Subgroups — #26370856

Order 12032.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689555	C₂	1	1	yes	maximal, characteristic, complemented
2	#20253108	C₂	1	2	yes	maximal, complemented
3	#20245518	C₂	1	4	yes	maximal, complemented
4	#20245513	C₂	1	2	yes	maximal, complemented
5	#20245508	C₂	1	2	yes	maximal, complemented
6	C₂×#15688953	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245138	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₇×#504	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689921	—	2	2	no	—
10	C₂×#13094046	C₂²	1	2	yes	—
11	#15689555	—	2	1	no	—
12	#15689555	—	2	1	no	—
13	#15689555	—	2	1	no	—
14	#15689555	—	2	1	no	—
15	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic, complemented
16	C₂×#13093969	C₂²	1	2	yes	complemented
17	C₂×D₇₅₂	C₂²	1	1	yes	characteristic, complemented
18	#15689398	C₂²	1	4	yes	—
19	#15689393	C₂²	1	4	yes	—
20	#15689386	—	2	4	no	—
21	C₂×#13093960	C₂²	1	2	yes	—
22	#15689105	—	2	2	no	—
23	#15688953	C₂²	1	2	yes	complemented
24	#15688953	C₂²	1	2	yes	complemented
25	#15688940	C₂²	1	4	yes	complemented
26	#15688937	C₂²	1	4	yes	complemented
27	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#504	—	2	1	no	—
29	C₄₇×#504	—	2	1	no	—
30	C₄₇×#504	C₂²	1	2	yes	—
31	C₄₇×#504	C₂²	1	2	yes	—
32	C₂×M₃₂×C₄₇	—	2	1	no	—
33	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
34	C₂²×C₁₆×C₄₇	—	2	1	no	—
35	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#10625350	—	2	2	no	—
37	#13094055	—	4	2	no	—
38	#13094054	—	2	4	no	—
39	#13094050	—	4	2	no	—
40	#13094049	—	2	4	no	—
41	#13094046	—	2	2	no	—
42	#13094046	—	2	2	no	—
43	#13094046	—	2	2	no	—
44	#13094046	—	2	2	no	—
45	#13094044	—	4	4	no	—
46	C₂×Dic₁₈₈	C₂³	1	2	yes	—
47	C₂×#10625325	—	2	2	no	—
48	C₂×D₃₇₆	C₂³	1	2	yes	—
49	C₂×#10625329	—	2	2	no	—
50	Dic₃₇₆	—	2	1	no	—
51	Dic₃₇₆	—	2	1	no	—
52	#13093969	—	2	2	no	—
53	#13093969	—	2	2	no	—
54	D₇₅₂	—	2	1	no	—
55	D₇₅₂	—	2	1	no	—
56	#13093960	—	2	2	no	—
57	#13093960	C₂³	1	4	yes	—
58	#13093942	—	2	4	no	—
59	#13093940	—	2	4	no	—
60	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
61	#131×C₄₇	—	2	1	no	—
62	#131×C₄₇	—	2	1	no	—
63	#131×C₄₇	D₄	1	2	yes	—
64	#131×C₄₇	C₂³	1	2	yes	—
65	C₂×M₁₆×C₄₇	—	2	1	no	—
66	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	—	2	1	no	—
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	M₃₂×C₄₇	—	2	1	no	—
70	M₃₂×C₄₇	—	2	2	no	—
71	M₃₂×C₄₇	—	2	2	no	—
72	M₃₂×C₄₇	C₂³	1	2	yes	—
73	C₂×C₁₆×C₄₇	—	2	1	no	—
74	C₂×C₁₆×C₄₇	—	2	2	no	—
75	C₂×C₁₆×C₄₇	—	2	2	no	—
76	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
77	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
78	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
79	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
80	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
82	#41×C₄₇	—	2	1	no	—
83	#41×C₄₇	—	2	1	no	—
84	#41×C₄₇	D₈	1	2	yes	—
85	#41×C₄₇	C₂×D₄	1	2	yes	—
