Subgroups — #26186137

Order 12032.

646 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252503	C₂	1	2	yes	maximal, complemented
2	C₄×#13094043	C₂	1	1	yes	maximal, characteristic
3	#20242302	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242298	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242296	C₂	1	1	yes	maximal, characteristic, complemented
6	#20242281	C₂	1	2	yes	maximal, complemented
7	#20242276	C₂	1	2	yes	maximal, complemented
8	#334×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#20241772	C₂	1	2	yes	maximal, complemented
10	#20241673	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1594	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094043	C₄	1	1	yes	characteristic
13	C₄×#10625352	C₂²	1	2	yes	—
14	#15689253	C₂²	1	2	yes	—
15	#15689250	C₂²	1	2	yes	—
16	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
17	#15689217	C₄	1	1	yes	characteristic
18	#15689214	C₄	1	2	yes	—
19	C₄×#10625335	C₂²	1	2	yes	—
20	D₈×Dic₄₇	C₄	1	1	yes	characteristic
21	#15689201	C₂²	1	2	yes	—
22	#15689197	—	2	2	no	—
23	#15689192	—	2	2	no	—
24	#15689177	—	2	2	no	—
25	#15689174	C₂²	1	1	yes	characteristic
26	#15689174	C₂²	1	1	yes	characteristic
27	#15689160	C₄	1	1	yes	characteristic
28	#15689146	—	2	2	no	—
29	C₄⋊C₈×D₄₇	C₂²	1	2	yes	—
30	#15689116	C₄	1	2	yes	—
31	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
32	#15688874	C₂²	1	1	yes	characteristic
33	#15688873	C₂²	1	2	yes	—
34	#15688855	C₂²	1	2	yes	—
35	#15688853	C₂²	1	2	yes	—
36	D₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
37	#15688844	C₂²	1	1	yes	characteristic
38	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
39	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₄₇×#436	C₂²	1	2	yes	—
42	C₄₇×#334	C₂²	1	1	yes	characteristic
43	C₄₇×D₄⋊C₈	C₂²	1	2	yes	—
44	C₄₇×#322	C₂²	1	1	yes	characteristic
45	C₂×#10625352	C₂×C₄	1	2	yes	—
46	#13094043	—	2	4	no	—
47	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
48	C₂×#10625335	C₂×C₄	1	2	yes	—
49	C₂×#10625329	—	2	2	no	—
50	C₂×C₈×D₄₇	—	2	2	no	—
51	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
53	C₄×Dic₉₄	C₂³	1	2	yes	—
54	#13093989	C₂×C₄	1	2	yes	—
55	D₄×Dic₄₇	C₂×C₄	1	2	yes	—
56	C₄×C₂²⋊D₄₇	—	2	2	no	—
57	(C₂²⋊D₄₇)⋊C₄	—	2	2	no	—
58	#13093982	C₂×C₄	1	2	yes	—
59	#13093980	—	2	2	no	—
60	C₂×C₄×Dic₄₇	—	2	2	no	—
61	C₄²×D₄₇	C₂³	1	1	yes	characteristic
62	#13093958	C₂×C₄	1	1	yes	characteristic
63	#13093955	C₂×C₄	1	2	yes	—
64	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
65	Dic₄₇⋊C₈	—	2	2	no	—
66	Dic₄₇⋊C₈	—	2	2	no	—
67	Dic₄₇⋊C₈	—	2	1	no	—
68	Dic₄₇⋊C₈	—	2	2	no	—
69	Dic₄₇⋊C₈	D₄	1	2	yes	—
70	#13093949	C₂×C₄	1	2	yes	—
71	#13093947	C₂×C₄	1	2	yes	—
72	#13093945	C₂×C₄	1	2	yes	—
73	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
74	(C₂×D₄₇)⋊C₈	—	2	1	no	—
75	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
76	#13093934	—	2	2	no	—
