Subgroups — #26369614

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×#15689559	C₂	1	1	yes	maximal, characteristic
2	#20252262	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252240	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245261	C₂	1	2	yes	maximal, complemented
5	#20245249	C₂	1	2	yes	maximal, complemented
6	#20245243	C₂	1	2	yes	maximal, complemented
7	#20245232	C₂	1	2	yes	maximal, complemented
8	#20245174	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245096	C₂	1	1	yes	maximal, characteristic, complemented
10	#20244984	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄₇×#357	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	1	yes	characteristic
13	#15689604	—	2	1	no	—
14	#15689598	—	2	1	no	—
15	#15689559	C₄	1	8	yes	complemented
16	C₂×#13093968	C₂²	1	2	yes	—
17	C₂×#13093965	C₂²	1	2	yes	—
18	C₂×#13093964	C₂²	1	1	yes	characteristic
19	#15689235	—	2	2	no	—
20	#15689222	—	2	2	no	—
21	#15689217	C₂²	1	2	yes	complemented
22	D₈×Dic₄₇	C₂²	1	2	yes	—
23	D₈×Dic₄₇	C₂²	1	2	yes	—
24	#15689205	C₂²	1	2	yes	complemented
25	#15689161	C₂²	1	1	yes	characteristic
26	#15689128	—	2	1	no	—
27	#15689122	—	2	1	no	—
28	#15689078	C₂²	1	1	yes	characteristic
29	#15688950	C₂²	1	2	yes	complemented
30	#15688948	C₂²	1	2	yes	complemented
31	#15688928	C₂²	1	1	yes	characteristic, complemented
32	#15688928	C₂²	1	1	yes	characteristic, complemented
33	#15688925	C₂²	1	1	yes	characteristic, complemented
34	#15688925	C₂²	1	1	yes	characteristic, complemented
35	#15688898	C₂²	1	2	yes	complemented
36	C₁₆×Dic₄₇	C₂²	1	2	yes	complemented
37	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#357	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	#13093989	—	2	2	no	—
50	D₄×Dic₄₇	—	2	2	no	—
51	D₄×Dic₄₇	—	2	2	no	—
52	D₄×Dic₄₇	—	4	1	no	—
53	D₄×Dic₄₇	—	4	1	no	—
54	#13093986	—	2	2	no	—
55	C₄×C₂²⋊D₄₇	—	4	2	no	—
56	#13093983	—	2	1	no	—
57	#13093983	D₄	1	1	yes	characteristic
58	#13093982	—	4	1	no	—
59	C₄⋊C₄×D₄₇	—	4	1	no	—
60	C₂×C₂²⋊Dic₄₇	—	4	2	no	—
61	#13093968	C₂×C₄	1	8	yes	complemented
62	#13093965	C₂×C₄	1	8	yes	complemented
63	#13093964	C₂×C₄	1	4	yes	complemented
64	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
65	#13093958	C₂³	1	1	yes	characteristic
66	#13093958	C₂³	1	1	yes	characteristic
67	#13093956	—	2	1	no	—
68	#13093947	—	2	2	no	—
69	#13093945	—	2	2	no	—
70	#13093945	—	2	2	no	—
71	#13093944	—	2	2	no	—
72	#13093934	D₄	1	2	yes	—
73	C₈×Dic₄₇	C₂³	1	2	yes	—
74	C₂²×D₄×C₄₇	—	2	1	no	—
75	C₂×D₈×C₄₇	—	2	1	no	—
76	C₂×D₈×C₄₇	—	2	2	no	—
77	C₂×D₈×C₄₇	C₂×C₄	1	4	yes	complemented
78	C₂×D₈×C₄₇	C₂³	1	2	yes	—
79	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
81	C₂×C₁₆×C₄₇	—	2	2	no	—
82	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
83	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
84	#107×C₄₇	—	2	1	no	—
85	#107×C₄₇	C₂³	1	2	yes	—
86	C₂⁴×C₄₇	—	4	1	no	—
87	C₂×D₄×C₄₇	—	2	4	no	—
88	C₂×D₄×C₄₇	—	2	2	no	—
89	C₂×D₄×C₄₇	—	4	1	no	—
90	C₂×D₄×C₄₇	—	4	2	no	—
91	C₂²×C₄×C₄₇	—	4	1	no	—
92	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₂²⋊D₄₇	—	4	1	no	—
94	C₂²×Dic₄₇	—	4	2	no	—
95	C₂²×Dic₄₇	—	4	2	no	—
96	#10625350	—	2	4	no	—
97	#10625350	—	4	2	no	—
98	C₂×D₁₈₈	—	2	1	no	—
99	C₂×C₄×D₄₇	—	4	1	no	—
100	C₂×C₄×D₄₇	—	4	2	no	—
101	C₂×Dic₉₄	—	2	1	no	—
102	D₈×C₄₇	—	2	4	no	—
103	D₈×C₄₇	—	2	2	no	—
104	D₈×C₄₇	C₂²×C₄	1	4	yes	—
105	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	#41	1	2	yes	—
108	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₄₇	—	2	2	no	—
110	C₄⋊C₄×C₄₇	—	4	1	no	—
111	C₂²⋊Dic₄₇	—	4	1	no	—
112	C₂²⋊Dic₄₇	—	4	2	no	—
113	C₂²⋊Dic₄₇	—	4	2	no	—
114	C₂²⋊Dic₄₇	D₈	1	2	yes	—
115	(C₂×D₄₇)⋊C₄	—	4	2	no	—
116	C₄⋊Dic₄₇	—	2	1	no	—
