Subgroups — #26316240

Order 12032.

608 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253911	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689244	C₂	1	1	yes	maximal, characteristic
3	#20244563	C₂	1	2	yes	maximal, complemented
4	#20244559	C₂	1	2	yes	maximal, complemented
5	#20244286	C₂	1	1	yes	maximal, characteristic
6	#20244284	C₂	1	1	yes	maximal, characteristic
7	#20243513	C₂	1	1	yes	maximal, characteristic, complemented
8	#20243441	C₂	1	2	yes	maximal, complemented
9	#20243433	C₂	1	2	yes	maximal, complemented
10	#20243273	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	1	yes	characteristic
13	C₂×#13094032	C₂²	1	2	yes	—
14	#15689676	C₂²	1	2	yes	—
15	#15689531	—	2	2	no	—
16	#15689530	—	2	2	no	—
17	#15689244	—	2	2	no	—
18	#15689244	—	2	2	no	—
19	C₂×#13093959	C₂²	1	1	yes	characteristic
20	#15689161	C₂²	1	1	yes	characteristic
21	#15689159	C₂²	1	1	yes	characteristic
22	#15689159	C₂²	1	2	yes	—
23	#15689141	C₂²	1	2	yes	—
24	#15689139	C₂²	1	2	yes	—
25	C₂×#13093950	C₂²	1	2	yes	—
26	#15689096	C₂²	1	2	yes	—
27	#15689096	C₂²	1	2	yes	—
28	#15689078	C₂²	1	1	yes	characteristic
29	C₂×#13093934	C₂²	1	1	yes	characteristic
30	#15688895	C₂²	1	2	yes	—
31	#15688884	C₂²	1	1	yes	characteristic
32	#15688882	C₂²	1	1	yes	characteristic
33	#15688880	C₂²	1	2	yes	—
34	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#427	C₂²	1	1	yes	characteristic
36	C₄₇×#405	C₂²	1	1	yes	characteristic
37	C₄₇×#344	C₂²	1	2	yes	—
38	C₄₇×H(3,ℤ/4)	C₂²	1	2	yes	—
39	C₂×#10625350	C₂³	1	2	yes	—
40	#13094046	—	2	1	no	—
41	#13094046	—	2	1	no	—
42	#13094046	—	2	1	no	—
43	#13094046	—	2	1	no	—
44	C₂×Dic₁₈₈	Q₈	1	1	yes	characteristic, complemented
45	C₂×#10625325	Q₈	1	2	yes	complemented
46	C₂×D₃₇₆	Q₈	1	1	yes	characteristic, complemented
47	#13094032	—	2	2	no	—
48	#13094032	—	2	2	no	—
49	#13094032	—	2	2	no	—
50	#13094032	—	2	2	no	—
51	C₂×#10625329	C₂³	1	1	yes	characteristic
52	C₂×#10625324	—	2	2	no	—
53	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
54	C₂×C₈×D₄₇	—	2	2	no	—
55	#13094023	—	2	2	no	—
56	#13094015	—	2	2	no	—
57	#13094001	—	4	2	no	—
58	C₄×Dic₉₄	—	4	2	no	—
59	#13093983	C₂³	1	1	yes	characteristic
60	#13093983	C₂³	1	2	yes	—
61	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
62	#13093959	—	2	2	no	—
63	#13093958	—	2	1	no	—
64	#13093957	—	2	2	no	—
65	#13093957	—	2	1	no	—
66	#13093956	—	2	1	no	—
67	#13093955	—	2	2	no	—
68	#13093955	—	2	1	no	—
69	#13093952	—	4	2	no	—
