Subgroups — #26316233

Order 12032.

612 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253902	C₂	1	1	yes	maximal, characteristic, complemented
2	#20244560	C₂	1	4	yes	maximal, complemented
3	#20244287	C₂	1	2	yes	maximal
4	#20243513	C₂	1	2	yes	maximal, complemented
5	#20243442	C₂	1	2	yes	maximal, complemented
6	#20243434	C₂	1	2	yes	maximal, complemented
7	#20243267	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#443	C₂	1	1	yes	maximal, characteristic
9	#15689680	C₂²	1	2	yes	—
10	#15689528	—	2	1	no	—
11	#15689525	—	2	1	no	—
12	#15689525	—	2	1	no	—
13	#15689521	—	2	1	no	—
14	C₂×#13093960	C₂²	1	1	yes	characteristic
15	#15689161	C₂²	1	2	yes	—
16	#15689161	C₂²	1	1	yes	characteristic
17	#15689161	C₂²	1	2	yes	—
18	#15689140	C₂²	1	4	yes	—
19	#15689097	C₂²	1	4	yes	—
20	#15689078	C₂²	1	2	yes	—
21	#15688895	C₂²	1	4	yes	—
22	#15688883	C₂²	1	2	yes	—
23	#15688880	C₂²	1	4	yes	—
24	C₂×#135×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#131×C₄₇	C₂²	1	2	yes	—
26	C₄₇×#443	—	2	1	no	—
27	C₄₇×#443	—	2	1	no	—
28	C₄₇×#443	—	2	1	no	—
29	C₄₇×#443	—	2	1	no	—
30	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#104×C₄₇	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×#10625329	C₂³	1	2	yes	—
34	#13094024	—	2	2	no	—
35	#13094016	—	2	2	no	—
36	#13094003	—	4	2	no	—
37	C₄×Dic₉₄	—	4	2	no	—
38	#13093983	C₂³	1	2	yes	—
39	#13093983	C₂³	1	4	yes	—
40	#13093960	—	2	1	no	—
41	#13093960	—	2	1	no	—
42	#13093958	—	2	2	no	—
43	#13093958	—	2	2	no	—
44	#13093958	—	2	1	no	—
45	#13093956	—	2	2	no	—
46	#13093956	—	2	2	no	—
47	#13093956	—	2	1	no	—
48	#13093951	—	4	4	no	—
49	#13093939	—	4	4	no	—
50	#13093934	—	2	2	no	—
51	C₈×Dic₄₇	—	2	2	no	—
52	C₂×#41×C₄₇	C₂³	1	2	yes	—
53	#135×C₄₇	—	2	1	no	—
54	#135×C₄₇	—	2	1	no	—
55	#135×C₄₇	—	2	1	no	—
56	#135×C₄₇	—	2	1	no	—
57	C₂×Q₁₆×C₄₇	Q₈	1	1	yes	characteristic, complemented
58	C₂×SD₁₆×C₄₇	Q₈	1	2	yes	complemented
59	C₂×D₈×C₄₇	Q₈	1	1	yes	characteristic, complemented
60	#131×C₄₇	—	2	2	no	—
61	#131×C₄₇	—	2	2	no	—
62	#131×C₄₇	—	2	2	no	—
63	#131×C₄₇	—	2	2	no	—
64	C₂×M₁₆×C₄₇	—	2	2	no	—
65	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
66	C₂²×C₈×C₄₇	—	2	2	no	—
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
70	#108×C₄₇	—	2	1	no	—
71	#108×C₄₇	—	2	1	no	—
72	#104×C₄₇	—	2	2	no	—
73	#104×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	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
78	#41×C₄₇	—	2	2	no	—
79	#41×C₄₇	—	2	2	no	—
80	#41×C₄₇	—	2	2	no	—
81	#41×C₄₇	—	2	2	no	—
82	C₂×Q₈×C₄₇	C₂×Q₈	1	2	yes	—
83	C₂×D₄×C₄₇	—	2	2	no	—
84	C₂×D₄×C₄₇	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₄₇	—	2	1	no	—
86	C₂²×C₄×C₄₇	—	2	2	no	—
87	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₉₄	—	4	2	no	—
89	Q₁₆×C₄₇	—	2	1	no	—
90	Q₁₆×C₄₇	—	2	1	no	—
91	SD₁₆×C₄₇	—	2	2	no	—
92	SD₁₆×C₄₇	—	2	2	no	—
93	D₈×C₄₇	—	2	1	no	—
94	D₈×C₄₇	—	2	1	no	—
95	M₁₆×C₄₇	—	2	2	no	—
96	M₁₆×C₄₇	—	2	2	no	—
97	M₁₆×C₄₇	—	4	2	no	—
98	M₁₆×C₄₇	—	4	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	4	2	no	—
101	C₂×C₈×C₄₇	—	4	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	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₈×C₄₇	#41	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	#41	1	1	yes	characteristic
112	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
113	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
114	C₄²×C₄₇	—	2	1	no	—
115	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₂²⋊Dic₄₇	—	2	2	no	—
118	C₂²⋊Dic₄₇	—	2	4	no	—
119	C₄⋊Dic₄₇	—	2	4	no	—
120	C₄⋊Dic₄₇	—	2	2	no	—
121	C₄⋊Dic₄₇	—	4	2	no	—
122	Dic₄₇⋊C₄	—	4	4	no	—
123	C₄×Dic₄₇	—	2	2	no	—
124	C₄×Dic₄₇	—	2	4	no	—
125	#10625329	—	2	2	no	—