117	C₄⋊Dic₄₇	—	2	1	no	—
118	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
119	C₄⋊Dic₄₇	D₈	1	1	yes	characteristic
120	Dic₄₇⋊C₄	—	4	2	no	—
121	C₄×Dic₄₇	—	4	1	no	—
122	C₄×Dic₄₇	—	4	2	no	—
123	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
124	#10625329	—	2	2	no	—
125	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
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	4	no	—
131	C₁₆×C₄₇	—	2	1	no	—
132	C₁₆×C₄₇	—	2	1	no	—
133	C₄₇⋊C₁₆	C₂²×C₄	1	4	yes	—
134	C₂³×C₄₇	—	4	2	no	—
135	C₂³×C₄₇	—	4	4	no	—
136	C₂³×C₄₇	—	16	1	no	—
137	C₂³×C₄₇	—	16	1	no	—
138	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
139	C₂²×D₄₇	—	4	1	no	—
140	D₄×C₄₇	—	2	4	no	—
141	D₄×C₄₇	—	4	2	no	—
142	D₄×C₄₇	—	4	4	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	—	16	1	no	—
145	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
146	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₄₇	#135	1	2	yes	—
148	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
149	C₂²⋊D₄₇	—	4	4	no	—
150	C₂×Dic₄₇	—	4	1	no	—
151	C₂×Dic₄₇	—	4	1	no	—
152	C₂×Dic₄₇	—	4	1	no	—
153	C₂×Dic₄₇	—	16	2	no	—
154	C₂×Dic₄₇	—	16	2	no	—
155	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
156	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
157	D₁₈₈	—	2	2	no	—
158	D₁₈₈	—	4	1	no	—
159	C₄×D₄₇	—	4	4	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	—	4	4	no	—
171	C₂²×C₄₇	—	8	2	no	—
172	C₂²×C₄₇	—	8	2	no	—
173	C₂²×C₄₇	—	16	4	no	—
174	C₂²×C₄₇	C₄×D₈	1	2	yes	—
175	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
176	C₂×D₄₇	—	4	2	no	—
177	C₂×D₄₇	—	8	1	no	—
178	C₂×D₄₇	—	8	1	no	—
179	C₄×C₄₇	—	8	2	no	—
180	C₄×C₄₇	C₄×D₈	1	2	yes	—
181	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
182	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
183	Dic₄₇	—	2	2	no	—
184	Dic₄₇	—	2	2	no	—
185	Dic₄₇	—	4	2	no	—
186	Dic₄₇	—	8	1	no	—
187	Dic₄₇	—	8	1	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	—	47	1	no	—
196	#2138	—	47	1	no	—
197	C₂×#357	D₄₇	1	1	yes	characteristic, complemented
198	#2108	—	47	1	no	—
199	C₂×C₄₇	—	2	2	no	—
200	C₂×C₄₇	—	8	4	no	—
201	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
202	C₂×C₄₇	#3415	1	1	yes	characteristic
203	C₂×C₄₇	#3416	1	1	yes	characteristic
204	D₄₇	—	8	2	no	—
205	C₂×#135	—	47	1	no	—
206	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
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	—	47	1	no	—
213	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
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	—	47	1	no	—
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	—	47	2	no	—
226	#366	—	47	2	no	—
227	#358	—	47	1	no	—
228	#358	—	47	1	no	—
229	#357	—	47	1	no	—
230	#357	—	47	1	no	—
231	#357	C₂×D₄₇	1	2	yes	—
232	#357	C₂×D₄₇	1	2	yes	—
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	—	2	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	—	2	1	no	—
243	C₂×D₈	—	2	2	no	—
244	C₂×D₈	—	47	1	no	—
245	C₂×D₈	C₄×D₄₇	1	4	yes	complemented
246	C₂×D₈	C₂²×D₄₇	1	2	yes	—
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	2	no	—
254	C₄×D₄	—	188	1	no	—
255	C₄×D₄	—	188	1	no	—
256	#117	—	47	1	no	—
257	#117	—	94	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	—	47	4	no	—
265	C₂×C₁₆	—	2	2	no	—
266	C₂×C₁₆	—	47	2	no	—
267	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
268	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
269	#107	—	2	1	no	—
270	#107	—	47	1	no	—
271	#107	—	47	1	no	—
272	#107	C₂²×D₄₇	1	2	yes	—
273	#106	—	94	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	—	4	1	no	—
281	#41	—	94	4	no	—
282	#41	—	188	2	no	—
283	C₂×Q₈	—	94	1	no	—
284	C₂×D₄	—	2	4	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	4	1	no	—
