Subgroups — #26369695

Order 12032.

642 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256522	C₂	1	1	yes	maximal, characteristic
2	#20252306	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245272	C₂	1	4	yes	maximal, complemented
4	#20245267	C₂	1	4	yes	maximal, complemented
5	#20245182	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688930	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15688929	C₂	1	1	yes	maximal, characteristic, complemented
8	#20244985	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2957	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094046	C₂²	1	1	yes	characteristic
11	#15689889	—	2	1	no	—
12	#15689564	C₄	1	2	yes	complemented
13	#15689563	C₄	1	2	yes	complemented
14	#15689559	C₄	1	2	yes	complemented
15	#15689558	C₄	1	2	yes	complemented
16	#15689553	C₂²	1	4	yes	—
17	C₂×#13093964	C₂²	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	#15689246	—	2	2	no	—
20	#15689241	C₂²	1	4	yes	complemented
21	C₂×#13093960	C₂²	1	1	yes	characteristic
22	#15689141	—	2	1	no	—
23	#15688952	C₂²	1	4	yes	complemented
24	#15688930	C₂²	1	2	yes	complemented
25	#15688930	C₂²	1	2	yes	complemented
26	#15688929	C₂²	1	2	yes	complemented
27	#15688929	C₂²	1	2	yes	complemented
28	#15688899	C₂²	1	2	yes	complemented
29	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	complemented
30	C₂×#135×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#568	—	2	1	no	—
32	C₄₇×#444	—	2	2	no	—
33	C₄₇×#443	C₂²	1	4	yes	complemented
34	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#421	—	2	1	no	—
36	C₄₇×#405	C₂²	1	1	yes	characteristic
37	C₂×#10625350	—	2	1	no	—
38	#13094046	—	2	1	no	—
39	#13094046	—	2	1	no	—
40	#13094046	C₂×C₄	1	2	yes	—
41	#13094046	C₂³	1	2	yes	—
42	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
43	C₂×#10625325	—	2	1	no	—
44	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
45	#13094032	—	4	2	no	—
46	#13094030	—	2	4	no	—
47	C₂×#10625329	—	2	1	no	—
48	M₁₆×D₄₇	—	4	2	no	—
49	#13093970	C₂×C₄	1	4	yes	—
50	#13093968	C₂×C₄	1	4	yes	—
51	#13093967	C₂×C₄	1	4	yes	—
52	#13093965	C₂×C₄	1	4	yes	—
53	#13093964	C₂×C₄	1	2	yes	—
54	#13093964	C₂×C₄	1	2	yes	—
55	C₂×C₄₇⋊C₁₆	—	2	1	no	—
56	C₂×C₄₇⋊C₁₆	—	2	1	no	—
57	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
60	#13093960	—	2	1	no	—
61	#13093960	C₂³	1	2	yes	—
62	#13093952	—	2	4	no	—
63	C₂×#41×C₄₇	—	2	1	no	—
64	#135×C₄₇	—	2	1	no	—
65	#135×C₄₇	—	2	1	no	—
66	#135×C₄₇	C₂×C₄	1	2	yes	—
67	#135×C₄₇	C₂³	1	2	yes	—
68	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
69	C₂×SD₁₆×C₄₇	—	2	1	no	—
70	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
71	#131×C₄₇	—	2	4	no	—
72	#131×C₄₇	—	4	2	no	—
73	C₂×M₁₆×C₄₇	—	2	1	no	—
74	C₂×M₁₆×C₄₇	—	4	2	no	—
75	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
77	#117×C₄₇	D₄	1	1	yes	characteristic
78	#108×C₄₇	—	2	1	no	—
79	#108×C₄₇	C₂³	1	2	yes	—
80	#104×C₄₇	—	2	4	no	—
81	#97×C₄₇	D₄	1	2	yes	—
82	C₄×C₈×C₄₇	C₂³	1	2	yes	—
83	#41×C₄₇	—	2	2	no	—
84	#41×C₄₇	—	2	2	no	—
85	#41×C₄₇	—	4	1	no	—
86	#41×C₄₇	—	4	1	no	—
87	C₂×Q₈×C₄₇	—	2	1	no	—
88	C₂×D₄×C₄₇	—	2	1	no	—
89	C₂×D₄×C₄₇	—	4	1	no	—
90	C₂²×C₄×C₄₇	—	4	1	no	—
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₄₇	—	4	1	no	—
93	#10625350	—	2	2	no	—
94	#10625350	—	2	2	no	—
95	#10625350	—	4	1	no	—
96	#10625350	—	4	1	no	—
97	C₂×D₁₈₈	—	2	1	no	—
98	C₂×C₄×D₄₇	—	4	1	no	—
99	C₂×Dic₉₄	—	2	1	no	—
100	Q₁₆×C₄₇	—	2	1	no	—
101	Q₁₆×C₄₇	C₂²×C₄	1	2	yes	—
102	SD₁₆×C₄₇	—	2	2	no	—
103	SD₁₆×C₄₇	—	2	2	no	—
104	D₈×C₄₇	—	2	1	no	—
105	D₈×C₄₇	C₂²×C₄	1	2	yes	—
106	M₁₆×C₄₇	—	2	2	no	—
107	M₁₆×C₄₇	—	4	1	no	—
108	M₁₆×C₄₇	—	4	4	no	—
109	M₁₆×C₄₇	D₈	1	2	yes	—
110	M₁₆×C₄₇	D₈	1	2	yes	—
111	C₂×C₈×C₄₇	—	4	1	no	—
112	C₂×C₈×C₄₇	—	4	4	no	—
113	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
118	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
119	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₄₇	—	2	1	no	—
