Subgroups — #26369618

Order 12032.

696 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	C₂×#15689561	C₂	1	1	yes	maximal, characteristic
2	#20252262	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252239	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245263	C₂	1	2	yes	maximal, complemented
5	#20245251	C₂	1	2	yes	maximal, complemented
6	#20245239	C₂	1	2	yes	maximal, complemented
7	#20245234	C₂	1	2	yes	maximal, complemented
8	#20245093	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15688923	C₂	1	1	yes	maximal, characteristic, complemented
10	#20244987	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2159	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	1	yes	characteristic
13	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
14	#15689604	—	2	1	no	—
15	#15689597	—	2	1	no	—
16	#15689561	C₄	1	8	yes	complemented
17	C₂×#13093969	C₂²	1	2	yes	—
18	C₂×D₇₅₂	C₂²	1	2	yes	—
19	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
20	#15689234	—	2	2	no	—
21	#15689222	—	2	2	no	—
22	#15689217	C₂²	1	2	yes	complemented
23	#15689205	C₂²	1	2	yes	—
24	#15689205	C₂²	1	2	yes	—
25	#15689205	C₂²	1	2	yes	complemented
26	#15689127	—	2	1	no	—
27	#15689122	—	2	1	no	—
28	#15689078	C₂²	1	1	yes	characteristic
29	#15688926	C₂²	1	1	yes	characteristic, complemented
30	#15688926	C₂²	1	1	yes	characteristic, complemented
31	#15688923	C₂²	1	2	yes	—
32	#15688923	C₂²	1	2	yes	—
33	#15688923	C₂²	1	1	yes	characteristic, complemented
34	#15688923	C₂²	1	1	yes	characteristic, complemented
35	#15688898	C₂²	1	2	yes	complemented
36	C₁₆×Dic₄₇	C₂²	1	2	yes	complemented
37	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#427	C₂²	1	1	yes	characteristic
39	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#367	C₂²	1	2	yes	complemented
41	C₄₇×#366	C₂²	1	2	yes	complemented
42	C₂×#10625350	—	2	1	no	—
43	C₂²×D₁₈₈	—	2	1	no	—
44	#13094046	—	2	2	no	—
45	#13094046	C₂×C₄	1	4	yes	complemented
46	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
47	C₂×#10625325	—	2	1	no	—
48	C₂×D₃₇₆	—	2	1	no	—
49	C₂×D₃₇₆	—	2	2	no	—
50	C₂×D₃₇₆	C₂×C₄	1	4	yes	complemented
51	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
52	C₂×D₃₇₆	C₂³	1	2	yes	—
53	C₂×#10625329	D₄	1	1	yes	characteristic
54	#13093989	—	2	2	no	—
55	C₄×D₁₈₈	—	4	1	no	—
56	C₄×D₁₈₈	—	4	1	no	—
57	#13093986	—	2	2	no	—
58	#13093986	—	2	2	no	—
59	#13093986	—	2	2	no	—
60	C₄×C₂²⋊D₄₇	—	4	2	no	—
61	#13093983	D₄	1	1	yes	characteristic
62	#13093982	—	4	1	no	—
63	C₄⋊C₄×D₄₇	—	4	1	no	—
64	C₂²⋊C₄×D₄₇	—	4	2	no	—
65	#13093969	C₂×C₄	1	8	yes	complemented