77	#13093934	—	2	2	no	—
78	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
83	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
84	C₂×M₁₆×C₄₇	—	2	2	no	—
85	C₄×D₄×C₄₇	C₂³	1	2	yes	—
86	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
88	#102×C₄₇	C₂×C₄	1	2	yes	—
89	C₂²⋊C₈×C₄₇	—	2	2	no	—
90	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₄₇	—	2	2	no	—
95	C₂×C₂²⋊D₄₇	—	2	2	no	—
96	C₂²×Dic₄₇	—	2	2	no	—
97	C₂²×Dic₄₇	—	2	2	no	—
98	#10625352	—	2	4	no	—
99	#10625352	—	2	4	no	—
100	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×D₄₇	#41	1	1	yes	characteristic
103	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
104	D₈×C₄₇	—	2	2	no	—
105	M₁₆×C₄₇	—	8	2	no	—
106	C₂×C₈×C₄₇	—	2	2	no	—
107	C₂×C₈×C₄₇	—	2	1	no	—
108	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
111	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
112	C₂²⋊C₄×C₄₇	—	2	2	no	—
113	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
114	C₂²⋊Dic₄₇	—	2	2	no	—
115	#10625335	—	2	4	no	—
116	(C₂×D₄₇)⋊C₄	—	2	2	no	—
117	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
118	Dic₄₇⋊C₄	—	2	2	no	—
119	Dic₄₇⋊C₄	—	2	2	no	—
120	C₄×Dic₄₇	—	2	2	no	—
121	C₄×Dic₄₇	—	2	2	no	—
122	C₄×Dic₄₇	—	2	2	no	—
123	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₄₇	#41	1	1	yes	characteristic
126	#10625329	—	8	2	no	—
127	C₂×C₄₇⋊C₈	—	2	2	no	—
128	C₂×C₄₇⋊C₈	—	2	1	no	—
129	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
132	Dic₁₈₈	—	2	2	no	—
133	C₈×D₄₇	—	2	2	no	—
134	C₈×D₄₇	—	8	2	no	—
135	C₈×D₄₇	M₁₆	1	4	yes	complemented
136	C₂³×C₄₇	—	2	2	no	—
137	C₂²×D₄₇	C₄×D₄	1	1	yes	characteristic
138	D₄×C₄₇	—	2	2	no	—
139	D₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
146	C₂²⋊D₄₇	—	4	4	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	—	4	4	no	—
150	C₂×Dic₄₇	—	8	2	no	—
151	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
154	C₄×D₄₇	—	2	2	no	—
155	C₄×D₄₇	—	4	1	no	—
156	C₄×D₄₇	C₂×M₁₆	1	2	yes	—
157	C₄×D₄₇	#131	1	2	yes	—
158	Dic₉₄	—	2	2	no	—
159	Dic₉₄	—	2	2	no	—
160	C₈×C₄₇	—	2	1	no	—
161	C₈×C₄₇	—	2	2	no	—
162	C₈×C₄₇	—	4	1	no	—
163	C₈×C₄₇	—	4	2	no	—
164	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
165	C₄₇⋊C₈	—	2	1	no	—
166	C₄₇⋊C₈	—	2	2	no	—
167	C₄₇⋊C₈	—	4	1	no	—
168	C₄₇⋊C₈	—	4	2	no	—
169	C₄₇⋊C₈	C₂×M₁₆	1	2	yes	—
170	#17899	—	47	1	no	maximal, Sylow
171	C₂²×C₄₇	—	4	2	no	—
172	C₂²×C₄₇	—	4	2	no	—
173	C₂²×C₄₇	—	4	2	no	—
174	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×D₄₇	—	2	1	no	—
176	C₂×D₄₇	—	2	1	no	—
177	C₂×D₄₇	#436	1	2	yes	—
178	C₄×C₄₇	—	2	1	no	—
179	C₄×C₄₇	—	4	2	no	—
180	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
181	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
182	C₄×C₄₇	#577	1	2	yes	—
183	Dic₄₇	—	2	1	no	—
184	Dic₄₇	—	2	1	no	—
185	Dic₄₇	—	4	2	no	—
186	Dic₄₇	—	4	2	no	—
187	Dic₄₇	#436	1	2	yes	—
188	#2989	—	47	2	no	—
189	C₄×#135	—	47	1	no	—
190	#1594	—	47	1	no	—
