Subgroups — #26318744

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252299	C₂	1	1	yes	maximal, characteristic, complemented
2	#20244963	C₂	1	1	yes	maximal, characteristic, complemented
3	#20244904	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244675	C₂	1	1	yes	maximal, characteristic, complemented
5	#20244455	C₂	1	1	yes	maximal, characteristic
6	#20244057	C₂	1	2	yes	maximal, complemented
7	#20243836	C₂	1	2	yes	maximal, complemented
8	#20243803	C₂	1	2	yes	maximal, complemented
9	#20243627	C₂	1	2	yes	maximal, complemented
10	#20243537	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689923	C₂²	1	1	yes	characteristic
13	#15689889	C₂²	1	2	yes	—
14	#15689634	C₂²	1	2	yes	—
15	#15689246	—	2	2	no	—
16	#15689241	—	2	2	no	—
17	#15689172	C₂²	1	1	yes	characteristic
18	#15689171	C₂²	1	2	yes	—
19	#15689139	C₂²	1	2	yes	—
20	C₂×#13093952	C₂²	1	2	yes	—
21	C₂×#13093952	C₂²	1	2	yes	—
22	#15689105	C₂²	1	2	yes	—
23	#15689087	C₂²	1	2	yes	—
24	#15688895	C₂²	1	2	yes	—
25	#15688894	C₂²	1	2	yes	—
26	#15688883	C₂²	1	2	yes	—
27	#15688883	C₂²	1	2	yes	—
28	C₄₇⋊#338	C₂²	1	1	yes	characteristic
29	#15688880	C₂²	1	1	yes	characteristic
30	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
31	C₄₇×#433	—	2	2	no	—
32	C₄₇×#433	—	2	2	no	—
33	C₄₇×#423	C₂²	1	1	yes	characteristic
34	C₄₇×#423	C₂²	1	1	yes	characteristic
35	C₄₇×#407	C₂²	1	1	yes	characteristic
36	C₄₇×#407	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×#10625350	C₂³	1	2	yes	—
40	#13094057	—	2	2	no	—
41	#13094052	—	2	2	no	—
42	#13094046	—	2	2	no	—
43	#13094032	—	4	2	no	—
44	#13094030	—	2	4	no	—
45	C₂×#10625329	C₂³	1	2	yes	—
46	C₂×#10625329	C₂³	1	2	yes	—
47	M₁₆×D₄₇	—	4	2	no	—
48	#13094007	—	2	2	no	—
49	#13094003	—	2	2	no	—
50	#13093998	—	2	2	no	—
51	#13093993	—	2	2	no	—
52	C₄×C₂²⋊D₄₇	—	4	2	no	—
53	#13093983	C₂³	1	2	yes	—
54	#13093960	—	2	2	no	—
55	#13093959	—	4	2	no	—
56	#13093952	—	2	2	no	—
57	#13093952	—	2	2	no	—
58	#13093952	—	2	4	no	—
59	#13093950	—	4	2	no	—
60	#13093942	—	2	2	no	—
61	#13093940	—	2	2	no	—
62	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
63	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
64	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
68	#117×C₄₇	—	2	1	no	—
69	#117×C₄₇	—	2	1	no	—
70	#117×C₄₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
73	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₄₇	—	2	2	no	—
76	C₄⋊C₈×C₄₇	—	2	2	no	—
77	C₂²⋊C₈×C₄₇	—	2	2	no	—
78	C₂²⋊C₈×C₄₇	—	2	2	no	—
79	#97×C₄₇	—	2	1	no	—
80	#97×C₄₇	D₄	1	2	yes	complemented
81	C₄×C₈×C₄₇	—	2	1	no	—
82	C₄×C₈×C₄₇	D₄	1	2	yes	complemented
83	C₂⁴×C₄₇	#41	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	—	2	1	no	—
85	C₂²×C₄×C₄₇	—	2	1	no	—
86	C₂²×C₄×C₄₇	—	2	1	no	—
87	C₂²×C₄×C₄₇	—	2	1	no	—
88	C₂²×C₄×C₄₇	—	2	2	no	—
89	C₂²×C₄×C₄₇	—	2	2	no	—
90	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₂×C₂²⋊D₄₇	—	2	2	no	—