70	#13093950	—	2	4	no	—
71	#13093950	—	4	2	no	—
72	#13093938	—	4	2	no	—
73	#13093938	—	4	2	no	—
74	#13093934	—	2	1	no	—
75	#13093934	D₄	1	2	yes	—
76	#13093934	D₄	1	2	yes	—
77	C₈×Dic₄₇	—	2	1	no	—
78	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
79	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
81	#117×C₄₇	C₂³	1	1	yes	characteristic
82	#117×C₄₇	C₂³	1	2	yes	—
83	#108×C₄₇	—	2	1	no	—
84	#108×C₄₇	—	2	1	no	—
85	#107×C₄₇	—	2	1	no	—
86	#106×C₄₇	—	2	1	no	—
87	#97×C₄₇	—	2	1	no	—
88	C₄×C₈×C₄₇	—	2	1	no	—
89	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₂×C₂²⋊D₄₇	—	2	2	no	—
91	C₂²×Dic₄₇	—	2	1	no	—
92	#10625350	—	2	2	no	—
93	#10625350	—	2	2	no	—
94	#10625350	—	2	2	no	—
95	#10625350	—	2	2	no	—
96	C₂×D₁₈₈	C₂×Q₈	1	2	yes	—
97	C₂×C₄×D₄₇	—	2	2	no	—
98	C₂×Dic₉₄	—	4	2	no	—
99	C₂×Dic₉₄	C₂×Q₈	1	2	yes	—
100	M₁₆×C₄₇	—	2	2	no	—
101	M₁₆×C₄₇	—	2	1	no	—
102	M₁₆×C₄₇	—	2	2	no	—
103	M₁₆×C₄₇	—	2	1	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	—	2	1	no	—
106	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
111	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
112	C₄⋊C₄×C₄₇	—	2	2	no	—
113	C₄⋊C₄×C₄₇	—	2	1	no	—
114	C₂²⋊C₄×C₄₇	—	2	1	no	—
115	C₂²⋊C₄×C₄₇	—	2	2	no	—
116	C₄²×C₄₇	—	2	1	no	—
117	C₄²×C₄₇	—	2	2	no	—
118	C₂²⋊Dic₄₇	—	2	1	no	—
119	C₂²⋊Dic₄₇	—	2	2	no	—
120	C₄⋊Dic₄₇	—	2	2	no	—
121	C₄⋊Dic₄₇	—	2	1	no	—
122	C₄⋊Dic₄₇	—	4	2	no	—
123	Dic₄₇⋊C₄	—	4	2	no	—
124	Dic₄₇⋊C₄	—	4	2	no	—
125	C₄×Dic₄₇	—	2	1	no	—
126	C₄×Dic₄₇	—	2	1	no	—
127	C₄×Dic₄₇	—	2	2	no	—
128	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
129	#10625329	—	2	1	no	—
130	#10625329	—	2	1	no	—
131	C₂×C₄₇⋊C₈	—	2	1	no	—
132	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
133	C₂×C₄₇⋊C₈	#41	1	1	yes	characteristic
134	C₂×C₄₇⋊C₈	#41	1	1	yes	characteristic
135	C₂×C₄₇⋊C₈	#41	1	2	yes	—
136	Dic₁₈₈	—	2	1	no	—
137	Dic₁₈₈	—	2	1	no	—
138	D₃₇₆	—	2	1	no	—
139	D₃₇₆	—	2	1	no	—
140	#10625325	—	2	2	no	—
141	#10625325	—	2	2	no	—
142	#10625324	—	4	2	no	—
143	#10625324	—	4	2	no	—
144	C₈×D₄₇	—	4	2	no	—
145	C₈×D₄₇	—	4	2	no	—
146	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂²×D₄₇	—	2	2	no	—
148	C₂×C₄×C₄₇	—	4	2	no	—
149	C₂×C₄×C₄₇	—	4	1	no	—
150	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²⋊D₄₇	—	4	2	no	—
157	C₂²⋊D₄₇	—	4	2	no	—
158	C₂×Dic₄₇	—	2	2	no	—
159	C₂×Dic₄₇	—	2	2	no	—
160	C₂×Dic₄₇	—	4	2	no	—
161	C₂×Dic₄₇	—	4	1	no	—
162	C₂×Dic₄₇	—	8	1	no	—
163	D₁₈₈	—	2	2	no	—
164	D₁₈₈	—	2	2	no	—
165	C₄×D₄₇	—	4	2	no	—
