Subgroups — #26233879

Order 12032.

692 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252042	C₂	1	2	yes	maximal
2	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
3	#20244141	C₂	1	2	yes	maximal, complemented
4	#20243988	C₂	1	4	yes	maximal, complemented
5	#20243488	C₂	1	2	yes	maximal
6	#20243426	C₂	1	1	yes	maximal, characteristic
7	#20243272	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1752	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093983	C₂²	1	2	yes	—
10	#15689181	—	2	4	no	—
11	#15689177	—	2	2	no	—
12	#15689173	—	2	2	no	—
13	C₂×#13093960	C₂²	1	2	yes	—
14	#15689161	C₄	1	1	yes	characteristic, complemented
15	#15689161	C₄	1	1	yes	characteristic, complemented
16	#15689161	C₄	1	2	yes	complemented
17	#15689161	C₂²	1	2	yes	—
18	#15689161	C₂²	1	1	yes	characteristic
19	#15689161	C₂²	1	1	yes	characteristic
20	C₂×#13093958	C₄	1	1	yes	characteristic, complemented
21	C₂×#13093958	C₄	1	1	yes	characteristic, complemented
22	C₂×#13093956	C₄	1	2	yes	complemented
23	#15689149	C₂²	1	4	yes	—
24	#15689087	C₂²	1	2	yes	—
25	#15689078	C₂²	1	4	yes	—
26	#15688888	C₂²	1	2	yes	—
27	#15688883	C₂²	1	4	yes	—
28	#15688883	C₂²	1	4	yes	—
29	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#97×C₄₇	C₂²	1	2	yes	—
32	C₄×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×#336	C₂²	1	2	yes	—
34	C₂×#10625329	C₂³	1	4	yes	—
35	#13093983	—	2	2	no	—
36	#13093983	—	2	2	no	—
37	#13093983	C₂×C₄	1	2	yes	—
38	#13093983	C₂×C₄	1	2	yes	—
39	#13093983	C₂³	1	2	yes	—
40	#13093983	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
44	C₂×C₄×Dic₄₇	—	2	2	no	—
45	#13093960	—	2	4	no	—
46	#13093958	C₂×C₄	1	2	yes	—
47	#13093958	C₂×C₄	1	2	yes	—
48	#13093958	C₂×C₄	1	1	yes	characteristic
49	#13093958	C₂×C₄	1	1	yes	characteristic
50	#13093958	C₂×C₄	1	1	yes	characteristic
51	#13093958	C₂×C₄	1	1	yes	characteristic
52	#13093956	C₂×C₄	1	2	yes	—
53	#13093956	C₂×C₄	1	2	yes	—
54	#13093956	C₂×C₄	1	2	yes	—
55	#13093956	C₂×C₄	1	2	yes	—
56	Dic₄₇⋊C₈	—	2	4	no	—
57	Dic₄₇⋊C₈	—	2	4	no	—
58	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
59	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
60	#13093934	—	2	4	no	—
61	C₈×Dic₄₇	—	2	4	no	—
62	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	—	2	4	no	—
64	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
70	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
72	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
73	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
74	C₂²⋊C₈×C₄₇	D₄	1	2	yes	—
75	#97×C₄₇	—	2	4	no	—
76	#95×C₄₇	—	2	1	no	—
77	C₂⁴×C₄₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₄₇	—	2	2	no	—
88	C₂²×Dic₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₄₇	#41	1	4	yes	—
102	C₂²⋊C₄×C₄₇	—	4	2	no	—
103	C₄²×C₄₇	—	2	2	no	—
104	C₄²×C₄₇	—	2	2	no	—
105	C₂²⋊Dic₄₇	—	2	2	no	—
106	C₂²⋊Dic₄₇	—	2	2	no	—
107	C₂²⋊Dic₄₇	—	2	2	no	—
108	C₂²⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	—	2	2	no	—
110	C₄⋊Dic₄₇	—	2	2	no	—
111	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
112	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
113	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
114	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
115	C₄×Dic₄₇	—	2	2	no	—
116	C₄×Dic₄₇	—	2	2	no	—
117	C₄×Dic₄₇	—	2	2	no	—
118	C₄×Dic₄₇	—	2	2	no	—
119	#10625329	—	2	4	no	—
120	#10625329	—	2	4	no	—
121	C₂×C₄₇⋊C₈	—	2	4	no	—
122	C₂³×C₄₇	—	2	2	no	—
123	C₂³×C₄₇	—	2	2	no	—
