Subgroups — #26131929

Order 12032.

954 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688927	C₂	1	2	yes	maximal, complemented
2	C₂×#15688924	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243417	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#135×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#13093958	C₂²	1	2	yes	—
8	#15688983	—	2	1	no	—
9	#15688927	—	2	4	no	—
10	#15688924	—	2	4	no	—
11	C₂²⋊(C₄₇⋊C₁₆)	—	2	2	no	—
12	#15688886	—	2	1	no	—
13	C₂²×#41×C₄₇	—	2	1	no	—
14	C₂×#135×C₄₇	—	2	4	no	—
15	C₂×#135×C₄₇	—	2	4	no	—
16	C₂×#135×C₄₇	—	2	2	no	—
17	C₂×#135×C₄₇	C₄	1	4	yes	complemented
18	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₄₇	—	2	1	no	—
20	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
24	C₂×C₄₇⋊C₁₆	—	2	2	no	—
25	C₂×C₄₇⋊C₁₆	—	2	2	no	—
26	C₂×C₄₇⋊C₁₆	—	2	2	no	—
27	#13093958	—	2	2	no	—
28	#13093958	—	2	2	no	—
29	C₄₇⋊#95	—	4	1	no	—
30	C₂×#41×C₄₇	—	2	8	no	—
31	C₂×#41×C₄₇	—	2	4	no	—
32	C₂×#41×C₄₇	—	4	4	no	—
33	C₂×#41×C₄₇	—	4	2	no	—
34	C₂²×Q₈×C₄₇	—	2	1	no	—
35	C₂²×D₄×C₄₇	—	2	1	no	—
36	C₂²×D₄×C₄₇	—	4	1	no	—
37	C₂³×C₄×C₄₇	—	4	1	no	—
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	#135×C₄₇	—	2	16	no	—
40	#135×C₄₇	—	2	8	no	—
41	#135×C₄₇	—	2	8	no	—
42	C₂×Q₁₆×C₄₇	—	2	2	no	—
43	C₂×Q₁₆×C₄₇	—	2	1	no	—
44	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×Q₁₆×C₄₇	D₄	1	4	yes	—
46	C₂×SD₁₆×C₄₇	—	2	8	no	—
47	C₂×SD₁₆×C₄₇	—	2	4	no	—
48	C₂×D₈×C₄₇	—	2	2	no	—
49	C₂×D₈×C₄₇	—	2	1	no	—
50	C₂×D₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₂×D₈×C₄₇	D₄	1	4	yes	—
52	C₂²×C₈×C₄₇	—	2	2	no	—
53	C₂²×C₈×C₄₇	—	2	2	no	—
54	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	D₄	1	2	yes	—
57	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
58	C₂⁴×C₄₇	—	4	1	no	—
59	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
60	#41×C₄₇	—	2	16	no	—
61	#41×C₄₇	—	4	8	no	—
62	#41×C₄₇	—	4	16	no	—
63	C₂×Q₈×C₄₇	—	2	2	no	—
64	C₂×Q₈×C₄₇	—	2	4	no	—
65	C₂×Q₈×C₄₇	—	4	1	no	—
66	C₂×Q₈×C₄₇	—	4	2	no	—
67	C₂×D₄×C₄₇	—	2	2	no	—
68	C₂×D₄×C₄₇	—	2	4	no	—
69	C₂×D₄×C₄₇	—	4	8	no	—
70	C₂×D₄×C₄₇	—	4	4	no	—
71	C₂×D₄×C₄₇	—	4	1	no	—
72	C₂×D₄×C₄₇	—	4	2	no	—
73	C₂²×C₄×C₄₇	—	2	2	no	—
74	C₂²×C₄×C₄₇	—	2	2	no	—
75	C₂²×C₄×C₄₇	—	4	8	no	—
76	C₂²×C₄×C₄₇	—	4	1	no	—
77	C₂²×C₄×C₄₇	—	4	4	no	—
78	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×Dic₄₇	—	4	2	no	—
83	Q₁₆×C₄₇	—	2	2	no	—
84	Q₁₆×C₄₇	—	2	4	no	—
85	Q₁₆×C₄₇	C₂²⋊C₄	1	4	yes	complemented
86	SD₁₆×C₄₇	—	2	16	no	—
87	D₈×C₄₇	—	2	2	no	—
88	D₈×C₄₇	—	2	4	no	—
89	D₈×C₄₇	C₂²⋊C₄	1	4	yes	complemented
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	2	4	no	—
94	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₂²⋊Dic₄₇	—	8	2	no	—
100	C₄⋊Dic₄₇	—	4	2	no	—
101	C₄⋊Dic₄₇	—	4	2	no	—
102	C₄₇⋊C₁₆	—	4	2	no	—
103	C₂³×C₄₇	—	2	2	no	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	—	2	1	no	—
106	C₂³×C₄₇	—	2	1	no	—
107	C₂³×C₄₇	—	4	2	no	—
108	C₂³×C₄₇	—	4	4	no	—
109	C₂³×C₄₇	—	8	2	no	—
110	C₂³×C₄₇	—	8	1	no	—
111	C₂³×C₄₇	—	8	1	no	—
112	C₂³×C₄₇	#102	1	2	yes	—
113	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₄₇	—	2	4	no	—
115	Q₈×C₄₇	—	4	2	no	—
116	Q₈×C₄₇	—	4	4	no	—
117	D₄×C₄₇	—	2	4	no	—
118	D₄×C₄₇	—	4	16	no	—
