	#15689553	1	4	yes	complemented
334	C₄	C₂×#13094020	1	1	yes	characteristic
335	C₄	C₂²×#10625334	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	16	2	no	—
338	C₂	—	752	2	no	—
339	C₂	C₂×#15689499	1	1	yes	characteristic
340	C₂	C₂×#15689553	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	#33713
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. #20245647	C₂	semidirect
4. #20245645	C₂	semidirect
5. #20245643	C₂	semidirect
6. #20245640	C₂	semidirect
7. #20245637	C₂	semidirect
8. C₂×C₄×C₄₇⋊C₁₆	C₂	semidirect
9. C₄₇×#3147	C₂	semidirect
54. C₂×C₄₇⋊C₁₆	D₄	semidirect
162. C₄×C₄₇	#508	semidirect
168. #3147	D₄₇	semidirect
206. C₄₇	#33713	semidirect
333. C₄	#15689553	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
#20245647	2
#20245645	2
#20245643	2
#20245640	2
#20245637	2
C₂×C₄×C₄₇⋊C₁₆	2
C₄₇×#3147	2
#33713	47