Subgroups — #26371136

Order 12032.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253044	C₂	1	1	yes	maximal, characteristic
2	#20251985	C₂	1	1	yes	maximal, characteristic
3	#20245599	C₂	1	2	yes	maximal, complemented
4	#20245598	C₂	1	2	yes	maximal, complemented
5	#20245592	C₂	1	2	yes	maximal, complemented
6	#20245587	C₂	1	2	yes	maximal, complemented
7	#20245177	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245107	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15688928	C₂	1	1	yes	maximal, characteristic, complemented
10	#20245001	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#2158	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
13	C₂×#13094030	C₂²	1	1	yes	characteristic
14	#15689699	—	2	1	no	—
15	#15689603	—	2	1	no	—
16	C₂×#13093964	C₂²	1	1	yes	characteristic
17	#15689382	—	2	2	no	—
18	#15689365	C₂²	1	2	yes	—
19	#15689365	C₂²	1	2	yes	—
20	#15689166	—	2	1	no	—
21	#15689162	—	2	2	no	—
22	#15689160	C₂²	1	2	yes	—
23	C₂×#13093958	C₂²	1	1	yes	characteristic
24	#107×D₄₇	C₂²	1	2	yes	—
25	#15689131	—	2	1	no	—
26	#15688950	C₂²	1	2	yes	—
27	#15688948	C₂²	1	2	yes	—
28	#15688928	C₂²	1	2	yes	—
29	#15688928	C₂²	1	2	yes	—
30	#15688926	C₂²	1	2	yes	—
31	#15688926	C₂²	1	2	yes	—
32	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
33	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
34	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#427	C₂²	1	1	yes	characteristic
36	C₄₇×#427	C₂²	1	1	yes	characteristic
37	C₄₇×#367	C₂²	1	1	yes	characteristic
38	C₄₇×#367	C₂²	1	1	yes	characteristic
39	C₄₇×#366	C₂²	1	1	yes	characteristic
40	C₄₇×#366	C₂²	1	1	yes	characteristic
41	C₂×#10625350	D₄	1	1	yes	characteristic
42	C₂²×Dic₉₄	—	2	1	no	—
43	C₂×Dic₁₈₈	—	2	2	no	—
44	C₂×Dic₁₈₈	—	2	2	no	—
45	C₂×Dic₁₈₈	—	2	1	no	—
46	C₂×Dic₁₈₈	C₂³	1	2	yes	—
47	#13094030	—	2	2	no	—
48	#13094030	—	2	2	no	—
49	C₂×#10625329	D₄	1	1	yes	characteristic
50	C₂×#10625324	D₄	1	2	yes	—
51	C₂×C₈×D₄₇	C₂³	1	2	yes	—
52	#13094024	—	4	1	no	—
53	#13094012	—	4	1	no	—
54	#13094008	—	4	2	no	—
55	#13094004	—	2	2	no	—
56	#13094004	—	2	2	no	—
57	C₄×Dic₉₄	—	4	1	no	—
58	C₄×D₁₈₈	—	4	1	no	—
59	(C₂²⋊D₄₇)⋊C₄	—	4	2	no	—
60	#13093982	—	2	2	no	—
61	C₂×C₄⋊Dic₄₇	—	2	1	no	—
62	C₄⋊C₄×D₄₇	—	2	2	no	—
63	#13093964	—	2	2	no	—
64	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
65	#13093958	—	2	1	no	—
66	#13093958	C₂³	1	2	yes	—
67	#13093957	—	2	2	no	—
68	#13093955	—	2	2	no	—
69	#13093947	—	2	2	no	—
70	#13093947	—	2	2	no	—
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	#117×C₄₇	—	2	1	no	—
73	#117×C₄₇	—	2	1	no	—
74	C₂×C₁₆×C₄₇	—	2	2	no	—
75	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
77	#107×C₄₇	C₂³	1	1	yes	characteristic
78	#107×C₄₇	C₂³	1	1	yes	characteristic