66	D₇₅₂	C₂×C₄	1	8	yes	complemented
67	C₂×C₄₇⋊C₁₆	—	2	2	no	—
68	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
70	#13093947	—	2	2	no	—
71	#13093944	—	2	2	no	—
72	#13093944	—	2	2	no	—
73	#13093944	—	2	2	no	—
74	#13093934	D₄	1	2	yes	—
75	C₈×Dic₄₇	C₂³	1	2	yes	—
76	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
77	#117×C₄₇	—	2	1	no	—
78	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
79	M₃₂×C₄₇	C₂×C₄	1	4	yes	complemented
80	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
81	#107×C₄₇	—	2	1	no	—
82	#107×C₄₇	C₂³	1	2	yes	—
83	#107×C₄₇	C₂³	1	1	yes	characteristic
84	#107×C₄₇	C₂³	1	1	yes	characteristic
85	#106×C₄₇	—	2	1	no	—
86	C₂³×D₄₇	—	4	1	no	—
87	C₂²×C₄×C₄₇	—	4	1	no	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₂²⋊D₄₇	—	4	1	no	—
90	#10625350	—	2	4	no	—
91	#10625350	—	4	2	no	—
92	C₂×D₁₈₈	—	2	4	no	—
93	C₂×D₁₈₈	—	2	2	no	—
94	C₂×D₁₈₈	—	2	1	no	—
95	C₂×D₁₈₈	—	4	1	no	—
96	C₂×D₁₈₈	—	4	2	no	—
97	C₂×C₄×D₄₇	—	4	1	no	—
98	C₂×C₄×D₄₇	—	4	2	no	—
99	C₂×C₄×D₄₇	—	4	2	no	—
100	C₂×C₄×D₄₇	—	4	2	no	—
101	C₂×Dic₉₄	—	2	1	no	—
102	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	#41	1	2	yes	—
105	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₄₇	—	2	1	no	—
107	C₄⋊C₄×C₄₇	—	2	1	no	—
108	C₄⋊C₄×C₄₇	—	2	2	no	—
109	C₄⋊C₄×C₄₇	—	4	1	no	—
110	C₂²⋊C₄×C₄₇	—	4	1	no	—
111	C₄²×C₄₇	—	4	1	no	—
112	C₂²⋊Dic₄₇	D₈	1	2	yes	—
113	(C₂×D₄₇)⋊C₄	—	4	2	no	—
114	(C₂×D₄₇)⋊C₄	—	4	2	no	—
115	(C₂×D₄₇)⋊C₄	—	4	2	no	—
116	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
117	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
118	Dic₄₇⋊C₄	—	4	2	no	—
119	C₄×Dic₄₇	—	4	2	no	—
120	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
121	#10625329	—	2	2	no	—
122	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
123	Dic₁₈₈	—	2	1	no	—
124	Dic₁₈₈	C₂²×C₄	1	2	yes	—
125	D₃₇₆	—	2	4	no	—
126	D₃₇₆	—	2	2	no	—
127	D₃₇₆	—	2	1	no	—
128	D₃₇₆	C₂²×C₄	1	2	yes	—
129	D₃₇₆	C₂²×C₄	1	4	yes	—
130	#10625325	—	2	4	no	—
131	C₁₆×C₄₇	C₂²×C₄	1	4	yes	—
132	C₄₇⋊C₁₆	—	2	1	no	—
133	C₄₇⋊C₁₆	—	2	1	no	—
134	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₄₇	—	4	1	no	—
136	C₂²×D₄₇	—	4	2	no	—
137	C₂²×D₄₇	—	4	4	no	—
138	C₂²×D₄₇	—	16	1	no	—
139	C₂²×D₄₇	—	16	1	no	—
140	C₂×C₄×C₄₇	—	4	2	no	—
141	C₂×C₄×C₄₇	—	4	1	no	—
142	C₂×C₄×C₄₇	—	4	1	no	—
143	C₂×C₄×C₄₇	—	16	1	no	—
144	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
145	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₄₇	#135	1	2	yes	—
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₄₇	—	4	4	no	—
149	C₂×Dic₄₇	—	4	1	no	—