121	C₂²⋊C₄×C₄₇	—	2	1	no	—
122	C₄²×C₄₇	C₂×D₄	1	2	yes	—
123	#10625329	—	2	2	no	—
124	#10625329	—	4	1	no	—
125	C₂×C₄₇⋊C₈	—	4	1	no	—
126	Dic₁₈₈	—	2	1	no	—
127	Dic₁₈₈	C₂²×C₄	1	2	yes	—
128	D₃₇₆	—	2	1	no	—
129	D₃₇₆	C₂²×C₄	1	2	yes	—
130	#10625325	—	2	2	no	—
131	#10625325	—	2	2	no	—
132	#10625324	—	4	4	no	—
133	C₈×D₄₇	—	4	4	no	—
134	C₄₇⋊C₁₆	—	2	1	no	—
135	C₄₇⋊C₁₆	—	2	1	no	—
136	C₄₇⋊C₁₆	C₂²×C₄	1	4	yes	—
137	C₂³×C₄₇	—	4	1	no	—
138	C₂³×C₄₇	#135	1	1	yes	characteristic
139	C₂²×D₄₇	—	4	1	no	—
140	Q₈×C₄₇	—	2	2	no	—
141	Q₈×C₄₇	—	4	1	no	—
142	D₄×C₄₇	—	2	2	no	—
143	D₄×C₄₇	—	4	2	no	—
144	D₄×C₄₇	—	4	2	no	—
145	D₄×C₄₇	—	4	1	no	—
146	C₂×C₄×C₄₇	—	2	2	no	—
147	C₂×C₄×C₄₇	—	4	1	no	—
148	C₂×C₄×C₄₇	—	4	2	no	—
149	C₂×C₄×C₄₇	—	4	2	no	—
150	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
155	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
156	C₂²⋊D₄₇	—	4	2	no	—
157	C₂²⋊D₄₇	—	4	2	no	—
158	C₂×Dic₄₇	—	4	1	no	—
159	D₁₈₈	—	2	2	no	—
160	D₁₈₈	—	4	1	no	—
161	C₄×D₄₇	—	4	2	no	—
162	C₄×D₄₇	—	4	2	no	—
163	Dic₉₄	—	2	2	no	—
164	Dic₉₄	—	4	1	no	—
165	C₈×C₄₇	—	4	2	no	—
166	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
167	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
168	C₈×C₄₇	C₂×D₈	1	4	yes	—
169	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
170	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
171	C₄₇⋊C₈	—	4	2	no	—
172	#33464	—	47	1	no	maximal, Sylow
173	C₂²×C₄₇	—	4	1	no	—
174	C₂²×C₄₇	—	4	2	no	—
175	C₂²×C₄₇	—	16	1	no	—
176	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
177	C₂²×C₄₇	C₄×D₈	1	1	yes	characteristic
178	C₂²×C₄₇	C₂×#135	1	1	yes	characteristic
179	C₂×D₄₇	—	4	2	no	—
180	C₂×D₄₇	—	16	1	no	—
181	C₄×C₄₇	—	4	2	no	—
182	C₄×C₄₇	—	4	2	no	—
183	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
184	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
185	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
187	Dic₄₇	—	4	2	no	—
188	#3414	—	47	1	no	—
189	#2957	—	47	1	no	—
190	#2957	D₄₇	1	1	yes	characteristic, complemented
191	#2179	—	47	4	no	—
192	#2178	—	47	4	no	—
193	#2161	—	47	1	no	—
194	C₂×#359	—	47	1	no	—
195	C₂×#359	—	47	1	no	—
196	#2108	—	47	1	no	—
197	C₂×C₄₇	—	2	1	no	—
198	C₂×C₄₇	—	2	1	no	—
199	C₂×C₄₇	—	2	1	no	—
200	C₂×C₄₇	—	8	2	no	—
201	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
202	D₄₇	—	8	2	no	—
203	C₂×#135	—	47	1	no	—
204	C₂×#135	C₂×D₄₇	1	1	yes	characteristic
205	#568	—	2	1	no	—
206	#568	—	94	1	no	—
207	#510	—	47	2	no	—
208	#510	—	47	2	no	—
209	#509	—	47	2	no	—
210	#509	—	47	2	no	—
211	#508	—	47	4	no	—
212	C₂×M₃₂	—	47	1	no	—
213	C₂²×C₁₆	—	47	1	no	—
214	#444	—	2	2	no	—
215	#444	—	94	2	no	—
216	#443	—	47	4	no	—
217	#443	C₂×D₄₇	1	4	yes	complemented
218	C₂×#108	—	47	1	no	—
219	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
220	#421	—	2	1	no	—
221	#421	—	94	1	no	—
222	#405	C₂×D₄₇	1	1	yes	characteristic
223	#368	—	47	4	no	—
224	#359	—	47	2	no	—
225	#359	—	47	2	no	—
226	#359	—	47	2	no	—
227	#359	—	47	2	no	—
228	#346	—	47	2	no	—
229	C₄×C₁₆	—	47	2	no	—
230	C₄₇	#33464	1	1	yes	characteristic, complemented, Sylow
231	C₂×#41	—	2	1	no	—
232	C₂×#41	—	94	1	no	—
233	#135	—	2	1	no	—
234	#135	—	2	1	no	—
235	#135	—	47	2	no	—
236	#135	—	47	2	no	—
237	#135	—	94	1	no	—
238	#135	—	94	1	no	—
239	#135	C₄×D₄₇	1	2	yes	—
240	#135	C₂²×D₄₇	1	2	yes	—
241	C₂×Q₁₆	—	47	1	no	—
242	C₂×Q₁₆	C₄×D₄₇	1	1	yes	characteristic
243	C₂×SD₁₆	—	2	1	no	—
244	C₂×SD₁₆	—	94	1	no	—
245	C₂×D₈	—	47	1	no	—
246	C₂×D₈	C₄×D₄₇	1	1	yes	characteristic
247	#131	—	2	4	no	—
248	#131	—	4	2	no	—
249	#131	—	94	4	no	—
250	#131	—	188	2	no	—
251	C₂×M₁₆	—	2	1	no	—
252	C₂×M₁₆	—	4	2	no	—
253	C₂×M₁₆	—	94	1	no	—
254	C₂×M₁₆	—	188	2	no	—
255	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
256	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
257	#117	C₂²⋊D₄₇	1	1	yes	characteristic
258	Q₃₂	—	47	4	no	—
259	SD₃₂	—	47	4	no	—
260	SD₃₂	—	47	4	no	—
261	D₁₆	—	47	4	no	—
