Subgroups — #26370772

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093965	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689540	C₂	1	1	yes	maximal, characteristic, complemented
3	#20253036	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245489	C₂	1	4	yes	maximal, complemented
5	#20245472	C₂	1	4	yes	maximal, complemented
6	#20245170	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245104	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245103	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#3058	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
11	#15689687	—	2	1	no	—
12	C₂×#13093965	—	2	4	no	—
13	C₂×#13093965	C₂²	1	2	yes	—
14	C₂×#13093965	C₂²	1	2	yes	—
15	#15689540	—	2	4	no	—
16	C₂×#13093964	C₂²	1	1	yes	characteristic
17	C₂×#13093964	C₂²	1	2	yes	complemented
18	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	complemented
20	#15689376	—	2	2	no	—
21	#15689359	C₂²	1	4	yes	complemented
22	#15689161	C₂²	1	1	yes	characteristic
23	#15689129	—	2	1	no	—
24	#15688950	C₂²	1	2	yes	complemented
25	#15688948	C₂²	1	2	yes	complemented
26	#15688924	C₂²	1	4	yes	complemented
27	#15688923	C₂²	1	4	yes	complemented
28	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#535	—	2	1	no	—
31	C₄₇×#469	—	2	2	no	—
32	C₄₇×#466	C₂²	1	4	yes	complemented
33	C₄₇×#427	C₂²	1	1	yes	characteristic
34	C₄₇×#418	—	2	1	no	—
35	C₂²×D₁₈₈	—	2	1	no	—
36	C₂×D₃₇₆	—	2	2	no	—
37	C₂×D₃₇₆	—	2	2	no	—
38	C₂×D₃₇₆	—	2	1	no	—
39	C₂×D₃₇₆	C₂³	1	2	yes	—
40	#13094020	—	4	1	no	—
41	#13094013	—	4	1	no	—
42	#13093998	—	2	4	no	—
43	#13093992	—	4	2	no	—
44	#13093983	—	2	1	no	—
45	#13093965	—	2	4	no	—
46	#13093965	—	2	4	no	—
47	#13093964	—	2	2	no	—
48	#13093964	—	2	4	no	—
49	C₂×C₄₇⋊C₁₆	—	2	4	no	—
50	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	complemented
51	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
54	#13093958	C₂³	1	2	yes	—
55	#13093956	—	2	1	no	—
56	#13093946	—	2	4	no	—
57	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
58	C₂²×D₄×C₄₇	—	2	1	no	—
59	C₂×D₈×C₄₇	—	2	2	no	—
60	C₂×D₈×C₄₇	—	2	1	no	—
61	C₂×D₈×C₄₇	—	2	2	no	—
62	C₂×D₈×C₄₇	C₂³	1	2	yes	—
63	#131×C₄₇	—	2	2	no	—
64	#131×C₄₇	—	2	2	no	—
65	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
66	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	#127×C₄₇	—	4	1	no	—
70	#124×C₄₇	—	4	1	no	—
71	#121×C₄₇	—	2	4	no	—
72	#120×C₄₇	—	4	2	no	—
73	#117×C₄₇	—	2	1	no	—
74	#107×C₄₇	C₂³	1	2	yes	—
75	#106×C₄₇	—	2	1	no	—
76	#102×C₄₇	—	2	4	no	—
77	C₂⁴×C₄₇	—	4	1	no	—
78	C₂³×D₄₇	—	4	1	no	—
79	#41×C₄₇	—	2	2	no	—
80	#41×C₄₇	—	2	2	no	—
81	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
82	C₂×D₄×C₄₇	—	2	2	no	—
83	C₂×D₄×C₄₇	—	4	4	no	—