191	#1594	D₄₇	1	1	yes	characteristic, complemented
192	#1593	—	47	1	no	—
193	#1593	—	47	1	no	—
194	#1590	—	47	2	no	—
195	#1590	—	47	2	no	—
196	C₂×#334	—	47	1	no	—
197	#1477	—	47	2	no	—
198	#1450	—	47	1	no	—
199	C₂×C₄₇	—	8	2	no	—
200	C₂×C₄₇	C₂×#436	1	1	yes	characteristic
201	C₂×C₄₇	#2952	1	1	yes	characteristic
202	C₂×C₄₇	#2957	1	1	yes	characteristic
203	D₄₇	—	2	2	no	—
204	C₂×#135	—	47	1	no	—
205	C₄×#41	—	47	2	no	—
206	#446	—	47	2	no	—
207	#446	—	47	2	no	—
208	C₄×Q₁₆	—	47	1	no	—
209	C₄×Q₁₆	—	47	1	no	—
210	C₄×SD₁₆	—	47	2	no	—
211	C₄×SD₁₆	—	47	2	no	—
212	C₄×D₈	—	47	1	no	—
213	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
214	#436	—	47	2	no	—
215	#436	C₂×D₄₇	1	2	yes	—
216	#435	—	94	2	no	—
217	#435	—	94	2	no	—
218	#432	—	94	2	no	—
219	#431	—	47	1	no	—
220	#431	—	47	1	no	—
221	#427	—	47	1	no	—
222	#423	—	94	2	no	—
223	C₂×C₄⋊C₈	—	47	2	no	—
224	#416	—	47	2	no	—
225	C₂×C₄×C₈	—	47	1	no	—
226	#334	—	47	2	no	—
227	#334	—	47	1	no	—
228	#334	C₂×D₄₇	1	1	yes	characteristic
229	Q₈⋊C₈	—	47	2	no	—
230	Q₈⋊C₈	—	47	2	no	—
231	D₄⋊C₈	—	47	2	no	—
232	D₄⋊C₈	C₂×D₄₇	1	2	yes	—
233	#322	—	47	1	no	—
234	#322	C₂×D₄₇	1	1	yes	characteristic
235	C₈²	—	47	1	no	—
236	C₈²	—	47	1	no	—
237	C₄₇	#17899	1	1	yes	characteristic, complemented, Sylow
238	C₂×#41	—	47	2	no	—
239	#135	—	94	4	no	—
240	C₂×Q₁₆	—	47	1	no	—
241	C₂×SD₁₆	—	47	2	no	—
242	C₂×D₈	C₄×D₄₇	1	1	yes	characteristic
243	C₂×M₁₆	—	2	2	no	—
244	C₂×M₁₆	—	94	2	no	—
245	C₂²×C₈	—	47	1	no	—
246	C₂²×C₈	—	47	1	no	—
247	C₂²×C₈	—	94	2	no	—
248	C₄×Q₈	—	47	2	no	—
249	C₄×Q₈	—	47	2	no	—
250	C₄×D₄	—	47	2	no	—
251	C₄×D₄	—	94	2	no	—
252	C₄×D₄	—	94	2	no	—
253	C₄×D₄	C₂²×D₄₇	1	2	yes	—
254	#117	—	47	2	no	—
255	#117	—	94	2	no	—
256	C₂×C₄²	—	47	1	no	—
257	C₂×C₄²	—	94	2	no	—
258	#107	—	47	1	no	—
259	#107	C₄×D₄₇	1	1	yes	characteristic
260	#106	—	47	2	no	—
261	C₄⋊C₈	—	47	2	no	—
262	C₄⋊C₈	—	47	2	no	—
263	C₄⋊C₈	—	94	2	no	—
264	C₄⋊C₈	—	94	1	no	—
265	C₄⋊C₈	—	94	2	no	—
266	C₄⋊C₈	—	94	2	no	—
267	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
268	#103	—	47	2	no	—
269	#103	—	47	2	no	—
270	#102	—	47	2	no	—
271	#102	C₄×D₄₇	1	2	yes	—
272	C₂²⋊C₈	—	2	2	no	—
273	C₂²⋊C₈	—	47	2	no	—
274	C₂²⋊C₈	—	94	1	no	—
275	C₂²⋊C₈	—	94	2	no	—
276	#97	—	94	2	no	—
277	#97	—	94	2	no	—
278	C₄×C₈	—	47	1	no	—
279	C₄×C₈	—	47	1	no	—
280	C₄×C₈	—	47	1	no	—
281	C₄×C₈	—	47	1	no	—
282	C₄×C₈	—	47	1	no	—
283	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
286	#41	—	94	4	no	—
287	#41	—	94	4	no	—
288	C₂×Q₈	—	47	2	no	—
289	C₂×D₄	—	94	2	no	—
290	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	47	1	no	—
293	C₂²×C₄	—	47	1	no	—
294	C₂²×C₄	—	47	1	no	—
295	C₂²×C₄	—	94	2	no	—
296	C₂²×C₄	—	94	2	no	—
297	Q₁₆	—	94	2	no	—
298	SD₁₆	—	94	4	no	—
299	D₈	—	2	2	no	—
300	M₁₆	—	8	2	no	—
301	M₁₆	—	376	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	47	1	no	—
305	C₂×C₈	—	47	4	no	—
306	C₂×C₈	—	47	1	no	—
