Subgroups — #26370554

Order 12032.

770 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689559	C₂	1	2	yes	maximal, complemented
2	#20253816	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245443	C₂	1	4	yes	maximal, complemented
4	#20245433	C₂	1	2	yes	maximal, complemented
5	#20245423	C₂	1	2	yes	maximal, complemented
6	#20245144	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245098	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₈×C₄₇	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	#15689559	—	2	8	no	—
13	C₂×#13093968	C₂²	1	4	yes	—
14	C₂×#13093965	C₂²	1	4	yes	—
15	C₂×#13093964	C₂²	1	2	yes	—
16	#15689507	—	2	1	no	—
17	#15689507	C₂²	1	2	yes	complemented
18	#15689503	C₂²	1	1	yes	characteristic, complemented
19	#15689500	C₂²	1	1	yes	characteristic, complemented
20	#15689496	—	2	1	no	—
21	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	complemented
22	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	complemented
23	#15688926	C₂²	1	4	yes	complemented
24	#15688923	C₂²	1	4	yes	complemented
25	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₄₇	—	2	1	no	—
27	C₄×D₈×C₄₇	—	2	1	no	—
28	C₄×D₈×C₄₇	—	2	1	no	—
29	C₄×D₈×C₄₇	C₂²	1	2	yes	—
30	C₄×D₈×C₄₇	C₂²	1	2	yes	—
31	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×#102×C₄₇	—	2	1	no	—
33	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₂×#10625350	—	2	2	no	—
35	#13094046	—	2	4	no	—
36	#13094046	—	2	4	no	—
37	#13094046	—	4	4	no	—
38	C₂×Dic₁₈₈	—	2	1	no	—
39	C₂×Dic₁₈₈	C₂³	1	2	yes	—
40	C₂×#10625325	—	2	2	no	—
41	C₂×#10625325	—	2	2	no	—
42	C₂×D₃₇₆	—	2	1	no	—
43	C₂×D₃₇₆	C₂³	1	2	yes	—
44	#13094024	—	2	1	no	—
45	#13094021	—	2	1	no	—
46	#13094012	—	2	2	no	—
47	#13093998	—	4	1	no	—
48	#13093998	—	4	1	no	—
49	C₄×D₁₈₈	—	4	1	no	—
50	C₄×D₁₈₈	—	4	1	no	—
51	#13093968	—	2	8	no	—
52	#13093965	—	2	8	no	—
53	#13093964	—	2	4	no	—
54	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
55	C₂²×D₄×C₄₇	—	2	1	no	—
56	C₂³×C₄×C₄₇	—	4	1	no	—
57	C₂×D₈×C₄₇	—	2	1	no	—
58	C₂×D₈×C₄₇	—	2	2	no	—
59	C₂×D₈×C₄₇	D₄	1	4	yes	complemented
60	C₂×D₈×C₄₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₄×D₄×C₄₇	—	2	2	no	—
64	C₄×D₄×C₄₇	—	2	2	no	—
65	C₄×D₄×C₄₇	—	4	1	no	—
66	C₄×D₄×C₄₇	—	4	1	no	—
67	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
69	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
70	#107×C₄₇	—	2	1	no	—
71	#107×C₄₇	C₂³	1	2	yes	—
72	#102×C₄₇	—	2	2	no	—
73	#102×C₄₇	—	2	2	no	—
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	C₂⁴×C₄₇	—	4	1	no	—
79	C₂×D₄×C₄₇	—	2	4	no	—
80	C₂×D₄×C₄₇	—	2	2	no	—
81	C₂×D₄×C₄₇	—	4	1	no	—
82	C₂×D₄×C₄₇	—	4	2	no	—
83	C₂²×C₄×C₄₇	—	4	1	no	—
84	C₂²×C₄×C₄₇	—	4	2	no	—
85	C₂²×C₄×C₄₇	—	4	2	no	—
86	C₂²×C₄×C₄₇	—	8	4	no	—
87	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₂²⋊D₄₇	—	4	2	no	—
91	#10625350	—	4	4	no	—
92	#10625350	—	4	4	no	—
93	C₂×D₁₈₈	—	2	2	no	—
94	C₂×D₁₈₈	—	4	1	no	—