119	D₄×C₄₇	—	4	2	no	—
120	D₄×C₄₇	—	4	4	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	4	no	—
125	C₂×C₄×C₄₇	—	4	16	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	4	no	—
128	C₂×C₄×C₄₇	#102	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#120	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×Dic₄₇	—	8	2	no	—
134	C₂×Dic₄₇	—	8	2	no	—
135	C₂×Dic₄₇	—	8	2	no	—
136	C₈×C₄₇	—	2	2	no	—
137	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₈×C₄₇	#120	1	2	yes	—
140	#12733	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	4	2	no	—
145	C₂²×C₄₇	—	4	2	no	—
146	C₂²×C₄₇	—	4	4	no	—
147	C₂²×C₄₇	—	8	2	no	—
148	C₂²×C₄₇	—	8	2	no	—
149	C₂²×C₄₇	—	8	4	no	—
150	C₂²×C₄₇	—	8	4	no	—
151	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
152	C₂²×C₄₇	#447	1	2	yes	—
153	C₂²×C₄₇	#508	1	2	yes	—
154	C₂²×C₄₇	#509	1	1	yes	characteristic
155	C₂²×C₄₇	#510	1	1	yes	characteristic
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	4	4	no	—
158	C₄×C₄₇	#379	1	1	yes	characteristic
159	C₄×C₄₇	#418	1	1	yes	characteristic
160	C₄×C₄₇	#450	1	2	yes	—
161	Dic₄₇	—	16	2	no	—
162	C₂²×#135	D₄₇	1	1	yes	characteristic
163	C₂×#358	—	47	2	no	—
164	C₂×#357	—	47	2	no	—
165	C₂×C₂²⋊C₁₆	—	47	1	no	—
166	#1808	—	47	1	no	—
167	C₂×C₄₇	—	4	2	no	—
168	C₂×C₄₇	—	8	4	no	—
169	C₂×C₄₇	#1786	1	1	yes	characteristic
170	C₂×C₄₇	#2138	1	1	yes	characteristic
171	C₂×C₄₇	#2139	1	1	yes	characteristic
172	C₂×C₄₇	#2186	1	2	yes	—
173	C₂×C₄₇	#2189	1	2	yes	—
174	C₂²×#41	—	2	1	no	—
175	C₂×#135	—	2	4	no	—
176	C₂×#135	—	2	4	no	—
177	C₂×#135	—	2	2	no	—
178	C₂×#135	Dic₄₇	1	4	yes	complemented
179	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
180	C₂²×SD₁₆	—	2	1	no	—
181	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
182	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
183	C₂²×C₁₆	—	47	2	no	—
184	C₂×#107	—	47	2	no	—
185	#380	—	94	1	no	—
186	#358	—	94	4	no	—
187	#357	—	94	4	no	—
188	C₂²⋊C₁₆	—	94	2	no	—
189	#340	—	94	1	no	—
190	C₄₇	#12733	1	1	yes	characteristic, complemented, Sylow
191	C₂×#41	—	2	8	no	—
192	C₂×#41	—	2	4	no	—
193	C₂×#41	—	4	4	no	—
194	C₂×#41	—	4	2	no	—
195	C₂²×Q₈	—	2	1	no	—
196	C₂²×D₄	—	2	1	no	—
197	C₂²×D₄	—	4	1	no	—
198	C₂³×C₄	—	4	1	no	—
199	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
200	#135	—	2	16	no	—
201	#135	—	2	8	no	—
202	#135	—	2	8	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	2	1	no	—
205	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
206	C₂×Q₁₆	C₂²⋊D₄₇	1	4	yes	—
207	C₂×SD₁₆	—	2	8	no	—
208	C₂×SD₁₆	—	2	4	no	—
209	C₂×D₈	—	2	2	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	C₂×Dic₄₇	1	2	yes	—
212	C₂×D₈	C₂²⋊D₄₇	1	4	yes	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	—	2	2	no	—
215	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
216	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
218	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	94	2	no	—
220	C₂×C₂²⋊C₄	—	188	1	no	—
221	C₂×C₁₆	—	94	2	no	—
222	C₂×C₁₆	—	94	2	no	—
223	C₂×C₁₆	—	94	2	no	—
224	#107	—	94	2	no	—
225	#107	—	94	2	no	—
226	#95	—	188	1	no	—
227	C₂⁴	—	4	1	no	—
228	C₂⁴	#10625334	1	1	yes	characteristic
229	#41	—	2	16	no	—
230	#41	—	4	16	no	—