93	C₂×C₂²⋊D₄₇	—	4	2	no	—
94	#10625350	—	2	2	no	—
95	#10625350	—	2	2	no	—
96	#10625350	—	2	2	no	—
97	#10625350	—	2	2	no	—
98	C₂×D₁₈₈	#41	1	2	yes	—
99	C₂×C₄×D₄₇	—	2	2	no	—
100	C₂×Dic₉₄	#41	1	2	yes	—
101	M₁₆×C₄₇	—	2	2	no	—
102	M₁₆×C₄₇	—	2	2	no	—
103	M₁₆×C₄₇	—	2	2	no	—
104	M₁₆×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	—	2	1	no	—
106	C₂×C₈×C₄₇	—	2	1	no	—
107	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
109	C₄⋊C₄×C₄₇	—	2	2	no	—
110	C₄⋊C₄×C₄₇	—	2	2	no	—
111	C₂²⋊C₄×C₄₇	—	2	2	no	—
112	C₂²⋊C₄×C₄₇	—	2	2	no	—
113	C₄²×C₄₇	—	2	1	no	—
114	C₄²×C₄₇	—	2	1	no	—
115	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₄₇	#41	1	2	yes	—
118	C₂²⋊Dic₄₇	—	2	2	no	—
119	C₂²⋊Dic₄₇	—	4	2	no	—
120	(C₂×D₄₇)⋊C₄	—	4	2	no	—
121	C₄⋊Dic₄₇	—	2	2	no	—
122	Dic₄₇⋊C₄	—	4	2	no	—
123	C₄×Dic₄₇	—	2	2	no	—
124	#10625329	—	2	2	no	—
125	#10625329	—	2	2	no	—
126	#10625329	—	2	2	no	—
127	#10625329	—	2	2	no	—
128	C₂×C₄₇⋊C₈	—	2	2	no	—
129	C₂×C₄₇⋊C₈	—	2	2	no	—
130	Dic₁₈₈	—	2	2	no	—
131	D₃₇₆	—	2	2	no	—
132	#10625325	—	2	4	no	—
133	#10625324	—	4	4	no	—
134	C₈×D₄₇	—	4	4	no	—
135	C₂³×C₄₇	—	2	1	no	—
136	C₂³×C₄₇	—	2	2	no	—
137	C₂³×C₄₇	—	8	1	no	—
138	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₂²×D₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	2	1	no	—
141	C₂×C₄×C₄₇	—	2	1	no	—
142	C₂×C₄×C₄₇	—	2	2	no	—
143	C₂×C₄×C₄₇	—	2	1	no	—
144	C₂×C₄×C₄₇	—	2	2	no	—
145	C₂×C₄×C₄₇	—	4	1	no	—
146	C₂×C₄×C₄₇	—	4	1	no	—
147	C₂×C₄×C₄₇	—	4	2	no	—
148	C₂×C₄×C₄₇	—	8	1	no	—
149	C₂×C₄×C₄₇	—	8	1	no	—
150	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
152	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
154	C₂²⋊D₄₇	—	4	2	no	—
155	C₂²⋊D₄₇	—	4	2	no	—
156	C₂²⋊D₄₇	—	8	4	no	—
157	C₂×Dic₄₇	—	2	2	no	—
158	C₂×Dic₄₇	—	4	2	no	—
159	D₁₈₈	—	2	2	no	—
160	D₁₈₈	—	2	2	no	—
161	C₄×D₄₇	—	4	2	no	—
162	C₄×D₄₇	—	4	2	no	—
163	Dic₉₄	—	2	2	no	—
164	Dic₉₄	—	2	2	no	—
165	C₈×C₄₇	—	2	2	no	—
166	C₈×C₄₇	—	2	2	no	—
167	C₈×C₄₇	—	4	1	no	—
168	C₈×C₄₇	—	4	1	no	—
169	C₄₇⋊C₈	—	4	2	no	—
170	C₄₇⋊C₈	—	4	2	no	—
171	#29385	—	47	1	no	maximal, Sylow
172	C₂²×C₄₇	—	4	1	no	—
173	C₂²×C₄₇	—	4	2	no	—
174	C₂²×C₄₇	—	4	2	no	—
175	C₂²×C₄₇	—	4	2	no	—
176	C₂²×C₄₇	—	4	1	no	—
177	C₂²×C₄₇	#525	1	1	yes	characteristic
178	C₂²×C₄₇	#551	1	2	yes	—
179	C₂×D₄₇	—	4	2	no	—
180	C₂×D₄₇	—	8	2	no	—
181	C₄×C₄₇	—	4	1	no	—
182	C₄×C₄₇	—	4	1	no	—
183	C₄×C₄₇	—	4	2	no	—
184	C₄×C₄₇	—	4	1	no	—
185	C₄×C₄₇	#525	1	1	yes	characteristic
186	C₄×C₄₇	#535	1	1	yes	characteristic
187	C₄×C₄₇	#553	1	2	yes	—
188	Dic₄₇	—	4	2	no	—
189	Dic₄₇	—	8	2	no	—
190	#2956	—	47	1	no	—
191	#2921	D₄₇	1	1	yes	characteristic, complemented
192	#2102	—	47	1	no	—
193	#2091	—	47	1	no	—
194	#2050	—	47	1	no	—
195	#2010	—	47	1	no	—
196	#1932	—	47	2	no	—
197	#1897	—	47	2	no	—
198	#1891	—	47	2	no	—
199	#1861	—	47	2	no	—
200	#1841	—	47	1	no	—
201	C₂×C₄₇	—	2	2	no	—
202	C₂×C₄₇	—	2	1	no	—
203	C₂×C₄₇	—	8	1	no	—