262	M₃₂	—	47	2	no	—
263	M₃₂	—	47	2	no	—
264	C₂×C₁₆	—	47	2	no	—
265	C₂×C₁₆	—	47	1	no	—
266	C₂×C₁₆	—	47	1	no	—
267	C₂×C₁₆	—	94	1	no	—
268	C₂×C₁₆	—	94	1	no	—
269	#108	—	2	1	no	—
270	#108	—	47	2	no	—
271	#108	—	94	1	no	—
272	#108	C₂²×D₄₇	1	2	yes	—
273	#104	—	2	4	no	—
274	#104	—	94	4	no	—
275	#97	C₂²⋊D₄₇	1	2	yes	—
276	C₄×C₈	C₂²×D₄₇	1	2	yes	—
277	#41	—	2	2	no	—
278	#41	—	2	2	no	—
279	#41	—	4	1	no	—
280	#41	—	4	1	no	—
281	#41	—	94	2	no	—
282	#41	—	94	2	no	—
283	#41	—	188	1	no	—
284	#41	—	188	1	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	94	1	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	94	1	no	—
290	C₂×D₄	—	188	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	188	1	no	—
293	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
294	Q₁₆	—	2	1	no	—
295	Q₁₆	—	47	2	no	—
296	Q₁₆	—	94	1	no	—
297	Q₁₆	C₂×C₄×D₄₇	1	2	yes	—
298	SD₁₆	—	2	2	no	—
299	SD₁₆	—	2	2	no	—
300	SD₁₆	—	94	2	no	—
301	SD₁₆	—	94	2	no	—
302	D₈	—	2	1	no	—
303	D₈	—	47	2	no	—
304	D₈	—	94	1	no	—
305	D₈	C₂×C₄×D₄₇	1	2	yes	—
306	M₁₆	—	2	2	no	—
307	M₁₆	—	4	1	no	—
308	M₁₆	—	4	4	no	—
309	M₁₆	—	94	2	no	—
310	M₁₆	—	188	1	no	—
311	M₁₆	—	188	4	no	—
312	M₁₆	#10625334	1	2	yes	—
313	M₁₆	#10625334	1	2	yes	—
314	C₂×C₈	—	4	1	no	—
315	C₂×C₈	—	4	4	no	—
316	C₂×C₈	—	188	1	no	—
317	C₂×C₈	—	188	4	no	—
318	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
319	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
320	C₂×C₈	#10625350	1	1	yes	characteristic
321	C₂×C₈	#10625350	1	1	yes	characteristic
322	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
323	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
324	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
325	C₄⋊C₄	—	2	1	no	—
326	C₂²⋊C₄	—	2	1	no	—
327	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
328	C₁₆	—	47	4	no	—
329	C₁₆	—	94	1	no	—
330	C₁₆	—	94	1	no	—
331	C₂³	—	4	1	no	—
332	C₂³	—	188	1	no	—
333	C₂³	#13094044	1	1	yes	characteristic
334	Q₈	—	2	2	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	94	2	no	—
337	Q₈	—	188	1	no	—
338	D₄	—	2	2	no	—
339	D₄	—	4	2	no	—
340	D₄	—	4	2	no	—
341	D₄	—	4	1	no	—
342	D₄	—	94	2	no	—
343	D₄	—	188	2	no	—
344	D₄	—	188	2	no	—
345	D₄	—	188	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	188	1	no	—
351	C₂×C₄	—	188	2	no	—
352	C₂×C₄	—	188	2	no	—
353	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
354	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
355	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
356	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
357	C₂×C₄	#13094044	1	1	yes	characteristic
358	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	188	2	no	—
361	C₈	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
362	C₈	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
363	C₈	C₂×#10625334	1	4	yes	—
364	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
365	C₈	C₂×#10625350	1	1	yes	characteristic
366	C₂²	—	4	1	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	16	1	no	—
369	C₂²	—	188	2	no	—
370	C₂²	—	752	1	no	—
371	C₂²	C₄×#10625334	1	1	yes	characteristic
372	C₂²	C₄×#10625334	1	1	yes	characteristic
373	C₂²	C₂×#13094044	1	1	yes	characteristic
374	C₄	—	4	2	no	—
375	C₄	—	4	2	no	—
376	C₄	—	188	2	no	—
377	C₄	C₄×#10625334	1	1	yes	characteristic
378	C₄	C₄×#10625334	1	1	yes	characteristic
379	C₄	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
380	C₄	C₂²×#10625334	1	1	yes	characteristic
381	C₂	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	—	2	1	no	—
384	C₂	—	8	2	no	—
385	C₂	—	376	2	no	—
386	C₂	C₂×C₄×#10625334	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×#10625334
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2957	C₂
socle	C₂×C₄₇	C₂×C₄×D₈
prime radical	#20256522	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2957	D₄₇
hypocenter	C₄₇	#33464
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405 ◁ #2957