231	#41	—	4	8	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	4	8	no	—
239	C₂×D₄	—	4	4	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	4	8	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	188	2	no	—
248	C₂²×C₄	#10625334	1	1	yes	characteristic
249	C₂²×C₄	#10625335	1	2	yes	—
250	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
251	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
252	Q₁₆	—	2	2	no	—
253	Q₁₆	—	2	4	no	—
254	Q₁₆	C₂²⋊Dic₄₇	1	4	yes	complemented
255	SD₁₆	—	2	16	no	—
256	D₈	—	2	2	no	—
257	D₈	—	2	4	no	—
258	D₈	C₂²⋊Dic₄₇	1	4	yes	complemented
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
264	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
266	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
267	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
268	C₄⋊C₄	—	188	2	no	—
269	C₄⋊C₄	—	188	2	no	—
270	C₂²⋊C₄	—	376	2	no	—
271	C₁₆	—	188	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	8	2	no	—
279	C₂³	—	8	1	no	—
280	C₂³	—	8	1	no	—
281	C₂³	#13093945	1	2	yes	—
282	C₂³	C₂×#10625334	1	1	yes	characteristic
283	Q₈	—	2	4	no	—
284	Q₈	—	4	4	no	—
285	Q₈	—	4	2	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	16	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	16	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	376	2	no	—
298	C₂×C₄	—	376	2	no	—
299	C₂×C₄	—	376	2	no	—
300	C₂×C₄	#13093945	1	2	yes	—
301	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
302	C₂×C₄	#13093993	1	2	yes	—
303	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
304	C₂×C₄	#13094050	1	2	yes	—
305	C₈	—	2	2	no	—
306	C₈	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
307	C₈	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
308	C₈	#13093993	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	8	4	no	—
316	C₂²	—	8	4	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	C₂×#13093945	1	1	yes	characteristic
320	C₂²	#15689257	1	2	yes	—
321	C₂²	#15689553	1	2	yes	—
322	C₂²	#15689558	1	1	yes	characteristic
323	C₂²	#15689563	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	4	4	no	—
326	C₄	—	752	2	no	—
327	C₄	#15688979	1	1	yes	characteristic
328	C₄	#15689128	1	1	yes	characteristic
329	C₄	#15689274	1	2	yes	—
330	C₂	—	4	2	no	—
331	C₂	—	8	4	no	—
332	C₂	#20243319	1	1	yes	characteristic
333	C₂	#20245095	1	1	yes	characteristic
334	C₂	#20245100	1	1	yes	characteristic
335	C₂	#20245311	1	2	yes	—
336	C₂	#20245328	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting, prime radical	C₂²×#135×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂²×#135	D₄₇
hypocenter	C₄₇	#12733
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#15688927	C₂	semidirect
2. C₂×#15688924	C₂	semidirect
3. C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
4. #20243417	C₂	semidirect
17. C₂×#135×C₄₇	C₄	semidirect
85. Q₁₆×C₄₇	C₂²⋊C₄	semidirect
89. D₈×C₄₇	C₂²⋊C₄	semidirect
178. C₂×#135	Dic₄₇	semidirect
190. C₄₇	#12733	semidirect
254. Q₁₆	C₂²⋊Dic₄₇	semidirect
258. D₈	C₂²⋊Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688927	2
C₂×#15688924	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
#20243417	2
C₂²×#135×C₄₇	2
#12733	47