126	#10625329	—	2	2	no	—
127	C₂×C₄₇⋊C₈	—	2	2	no	—
128	C₂³×C₄₇	—	2	2	no	—
129	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
130	Q₈×C₄₇	—	2	2	no	—
131	Q₈×C₄₇	—	2	2	no	—
132	D₄×C₄₇	—	2	2	no	—
133	D₄×C₄₇	—	2	2	no	—
134	D₄×C₄₇	—	4	2	no	—
135	D₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	2	1	no	—
137	C₂×C₄×C₄₇	—	2	1	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	4	2	no	—
140	C₂×C₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×Dic₄₇	—	4	4	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	Dic₉₄	—	8	4	no	—
151	C₈×C₄₇	—	4	2	no	—
152	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
153	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
155	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
156	C₈×C₄₇	#124	1	1	yes	characteristic
157	C₈×C₄₇	#124	1	1	yes	characteristic
158	C₈×C₄₇	#124	1	1	yes	characteristic
159	C₈×C₄₇	#124	1	1	yes	characteristic
160	C₄₇⋊C₈	—	4	2	no	—
161	#29154	—	47	1	no	maximal, Sylow
162	C₂²×C₄₇	—	4	1	no	—
163	C₂²×C₄₇	—	4	2	no	—
164	C₂²×C₄₇	—	8	2	no	—
165	C₂²×C₄₇	#535	1	1	yes	characteristic
166	C₂²×C₄₇	#554	1	1	yes	characteristic
167	C₂²×C₄₇	#555	1	1	yes	characteristic
168	C₄×C₄₇	—	2	2	no	—
169	C₄×C₄₇	—	4	1	no	—
170	C₄×C₄₇	—	4	2	no	—
171	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
173	C₄×C₄₇	#554	1	1	yes	characteristic
174	C₄×C₄₇	#555	1	1	yes	characteristic
175	Dic₄₇	—	8	4	no	—
176	Dic₄₇	—	8	2	no	—
177	#3160	—	47	1	no	—
178	C₂×#443	D₄₇	1	1	yes	characteristic
179	#2031	—	47	4	no	—
180	#1976	—	47	2	no	—
181	#1833	—	47	2	no	—
182	#1814	—	47	2	no	—
183	#1812	—	47	2	no	—
184	#1772	—	47	1	no	—
185	C₂×C₄₇	—	2	1	no	—
186	C₂×C₄₇	—	2	1	no	—
187	C₂×C₄₇	—	2	1	no	—
188	C₂×C₄₇	—	8	2	no	—
189	C₂×C₄₇	#2563	1	1	yes	characteristic
190	C₂×#135	C₂×D₄₇	1	1	yes	characteristic
191	C₂×#131	C₂×D₄₇	1	2	yes	—
192	#533	—	47	2	no	—
193	#500	—	94	1	no	—
194	#499	—	94	1	no	—
195	#499	—	94	1	no	—
196	#498	—	94	1	no	—
197	#443	—	2	1	no	—
198	#443	—	2	1	no	—
199	#443	—	2	1	no	—
200	#443	—	2	1	no	—
201	C₂×#108	—	47	1	no	—
202	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
203	#427	—	47	1	no	—
204	#427	—	47	2	no	—
205	#427	—	47	2	no	—
206	#421	—	47	4	no	—
207	C₂×#104	C₂×D₄₇	1	2	yes	—
208	#410	—	47	4	no	—
209	#405	—	47	2	no	—
210	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
211	#344	—	47	4	no	—
212	#339	—	47	2	no	—
213	H(3,ℤ/4)	—	47	4	no	—
214	C₄₇	#29154	1	1	yes	characteristic, complemented, Sylow
215	C₂×#41	C₂²×D₄₇	1	2	yes	—
216	#135	—	2	1	no	—
217	#135	—	2	1	no	—
218	#135	—	2	1	no	—
219	#135	—	2	1	no	—
220	C₂×Q₁₆	Dic₉₄	1	1	yes	characteristic, complemented
221	C₂×SD₁₆	Dic₉₄	1	2	yes	complemented
222	C₂×D₈	Dic₉₄	1	1	yes	characteristic, complemented
223	#131	—	2	2	no	—
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	#131	—	2	2	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	—	47	2	no	—
229	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
230	C₂²×C₈	—	2	2	no	—
231	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
233	C₄⋊Q₈	—	94	2	no	—
234	#125	—	94	2	no	—
235	C₂²⋊Q₈	—	188	2	no	—
236	C₄×Q₈	—	188	2	no	—
237	#117	—	47	4	no	—
238	#117	—	47	2	no	—
239	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
240	#108	—	2	1	no	—
241	#108	—	2	1	no	—
242	#108	—	94	1	no	—
243	#108	—	94	1	no	—
244	#107	—	94	1	no	—
245	#107	—	94	2	no	—
246	#107	—	94	2	no	—
247	#106	—	94	1	no	—
248	#106	—	94	2	no	—
249	#106	—	94	2	no	—
250	#104	—	2	2	no	—
251	#104	—	2	2	no	—
252	#104	—	188	4	no	—
253	#99	—	188	4	no	—
254	#97	—	94	2	no	—
255	C₄×C₈	—	94	2	no	—
256	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
257	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