86	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
87	C₂×D₄×C₄₇	—	2	1	no	—
88	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	—	2	1	no	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₄₇	—	4	2	no	—
92	#10625350	—	4	2	no	—
93	#10625350	—	4	2	no	—
94	#10625350	—	4	2	no	—
95	#10625350	—	4	2	no	—
96	C₂×D₁₈₈	—	2	2	no	—
97	C₂×C₄×D₄₇	—	4	2	no	—
98	C₂×Dic₉₄	—	2	2	no	—
99	M₁₆×C₄₇	—	2	1	no	—
100	M₁₆×C₄₇	—	2	2	no	—
101	M₁₆×C₄₇	—	2	2	no	—
102	M₁₆×C₄₇	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₄₇	—	2	1	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	—	2	2	no	—
106	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
110	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
111	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
112	#10625337	—	4	4	no	—
113	#10625336	—	4	4	no	—
114	#10625335	—	4	4	no	—
115	#10625334	—	4	4	no	—
116	#10625329	—	2	4	no	—
117	#10625329	—	4	2	no	—
118	C₂×C₄₇⋊C₈	—	4	2	no	—
119	Dic₁₈₈	—	2	2	no	—
120	Dic₁₈₈	—	2	2	no	—
121	D₃₇₆	—	2	2	no	—
122	D₃₇₆	—	2	2	no	—
123	#10625325	—	4	2	no	—
124	#10625325	—	4	2	no	—
125	C₁₆×C₄₇	—	2	2	no	—
126	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
129	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
130	C₂³×C₄₇	—	2	1	no	—
131	C₂³×C₄₇	#135	1	1	yes	characteristic
132	C₂²×D₄₇	—	4	2	no	—
133	Q₈×C₄₇	—	2	1	no	—
134	Q₈×C₄₇	C₂×D₈	1	2	yes	—
135	D₄×C₄₇	—	2	2	no	—
136	D₄×C₄₇	—	2	2	no	—
137	D₄×C₄₇	—	2	1	no	—
138	D₄×C₄₇	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₄₇	—	2	1	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
143	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₄₇	—	8	2	no	—
149	C₂²⋊D₄₇	—	8	2	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	D₁₈₈	—	4	2	no	—
152	D₁₈₈	—	4	2	no	—
153	C₄×D₄₇	—	8	2	no	—
154	C₄×D₄₇	—	8	2	no	—
155	Dic₉₄	—	4	2	no	—
156	Dic₉₄	—	4	2	no	—
157	C₈×C₄₇	—	2	2	no	—
158	C₈×C₄₇	#121	1	1	yes	characteristic
159	C₈×C₄₇	#121	1	1	yes	characteristic
160	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
162	C₄₇⋊C₈	—	4	4	no	—
163	#33617	—	47	1	no	maximal, Sylow
164	C₂²×C₄₇	—	2	2	no	—
165	C₂²×C₄₇	—	4	1	no	—
166	C₂²×C₄₇	—	4	2	no	—
167	C₂²×C₄₇	#466	1	1	yes	characteristic
168	C₂²×C₄₇	#466	1	1	yes	characteristic
169	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
170	C₂×D₄₇	—	8	2	no	—
171	C₂×D₄₇	—	16	2	no	—
172	C₄×C₄₇	—	2	2	no	—
173	C₄×C₄₇	#466	1	1	yes	characteristic
174	C₄×C₄₇	#466	1	1	yes	characteristic
175	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
176	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
177	Dic₄₇	—	8	2	no	—
178	C₂×#508	—	47	1	no	—
179	C₂×#504	D₄₇	1	1	yes	characteristic, complemented
180	#3067	—	47	2	no	—
181	#2223	—	47	4	no	—
182	#2222	—	47	2	no	—
183	#2221	—	47	2	no	—
184	C₂×#368	—	47	1	no	—
185	#2148	—	47	2	no	—
186	C₂×C₄₇	—	2	1	no	—
187	C₂×C₄₇	—	2	1	no	—
188	C₂×C₄₇	—	2	1	no	—
189	C₂×C₄₇	—	4	2	no	—
190	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
191	D₄₇	—	16	2	no	—