95	C₂×D₁₈₈	—	4	1	no	—
96	C₂×C₄×D₄₇	—	4	2	no	—
97	C₂×Dic₉₄	—	2	2	no	—
98	D₈×C₄₇	—	2	4	no	—
99	D₈×C₄₇	—	2	2	no	—
100	D₈×C₄₇	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₄₇	—	4	1	no	—
110	C₂²⋊C₄×C₄₇	—	4	2	no	—
111	C₂²⋊C₄×C₄₇	—	4	2	no	—
112	C₄²×C₄₇	D₈	1	1	yes	characteristic
113	C₄²×C₄₇	D₈	1	1	yes	characteristic
114	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
116	(C₂×D₄₇)⋊C₄	—	4	2	no	—
117	(C₂×D₄₇)⋊C₄	—	4	2	no	—
118	C₄⋊Dic₄₇	—	4	1	no	—
119	C₄⋊Dic₄₇	—	4	1	no	—
120	Dic₁₈₈	—	2	2	no	—
121	Dic₁₈₈	—	2	2	no	—
122	Dic₁₈₈	—	4	2	no	—
123	D₃₇₆	—	2	2	no	—
124	D₃₇₆	—	2	2	no	—
125	D₃₇₆	—	4	2	no	—
126	#10625325	—	4	4	no	—
127	#10625325	—	4	4	no	—
128	C₄₇⋊C₁₆	—	2	4	no	—
129	C₂³×C₄₇	—	4	2	no	—
130	C₂³×C₄₇	—	4	4	no	—
131	C₂³×C₄₇	—	8	2	no	—
132	C₂³×C₄₇	—	8	2	no	—
133	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
134	C₂²×D₄₇	—	4	2	no	—
135	D₄×C₄₇	—	2	4	no	—
136	D₄×C₄₇	—	4	2	no	—
137	D₄×C₄₇	—	4	4	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	4	1	no	—
140	C₂×C₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	—	8	4	no	—
142	C₂×C₄×C₄₇	—	8	2	no	—
143	C₂×C₄×C₄₇	—	8	4	no	—
144	C₂×C₄×C₄₇	#121	1	2	yes	—
145	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
151	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
152	C₂²⋊D₄₇	—	8	4	no	—
153	C₂×Dic₄₇	—	4	2	no	—
154	D₁₈₈	—	4	2	no	—
155	D₁₈₈	—	4	2	no	—
156	D₁₈₈	—	8	2	no	—
157	D₁₈₈	—	8	2	no	—
158	C₄×D₄₇	—	8	4	no	—
159	Dic₉₄	—	4	2	no	—
160	Dic₉₄	—	4	2	no	—
161	C₈×C₄₇	—	2	1	no	—
162	C₈×C₄₇	—	2	1	no	—
163	C₈×C₄₇	#121	1	2	yes	—
164	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
165	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
166	#33580	—	47	1	no	maximal, Sylow
167	C₂²×C₄₇	—	2	1	no	—
168	C₂²×C₄₇	—	2	1	no	—
169	C₂²×C₄₇	—	4	4	no	—
170	C₂²×C₄₇	—	8	8	no	—
171	C₂²×C₄₇	—	8	2	no	—
172	C₂²×C₄₇	—	8	2	no	—
173	C₂²×C₄₇	#459	1	2	yes	—
174	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
175	C₂×D₄₇	—	8	2	no	—
176	C₂×D₄₇	—	8	2	no	—
177	C₂×D₄₇	—	8	2	no	—
178	C₄×C₄₇	—	2	1	no	—
179	C₄×C₄₇	—	2	1	no	—
180	C₄×C₄₇	—	2	1	no	—
181	C₄×C₄₇	—	8	2	no	—
182	C₄×C₄₇	#459	1	2	yes	—
183	C₄×C₄₇	#509	1	2	yes	complemented
184	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
185	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	Dic₄₇	—	8	2	no	—
187	C₂×#509	—	47	2	no	—
188	#3147	—	47	1	no	—
189	C₂×C₄×D₈	D₄₇	1	1	yes	characteristic, complemented
190	#2210	—	47	4	no	—
191	#2208	—	47	2	no	—
192	#2206	—	47	2	no	—
193	#2150	—	47	1	no	—
194	#2139	—	47	2	no	—
195	C₂×C₄₇	—	2	2	no	—
196	C₂×C₄₇	—	8	4	no	—
197	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
198	C₂×C₄₇	C₂×#509	1	1	yes	characteristic
199	C₂×C₄₇	#3414	1	1	yes	characteristic
200	D₄₇	—	16	2	no	—
201	C₂×#135	—	47	2	no	—
202	C₂×#135	—	94	1	no	—
203	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
204	#532	—	94	1	no	—
205	C₂×C₄×D₄	—	2	1	no	—
206	#509	—	94	8	no	—