Complements

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

N	H ≅ G/N	type
2. #20252306	C₂	semidirect
3. #20245272	C₂	semidirect
4. #20245267	C₂	semidirect
5. #20245182	C₂	semidirect
6. C₂×#15688930	C₂	semidirect
7. C₂×#15688929	C₂	semidirect
8. #20244985	C₂	semidirect
9. C₄₇×#2957	C₂	semidirect
12. #15689564	C₄	semidirect
13. #15689563	C₄	semidirect
14. #15689559	C₄	semidirect
15. #15689558	C₄	semidirect
20. #15689241	C₂²	semidirect
23. #15688952	C₂²	semidirect
24. #15688930	C₂²	semidirect
25. #15688930	C₂²	semidirect
26. #15688929	C₂²	semidirect
27. #15688929	C₂²	semidirect
28. #15688899	C₂²	semidirect
29. C₄×C₄₇⋊C₁₆	C₂²	semidirect
33. C₄₇×#443	C₂²	semidirect
190. #2957	D₄₇	semidirect
217. #443	C₂×D₄₇	semidirect
230. C₄₇	#33464	semidirect

Maximal subgroups

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

subgroup	index
#20256522	2
#20252306	2
#20245272	2
#20245267	2
#20245182	2
C₂×#15688930	2
C₂×#15688929	2
#20244985	2
C₄₇×#2957	2
#33464	47