192	#575	—	94	2	no	—
193	C₂×#135	—	47	2	no	—
194	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
195	#508	—	94	1	no	—
196	#508	—	94	1	no	—
197	#508	—	94	1	no	—
198	#508	—	94	1	no	—
199	C₂×Q₃₂	—	47	1	no	—
200	C₂×SD₃₂	—	47	2	no	—
201	C₂×D₁₆	—	47	1	no	—
202	#504	—	2	1	no	—
203	#504	—	2	1	no	—
204	#504	C₂×D₄₇	1	2	yes	—
205	#504	C₂×D₄₇	1	2	yes	—
206	C₂×M₃₂	—	2	1	no	—
207	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
208	C₂²×C₁₆	—	2	1	no	—
209	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
210	#473	—	47	4	no	—
211	#472	—	47	4	no	—
212	#471	—	94	4	no	—
213	C₂×#108	—	47	2	no	—
214	#413	—	94	2	no	—
215	#368	—	47	2	no	—
216	#368	—	47	2	no	—
217	#362	—	47	4	no	—
218	#361	—	47	4	no	—
219	C₄₇	#33617	1	1	yes	characteristic, complemented, Sylow
220	C₂×#41	—	94	2	no	—
221	C₂×#41	D₁₈₈	1	1	yes	characteristic
222	#137	—	94	4	no	—
223	#137	—	188	2	no	—
224	AGL(1,ℤ/8)	—	94	4	no	—
225	AGL(1,ℤ/8)	—	188	2	no	—
226	#135	—	94	2	no	—
227	#135	—	94	2	no	—
228	#135	—	94	2	no	—
229	#135	—	94	2	no	—
230	#135	—	188	4	no	—
231	C₂×Q₁₆	—	47	2	no	—
232	C₂×SD₁₆	—	94	2	no	—
233	C₂×D₈	—	47	2	no	—
234	#131	—	2	1	no	—
235	#131	—	2	1	no	—
236	#131	D₁₈₈	1	2	yes	—
237	#131	C₂²×D₄₇	1	2	yes	—
238	C₂×M₁₆	—	2	1	no	—
239	C₂×M₁₆	—	94	2	no	—
240	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
241	C₂²×C₈	—	2	1	no	—
242	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
243	Q₃₂	—	94	1	no	—
244	Q₃₂	—	94	1	no	—
245	SD₃₂	—	94	2	no	—
246	SD₃₂	—	94	2	no	—
247	D₁₆	—	94	1	no	—
248	D₁₆	—	94	1	no	—
249	M₃₂	—	2	1	no	—
250	M₃₂	—	2	2	no	—
251	M₃₂	—	2	2	no	—
252	M₃₂	C₂²×D₄₇	1	2	yes	—
253	C₂×C₁₆	—	2	1	no	—
254	C₂×C₁₆	—	2	2	no	—
255	C₂×C₁₆	—	2	2	no	—
256	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
257	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
258	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
259	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
261	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
262	#108	—	47	4	no	—
263	#108	—	94	2	no	—
264	#101	—	94	4	no	—
265	#100	—	94	4	no	—
266	#41	—	2	1	no	—
267	#41	—	2	1	no	—
268	#41	—	188	2	no	—
269	#41	—	188	2	no	—
270	#41	—	188	2	no	—
271	#41	—	188	2	no	—
272	#41	D₃₇₆	1	2	yes	—
273	#41	C₂×D₁₈₈	1	2	yes	—
274	C₂×Q₈	—	94	2	no	—
275	C₂×Q₈	D₃₇₆	1	1	yes	characteristic
276	C₂×D₄	—	2	1	no	—
277	C₂×D₄	—	94	2	no	—
278	C₂×D₄	—	188	2	no	—
279	C₂×D₄	D₃₇₆	1	1	yes	characteristic
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	188	2	no	—
282	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
283	Q₁₆	—	94	2	no	—
284	Q₁₆	—	94	2	no	—
285	Q₁₆	—	188	4	no	—
286	SD₁₆	—	188	4	no	—
287	SD₁₆	—	188	4	no	—
288	SD₁₆	—	188	2	no	—
289	SD₁₆	—	188	2	no	—
290	D₈	—	94	2	no	—
291	D₈	—	94	2	no	—
292	D₈	—	188	4	no	—
293	M₁₆	—	2	1	no	—
294	M₁₆	—	2	2	no	—
295	M₁₆	—	2	2	no	—
296	M₁₆	—	94	4	no	—
297	M₁₆	—	188	2	no	—
298	M₁₆	C₂×D₁₈₈	1	2	yes	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	188	2	no	—