207	C₂×SD₃₂	—	47	4	no	—
208	C₂×D₁₆	—	47	4	no	—
209	C₂×M₃₂	—	47	2	no	—
210	#495	—	47	2	no	—
211	#495	—	94	1	no	—
212	#494	—	47	1	no	—
213	#493	—	47	1	no	—
214	#492	—	94	1	no	—
215	C₄×D₈	—	2	1	no	—
216	C₄×D₈	—	2	1	no	—
217	C₄×D₈	C₂×D₄₇	1	2	yes	—
218	C₄×D₈	C₂×D₄₇	1	2	yes	—
219	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
220	C₂×#102	—	2	1	no	—
221	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
222	C₄⋊C₁₆	—	47	2	no	—
223	C₄⋊C₁₆	—	47	2	no	—
224	#358	—	47	4	no	—
225	#357	—	47	4	no	—
226	C₄₇	#33580	1	1	yes	characteristic, complemented, Sylow
227	C₂×#41	—	94	2	no	—
228	C₂²×D₄	—	2	1	no	—
229	C₂³×C₄	—	4	1	no	—
230	#135	—	94	4	no	—
231	#135	—	94	4	no	—
232	#135	—	188	4	no	—
233	C₂×Q₁₆	—	47	2	no	—
234	C₂×Q₁₆	—	94	1	no	—
235	C₂×SD₁₆	—	94	2	no	—
236	C₂×SD₁₆	—	94	2	no	—
237	C₂×D₈	—	2	2	no	—
238	C₂×D₈	—	2	1	no	—
239	C₂×D₈	—	47	2	no	—
240	C₂×D₈	—	94	1	no	—
241	C₂×D₈	D₁₈₈	1	4	yes	complemented
242	C₂×D₈	C₂²×D₄₇	1	2	yes	—
243	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
244	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
245	C₄⋊Q₈	—	94	1	no	—
246	#127	—	94	1	no	—
247	#124	—	94	2	no	—
248	#121	—	188	1	no	—
249	#121	—	188	1	no	—
250	C₄×D₄	—	2	2	no	—
251	C₄×D₄	—	2	2	no	—
252	C₄×D₄	—	4	1	no	—
253	C₄×D₄	—	4	1	no	—
254	C₄×D₄	—	188	1	no	—
255	C₄×D₄	—	188	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₂²⋊C₄	—	4	1	no	—
258	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
259	SD₃₂	—	94	8	no	—
260	D₁₆	—	94	8	no	—
261	M₃₂	—	94	4	no	—
262	C₂×C₁₆	—	47	4	no	—
263	#107	—	2	1	no	—
264	#107	C₂²×D₄₇	1	2	yes	—
265	#102	—	2	2	no	—
266	#102	—	2	2	no	—
267	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
268	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
271	C₂⁴	—	4	1	no	—
272	#41	—	188	4	no	—
273	#41	—	188	4	no	—
274	C₂×Q₈	—	94	2	no	—
275	C₂×D₄	—	2	4	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	2	no	—
279	C₂×D₄	—	94	2	no	—
280	C₂×D₄	—	188	2	no	—
281	C₂×D₄	—	188	1	no	—
282	C₂×D₄	—	188	1	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	4	2	no	—
285	C₂²×C₄	—	4	2	no	—
286	C₂²×C₄	—	8	4	no	—
287	C₂²×C₄	—	188	2	no	—
288	C₂²×C₄	#10625334	1	1	yes	characteristic
289	C₂²×C₄	#10625334	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
291	Q₁₆	—	94	2	no	—
292	Q₁₆	—	94	2	no	—
293	Q₁₆	—	188	2	no	—
294	SD₁₆	—	188	4	no	—
295	SD₁₆	—	188	4	no	—
296	D₈	—	2	4	no	—
297	D₈	—	2	2	no	—
298	D₈	—	94	2	no	—
299	D₈	—	94	2	no	—
300	D₈	—	188	2	no	—
301	D₈	C₂×D₁₈₈	1	4	yes	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
304	C₂×C₈	#10625350	1	2	yes	—
305	C₂×C₈	C₂×C₂²⋊D₄₇	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	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	188	1	no	—
312	C₄⋊C₄	—	188	1	no	—
313	C₂²⋊C₄	—	4	2	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₂²⋊C₄	—	188	2	no	—
316	C₂²⋊C₄	—	188	2	no	—
317	C₄²	#10625334	1	1	yes	characteristic