150	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
151	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
152	D₁₈₈	—	2	2	no	—
153	D₁₈₈	—	2	4	no	—
154	D₁₈₈	—	4	2	no	—
155	D₁₈₈	—	4	1	no	—
156	D₁₈₈	—	4	4	no	—
157	C₄×D₄₇	—	4	4	no	—
158	C₄×D₄₇	—	16	2	no	—
159	C₄×D₄₇	—	16	2	no	—
160	C₄×D₄₇	—	16	2	no	—
161	Dic₉₄	—	2	2	no	—
162	Dic₉₄	—	4	1	no	—
163	C₈×C₄₇	C₄×D₄	1	2	yes	—
164	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
165	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
166	C₄₇⋊C₈	—	2	2	no	—
167	#33454	—	47	1	no	maximal, Sylow
168	C₂²×C₄₇	—	2	1	no	—
169	C₂²×C₄₇	—	2	1	no	—
170	C₂²×C₄₇	C₄×D₈	1	2	yes	—
171	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
172	C₂×D₄₇	—	4	2	no	—
173	C₂×D₄₇	—	4	4	no	—
174	C₂×D₄₇	—	8	1	no	—
175	C₂×D₄₇	—	8	2	no	—
176	C₂×D₄₇	—	8	2	no	—
177	C₂×D₄₇	—	8	1	no	—
178	C₂×D₄₇	—	16	4	no	—
179	C₄×C₄₇	—	8	1	no	—
180	C₄×C₄₇	—	8	2	no	—
181	C₄×C₄₇	—	8	1	no	—
182	C₄×C₄₇	C₄×D₈	1	2	yes	—
183	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
184	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
185	Dic₄₇	—	2	2	no	—
186	Dic₄₇	—	2	2	no	—
187	Dic₄₇	—	4	2	no	—
188	C₂×#509	—	47	1	no	—
189	#2951	—	47	1	no	—
190	#2948	—	47	1	no	—
191	#2177	—	47	2	no	—
192	#2175	—	47	2	no	—
193	C₄×SD₃₂	—	47	2	no	—
194	C₄×D₁₆	—	47	2	no	—
195	#2159	D₄₇	1	1	yes	characteristic, complemented
196	#2138	—	47	1	no	—
197	C₂×#357	—	47	1	no	—
198	#2108	—	47	1	no	—
199	C₂×C₄₇	—	2	2	no	—
200	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
201	C₂×C₄₇	#3415	1	1	yes	characteristic
202	C₂×C₄₇	#3416	1	1	yes	characteristic
203	D₄₇	—	8	2	no	—
204	D₄₇	—	8	4	no	—
205	C₂×#135	—	47	1	no	—
206	C₂²×D₈	—	47	1	no	—
207	#520	—	94	1	no	—
208	#518	—	94	1	no	—
209	#509	—	47	8	no	—
210	C₂×SD₃₂	—	47	2	no	—
211	C₂×D₁₆	—	47	2	no	—
212	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
213	C₂²×C₁₆	—	47	1	no	—
214	#442	—	94	2	no	—
215	#440	—	94	2	no	—
216	C₄×Q₁₆	—	47	2	no	—
217	C₄×D₈	—	47	2	no	—
218	C₄×D₈	—	47	2	no	—
219	C₄×D₈	—	47	2	no	—
220	#427	C₂×D₄₇	1	1	yes	characteristic
221	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
222	#418	—	94	1	no	—
223	#417	—	94	1	no	—
224	#405	—	47	1	no	—
225	#367	C₂×D₄₇	1	2	yes	complemented
226	#366	C₂×D₄₇	1	2	yes	complemented
227	#358	—	47	1	no	—
228	#358	—	47	1	no	—
229	#357	—	47	2	no	—
230	#357	—	47	2	no	—
231	#357	—	47	1	no	—
232	#357	—	47	1	no	—
233	#346	—	47	2	no	—
234	C₄×C₁₆	—	47	2	no	—
235	C₄₇	#33454	1	1	yes	characteristic, complemented, Sylow
236	C₂×#41	—	94	1	no	—
237	C₂²×D₄	—	94	1	no	—
238	#135	—	47	4	no	—
239	#135	—	94	2	no	—
