Subgroups — #26369667

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689553	C₂	1	1	yes	maximal, characteristic
2	#20252264	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252262	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245261	C₂	1	2	yes	maximal, complemented
5	#20245256	C₂	1	2	yes	maximal, complemented
6	#20245243	C₂	1	2	yes	maximal, complemented
7	#20245237	C₂	1	2	yes	maximal, complemented
8	C₂×#15688950	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245101	C₂	1	1	yes	maximal, characteristic, complemented
10	#20244987	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2139	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	1	yes	characteristic
13	#15689606	—	2	1	no	—
14	#15689604	—	2	1	no	—
15	#15689553	C₄	1	8	yes	complemented
16	C₂×#13093970	C₂²	1	1	yes	characteristic
17	C₂×#13093968	C₂²	1	1	yes	characteristic
18	C₂×#13093967	C₂²	1	1	yes	characteristic
19	C₂×#13093965	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
21	#15689224	—	2	2	no	—
22	#15689222	—	2	2	no	—
23	Q₁₆×Dic₄₇	C₂²	1	2	yes	—
24	#15689217	C₂²	1	2	yes	—
25	D₈×Dic₄₇	C₂²	1	2	yes	—
26	#15689205	C₂²	1	2	yes	—
27	C₂×#13093958	C₂²	1	1	yes	characteristic
28	#15689124	—	2	1	no	—
29	#15689122	—	2	1	no	—
30	#15689078	C₂²	1	1	yes	characteristic
31	#15688950	C₂²	1	2	yes	—
32	#15688950	C₂²	1	2	yes	—
33	#15688928	C₂²	1	2	yes	—
34	#15688925	C₂²	1	2	yes	—
35	#15688898	C₂²	1	2	yes	—
36	C₁₆×Dic₄₇	C₂²	1	2	yes	—
37	C₂×#135×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#358	C₂²	1	2	yes	—
41	C₄₇×#357	C₂²	1	2	yes	—
42	C₂×#10625350	—	2	1	no	—
43	#13094046	—	2	2	no	—
44	#13094046	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
46	C₂×#10625325	—	2	1	no	—
47	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
48	C₂×#10625329	D₄	1	1	yes	characteristic
49	Q₈×Dic₄₇	—	2	2	no	—
50	#13093989	—	2	2	no	—
51	D₄×Dic₄₇	—	2	2	no	—
52	D₄×Dic₄₇	—	4	2	no	—
53	#13093986	—	2	2	no	—
54	C₄×C₂²⋊D₄₇	—	4	2	no	—
55	#13093983	—	4	1	no	—
56	#13093983	D₄	1	1	yes	characteristic
57	#13093982	—	4	1	no	—
58	C₂×C₄⋊Dic₄₇	—	2	1	no	—
59	C₂×C₄⋊Dic₄₇	—	4	1	no	—
60	C₄⋊C₄×D₄₇	—	4	1	no	—
61	#13093970	C₂×C₄	1	4	yes	complemented
62	#13093968	C₂×C₄	1	4	yes	complemented
63	#13093967	C₂×C₄	1	4	yes	complemented
64	#13093965	C₂×C₄	1	4	yes	complemented
65	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
66	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
68	#13093958	—	2	1	no	—
69	#13093958	C₂³	1	2	yes	—
70	#13093948	—	2	2	no	—
71	#13093947	—	2	2	no	—
72	#13093945	—	2	2	no	—
73	#13093944	—	2	2	no	—
74	#13093934	D₄	1	2	yes	—
75	C₈×Dic₄₇	C₂³	1	2	yes	—
76	C₂×#41×C₄₇	—	2	1	no	—
77	#135×C₄₇	—	2	2	no	—
78	#135×C₄₇	C₂×C₄	1	4	yes	complemented
79	C₂×Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
80	C₂×SD₁₆×C₄₇	—	2	1	no	—
81	C₂×D₈×C₄₇	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
84	M₃₂×C₄₇	—	2	2	no	—
85	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
86	#107×C₄₇	—	2	1	no	—
87	#107×C₄₇	C₂³	1	2	yes	—
88	#41×C₄₇	—	2	4	no	—
89	#41×C₄₇	—	4	2	no	—
90	C₂×Q₈×C₄₇	—	2	1	no	—
91	C₂×D₄×C₄₇	—	2	1	no	—
92	C₂×D₄×C₄₇	—	4	1	no	—
93	C₂²×C₄×C₄₇	—	4	1	no	—