204	C₂×C₄₇	#2598	1	1	yes	characteristic
205	D₄₇	—	8	2	no	—
206	#575	—	47	1	no	—
207	#568	—	47	2	no	—
208	#525	—	47	2	no	—
209	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
210	#444	—	94	2	no	—
211	#443	—	94	2	no	—
212	#433	—	2	2	no	—
213	#433	—	2	2	no	—
214	#430	—	47	1	no	—
215	#430	—	47	2	no	—
216	#423	C₂×D₄₇	1	1	yes	characteristic
217	#423	C₂×D₄₇	1	1	yes	characteristic
218	#421	—	47	2	no	—
219	C₂×#104	—	47	2	no	—
220	C₂×#104	—	47	2	no	—
221	#413	—	47	2	no	—
222	#407	—	47	2	no	—
223	#407	C₂×D₄₇	1	1	yes	characteristic
224	#407	C₂×D₄₇	1	1	yes	characteristic
225	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
226	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
227	#344	—	47	2	no	—
228	#344	—	47	2	no	—
229	#339	—	47	2	no	—
230	#339	—	47	2	no	—
231	#338	—	47	1	no	—
232	H(3,ℤ/4)	—	47	1	no	—
233	C₄₇	#29385	1	1	yes	characteristic, complemented, Sylow
234	C₂×#41	—	47	2	no	—
235	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
236	#137	—	94	2	no	—
237	AGL(1,ℤ/8)	—	94	2	no	—
238	#135	—	94	2	no	—
239	#131	—	94	4	no	—
240	#131	—	188	2	no	—
241	C₂×M₁₆	—	47	2	no	—
242	C₂×M₁₆	—	47	2	no	—
243	C₂×M₁₆	—	188	2	no	—
244	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
245	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
246	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
247	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
248	#123	—	94	2	no	—
249	C₂²⋊Q₈	—	94	2	no	—
250	#121	—	94	2	no	—
251	#120	—	94	2	no	—
252	C₄×D₄	—	188	2	no	—
253	#117	—	2	2	no	—
254	#117	—	2	1	no	—
255	#117	—	2	1	no	—
256	#117	—	47	2	no	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₂²⋊C₄	—	2	1	no	—
259	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
260	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
261	#108	—	94	2	no	—
262	#108	—	188	2	no	—
263	C₄⋊C₈	—	2	2	no	—
264	C₄⋊C₈	—	2	2	no	—
265	#104	—	94	4	no	—
266	#104	—	94	2	no	—
267	#104	—	94	2	no	—
268	#104	—	188	2	no	—
269	#101	—	94	2	no	—
270	#100	—	94	2	no	—
271	C₂²⋊C₈	—	2	2	no	—
272	C₂²⋊C₈	—	2	2	no	—
273	C₂²⋊C₈	—	188	2	no	—
274	#97	—	2	1	no	—
275	#97	C₂²⋊D₄₇	1	2	yes	complemented
276	C₄×C₈	—	2	1	no	—
277	C₄×C₈	C₂²⋊D₄₇	1	2	yes	complemented
278	C₂⁴	#10625350	1	1	yes	characteristic
279	#41	—	94	2	no	—
280	#41	—	94	2	no	—
281	#41	—	94	2	no	—
282	#41	—	94	2	no	—
283	C₂×Q₈	—	47	2	no	—
284	C₂×D₄	—	47	2	no	—
285	C₂×D₄	—	94	2	no	—
286	C₂×D₄	—	188	2	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	2	1	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	1	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	94	2	no	—
294	C₂²×C₄	#10625350	1	1	yes	characteristic
295	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
296	Q₁₆	—	94	2	no	—
297	SD₁₆	—	94	4	no	—
298	D₈	—	94	2	no	—
299	M₁₆	—	2	2	no	—
300	M₁₆	—	2	2	no	—
301	M₁₆	—	2	2	no	—
302	M₁₆	—	2	2	no	—
303	M₁₆	—	94	2	no	—
304	M₁₆	—	94	2	no	—
