Subgroups — #26371202

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253967	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245568	C₂	1	2	yes	maximal, complemented
3	#20245525	C₂	1	2	yes	maximal, complemented
4	#20245423	C₂	1	2	yes	maximal, complemented
5	#20245300	C₂	1	2	yes	maximal, complemented
6	C₄×#13093965	C₂	1	4	yes	maximal, complemented
7	#20245187	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#15689673	—	2	1	no	—
10	C₂×#13093965	C₂²	1	4	yes	—
11	#15689528	C₂²	1	1	yes	characteristic, complemented
12	#15689500	C₂²	1	1	yes	characteristic, complemented
13	#15689470	—	2	1	no	—
14	#15689359	—	2	1	no	—
15	#15689359	C₂²	1	2	yes	—
16	C₄×D₃₇₆	—	2	1	no	—
17	C₄×D₃₇₆	C₂²	1	2	yes	complemented
18	#15688948	C₂²	1	4	yes	complemented
19	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	complemented
20	#15688924	C₂²	1	4	yes	—
21	#15688923	C₂²	1	4	yes	complemented
22	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
23	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	complemented
24	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₄₇	—	2	1	no	—
26	C₄×D₈×C₄₇	—	2	1	no	—
27	C₄×D₈×C₄₇	—	2	1	no	—
28	C₄×D₈×C₄₇	C₂²	1	2	yes	—
29	C₄×D₈×C₄₇	C₂²	1	2	yes	—
30	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#102×C₄₇	—	2	1	no	—
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×#10625350	—	4	1	no	—
34	C₂×D₃₇₆	—	2	1	no	—
35	C₂×D₃₇₆	C₂³	1	2	yes	—
36	#13094024	—	2	1	no	—
37	#13094021	—	2	1	no	—
38	#13094012	—	4	1	no	—
39	#13093998	—	2	2	no	—
40	#13093998	—	4	1	no	—
41	C₄×D₁₈₈	—	2	2	no	—
42	C₄×D₁₈₈	—	4	1	no	—
43	#13093965	—	2	8	no	—
44	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
45	#13093958	—	2	1	no	—
46	#13093958	C₂³	1	2	yes	—
47	#13093946	—	2	2	no	—
48	#13093946	—	2	2	no	—
49	C₂²×D₄×C₄₇	—	2	1	no	—
50	C₂³×C₄×C₄₇	—	4	1	no	—
51	C₂×D₈×C₄₇	—	2	4	no	—
52	C₂×D₈×C₄₇	—	2	1	no	—
53	C₂×D₈×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₄×D₄×C₄₇	—	2	2	no	—
57	C₄×D₄×C₄₇	—	2	2	no	—
58	C₄×D₄×C₄₇	—	4	1	no	—
59	C₄×D₄×C₄₇	—	4	1	no	—
60	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
61	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
62	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
63	#107×C₄₇	—	2	1	no	—
64	#107×C₄₇	C₂³	1	2	yes	—
65	#102×C₄₇	—	2	2	no	—
66	#102×C₄₇	—	2	2	no	—
67	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
68	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
69	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₂⁴×C₄₇	—	4	1	no	—
72	C₂×D₄×C₄₇	—	2	4	no	—
73	C₂×D₄×C₄₇	—	2	2	no	—
74	C₂×D₄×C₄₇	—	4	2	no	—
75	C₂×D₄×C₄₇	—	4	1	no	—
76	C₂²×C₄×C₄₇	—	4	1	no	—
77	C₂²×C₄×C₄₇	—	4	2	no	—
78	C₂²×C₄×C₄₇	—	4	2	no	—
79	C₂²×C₄×C₄₇	—	8	4	no	—
80	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₂²⋊D₄₇	—	4	2	no	—
84	#10625350	—	8	4	no	—
85	C₂×D₁₈₈	—	2	2	no	—
86	C₂×D₁₈₈	—	4	1	no	—
87	C₂×D₁₈₈	—	4	1	no	—
88	C₂×C₄×D₄₇	—	4	2	no	—
89	C₂×Dic₉₄	—	4	1	no	—
90	D₈×C₄₇	—	2	4	no	—
91	D₈×C₄₇	—	2	2	no	—
92	D₈×C₄₇	#41	1	4	yes	complemented
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	4	1	no	—
95	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₄₇	—	2	2	no	—
100	C₄⋊C₄×C₄₇	—	4	1	no	—
101	C₂²⋊C₄×C₄₇	—	4	2	no	—
102	C₂²⋊C₄×C₄₇	—	4	2	no	—
103	C₄²×C₄₇	D₈	1	1	yes	characteristic
104	C₄²×C₄₇	D₈	1	1	yes	characteristic
105	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
107	(C₂×D₄₇)⋊C₄	—	4	2	no	—
108	(C₂×D₄₇)⋊C₄	—	4	2	no	—
109	C₄⋊Dic₄₇	—	2	2	no	—
110	C₄⋊Dic₄₇	—	4	1	no	—
111	D₃₇₆	—	2	4	no	—
112	D₃₇₆	—	4	2	no	—
113	C₄₇⋊C₁₆	—	2	4	no	—