166	C₄×D₄₇	—	4	2	no	—
167	Dic₉₄	—	2	2	no	—
168	Dic₉₄	—	2	2	no	—
169	Dic₉₄	—	8	4	no	—
170	C₈×C₄₇	—	4	2	no	—
171	C₈×C₄₇	—	4	1	no	—
172	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
173	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
174	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
175	C₈×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
176	C₄₇⋊C₈	—	4	1	no	—
177	C₄₇⋊C₈	#124	1	2	yes	—
178	C₄₇⋊C₈	#124	1	2	yes	—
179	#29155	—	47	1	no	maximal, Sylow
180	C₂²×C₄₇	—	4	1	no	—
181	C₂²×C₄₇	#535	1	1	yes	characteristic
182	C₂²×C₄₇	#554	1	1	yes	characteristic
183	C₂²×C₄₇	#555	1	1	yes	characteristic
184	C₂×D₄₇	—	4	2	no	—
185	C₂×D₄₇	—	8	2	no	—
186	C₄×C₄₇	—	8	2	no	—
187	C₄×C₄₇	—	8	1	no	—
188	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
189	C₄×C₄₇	C₂×#124	1	1	yes	characteristic
190	C₄×C₄₇	#554	1	1	yes	characteristic
191	C₄×C₄₇	#555	1	1	yes	characteristic
192	Dic₄₇	—	4	2	no	—
193	Dic₄₇	—	4	2	no	—
194	Dic₄₇	—	8	2	no	—
195	Dic₄₇	—	8	1	no	—
196	#3162	—	47	1	no	—
197	C₂×#444	—	47	1	no	—
198	#2031	—	47	2	no	—
199	#2031	—	47	2	no	—
200	#1976	—	47	1	no	—
201	#1976	—	47	1	no	—
202	#1833	—	47	1	no	—
203	#1833	D₄₇	1	1	yes	characteristic, complemented
204	#1814	—	47	2	no	—
205	#1812	—	47	2	no	—
206	#1773	—	47	1	no	—
207	C₂×C₄₇	—	2	1	no	—
208	C₂×C₄₇	—	2	1	no	—
209	C₂×C₄₇	—	2	1	no	—
210	C₂×C₄₇	#2563	1	1	yes	characteristic
211	D₄₇	—	8	2	no	—
212	C₂×#135	—	47	1	no	—
213	C₂×#131	—	47	2	no	—
214	#533	—	47	2	no	—
215	#501	—	94	2	no	—
216	#501	—	94	2	no	—
217	#444	—	94	2	no	—
218	#444	—	94	2	no	—
219	C₂×#108	—	47	1	no	—
220	C₂×#108	C₂×D₄₇	1	1	yes	characteristic
221	#427	—	47	1	no	—
222	#427	—	47	1	no	—
223	#427	—	47	2	no	—
224	#427	C₂×D₄₇	1	1	yes	characteristic
225	#421	—	47	2	no	—
226	#421	—	47	2	no	—
227	C₂×#104	—	47	2	no	—
228	#410	—	47	2	no	—
229	#410	—	47	2	no	—
230	#405	—	47	1	no	—
231	#405	C₂×D₄₇	1	1	yes	characteristic
232	C₂×#97	—	47	1	no	—
233	#344	—	47	2	no	—
234	#344	C₂×D₄₇	1	2	yes	—
235	#339	—	47	1	no	—
236	#339	—	47	1	no	—
237	H(3,ℤ/4)	—	47	2	no	—
238	H(3,ℤ/4)	C₂×D₄₇	1	2	yes	—
239	C₄₇	#29155	1	1	yes	characteristic, complemented, Sylow
240	C₂×#41	—	47	2	no	—
241	#135	—	94	1	no	—
242	#135	—	94	1	no	—
243	#135	—	94	1	no	—
244	#135	—	94	1	no	—
245	C₂×Q₁₆	—	47	1	no	—
246	C₂×SD₁₆	—	47	2	no	—
247	C₂×D₈	—	47	1	no	—
248	#131	—	94	2	no	—
249	#131	—	94	2	no	—
250	#131	—	94	2	no	—
251	#131	—	94	2	no	—
252	C₂×M₁₆	—	47	1	no	—
253	C₂×M₁₆	—	94	2	no	—