94	C₂²×C₄×C₄₇	—	4	1	no	—
95	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₂²⋊D₄₇	—	4	1	no	—
97	C₂²×Dic₄₇	—	4	1	no	—
98	C₂²×Dic₄₇	—	4	2	no	—
99	#10625350	—	2	4	no	—
100	#10625350	—	4	2	no	—
101	C₂×D₁₈₈	—	2	1	no	—
102	C₂×C₄×D₄₇	—	4	1	no	—
103	C₂×C₄×D₄₇	—	4	2	no	—
104	C₂×Dic₉₄	—	2	1	no	—
105	Q₁₆×C₄₇	—	2	1	no	—
106	Q₁₆×C₄₇	C₂²×C₄	1	2	yes	—
107	SD₁₆×C₄₇	—	2	4	no	—
108	D₈×C₄₇	—	2	1	no	—
109	D₈×C₄₇	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₄₇	#41	1	2	yes	—
113	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₄₇	—	2	2	no	—
115	C₄⋊C₄×C₄₇	—	4	1	no	—
116	C₂²⋊Dic₄₇	—	4	2	no	—
117	C₂²⋊Dic₄₇	D₈	1	2	yes	—
118	(C₂×D₄₇)⋊C₄	—	4	2	no	—
119	C₄⋊Dic₄₇	—	2	2	no	—
120	C₄⋊Dic₄₇	—	4	1	no	—
121	C₄⋊Dic₄₇	—	4	2	no	—
122	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
123	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
124	Dic₄₇⋊C₄	—	4	2	no	—
125	C₄×Dic₄₇	—	4	2	no	—
126	C₄×Dic₄₇	—	4	2	no	—
127	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
128	#10625329	—	2	2	no	—
129	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
130	Dic₁₈₈	—	2	1	no	—
131	Dic₁₈₈	C₂²×C₄	1	2	yes	—
132	D₃₇₆	—	2	1	no	—
133	D₃₇₆	C₂²×C₄	1	2	yes	—
134	#10625325	—	2	4	no	—
135	C₁₆×C₄₇	—	2	2	no	—
136	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
137	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
138	C₂³×C₄₇	—	4	1	no	—
139	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
140	C₂²×D₄₇	—	4	1	no	—
141	Q₈×C₄₇	—	2	2	no	—
142	Q₈×C₄₇	—	4	1	no	—
143	D₄×C₄₇	—	2	2	no	—
144	D₄×C₄₇	—	4	4	no	—
145	D₄×C₄₇	—	4	1	no	—
146	C₂×C₄×C₄₇	—	4	1	no	—
147	C₂×C₄×C₄₇	—	4	4	no	—
148	C₂×C₄×C₄₇	—	4	2	no	—
149	C₂×C₄×C₄₇	—	16	1	no	—
150	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
151	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₄₇	#135	1	2	yes	—
153	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
154	C₂²⋊D₄₇	—	4	4	no	—
155	C₂×Dic₄₇	—	4	1	no	—
156	C₂×Dic₄₇	—	4	2	no	—
157	C₂×Dic₄₇	—	16	2	no	—
158	C₂×Dic₄₇	—	16	1	no	—
159	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
160	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
161	D₁₈₈	—	2	2	no	—
162	D₁₈₈	—	4	1	no	—
163	C₄×D₄₇	—	4	4	no	—
164	C₄×D₄₇	—	16	2	no	—
165	Dic₉₄	—	2	2	no	—
166	Dic₉₄	—	4	1	no	—
167	C₈×C₄₇	C₄×D₄	1	2	yes	—
168	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
169	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
170	C₄₇⋊C₈	—	2	2	no	—
171	#33460	—	47	1	no	maximal, Sylow
172	C₂²×C₄₇	—	2	1	no	—
173	C₂²×C₄₇	—	2	1	no	—
174	C₂²×C₄₇	—	4	2	no	—
175	C₂²×C₄₇	—	8	1	no	—
176	C₂²×C₄₇	—	8	1	no	—
177	C₂²×C₄₇	C₄×D₈	1	2	yes	—
178	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
179	C₂×D₄₇	—	4	2	no	—
180	C₂×D₄₇	—	8	1	no	—
181	C₂×D₄₇	—	8	1	no	—
182	C₄×C₄₇	—	4	2	no	—
183	C₄×C₄₇	—	8	2	no	—
184	C₄×C₄₇	C₄×D₈	1	2	yes	—
185	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
187	Dic₄₇	—	2	2	no	—
188	Dic₄₇	—	2	2	no	—
189	Dic₄₇	—	4	2	no	—
190	Dic₄₇	—	8	2	no	—
191	C₂×#508	—	47	1	no	—
192	#2951	—	47	1	no	—
193	#2951	—	47	1	no	—
194	#2177	—	47	2	no	—
195	#2176	—	47	2	no	—
196	C₄×SD₃₂	—	47	2	no	—
197	C₄×SD₃₂	—	47	2	no	—
198	C₂×#367	—	47	1	no	—
199	#2139	—	47	1	no	—