114	C₂³×C₄₇	—	4	4	no	—
115	C₂³×C₄₇	—	4	2	no	—
116	C₂³×C₄₇	—	8	2	no	—
117	C₂³×C₄₇	—	8	2	no	—
118	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
119	C₂²×D₄₇	—	4	2	no	—
120	D₄×C₄₇	—	2	4	no	—
121	D₄×C₄₇	—	4	4	no	—
122	D₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	1	no	—
125	C₂×C₄×C₄₇	—	4	1	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	8	2	no	—
128	C₂×C₄×C₄₇	—	8	4	no	—
129	C₂×C₄×C₄₇	—	8	4	no	—
130	C₂×C₄×C₄₇	D₁₆	1	4	yes	complemented
131	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₄₇	—	8	4	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	D₁₈₈	—	4	2	no	—
140	D₁₈₈	—	4	2	no	—
141	D₁₈₈	—	8	2	no	—
142	D₁₈₈	—	8	2	no	—
143	C₄×D₄₇	—	8	4	no	—
144	Dic₉₄	—	8	2	no	—
145	C₈×C₄₇	—	2	1	no	—
146	C₈×C₄₇	—	2	1	no	—
147	C₈×C₄₇	—	2	1	no	—
148	C₈×C₄₇	C₂×#41	1	2	yes	—
149	#33660	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	1	no	—
152	C₂²×C₄₇	—	4	4	no	—
153	C₂²×C₄₇	—	8	8	no	—
154	C₂²×C₄₇	—	8	2	no	—
155	C₂²×C₄₇	—	8	2	no	—
156	C₂²×C₄₇	C₂×D₁₆	1	2	yes	—
157	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
158	C₂×D₄₇	—	8	2	no	—
159	C₂×D₄₇	—	8	2	no	—
160	C₂×D₄₇	—	8	2	no	—
161	C₄×C₄₇	—	2	1	no	—
162	C₄×C₄₇	—	2	1	no	—
163	C₄×C₄₇	—	2	1	no	—
164	C₄×C₄₇	—	8	2	no	—
165	C₄×C₄₇	C₂×D₁₆	1	2	yes	—
166	C₄×C₄₇	C₂×D₁₆	1	2	yes	—
167	C₄×C₄₇	#525	1	1	yes	characteristic
168	C₄×C₄₇	#575	1	1	yes	characteristic
169	Dic₄₇	—	8	2	no	—
170	#3169	—	47	1	no	—
171	C₂×C₄×D₈	D₄₇	1	1	yes	characteristic, complemented
172	#2232	—	47	2	no	—
173	#2224	—	47	2	no	—
174	#2206	—	47	2	no	—
175	#2184	—	47	2	no	—
176	C₄×D₁₆	—	47	4	no	—
177	#2162	—	47	1	no	—
178	C₂×C₄₇	—	2	2	no	—
179	C₂×C₄₇	—	8	4	no	—
180	C₂×C₄₇	#3100	1	1	yes	characteristic
181	C₂×C₄₇	C₂²×D₁₆	1	1	yes	characteristic
182	C₂×C₄₇	#3414	1	1	yes	characteristic
183	D₄₇	—	16	2	no	—
184	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
185	#532	—	94	1	no	—
186	C₂×C₄×D₄	—	2	1	no	—
187	C₂×D₁₆	—	47	4	no	—
188	#500	—	47	1	no	—
189	#493	—	47	1	no	—
190	#486	—	94	1	no	—
191	#466	—	47	2	no	—
192	#466	—	94	1	no	—
193	C₄×D₈	—	2	1	no	—
194	C₄×D₈	—	2	1	no	—
195	C₄×D₈	—	47	2	no	—
196	C₄×D₈	—	94	1	no	—
197	C₄×D₈	C₂×D₄₇	1	2	yes	—
198	C₄×D₈	C₂×D₄₇	1	2	yes	—
199	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
200	C₂×#102	—	2	1	no	—
201	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
202	#366	—	47	4	no	—
203	C₄⋊C₁₆	—	47	2	no	—
204	#357	—	47	4	no	—
205	#357	—	47	4	no	—
206	C₂²⋊C₁₆	—	47	2	no	—
207	C₄×C₁₆	—	47	2	no	—
208	C₄₇	#33660	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	188	1	no	—
210	C₂²×D₄	—	2	1	no	—
211	C₂³×C₄	—	4	1	no	—
212	C₂×D₈	—	2	4	no	—
213	C₂×D₈	—	2	1	no	—
214	C₂×D₈	—	47	2	no	—
215	C₂×D₈	—	94	1	no	—
216	C₂×D₈	C₂²×D₄₇	1	2	yes	—
217	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
219	C₄⋊Q₈	—	94	1	no	—
220	#127	—	94	1	no	—
221	#124	—	188	1	no	—
222	#121	—	94	2	no	—
223	#121	—	188	1	no	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	—	4	1	no	—
227	C₄×D₄	—	4	1	no	—
228	C₄×D₄	—	94	2	no	—
229	C₄×D₄	—	188	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₂²⋊C₄	—	4	1	no	—
232	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
233	D₁₆	—	94	8	no	—
234	C₂×C₁₆	—	47	4	no	—
235	#107	—	2	1	no	—
236	#107	—	47	2	no	—
237	#107	—	94	1	no	—
238	#107	C₂²×D₄₇	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	#102	—	94	2	no	—
242	#102	—	94	2	no	—
243	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
244	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
247	C₂⁴	—	4	1	no	—
248	#41	—	376	4	no	—