254	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
255	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
256	C₂²×C₈	—	47	1	no	—
257	C₂²×C₈	—	94	2	no	—
258	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
259	C₄⋊Q₈	—	94	2	no	—
260	#125	—	94	2	no	—
261	C₂²⋊Q₈	—	188	2	no	—
262	C₄×Q₈	—	188	2	no	—
263	#117	—	47	2	no	—
264	#117	—	47	1	no	—
265	#117	C₂²×D₄₇	1	2	yes	—
266	#117	C₂²×D₄₇	1	1	yes	characteristic
267	C₂×C₄²	—	47	1	no	—
268	#108	—	2	1	no	—
269	#108	—	2	1	no	—
270	#108	—	94	2	no	—
271	#107	—	2	1	no	—
272	#107	—	94	1	no	—
273	#107	—	94	2	no	—
274	#107	—	94	1	no	—
275	#106	—	2	1	no	—
276	#106	—	94	1	no	—
277	#106	—	94	2	no	—
278	#106	—	94	1	no	—
279	#104	—	94	4	no	—
280	#104	—	188	2	no	—
281	#104	—	188	2	no	—
282	#99	—	188	2	no	—
283	#99	—	188	2	no	—
284	#97	—	2	1	no	—
285	#97	—	47	2	no	—
286	#97	—	47	2	no	—
287	#97	—	94	1	no	—
288	C₄×C₈	—	2	1	no	—
289	C₄×C₈	—	94	1	no	—
290	#41	—	94	2	no	—
291	#41	—	94	2	no	—
292	#41	—	94	2	no	—
293	#41	—	94	2	no	—
294	C₂×Q₈	—	47	2	no	—
295	C₂×Q₈	—	188	2	no	—
296	C₂×D₄	—	47	2	no	—
297	C₂×D₄	—	94	2	no	—
298	C₂²×C₄	—	94	1	no	—
299	C₂²×C₄	—	94	2	no	—
300	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
301	Q₁₆	—	94	1	no	—
302	Q₁₆	—	94	1	no	—
303	SD₁₆	—	94	2	no	—
304	SD₁₆	—	94	2	no	—
305	D₈	—	94	1	no	—
306	D₈	—	94	1	no	—
307	M₁₆	—	2	2	no	—
308	M₁₆	—	2	1	no	—
309	M₁₆	—	2	2	no	—
310	M₁₆	—	2	1	no	—
311	M₁₆	—	94	1	no	—
312	M₁₆	—	94	1	no	—
313	M₁₆	—	188	2	no	—
314	M₁₆	—	188	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	47	2	no	—
318	C₂×C₈	—	47	1	no	—
319	C₂×C₈	—	47	1	no	—
320	C₂×C₈	—	47	2	no	—
321	C₂×C₈	—	94	1	no	—
322	C₂×C₈	—	188	2	no	—
323	C₂×C₈	—	188	2	no	—
324	C₂×C₈	#10625350	1	1	yes	characteristic
325	C₂×C₈	#10625350	1	1	yes	characteristic
326	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
327	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
328	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
329	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	1	no	—
332	C₄⋊C₄	—	94	2	no	—
333	C₄⋊C₄	—	94	1	no	—
334	C₄⋊C₄	—	188	2	no	—
335	C₄⋊C₄	—	188	2	no	—
336	C₄⋊C₄	—	188	2	no	—
337	C₂²⋊C₄	—	2	1	no	—
338	C₂²⋊C₄	—	2	2	no	—
339	C₂²⋊C₄	—	94	1	no	—
340	C₂²⋊C₄	—	94	2	no	—
341	C₄²	—	2	1	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	47	2	no	—
344	C₄²	—	94	1	no	—
345	C₄²	—	94	1	no	—
346	C₄²	—	94	2	no	—
347	C₂³	—	94	2	no	—
348	C₂³	#13094070	1	1	yes	characteristic
349	Q₈	—	94	2	no	—