200	#2139	D₄₇	1	1	yes	characteristic, complemented
201	#2108	—	47	1	no	—
202	C₂×C₄₇	—	2	2	no	—
203	C₂×C₄₇	—	8	2	no	—
204	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
205	C₂×C₄₇	#3416	1	1	yes	characteristic
206	C₂×C₄₇	#3416	1	1	yes	characteristic
207	D₄₇	—	8	2	no	—
208	C₂×#135	—	47	1	no	—
209	C₂×#135	C₂×D₄₇	1	1	yes	characteristic
210	#520	—	94	1	no	—
211	#520	—	94	1	no	—
212	#508	—	47	8	no	—
213	C₂×Q₃₂	—	47	1	no	—
214	C₂×SD₃₂	—	47	1	no	—
215	C₂×SD₃₂	—	47	1	no	—
216	C₂×D₁₆	—	47	1	no	—
217	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
218	C₂²×C₁₆	—	47	1	no	—
219	#440	—	94	2	no	—
220	#440	—	94	2	no	—
221	C₄×Q₁₆	—	47	2	no	—
222	C₄×Q₁₆	—	47	2	no	—
223	C₄×D₈	—	47	2	no	—
224	C₄×D₈	—	47	2	no	—
225	C₂×#107	—	47	1	no	—
226	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
227	#417	—	94	1	no	—
228	#417	—	94	1	no	—
229	#405	—	47	1	no	—
230	#367	—	47	2	no	—
231	#367	—	47	2	no	—
232	#358	—	47	2	no	—
233	#358	C₂×D₄₇	1	2	yes	—
234	#357	—	47	2	no	—
235	#357	C₂×D₄₇	1	2	yes	—
236	#346	—	47	2	no	—
237	C₄×C₁₆	—	47	2	no	—
238	C₄₇	#33460	1	1	yes	characteristic, complemented, Sylow
239	C₂×#41	—	2	1	no	—
240	C₂×#41	—	94	1	no	—
241	#135	—	2	2	no	—
242	#135	—	47	4	no	—
243	#135	—	94	2	no	—
244	#135	C₄×D₄₇	1	4	yes	complemented
245	C₂×Q₁₆	—	47	1	no	—
246	C₂×Q₁₆	C₂²×D₄₇	1	1	yes	characteristic
247	C₂×SD₁₆	—	2	1	no	—
248	C₂×SD₁₆	—	94	1	no	—
249	C₂×D₈	—	47	1	no	—
250	C₂×D₈	C₂²×D₄₇	1	1	yes	characteristic
251	C₂×M₁₆	—	47	1	no	—
252	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
253	C₄×Q₈	—	94	2	no	—
254	C₄×Q₈	—	94	2	no	—
255	C₄×D₄	—	94	2	no	—
256	C₄×D₄	—	94	2	no	—
257	C₄×D₄	—	188	2	no	—
258	C₄×D₄	—	188	2	no	—
259	#117	—	47	1	no	—
260	#117	—	188	1	no	—
261	#117	—	188	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	—	94	1	no	—
264	C₂×C₄⋊C₄	—	188	1	no	—
265	C₂×C₄⋊C₄	—	188	1	no	—
266	Q₃₂	—	47	4	no	—
267	SD₃₂	—	47	4	no	—
268	SD₃₂	—	47	4	no	—
269	D₁₆	—	47	4	no	—
270	M₃₂	—	2	2	no	—
271	C₂×C₁₆	—	47	4	no	—
272	C₂×C₁₆	—	47	1	no	—
273	C₂×C₁₆	—	47	1	no	—
274	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
275	#107	—	2	1	no	—
276	#107	—	47	2	no	—
277	#107	—	94	1	no	—
278	#107	C₂²×D₄₇	1	2	yes	—
279	#103	—	94	2	no	—
280	#103	—	94	2	no	—
281	#102	—	94	2	no	—
282	#102	—	94	2	no	—
283	#97	—	47	2	no	—
284	C₄×C₈	—	47	2	no	—
285	#41	—	2	4	no	—
286	#41	—	4	2	no	—
287	#41	—	94	4	no	—
288	#41	—	188	2	no	—
289	C₂×Q₈	—	2	1	no	—
290	C₂×Q₈	—	94	1	no	—
291	C₂×D₄	—	2	1	no	—
292	C₂×D₄	—	4	1	no	—
293	C₂×D₄	—	94	1	no	—
294	C₂×D₄	—	188	1	no	—
295	C₂²×C₄	—	4	1	no	—
296	C₂²×C₄	—	4	1	no	—
297	C₂²×C₄	—	188	2	no	—
298	C₂²×C₄	—	188	2	no	—
299	C₂²×C₄	—	188	1	no	—
300	C₂²×C₄	—	188	1	no	—
301	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
302	Q₁₆	—	2	1	no	—
303	Q₁₆	—	47	2	no	—
304	Q₁₆	—	94	1	no	—
305	Q₁₆	C₂×C₄×D₄₇	1	2	yes	—
306	SD₁₆	—	2	4	no	—
307	SD₁₆	—	94	4	no	—
308	D₈	—	2	1	no	—
309	D₈	—	47	2	no	—
310	D₈	—	94	1	no	—
311	D₈	C₂×C₄×D₄₇	1	2	yes	—
312	M₁₆	—	94	2	no	—
313	C₂×C₈	—	47	2	no	—
314	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
315	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
316	C₂×C₈	#10625352	1	2	yes	—