305	M₁₆	—	94	2	no	—
306	M₁₆	—	94	2	no	—
307	M₁₆	—	188	4	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	94	2	no	—
311	C₂×C₈	—	94	2	no	—
312	C₂×C₈	—	188	4	no	—
313	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
314	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	94	2	no	—
318	C₄⋊C₄	—	188	2	no	—
319	C₂²⋊C₄	—	2	2	no	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	94	2	no	—
322	C₂²⋊C₄	—	188	2	no	—
323	C₂²⋊C₄	—	188	2	no	—
324	C₄²	—	2	1	no	—
325	C₄²	—	2	1	no	—
326	C₄²	—	94	2	no	—
327	C₄²	#10625350	1	2	yes	—
328	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
329	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
330	C₂³	—	2	2	no	—
331	C₂³	—	2	1	no	—
332	C₂³	—	8	1	no	—
333	C₂³	—	94	2	no	—
334	C₂³	C₂×#10625350	1	1	yes	characteristic
335	Q₈	—	94	2	no	—
336	Q₈	—	94	2	no	—
337	D₄	—	94	2	no	—
338	D₄	—	94	2	no	—
339	D₄	—	376	4	no	—
340	D₄	—	188	2	no	—
341	D₄	—	188	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	8	1	no	—
351	C₂×C₄	—	8	1	no	—
352	C₂×C₄	—	94	2	no	—
353	C₂×C₄	—	188	2	no	—
354	C₂×C₄	—	188	2	no	—
355	C₂×C₄	—	188	2	no	—
356	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
357	C₂×C₄	#41×D₄₇	1	2	yes	—
358	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
359	C₂×C₄	#13094070	1	2	yes	—
360	C₈	—	2	2	no	—
361	C₈	—	2	2	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	188	2	no	—
365	C₈	—	188	2	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	—	4	1	no	—
371	C₂²	—	376	2	no	—
372	C₂²	—	188	2	no	—
373	C₂²	#15689634	1	1	yes	characteristic
374	C₂²	#15689797	1	2	yes	—
375	C₄	—	4	2	no	—
376	C₄	—	4	1	no	—
377	C₄	—	4	1	no	—
378	C₄	—	4	1	no	—
379	C₄	—	376	2	no	—
380	C₄	—	188	2	no	—
381	C₄	#15689630	1	1	yes	characteristic
382	C₄	#15689687	1	1	yes	characteristic
383	C₄	#15689817	1	2	yes	—
384	C₂	—	2	1	no	—
385	C₂	—	2	2	no	—
386	C₂	—	8	1	no	—
387	C₂	—	376	2	no	—
388	C₂	#20248954	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15689687
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2921	C₂
socle	C₂×C₄₇	#2598
prime radical	#20244455	C₂
hypercenter	#2921	D₄₇
hypocenter	C₄₇	#29385
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2921

Complements

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

N	H ≅ G/N	type
1. #20252299	C₂	semidirect
2. #20244963	C₂	semidirect
3. #20244904	C₂	semidirect
4. #20244675	C₂	semidirect
6. #20244057	C₂	semidirect
7. #20243836	C₂	semidirect
8. #20243803	C₂	semidirect
9. #20243627	C₂	semidirect
10. #20243537	C₂	semidirect
11. C₄₇×#2921	C₂	semidirect
80. #97×C₄₇	D₄	semidirect
82. C₄×C₈×C₄₇	D₄	semidirect
191. #2921	D₄₇	semidirect
233. C₄₇	#29385	semidirect
275. #97	C₂²⋊D₄₇	semidirect
277. C₄×C₈	C₂²⋊D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20252299	2
#20244963	2
#20244904	2
#20244675	2
#20244455	2
#20244057	2
#20243836	2
#20243803	2
#20243627	2
#20243537	2
C₄₇×#2921	2
#29385	47