240	C₂×Q₁₆	—	47	1	no	—
241	C₂×SD₁₆	—	94	1	no	—
242	C₂×D₈	—	47	4	no	—
243	C₂×D₈	—	47	2	no	—
244	C₂×D₈	—	47	1	no	—
245	C₂×D₈	—	94	1	no	—
246	C₂×D₈	—	94	2	no	—
247	C₂×M₁₆	—	47	1	no	—
248	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
249	C₄×Q₈	—	94	2	no	—
250	C₄×D₄	—	94	2	no	—
251	C₄×D₄	—	94	2	no	—
252	C₄×D₄	—	94	2	no	—
253	C₄×D₄	—	188	1	no	—
254	C₄×D₄	—	188	1	no	—
255	C₄×D₄	—	188	2	no	—
256	#117	—	2	1	no	—
257	#117	—	47	1	no	—
258	#117	—	188	1	no	—
259	C₂×C₄⋊C₄	—	2	1	no	—
260	C₂×C₄⋊C₄	—	188	1	no	—
261	C₂×C₂²⋊C₄	—	188	2	no	—
262	SD₃₂	—	47	8	no	—
263	D₁₆	—	47	8	no	—
264	M₃₂	C₄×D₄₇	1	4	yes	complemented
265	C₂×C₁₆	—	47	1	no	—
266	C₂×C₁₆	—	47	1	no	—
267	C₂×C₁₆	—	94	2	no	—
268	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
269	#107	—	2	1	no	—
270	#107	C₂²×D₄₇	1	2	yes	—
271	#107	C₂²×D₄₇	1	1	yes	characteristic
272	#107	C₂²×D₄₇	1	1	yes	characteristic
273	#106	—	2	1	no	—
274	#103	—	94	2	no	—
275	#102	—	94	2	no	—
276	#102	—	94	2	no	—
277	#102	—	94	2	no	—
278	#97	—	47	2	no	—
279	C₄×C₈	—	47	2	no	—
280	C₂⁴	—	188	1	no	—
281	#41	—	94	4	no	—
282	#41	—	188	2	no	—
283	C₂×Q₈	—	94	1	no	—
284	C₂×D₄	—	94	4	no	—
285	C₂×D₄	—	94	2	no	—
286	C₂×D₄	—	94	1	no	—
287	C₂×D₄	—	188	1	no	—
288	C₂×D₄	—	188	1	no	—
289	C₂×D₄	—	188	2	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	188	2	no	—
292	C₂²×C₄	—	188	2	no	—
293	C₂²×C₄	—	188	2	no	—
294	C₂²×C₄	—	188	1	no	—
295	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
296	Q₁₆	—	47	2	no	—
297	Q₁₆	—	94	1	no	—
298	SD₁₆	—	94	4	no	—
299	D₈	—	47	2	no	—
300	D₈	—	47	4	no	—
301	D₈	—	94	4	no	—
302	D₈	—	94	2	no	—
303	D₈	—	94	1	no	—
304	M₁₆	—	94	2	no	—
305	C₂×C₈	—	47	2	no	—
306	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
307	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
308	C₂×C₈	#10625354	1	2	yes	—
309	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	47	1	no	—
315	C₄⋊C₄	—	47	1	no	—
316	C₄⋊C₄	—	188	2	no	—
317	C₂²⋊C₄	—	4	1	no	—
318	C₂²⋊C₄	—	47	2	no	—
319	C₂²⋊C₄	—	188	2	no	—
320	C₂²⋊C₄	—	188	2	no	—
321	C₂²⋊C₄	—	188	2	no	—
322	C₄²	—	4	1	no	—
323	C₄²	—	47	2	no	—
324	C₄²	—	188	2	no	—
325	C₁₆	—	94	1	no	—
326	C₁₆	—	94	1	no	—
327	C₁₆	C₂×C₄×D₄₇	1	4	yes	—
328	C₂³	—	188	1	no	—
329	C₂³	—	188	2	no	—
330	C₂³	—	188	4	no	—
331	C₂³	—	752	1	no	—
332	C₂³	—	752	1	no	—
333	C₂³	D₈×D₄₇	1	1	yes	characteristic
334	Q₈	—	94	2	no	—
335	Q₈	—	188	1	no	—
336	D₄	—	94	4	no	—
337	D₄	—	94	2	no	—
338	D₄	—	188	4	no	—
339	D₄	—	188	2	no	—
340	D₄	—	188	1	no	—
341	D₄	—	188	4	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	16	1	no	—
346	C₂×C₄	—	47	2	no	—