317	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	47	1	no	—
321	C₄⋊C₄	—	47	1	no	—
322	C₄⋊C₄	—	94	2	no	—
323	C₄⋊C₄	—	188	2	no	—
324	C₄⋊C₄	—	188	2	no	—
325	C₄⋊C₄	—	188	1	no	—
326	C₂²⋊C₄	—	47	2	no	—
327	C₂²⋊C₄	—	188	2	no	—
328	C₂²⋊C₄	—	188	2	no	—
329	C₄²	—	47	2	no	—
330	C₄²	—	188	2	no	—
331	C₄²	—	188	2	no	—
332	C₁₆	—	2	2	no	—
333	C₁₆	—	47	2	no	—
334	C₁₆	—	47	2	no	—
335	C₂³	—	4	1	no	—
336	C₂³	—	188	1	no	—
337	C₂³	D₈×D₄₇	1	1	yes	characteristic
338	Q₈	—	2	2	no	—
339	Q₈	—	4	1	no	—
340	Q₈	—	94	2	no	—
341	Q₈	—	188	1	no	—
342	D₄	—	2	2	no	—
343	D₄	—	4	4	no	—
344	D₄	—	4	1	no	—
345	D₄	—	94	2	no	—
346	D₄	—	188	4	no	—
347	D₄	—	188	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	4	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	16	1	no	—
352	C₂×C₄	—	47	2	no	—
353	C₂×C₄	—	47	2	no	—
354	C₂×C₄	—	188	2	no	—
355	C₂×C₄	—	188	1	no	—
356	C₂×C₄	—	188	4	no	—
357	C₂×C₄	—	752	2	no	—
358	C₂×C₄	—	752	2	no	—
359	C₂×C₄	—	752	1	no	—
360	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
361	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
362	C₂×C₄	#13094045	1	2	yes	—
363	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
364	C₈	—	94	2	no	—
365	C₈	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
366	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
367	C₈	C₂×#10625352	1	1	yes	characteristic
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	8	1	no	—
372	C₂²	—	8	1	no	—
373	C₂²	—	376	1	no	—
374	C₂²	—	376	1	no	—
375	C₂²	—	188	2	no	—
376	C₂²	#15689203	1	2	yes	—
377	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
378	C₄	—	4	2	no	—
379	C₄	—	8	2	no	—
380	C₄	—	94	2	no	—
381	C₄	—	94	2	no	—
382	C₄	—	376	2	no	—
383	C₄	—	188	2	no	—
384	C₄	#15689203	1	2	yes	—
385	C₄	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
386	C₄	C₂×#13094045	1	1	yes	characteristic
387	C₂	—	2	2	no	—
388	C₂	—	8	2	no	—
389	C₂	—	376	2	no	—
390	C₂	C₂×#15689203	1	1	yes	characteristic
391	C₂	#20256533	1	1	yes	characteristic
392	C₂	#20256534	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2139	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#15689553	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2139	D₄₇
hypocenter	C₄₇	#33460
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2139

Complements

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

N	H ≅ G/N	type
2. #20252264	C₂	semidirect
3. #20252262	C₂	semidirect
4. #20245261	C₂	semidirect
5. #20245256	C₂	semidirect
6. #20245243	C₂	semidirect
7. #20245237	C₂	semidirect
8. C₂×#15688950	C₂	semidirect
9. #20245101	C₂	semidirect
10. #20244987	C₂	semidirect
11. C₄₇×#2139	C₂	semidirect
15. #15689553	C₄	semidirect
44. #13094046	C₂×C₄	semidirect
61. #13093970	C₂×C₄	semidirect
62. #13093968	C₂×C₄	semidirect
63. #13093967	C₂×C₄	semidirect
64. #13093965	C₂×C₄	semidirect
65. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
78. #135×C₄₇	C₂×C₄	semidirect
200. #2139	D₄₇	semidirect
238. C₄₇	#33460	semidirect
244. #135	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×#15689553	2
#20252264	2
#20252262	2
#20245261	2
#20245256	2
#20245243	2
#20245237	2
C₂×#15688950	2
#20245101	2
#20244987	2
C₄₇×#2139	2
#33460	47