124	C₂³×C₄₇	—	4	1	no	—
125	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
126	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
127	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	2	1	no	—
130	C₂×C₄×C₄₇	—	2	1	no	—
131	C₂×C₄×C₄₇	—	2	1	no	—
132	C₂×C₄×C₄₇	—	2	1	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
143	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
144	C₂×Dic₄₇	—	2	2	no	—
145	C₂×Dic₄₇	—	2	2	no	—
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	4	4	no	—
149	C₂×Dic₄₇	—	8	2	no	—
150	C₈×C₄₇	—	2	1	no	—
151	C₈×C₄₇	—	2	1	no	—
152	C₈×C₄₇	—	2	4	no	—
153	C₈×C₄₇	#117	1	2	yes	—
154	C₈×C₄₇	#117	1	2	yes	—
155	C₄₇⋊C₈	—	4	4	no	—
156	#22607	—	47	1	no	maximal, Sylow
157	C₂²×C₄₇	—	2	1	no	—
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	2	1	no	—
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	—	2	1	no	—
162	C₂²×C₄₇	—	2	1	no	—
163	C₂²×C₄₇	—	2	1	no	—
164	C₂²×C₄₇	—	4	4	no	—
165	C₂²×C₄₇	#444	1	2	yes	—
166	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₄₇	#552	1	2	yes	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	—	8	2	no	—
171	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
172	C₄×C₄₇	#519	1	1	yes	characteristic
173	C₄×C₄₇	#547	1	2	yes	—
174	Dic₄₇	—	4	2	no	—
175	Dic₄₇	—	4	2	no	—
176	Dic₄₇	—	4	2	no	—
177	Dic₄₇	—	4	2	no	—
178	#2924	—	47	2	no	—
179	C₂×#427	—	47	1	no	—
180	#1946	—	47	2	no	—
181	#1922	—	47	4	no	—
182	#1826	—	47	2	no	—
183	#1810	—	47	1	no	—
184	#1773	—	47	2	no	—
185	#1752	D₄₇	1	1	yes	characteristic
186	C₂×C₄₇	—	2	2	no	—
187	C₂×C₄₇	—	2	2	no	—
188	C₂×C₄₇	—	4	1	no	—
189	C₂×C₄₇	#2344	1	1	yes	characteristic
190	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
191	C₂×C₄₇	#2957	1	1	yes	characteristic
192	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
193	C₂×#117	—	47	2	no	—
194	#433	—	94	4	no	—
195	#432	—	94	2	no	—
196	#431	—	94	2	no	—
197	C₂×#108	—	47	2	no	—
198	#427	—	47	1	no	—
199	#427	—	47	1	no	—
200	#427	—	47	2	no	—
201	#427	—	47	2	no	—
202	#427	—	47	1	no	—
203	#427	—	47	1	no	—
204	C₂×#107	—	47	1	no	—
205	C₂×#107	—	47	1	no	—
206	C₂×#106	—	47	2	no	—
207	#424	—	47	4	no	—
208	#407	—	47	2	no	—
209	C₂×C₂²⋊C₈	C₂×D₄₇	1	1	yes	characteristic
210	#405	—	47	4	no	—
211	C₂×#97	C₂×D₄₇	1	2	yes	—
212	C₄×C₂²⋊C₄	C₂×D₄₇	1	1	yes	characteristic
213	#341	—	47	2	no	—
214	#339	—	47	4	no	—
215	#339	—	47	4	no	—
216	#336	C₂×D₄₇	1	2	yes	—
217	C₄₇	#22607	1	1	yes	characteristic, complemented, Sylow
218	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
219	C₂×M₁₆	—	47	4	no	—
220	C₂²×C₈	—	2	4	no	—
221	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
222	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
223	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
224	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
227	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
228	#117	—	47	2	no	—
229	#117	—	47	2	no	—
230	#117	—	47	2	no	—
231	#117	—	47	2	no	—
232	#117	—	94	2	no	—
233	#117	—	94	2	no	—
234	C₂×C₄⋊C₄	—	47	2	no	—
235	C₂×C₄⋊C₄	—	47	2	no	—
236	C₂×C₂²⋊C₄	—	2	1	no	—
237	C₂×C₂²⋊C₄	—	94	2	no	—
238	C₂×C₄²	—	94	2	no	—
239	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
240	#108	—	94	4	no	—
241	#107	—	47	2	no	—
242	#107	—	47	2	no	—
243	#107	—	47	1	no	—
244	#107	—	47	1	no	—
245	#107	—	47	1	no	—
246	#107	—	47	1	no	—
247	#106	—	47	2	no	—
248	#106	—	47	2	no	—
249	#106	—	47	2	no	—
250	#106	—	47	2	no	—
251	C₄⋊C₈	—	94	4	no	—