79	#107×C₄₇	C₂³	1	1	yes	characteristic
80	#107×C₄₇	C₂³	1	1	yes	characteristic
81	#106×C₄₇	—	2	1	no	—
82	#106×C₄₇	—	2	1	no	—
83	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₄₇	D₈	1	2	yes	—
85	C₂²×Dic₄₇	—	4	1	no	—
86	C₂²×Dic₄₇	—	4	1	no	—
87	#10625350	—	2	2	no	—
88	#10625350	—	2	2	no	—
89	C₂×D₁₈₈	D₈	1	1	yes	characteristic
90	C₂×C₄×D₄₇	—	4	2	no	—
91	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
92	C₂×Dic₉₄	—	2	2	no	—
93	C₂×Dic₉₄	—	4	2	no	—
94	C₂×Dic₉₄	—	4	2	no	—
95	C₂×Dic₉₄	—	4	1	no	—
96	C₂×Dic₉₄	D₈	1	1	yes	characteristic
97	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₇	#41	1	2	yes	—
99	C₂×C₈×C₄₇	#41	1	2	yes	—
100	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₄₇	—	2	1	no	—
102	C₄⋊C₄×C₄₇	—	2	1	no	—
103	C₄⋊C₄×C₄₇	—	2	1	no	—
104	C₄⋊C₄×C₄₇	—	2	1	no	—
105	C₂²⋊C₄×C₄₇	—	4	1	no	—
106	C₂²⋊C₄×C₄₇	—	4	1	no	—
107	C₄²×C₄₇	—	4	1	no	—
108	C₄²×C₄₇	—	4	1	no	—
109	(C₂×D₄₇)⋊C₄	—	4	2	no	—
110	(C₂×D₄₇)⋊C₄	—	4	2	no	—
111	(C₂×D₄₇)⋊C₄	—	4	2	no	—
112	C₄⋊Dic₄₇	—	2	2	no	—
113	C₄⋊Dic₄₇	—	4	1	no	—
114	C₄⋊Dic₄₇	—	4	2	no	—
115	C₄⋊Dic₄₇	—	4	2	no	—
116	Dic₄₇⋊C₄	—	4	2	no	—
117	C₄×Dic₄₇	—	4	2	no	—
118	#10625329	—	2	2	no	—
119	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
120	Dic₁₈₈	—	2	2	no	—
121	Dic₁₈₈	—	2	2	no	—
122	Dic₁₈₈	—	4	2	no	—
123	#10625324	—	2	4	no	—
124	C₈×D₄₇	—	2	2	no	—
125	C₈×D₄₇	—	2	2	no	—
126	C₁₆×C₄₇	—	2	1	no	—
127	C₁₆×C₄₇	—	2	1	no	—
128	C₄₇⋊C₁₆	—	2	2	no	—
129	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
130	C₂²×D₄₇	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₄₇	—	4	1	no	—
132	C₂×C₄×C₄₇	—	4	1	no	—
133	C₂×C₄×C₄₇	—	4	1	no	—
134	C₂×C₄×C₄₇	—	4	1	no	—
135	C₂×C₄×C₄₇	#123	1	2	yes	—
136	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#137	1	2	yes	—
138	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₄₇	—	2	4	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	8	2	no	—
143	C₂×Dic₄₇	—	16	1	no	—
144	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
145	D₁₈₈	—	2	2	no	—
146	C₄×D₄₇	—	2	2	no	—
147	C₄×D₄₇	—	2	2	no	—
148	C₄×D₄₇	—	16	2	no	—
149	Dic₉₄	—	2	2	no	—
150	Dic₉₄	—	4	2	no	—
151	Dic₉₄	—	4	2	no	—
152	Dic₉₄	—	8	2	no	—
153	C₈×C₄₇	#123	1	2	yes	—
154	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
155	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
156	C₄₇⋊C₈	—	2	2	no	—
157	#33651	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	2	1	no	—
160	C₂²×C₄₇	#480	1	2	yes	—
161	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₄₇	—	2	2	no	—
163	C₂×D₄₇	—	2	2	no	—
164	C₂×D₄₇	—	2	2	no	—
165	C₄×C₄₇	—	8	1	no	—
166	C₄×C₄₇	—	8	1	no	—
167	C₄×C₄₇	—	8	1	no	—
168	C₄×C₄₇	—	8	1	no	—
169	C₄×C₄₇	#480	1	2	yes	—
170	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