347	C₂×C₄	—	47	2	no	—
348	C₂×C₄	—	188	1	no	—
349	C₂×C₄	—	188	4	no	—
350	C₂×C₄	—	752	2	no	—
351	C₂×C₄	—	752	2	no	—
352	C₂×C₄	—	752	2	no	—
353	C₂×C₄	#13093986	1	2	yes	—
354	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
355	C₂×C₄	#13094047	1	2	yes	—
356	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
357	C₈	—	94	2	no	—
358	C₈	#13093986	1	2	yes	—
359	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
360	C₈	C₂×#10625354	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	376	1	no	—
364	C₂²	—	376	2	no	—
365	C₂²	—	376	2	no	—
366	C₂²	—	376	1	no	—
367	C₂²	—	188	2	no	—
368	C₂²	—	188	4	no	—
369	C₂²	—	752	4	no	—
370	C₂²	#15689205	1	2	yes	—
371	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
372	C₄	—	8	2	no	—
373	C₄	—	8	1	no	—
374	C₄	—	8	1	no	—
375	C₄	—	94	2	no	—
376	C₄	—	94	2	no	—
377	C₄	—	188	2	no	—
378	C₄	#15689205	1	2	yes	—
379	C₄	C₂×#13093986	1	1	yes	characteristic
380	C₄	C₂×#13094047	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	376	4	no	—
383	C₂	—	376	2	no	—
384	C₂	C₂×#15689205	1	1	yes	characteristic
385	C₂	#20256528	1	1	yes	characteristic
386	C₂	#20256533	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2159	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#15689561	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2159	D₄₇
hypocenter	C₄₇	#33454
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2159

Complements

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

N	H ≅ G/N	type
2. #20252262	C₂	semidirect
3. #20252239	C₂	semidirect
4. #20245263	C₂	semidirect
5. #20245251	C₂	semidirect
6. #20245239	C₂	semidirect
7. #20245234	C₂	semidirect
8. #20245093	C₂	semidirect
9. C₂×#15688923	C₂	semidirect
10. #20244987	C₂	semidirect
11. C₄₇×#2159	C₂	semidirect
16. #15689561	C₄	semidirect
22. #15689217	C₂²	semidirect
25. #15689205	C₂²	semidirect
29. #15688926	C₂²	semidirect
30. #15688926	C₂²	semidirect
33. #15688923	C₂²	semidirect
34. #15688923	C₂²	semidirect
35. #15688898	C₂²	semidirect
36. C₁₆×Dic₄₇	C₂²	semidirect
40. C₄₇×#367	C₂²	semidirect
41. C₄₇×#366	C₂²	semidirect
45. #13094046	C₂×C₄	semidirect
50. C₂×D₃₇₆	C₂×C₄	semidirect
65. #13093969	C₂×C₄	semidirect
66. D₇₅₂	C₂×C₄	semidirect
79. M₃₂×C₄₇	C₂×C₄	semidirect
195. #2159	D₄₇	semidirect
225. #367	C₂×D₄₇	semidirect
226. #366	C₂×D₄₇	semidirect
235. C₄₇	#33454	semidirect
264. M₃₂	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689561	2
#20252262	2
#20252239	2
#20245263	2
#20245251	2
#20245239	2
#20245234	2
#20245093	2
C₂×#15688923	2
#20244987	2
C₄₇×#2159	2
#33454	47