84	C₂×D₄×C₄₇	—	4	2	no	—
85	C₂×D₄×C₄₇	—	4	1	no	—
86	C₂×D₄×C₄₇	—	4	2	no	—
87	C₂×D₄×C₄₇	D₈	1	2	yes	—
88	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₄₇	—	4	4	no	—
92	C₂×D₁₈₈	—	2	2	no	—
93	C₂×D₁₈₈	—	4	2	no	—
94	C₂×D₁₈₈	—	4	2	no	—
95	C₂×D₁₈₈	—	4	1	no	—
96	D₈×C₄₇	—	2	2	no	—
97	D₈×C₄₇	—	2	2	no	—
98	D₈×C₄₇	—	4	2	no	—
99	M₁₆×C₄₇	—	2	2	no	—
100	M₁₆×C₄₇	—	2	4	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₄₇	#41	1	2	yes	—
107	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₄₇	—	2	2	no	—
109	C₂²⋊C₄×C₄₇	—	4	1	no	—
110	C₂²⋊C₄×C₄₇	—	4	4	no	—
111	C₄²×C₄₇	—	4	1	no	—
112	C₂²⋊Dic₄₇	—	4	1	no	—
113	(C₂×D₄₇)⋊C₄	—	4	4	no	—
114	C₄⋊Dic₄₇	—	2	2	no	—
115	C₄×Dic₄₇	—	4	1	no	—
116	D₃₇₆	—	2	2	no	—
117	D₃₇₆	—	2	2	no	—
118	D₃₇₆	—	4	2	no	—
119	C₄₇⋊C₁₆	—	2	2	no	—
120	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
121	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
122	C₂³×C₄₇	—	4	2	no	—
123	C₂³×C₄₇	—	8	2	no	—
124	C₂³×C₄₇	—	16	2	no	—
125	C₂³×C₄₇	C₂×D₈	1	2	yes	—
126	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
127	C₂²×D₄₇	—	4	2	no	—
128	C₂²×D₄₇	—	8	2	no	—
129	C₂²×D₄₇	—	16	2	no	—
130	Q₈×C₄₇	—	2	2	no	—
131	D₄×C₄₇	—	2	4	no	—
132	D₄×C₄₇	—	2	2	no	—
133	D₄×C₄₇	—	4	2	no	—
134	D₄×C₄₇	—	4	2	no	—
135	D₄×C₄₇	—	8	2	no	—
136	D₄×C₄₇	—	16	4	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	4	2	no	—
140	C₂×C₄×C₄₇	#121	1	2	yes	—
141	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₄₇	#135	1	2	yes	—
144	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₄₇	—	16	4	no	—
146	C₂×Dic₄₇	—	4	2	no	—
147	D₁₈₈	—	4	2	no	—
148	D₁₈₈	—	4	2	no	—
149	D₁₈₈	—	8	2	no	—
150	C₈×C₄₇	—	2	2	no	—
151	C₈×C₄₇	#121	1	2	yes	—
152	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
154	#33607	—	47	1	no	maximal, Sylow
155	C₂²×C₄₇	—	2	2	no	—
156	C₂²×C₄₇	—	2	4	no	—
157	C₂²×C₄₇	—	2	2	no	—
158	C₂²×C₄₇	—	8	4	no	—
159	C₂²×C₄₇	—	8	2	no	—
160	C₂²×C₄₇	—	16	4	no	—
161	C₂²×C₄₇	#466	1	2	yes	—
162	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
163	C₂×D₄₇	—	8	4	no	—
164	C₂×D₄₇	—	8	2	no	—
165	C₂×D₄₇	—	16	4	no	—
166	C₄×C₄₇	—	2	2	no	—
167	C₄×C₄₇	—	8	2	no	—
168	C₄×C₄₇	#466	1	2	yes	—
169	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
170	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
171	Dic₄₇	—	8	2	no	—
172	C₂²×D₁₆	—	47	1	no	—
173	C₂×#504	—	47	1	no	—
174	#3058	—	47	1	no	—
175	#3058	D₄₇	1	1	yes	characteristic, complemented
176	#2218	—	47	4	no	—
177	#2215	—	47	4	no	—
178	#2158	—	47	1	no	—
179	#2140	—	47	1	no	—
180	#2140	—	47	1	no	—
181	C₂×C₄₇	—	2	2	no	—
182	C₂×C₄₇	—	4	2	no	—
183	C₂×C₄₇	—	16	2	no	—
184	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
185	C₂×C₄₇	#3415	1	2	yes	—
186	D₄₇	—	16	2	no	—
187	C₂²×D₈	—	47	1	no	—