249	C₂×Q₈	—	188	1	no	—
250	C₂×D₄	—	2	4	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	94	2	no	—
255	C₂×D₄	—	188	2	no	—
256	C₂×D₄	—	188	1	no	—
257	C₂×D₄	—	188	1	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	8	4	no	—
262	C₂²×C₄	—	188	2	no	—
263	C₂²×C₄	#10625334	1	1	yes	characteristic
264	C₂²×C₄	#10625334	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
266	D₈	—	2	4	no	—
267	D₈	—	2	2	no	—
268	D₈	—	94	4	no	—
269	D₈	—	188	2	no	—
270	D₈	#10625350	1	4	yes	complemented
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
274	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
275	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	94	2	no	—
280	C₄⋊C₄	—	188	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	188	2	no	—
284	C₂²⋊C₄	—	188	2	no	—
285	C₄²	#10625334	1	1	yes	characteristic
286	C₄²	#10625334	1	1	yes	characteristic
287	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
289	C₁₆	—	94	4	no	—
290	C₂³	—	4	4	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	8	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	—	188	2	no	—
295	C₂³	C₂×#10625334	1	1	yes	characteristic
296	Q₈	—	376	2	no	—
297	D₄	—	2	4	no	—
298	D₄	—	4	4	no	—
299	D₄	—	4	2	no	—
300	D₄	—	376	2	no	—
301	D₄	—	376	4	no	—
302	D₄	—	376	2	no	—
303	D₄	—	188	2	no	—
304	D₄	—	188	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	—	8	4	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	—	376	4	no	—
313	C₂×C₄	—	188	2	no	—
314	C₂×C₄	#13093965	1	4	yes	complemented
315	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
316	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
317	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
318	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
319	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	C₂×#10625350	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	4	4	no	—
328	C₂²	—	8	8	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	376	2	no	—
332	C₂²	—	376	2	no	—
333	C₂²	—	376	2	no	—
334	C₂²	C₂×#13093965	1	2	yes	—
335	C₂²	C₂²×#10625334	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	8	2	no	—
340	C₄	—	376	2	no	—
341	C₄	C₂×#13093965	1	2	yes	—
342	C₄	C₂×#13093965	1	2	yes	—
343	C₄	#15689634	1	1	yes	characteristic
344	C₄	#15689921	1	1	yes	characteristic
345	C₂	—	2	2	no	—
346	C₂	—	8	4	no	—
347	C₂	—	752	2	no	—
348	C₂	#20253388	1	1	yes	characteristic
349	C₂	C₂²×#13093965	1	1	yes	characteristic
350	C₂	#20256522	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#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	#33660
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ 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. #20253967	C₂	semidirect
2. #20245568	C₂	semidirect
3. #20245525	C₂	semidirect
4. #20245423	C₂	semidirect
5. #20245300	C₂	semidirect
6. C₄×#13093965	C₂	semidirect
7. #20245187	C₂	semidirect
8. C₂×C₄×D₈×C₄₇	C₂	semidirect
11. #15689528	C₂²	semidirect
12. #15689500	C₂²	semidirect
17. C₄×D₃₇₆	C₂²	semidirect
18. #15688948	C₂²	semidirect
19. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
21. #15688923	C₂²	semidirect
23. C₄×C₄₇⋊C₁₆	C₂²	semidirect
92. D₈×C₄₇	#41	semidirect
130. C₂×C₄×C₄₇	D₁₆	semidirect
171. C₂×C₄×D₈	D₄₇	semidirect
208. C₄₇	#33660	semidirect
270. D₈	#10625350	semidirect
314. C₂×C₄	#13093965	semidirect

Maximal subgroups

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

subgroup	index
#20253967	2
#20245568	2
#20245525	2
#20245423	2
#20245300	2
C₄×#13093965	2
#20245187	2
C₂×C₄×D₈×C₄₇	2
#33660	47