303	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
304	C₂×C₈	#10625350	1	1	yes	characteristic
305	C₂×C₈	#10625350	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
307	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
309	C₁₆	—	2	2	no	—
310	C₁₆	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
311	C₁₆	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
312	C₁₆	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
313	C₁₆	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
314	C₂³	—	2	1	no	—
315	C₂³	—	188	2	no	—
316	C₂³	#13094046	1	1	yes	characteristic
317	Q₈	—	2	1	no	—
318	Q₈	—	188	2	no	—
319	Q₈	—	188	2	no	—
320	Q₈	C₂×D₃₇₆	1	2	yes	—
321	D₄	—	2	2	no	—
322	D₄	—	2	2	no	—
323	D₄	—	2	1	no	—
324	D₄	—	376	2	no	—
325	D₄	—	376	2	no	—
326	D₄	—	188	2	no	—
327	D₄	—	188	2	no	—
328	D₄	C₂×D₃₇₆	1	2	yes	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	376	2	no	—
333	C₂×C₄	—	376	2	no	—
334	C₂×C₄	—	188	2	no	—
335	C₂×C₄	#13093998	1	1	yes	characteristic
336	C₂×C₄	#13093998	1	1	yes	characteristic
337	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
338	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
339	C₂×C₄	#13094046	1	1	yes	characteristic
340	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
341	C₈	—	2	2	no	—
342	C₈	—	188	4	no	—
343	C₈	#13093998	1	1	yes	characteristic
344	C₈	#13093998	1	1	yes	characteristic
345	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
346	C₈	C₂×#10625350	1	1	yes	characteristic
347	C₂²	—	2	2	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	2	no	—
350	C₂²	—	376	2	no	—
351	C₂²	—	752	2	no	—
352	C₂²	#15689359	1	1	yes	characteristic
353	C₂²	#15689359	1	1	yes	characteristic
354	C₂²	C₂×#13094046	1	1	yes	characteristic
355	C₄	—	2	2	no	—
356	C₄	—	376	2	no	—
357	C₄	#15689359	1	1	yes	characteristic
358	C₄	#15689359	1	1	yes	characteristic
359	C₄	C₂×#13093998	1	1	yes	characteristic
360	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	4	2	no	—
365	C₂	—	752	2	no	—
366	C₂	C₂×#15689359	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15689359
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#504	C₂
socle	C₂×C₄₇	C₂×#466
hypercenter	C₂×#504	D₄₇
hypocenter	C₄₇	#33617
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#15689555	C₂	semidirect
2. #20253108	C₂	semidirect
3. #20245518	C₂	semidirect
4. #20245513	C₂	semidirect
5. #20245508	C₂	semidirect
6. C₂×#15688953	C₂	semidirect
7. #20245138	C₂	semidirect
8. C₂×C₄₇×#504	C₂	semidirect
15. C₂×Dic₃₇₆	C₂²	semidirect
16. C₂×#13093969	C₂²	semidirect
17. C₂×D₇₅₂	C₂²	semidirect
23. #15688953	C₂²	semidirect
24. #15688953	C₂²	semidirect
25. #15688940	C₂²	semidirect
26. #15688937	C₂²	semidirect
179. C₂×#504	D₄₇	semidirect
219. C₄₇	#33617	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689555	2
#20253108	2
#20245518	2
#20245513	2
#20245508	2
C₂×#15688953	2
#20245138	2
C₂×C₄₇×#504	2
#33617	47