188	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
189	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
190	#535	—	2	1	no	—
191	#535	—	94	1	no	—
192	C₂×D₁₆	—	47	2	no	—
193	C₂×D₁₆	—	47	2	no	—
194	C₂×D₁₆	—	94	4	no	—
195	#504	—	94	4	no	—
196	C₂×M₃₂	—	47	1	no	—
197	C₂×M₃₂	—	47	2	no	—
198	C₂²×C₁₆	—	47	1	no	—
199	C₂²×C₁₆	—	47	2	no	—
200	#469	—	2	2	no	—
201	#469	—	94	2	no	—
202	#466	—	47	4	no	—
203	#466	C₂×D₄₇	1	4	yes	complemented
204	#427	—	47	1	no	—
205	#427	C₂×D₄₇	1	1	yes	characteristic
206	#418	—	2	1	no	—
207	#418	—	94	1	no	—
208	#367	—	47	2	no	—
209	#366	—	47	2	no	—
210	#357	—	47	4	no	—
211	#357	—	47	4	no	—
212	C₄₇	#33607	1	1	yes	characteristic, complemented, Sylow
213	C₂×#41	C₂²⋊D₄₇	1	1	yes	characteristic
214	C₂²×D₄	—	2	1	no	—
215	C₂²×D₄	—	94	1	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂×D₈	—	2	2	no	—
218	C₂×D₈	—	2	1	no	—
219	C₂×D₈	—	47	2	no	—
220	C₂×D₈	—	94	2	no	—
221	C₂×D₈	—	94	2	no	—
222	C₂×D₈	—	94	1	no	—
223	C₂×D₈	C₂²×D₄₇	1	2	yes	—
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
227	C₂×M₁₆	C₂²⋊D₄₇	1	2	yes	—
228	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
229	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
230	#127	—	4	1	no	—
231	#127	—	188	1	no	—
232	#124	—	4	1	no	—
233	#124	—	188	1	no	—
234	#121	—	2	4	no	—
235	#121	—	94	4	no	—
236	#120	—	4	2	no	—
237	#120	—	188	2	no	—
238	#117	—	2	1	no	—
239	#117	—	94	1	no	—
240	D₁₆	—	94	4	no	—
241	D₁₆	—	94	4	no	—
242	M₃₂	—	94	4	no	—
243	M₃₂	—	94	2	no	—
244	C₂×C₁₆	—	47	4	no	—
245	C₂×C₁₆	—	47	1	no	—
246	C₂×C₁₆	—	47	1	no	—
247	C₂×C₁₆	—	47	2	no	—
248	C₂×C₁₆	—	94	4	no	—
249	#107	—	47	2	no	—
250	#107	C₂²×D₄₇	1	2	yes	—
251	#106	—	2	1	no	—
252	#106	—	94	1	no	—
253	#102	—	2	4	no	—
254	#102	—	94	4	no	—
255	C₂⁴	—	4	1	no	—
256	C₂⁴	—	188	1	no	—
257	#41	—	2	2	no	—
258	#41	—	2	2	no	—
259	C₂×Q₈	#10625334	1	1	yes	characteristic
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	4	4	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	94	2	no	—
266	C₂×D₄	—	188	4	no	—
267	C₂×D₄	—	188	2	no	—
268	C₂×D₄	—	188	1	no	—
269	C₂×D₄	—	188	2	no	—
270	C₂×D₄	#10625334	1	2	yes	—
271	C₂×D₄	#10625334	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
273	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
274	D₈	—	2	2	no	—
275	D₈	—	2	2	no	—
276	D₈	—	4	2	no	—
277	D₈	—	94	2	no	—
278	D₈	—	94	2	no	—
279	D₈	—	188	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	2	4	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	D₄×D₄₇	1	2	yes	—
285	C₂×C₈	#10625352	1	2	yes	—
286	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
287	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
288	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	94	2	no	—