318	C₄²	#10625334	1	1	yes	characteristic
319	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
320	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
321	C₁₆	—	94	4	no	—
322	C₂³	—	4	4	no	—
323	C₂³	—	4	2	no	—
324	C₂³	—	8	2	no	—
325	C₂³	—	8	2	no	—
326	C₂³	—	188	2	no	—
327	C₂³	C₂×#10625334	1	1	yes	characteristic
328	Q₈	—	188	2	no	—
329	Q₈	—	188	2	no	—
330	D₄	—	2	4	no	—
331	D₄	—	4	4	no	—
332	D₄	—	4	2	no	—
333	D₄	—	376	2	no	—
334	D₄	—	376	4	no	—
335	D₄	—	376	2	no	—
336	D₄	—	188	2	no	—
337	D₄	—	188	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	8	4	no	—
342	C₂×C₄	—	8	2	no	—
343	C₂×C₄	—	8	4	no	—
344	C₂×C₄	—	376	4	no	—
345	C₂×C₄	—	188	2	no	—
346	C₂×C₄	#13093998	1	2	yes	—
347	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
348	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
349	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
350	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
351	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
352	C₂×C₄	#13094049	1	2	yes	—
353	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	#13093998	1	2	yes	—
357	C₈	C₂²×D₁₈₈	1	1	yes	characteristic
358	C₈	C₂×#10625350	1	1	yes	characteristic
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	4	4	no	—
362	C₂²	—	8	8	no	—
363	C₂²	—	8	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	376	2	no	—
366	C₂²	—	376	2	no	—
367	C₂²	—	376	2	no	—
368	C₂²	#15689318	1	2	yes	—
369	C₂²	C₂²×#10625334	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	8	2	no	—
374	C₄	—	376	2	no	—
375	C₄	#15689318	1	2	yes	—
376	C₄	#15689559	1	2	yes	complemented
377	C₄	C₂×#13093998	1	1	yes	characteristic
378	C₄	C₂×#13094049	1	1	yes	characteristic
379	C₂	—	2	2	no	—
380	C₂	—	8	4	no	—
381	C₂	—	752	2	no	—
382	C₂	C₂×#15689318	1	1	yes	characteristic
383	C₂	C₂×#15689559	1	1	yes	characteristic
384	C₂	#20256522	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄×D₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	C₂×C₄×D₈	D₄₇
hypocenter	C₄₇	#33580
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15689559	C₂	semidirect
2. #20253816	C₂	semidirect
3. #20245443	C₂	semidirect
4. #20245433	C₂	semidirect
5. #20245423	C₂	semidirect
6. #20245144	C₂	semidirect
7. #20245098	C₂	semidirect
8. C₂×C₄×D₈×C₄₇	C₂	semidirect
17. #15689507	C₂²	semidirect
18. #15689503	C₂²	semidirect
19. #15689500	C₂²	semidirect
21. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
22. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
23. #15688926	C₂²	semidirect
24. #15688923	C₂²	semidirect
59. C₂×D₈×C₄₇	D₄	semidirect
183. C₄×C₄₇	#509	semidirect
189. C₂×C₄×D₈	D₄₇	semidirect
226. C₄₇	#33580	semidirect
241. C₂×D₈	D₁₈₈	semidirect
376. C₄	#15689559	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689559	2
#20253816	2
#20245443	2
#20245433	2
#20245423	2
#20245144	2
#20245098	2
C₂×C₄×D₈×C₄₇	2
#33580	47