287	C₂×D₄	—	4	2	no	—
288	C₂×D₄	—	94	1	no	—
289	C₂×D₄	—	188	1	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	—	2	4	no	—
300	D₈	—	2	2	no	—
301	D₈	—	47	2	no	—
302	D₈	—	94	1	no	—
303	D₈	C₂×C₄×D₄₇	1	4	yes	—
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	#10625352	1	2	yes	—
309	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	47	1	no	—
313	C₄⋊C₄	—	47	1	no	—
314	C₄⋊C₄	—	94	1	no	—
315	C₄⋊C₄	—	94	1	no	—
316	C₄⋊C₄	—	188	2	no	—
317	C₂²⋊C₄	—	47	2	no	—
318	C₂²⋊C₄	—	188	2	no	—
319	C₂²⋊C₄	—	188	2	no	—
320	C₂²⋊C₄	—	188	2	no	—
321	C₂²⋊C₄	—	188	1	no	—
322	C₄²	—	47	2	no	—
323	C₄²	—	188	2	no	—
324	C₄²	—	188	1	no	—
325	C₁₆	—	2	1	no	—
326	C₁₆	—	2	1	no	—
327	C₁₆	—	47	4	no	—
328	C₂³	—	4	2	no	—
329	C₂³	—	4	4	no	—
330	C₂³	—	16	1	no	—
331	C₂³	—	16	1	no	—
332	C₂³	—	188	1	no	—
333	C₂³	D₈×D₄₇	1	1	yes	characteristic
334	Q₈	—	94	2	no	—
335	Q₈	—	188	1	no	—
336	D₄	—	2	4	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	4	no	—
339	D₄	—	94	2	no	—
340	D₄	—	188	4	no	—
341	D₄	—	188	1	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	16	1	no	—
344	C₂×C₄	—	47	2	no	—
345	C₂×C₄	—	47	2	no	—
346	C₂×C₄	—	188	1	no	—
347	C₂×C₄	—	188	1	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	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
354	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
355	C₂×C₄	#13094045	1	2	yes	—
356	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
357	C₈	—	94	2	no	—
358	C₈	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
359	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
360	C₈	C₂×#10625352	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	4	4	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	16	4	no	—
367	C₂²	—	376	1	no	—
368	C₂²	—	376	1	no	—
369	C₂²	—	188	2	no	—
370	C₂²	#15689203	1	2	yes	—
371	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
372	C₄	—	8	2	no	—
373	C₄	—	94	2	no	—
374	C₄	—	94	2	no	—
375	C₄	—	376	1	no	—
376	C₄	—	376	1	no	—
377	C₄	—	188	2	no	—
378	C₄	#15689203	1	2	yes	—
379	C₄	C₂×(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
380	C₄	C₂×#13094045	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	8	4	no	—
383	C₂	—	376	2	no	—
384	C₂	C₂×#15689203	1	1	yes	characteristic
385	C₂	#20256525	1	1	yes	characteristic
386	C₂	#20256530	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#357	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#15689559	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#357	D₄₇
hypocenter	C₄₇	#33454
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#357

Complements

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

N	H ≅ G/N	type
2. #20252262	C₂	semidirect
3. #20252240	C₂	semidirect
4. #20245261	C₂	semidirect
5. #20245249	C₂	semidirect
6. #20245243	C₂	semidirect
7. #20245232	C₂	semidirect
8. #20245174	C₂	semidirect
9. #20245096	C₂	semidirect
10. #20244984	C₂	semidirect
11. C₂×C₄₇×#357	C₂	semidirect
15. #15689559	C₄	semidirect
21. #15689217	C₂²	semidirect
24. #15689205	C₂²	semidirect
29. #15688950	C₂²	semidirect
30. #15688948	C₂²	semidirect
31. #15688928	C₂²	semidirect
32. #15688928	C₂²	semidirect
33. #15688925	C₂²	semidirect
34. #15688925	C₂²	semidirect
35. #15688898	C₂²	semidirect
36. C₁₆×Dic₄₇	C₂²	semidirect
44. #13094046	C₂×C₄	semidirect
61. #13093968	C₂×C₄	semidirect
62. #13093965	C₂×C₄	semidirect
63. #13093964	C₂×C₄	semidirect
77. C₂×D₈×C₄₇	C₂×C₄	semidirect
197. C₂×#357	D₄₇	semidirect
235. C₄₇	#33454	semidirect
245. C₂×D₈	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×#15689559	2
#20252262	2
#20252240	2
#20245261	2
#20245249	2
#20245243	2
#20245232	2
#20245174	2
#20245096	2
#20244984	2
C₂×C₄₇×#357	2
#33454	47