350	Q₈	—	94	2	no	—
351	Q₈	—	376	4	no	—
352	D₄	—	94	2	no	—
353	D₄	—	94	2	no	—
354	D₄	—	188	2	no	—
355	D₄	—	188	2	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	94	2	no	—
359	C₂×C₄	—	94	2	no	—
360	C₂×C₄	—	376	1	no	—
361	C₂×C₄	—	188	1	no	—
362	C₂×C₄	—	188	2	no	—
363	C₂×C₄	—	188	2	no	—
364	C₂×C₄	—	188	2	no	—
365	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
366	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
367	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
368	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
369	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
370	C₂×C₄	#13094070	1	1	yes	characteristic
371	C₈	—	4	1	no	—
372	C₈	—	4	2	no	—
373	C₈	—	47	2	no	—
374	C₈	—	47	2	no	—
375	C₈	—	188	1	no	—
376	C₈	#13094002	1	1	yes	characteristic
377	C₈	#13094002	1	1	yes	characteristic
378	C₈	#13094002	1	1	yes	characteristic
379	C₈	#13094002	1	1	yes	characteristic
380	C₂²	—	4	1	no	—
381	C₂²	—	376	2	no	—
382	C₂²	—	188	2	no	—
383	C₂²	#15689691	1	1	yes	characteristic
384	C₂²	#15689829	1	1	yes	characteristic
385	C₂²	#15689836	1	1	yes	characteristic
386	C₄	—	8	2	no	—
387	C₄	—	8	1	no	—
388	C₄	—	376	2	no	—
389	C₄	—	376	1	no	—
390	C₄	—	188	2	no	—
391	C₄	—	188	2	no	—
392	C₄	C₂×#13094002	1	1	yes	characteristic
393	C₄	#124×D₄₇	1	1	yes	characteristic
394	C₄	#15689829	1	1	yes	characteristic
395	C₄	#15689836	1	1	yes	characteristic
396	C₂	—	2	1	no	—
397	C₂	—	2	1	no	—
398	C₂	—	2	1	no	—
399	C₂	—	376	2	no	—
400	C₂	#20248560	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15689836
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1833	C₂
socle	C₂×C₄₇	#2563
prime radical	C₂×#13094046	C₂²
hypercenter	#1833	D₄₇
hypocenter	C₄₇	#29155
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1833

Complements

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

N	H ≅ G/N	type
1. #20253911	C₂	semidirect
3. #20244563	C₂	semidirect
4. #20244559	C₂	semidirect
7. #20243513	C₂	semidirect
8. #20243441	C₂	semidirect
9. #20243433	C₂	semidirect
10. #20243273	C₂	semidirect
11. C₄₇×#1833	C₂	semidirect
44. C₂×Dic₁₈₈	Q₈	semidirect
45. C₂×#10625325	Q₈	semidirect
46. C₂×D₃₇₆	Q₈	semidirect
203. #1833	D₄₇	semidirect
239. C₄₇	#29155	semidirect

Maximal subgroups

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

subgroup	index
#20253911	2
C₂×#15689244	2
#20244563	2
#20244559	2
#20244286	2
#20244284	2
#20243513	2
#20243441	2
#20243433	2
#20243273	2
C₄₇×#1833	2
#29155	47