307	C₂×C₈	—	47	2	no	—
308	C₂×C₈	—	94	2	no	—
309	C₂×C₈	—	94	2	no	—
310	C₂×C₈	—	94	1	no	—
311	C₂×C₈	—	376	2	no	—
312	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
313	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
314	C₂×C₈	D₄×D₄₇	1	2	yes	—
315	C₄⋊C₄	—	47	2	no	—
316	C₄⋊C₄	—	94	2	no	—
317	C₄⋊C₄	—	94	2	no	—
318	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
319	C₂²⋊C₄	—	2	2	no	—
320	C₂²⋊C₄	—	94	2	no	—
321	C₂²⋊C₄	—	94	2	no	—
322	C₄²	—	47	1	no	—
323	C₄²	—	47	1	no	—
324	C₄²	—	47	1	no	—
325	C₄²	—	94	2	no	—
326	C₄²	—	94	2	no	—
327	C₄²	—	94	2	no	—
328	C₄²	C₂³×D₄₇	1	1	yes	characteristic
329	C₂³	—	2	2	no	—
330	C₂³	—	47	1	no	—
331	Q₈	—	94	2	no	—
332	Q₈	—	94	2	no	—
333	D₄	—	2	2	no	—
334	D₄	—	2	2	no	—
335	D₄	—	188	4	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	47	1	no	—
340	C₂×C₄	—	47	2	no	—
341	C₂×C₄	—	47	2	no	—
342	C₂×C₄	—	47	1	no	—
343	C₂×C₄	—	47	1	no	—
344	C₂×C₄	—	94	2	no	—
345	C₂×C₄	—	94	2	no	—
346	C₂×C₄	—	94	2	no	—
347	C₂×C₄	—	376	2	no	—
348	C₂×C₄	—	188	1	no	—
349	C₂×C₄	—	188	4	no	—
350	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
351	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
352	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	2	no	—
355	C₈	—	4	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	47	2	no	—
358	C₈	—	94	1	no	—
359	C₈	—	94	2	no	—
360	C₈	—	188	1	no	—
361	C₈	—	188	2	no	—
362	C₈	M₁₆×D₄₇	1	2	yes	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	47	2	no	—
367	C₂²	—	94	1	no	—
368	C₂²	—	94	1	no	—
369	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	47	2	no	—
373	C₄	—	94	1	no	—
374	C₄	—	94	1	no	—
375	C₄	—	188	2	no	—
376	C₄	—	188	2	no	—
377	C₄	C₂×M₁₆×D₄₇	1	1	yes	characteristic
378	C₄	#131×D₄₇	1	1	yes	characteristic
379	C₄	#15689933	1	2	yes	—
380	C₂	—	8	2	no	—
381	C₂	—	94	2	no	—
382	C₂	#436×D₄₇	1	1	yes	characteristic
383	C₂	#20252270	1	1	yes	characteristic
384	C₂	#20252302	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1594	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#13094043	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1594	D₄₇
hypocenter	C₄₇	#17899
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1594

Complements

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

N	H ≅ G/N	type
1. #20252503	C₂	semidirect
3. #20242302	C₂	semidirect
4. #20242298	C₂	semidirect
5. #20242296	C₂	semidirect
6. #20242281	C₂	semidirect
7. #20242276	C₂	semidirect
8. #334×D₄₇	C₂	semidirect
9. #20241772	C₂	semidirect
10. #20241673	C₂	semidirect
11. C₄₇×#1594	C₂	semidirect
135. C₈×D₄₇	M₁₆	semidirect
191. #1594	D₄₇	semidirect
237. C₄₇	#17899	semidirect

Maximal subgroups

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

subgroup	index
#20252503	2
C₄×#13094043	2
#20242302	2
#20242298	2
#20242296	2
#20242281	2
#20242276	2
#334×D₄₇	2
#20241772	2
#20241673	2
C₄₇×#1594	2
#17899	47