252	C₄⋊C₈	—	94	4	no	—
253	C₂²⋊C₈	—	94	2	no	—
254	C₂²⋊C₈	—	94	2	no	—
255	C₂²⋊C₈	D₁₈₈	1	2	yes	—
256	C₂²⋊C₈	D₁₈₈	1	2	yes	—
257	#97	—	2	4	no	—
258	#97	—	94	4	no	—
259	C₄×C₈	—	94	4	no	—
260	#95	—	2	1	no	—
261	C₂⁴	#10625350	1	1	yes	characteristic
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	94	2	no	—
266	C₂²×C₄	—	94	2	no	—
267	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
271	C₂²×C₄	#10625350	1	1	yes	characteristic
272	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
273	M₁₆	—	94	4	no	—
274	M₁₆	—	94	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	94	4	no	—
283	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
284	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
285	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
286	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
287	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
288	C₂×C₈	#10625352	1	4	yes	—
289	C₄⋊C₄	—	47	2	no	—
290	C₄⋊C₄	—	47	2	no	—
291	C₄⋊C₄	—	47	2	no	—
292	C₄⋊C₄	—	47	2	no	—
293	C₄⋊C₄	—	94	2	no	—
294	C₄⋊C₄	—	94	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	94	2	no	—
297	C₂²⋊C₄	—	94	2	no	—
298	C₂²⋊C₄	—	94	2	no	—
299	C₂²⋊C₄	—	94	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	94	2	no	—
303	C₄²	—	94	2	no	—
304	C₄²	—	94	2	no	—
305	C₄²	—	94	2	no	—
306	C₂³	—	2	2	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	4	1	no	—
309	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
310	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
311	C₂³	C₂×#10625350	1	1	yes	characteristic
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	94	2	no	—
323	C₂×C₄	—	94	2	no	—
324	C₂×C₄	—	94	2	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	376	2	no	—
327	C₂×C₄	—	188	4	no	—
328	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
329	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
332	C₂×C₄	C₂×#10625352	1	2	yes	—
333	C₂×C₄	#13094071	1	2	yes	—
334	C₈	—	2	4	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	188	4	no	—
338	C₈	#13093980	1	2	yes	—
339	C₈	#13093980	1	2	yes	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	#15689246	1	2	yes	—
349	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
350	C₂²	#15689813	1	2	yes	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	8	2	no	—
354	C₄	—	188	2	no	—
355	C₄	—	188	2	no	—
356	C₄	—	188	2	no	—
357	C₄	—	188	2	no	—
358	C₄	C₂×#13093980	1	1	yes	characteristic
359	C₄	#15689600	1	1	yes	characteristic
360	C₄	#15689777	1	2	yes	—
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	#20246255	1	1	yes	characteristic
365	C₂	C₂×#15689246	1	1	yes	characteristic
366	C₂	#20252306	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1752	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1752	D₄₇
hypocenter	C₄₇	#22607
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈ ◁ #1752

Complements

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

N	H ≅ G/N	type
3. #20244141	C₂	semidirect
4. #20243988	C₂	semidirect
7. #20243272	C₂	semidirect
14. #15689161	C₄	semidirect
15. #15689161	C₄	semidirect
16. #15689161	C₄	semidirect
20. C₂×#13093958	C₄	semidirect
21. C₂×#13093958	C₄	semidirect
22. C₂×#13093956	C₄	semidirect
217. C₄₇	#22607	semidirect

Maximal subgroups

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

subgroup	index
#20252042	2
C₂×#15689161	2
#20244141	2
#20243988	2
#20243488	2
#20243426	2
#20243272	2
C₄₇×#1752	2
#22607	47