291	C₂²⋊C₄	—	4	1	no	—
292	C₂²⋊C₄	—	4	4	no	—
293	C₂²⋊C₄	—	188	1	no	—
294	C₂²⋊C₄	—	188	4	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	188	1	no	—
297	C₁₆	—	47	2	no	—
298	C₁₆	—	47	2	no	—
299	C₁₆	—	94	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	8	2	no	—
302	C₂³	—	16	2	no	—
303	C₂³	—	376	2	no	—
304	C₂³	—	188	2	no	—
305	C₂³	—	752	2	no	—
306	C₂³	C₂×#10625334	1	2	yes	—
307	C₂³	C₂×#10625334	1	1	yes	characteristic
308	Q₈	—	2	2	no	—
309	D₄	—	2	4	no	—
310	D₄	—	2	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	8	2	no	—
314	D₄	—	16	4	no	—
315	D₄	—	376	2	no	—
316	D₄	—	188	2	no	—
317	D₄	—	188	2	no	—
318	D₄	—	752	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	188	2	no	—
323	C₂×C₄	#13093996	1	2	yes	—
324	C₂×C₄	C₂×#10625334	1	2	yes	—
325	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
326	C₂×C₄	#13094044	1	2	yes	—
327	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
328	C₈	—	2	2	no	—
329	C₈	#13093996	1	2	yes	—
330	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
331	C₈	C₂×#10625352	1	1	yes	characteristic
332	C₂²	—	2	2	no	—
333	C₂²	—	2	4	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	16	4	no	—
338	C₂²	—	376	4	no	—
339	C₂²	—	376	2	no	—
340	C₂²	—	752	4	no	—
341	C₂²	#15689357	1	2	yes	—
342	C₂²	C₂²×#10625334	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	8	2	no	—
345	C₄	—	376	2	no	—
346	C₄	#15689357	1	2	yes	—
347	C₄	C₂×#13093996	1	1	yes	characteristic
348	C₄	C₂×#13094044	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	2	no	—
351	C₂	—	16	2	no	—
352	C₂	—	752	2	no	—
353	C₂	C₂×#15689357	1	1	yes	characteristic
354	C₂	#20256527	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3058	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3058	D₄₇
hypocenter	C₄₇	#33607
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ #3058

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093965	C₂	semidirect
2. C₂×#15689540	C₂	semidirect
3. #20253036	C₂	semidirect
4. #20245489	C₂	semidirect
5. #20245472	C₂	semidirect
6. #20245170	C₂	semidirect
7. #20245104	C₂	semidirect
8. #20245103	C₂	semidirect
9. C₄₇×#3058	C₂	semidirect
17. C₂×#13093964	C₂²	semidirect
19. C₂²×C₄₇⋊C₁₆	C₂²	semidirect
21. #15689359	C₂²	semidirect
24. #15688950	C₂²	semidirect
25. #15688948	C₂²	semidirect
26. #15688924	C₂²	semidirect
27. #15688923	C₂²	semidirect
32. C₄₇×#466	C₂²	semidirect
50. C₂×C₄₇⋊C₁₆	D₄	semidirect
175. #3058	D₄₇	semidirect
203. #466	C₂×D₄₇	semidirect
212. C₄₇	#33607	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093965	2
C₂×#15689540	2
#20253036	2
#20245489	2
#20245472	2
#20245170	2
#20245104	2
#20245103	2
C₄₇×#3058	2
#33607	47