258	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
259	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
260	#41	—	2	2	no	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	2	2	no	—
264	C₂×Q₈	—	188	2	no	—
265	C₂×Q₈	C₂×Dic₉₄	1	2	yes	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	C₂×Dic₉₄	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	1	no	—
273	SD₁₆	—	2	2	no	—
274	SD₁₆	—	2	2	no	—
275	D₈	—	2	1	no	—
276	D₈	—	2	1	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	4	2	no	—
280	M₁₆	—	4	2	no	—
281	M₁₆	—	94	2	no	—
282	M₁₆	—	94	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	94	2	no	—
287	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
288	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
289	C₂×C₈	#10625350	1	1	yes	characteristic
290	C₂×C₈	#10625350	1	1	yes	characteristic
291	C₂×C₈	#10625352	1	1	yes	characteristic
292	C₂×C₈	#10625352	1	1	yes	characteristic
293	C₂×C₈	#10625352	1	1	yes	characteristic
294	C₂×C₈	#10625352	1	1	yes	characteristic
295	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
296	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
297	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
298	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
299	C₄⋊C₄	—	94	4	no	—
300	C₄⋊C₄	—	94	2	no	—
301	C₄⋊C₄	—	188	4	no	—
302	C₄⋊C₄	—	188	2	no	—
303	C₂²⋊C₄	—	94	2	no	—
304	C₂²⋊C₄	—	94	4	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	94	2	no	—
307	C₄²	—	94	4	no	—
308	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
309	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	#13094070	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	376	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	188	2	no	—
326	C₂×C₄	—	188	4	no	—
327	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
328	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
329	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
330	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
331	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
332	C₂×C₄	#13094070	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	188	2	no	—
335	C₈	#13094003	1	1	yes	characteristic
336	C₈	#13094003	1	1	yes	characteristic
337	C₈	#13094003	1	1	yes	characteristic
338	C₈	#13094003	1	1	yes	characteristic
339	C₈	#13094011	1	1	yes	characteristic
340	C₈	#13094011	1	1	yes	characteristic
341	C₈	#13094011	1	1	yes	characteristic
342	C₈	#13094011	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	#15689687	1	1	yes	characteristic
347	C₂²	#15689830	1	1	yes	characteristic
348	C₂²	#15689838	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	376	4	no	—
353	C₄	—	376	2	no	—
354	C₄	C₂×#13094003	1	1	yes	characteristic
355	C₄	C₂×#13094011	1	1	yes	characteristic
356	C₄	#15689830	1	1	yes	characteristic
357	C₄	#15689838	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	#20248561	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15689838
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#443	C₂
socle	C₂×C₄₇	#2563
prime radical	C₂×#135×C₄₇	C₂²
hypercenter	C₂×#443	D₄₇
hypocenter	C₄₇	#29154
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. #20253902	C₂	semidirect
2. #20244560	C₂	semidirect
4. #20243513	C₂	semidirect
5. #20243442	C₂	semidirect
6. #20243434	C₂	semidirect
7. #20243267	C₂	semidirect
57. C₂×Q₁₆×C₄₇	Q₈	semidirect
58. C₂×SD₁₆×C₄₇	Q₈	semidirect
59. C₂×D₈×C₄₇	Q₈	semidirect
214. C₄₇	#29154	semidirect
220. C₂×Q₁₆	Dic₉₄	semidirect
221. C₂×SD₁₆	Dic₉₄	semidirect
222. C₂×D₈	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20253902	2
#20244560	2
#20244287	2
#20243513	2
#20243442	2
#20243434	2
#20243267	2
C₂×C₄₇×#443	2
#29154	47