171	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
172	Dic₄₇	—	2	2	no	—
173	Dic₄₇	—	8	2	no	—
174	Dic₄₇	—	8	2	no	—
175	#3059	—	47	1	no	—
176	#2913	—	47	1	no	—
177	#2237	—	47	2	no	—
178	#2237	—	47	2	no	—
179	#2236	—	47	2	no	—
180	#2235	—	47	2	no	—
181	#2159	—	47	1	no	—
182	#2158	D₄₇	1	1	yes	characteristic, complemented
183	#2141	—	47	1	no	—
184	C₂×#358	—	47	1	no	—
185	#2113	—	47	1	no	—
186	C₂×C₄₇	—	2	2	no	—
187	C₂×C₄₇	C₂×#480	1	1	yes	characteristic
188	C₂×C₄₇	#3416	1	1	yes	characteristic
189	C₂×C₄₇	#3417	1	1	yes	characteristic
190	D₄₇	—	4	2	no	—
191	C₂²×Q₁₆	—	47	1	no	—
192	C₂×#131	—	47	1	no	—
193	#536	—	94	1	no	—
194	#520	—	94	1	no	—
195	C₂×M₃₂	—	47	1	no	—
196	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
197	#470	—	94	2	no	—
198	#467	—	47	2	no	—
199	#467	—	47	2	no	—
200	#428	—	94	2	no	—
201	#428	—	94	1	no	—
202	#427	—	47	2	no	—
203	#427	C₂×D₄₇	1	1	yes	characteristic
204	#427	C₂×D₄₇	1	1	yes	characteristic
205	C₂×#107	—	47	1	no	—
206	C₂×#107	—	47	2	no	—
207	#419	—	94	1	no	—
208	#367	—	47	2	no	—
209	#367	C₂×D₄₇	1	1	yes	characteristic
210	#367	C₂×D₄₇	1	1	yes	characteristic
211	#366	—	47	2	no	—
212	#366	C₂×D₄₇	1	1	yes	characteristic
213	#366	C₂×D₄₇	1	1	yes	characteristic
214	#358	—	47	2	no	—
215	#358	—	47	2	no	—
216	#358	—	47	2	no	—
217	#358	—	47	2	no	—
218	C₂²⋊C₁₆	—	47	2	no	—
219	C₂²⋊C₁₆	—	47	2	no	—
220	C₄₇	#33651	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	47	1	no	—
222	C₂²×Q₈	—	94	1	no	—
223	C₂×Q₁₆	—	47	2	no	—
224	C₂×Q₁₆	—	94	2	no	—
225	C₂×Q₁₆	—	94	1	no	—
226	C₂×Q₁₆	—	94	2	no	—
227	#131	—	94	2	no	—
228	#131	—	94	2	no	—
229	C₂×M₁₆	—	47	1	no	—
230	C₂×M₁₆	—	47	2	no	—
231	C₂²×C₈	—	47	2	no	—
232	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
233	C₄⋊Q₈	—	188	1	no	—
234	#124	—	188	1	no	—
235	#123	—	188	2	no	—
236	C₂²⋊Q₈	—	94	2	no	—
237	C₂²⋊Q₈	—	94	2	no	—
238	C₄×Q₈	—	188	1	no	—
239	C₄×D₄	—	188	1	no	—
240	C₄×D₄	—	188	2	no	—
241	#117	—	2	1	no	—
242	#117	—	2	1	no	—
243	#117	—	94	2	no	—
244	C₂×C₄⋊C₄	—	94	2	no	—
245	C₂×C₄⋊C₄	—	94	1	no	—
246	M₃₂	—	94	2	no	—
247	C₂×C₁₆	—	2	2	no	—
248	C₂×C₁₆	—	47	2	no	—
249	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
250	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
251	#107	—	47	2	no	—
252	#107	—	94	2	no	—
253	#107	—	94	1	no	—
254	#107	C₂²×D₄₇	1	1	yes	characteristic
255	#107	C₂²×D₄₇	1	1	yes	characteristic
256	#107	C₂²×D₄₇	1	1	yes	characteristic
257	#107	C₂²×D₄₇	1	1	yes	characteristic
258	#106	—	2	1	no	—
259	#106	—	2	1	no	—
260	#106	—	94	2	no	—
261	#103	—	94	2	no	—
262	#103	—	94	2	no	—
263	#41	—	94	2	no	—
264	#41	—	94	2	no	—
265	C₂×Q₈	—	47	1	no	—
266	C₂×Q₈	—	94	2	no	—
267	C₂×Q₈	—	188	2	no	—
268	C₂×Q₈	—	188	2	no	—
269	C₂×Q₈	—	188	1	no	—
270	C₂×D₄	—	47	2	no	—
271	C₂×D₄	—	47	1	no	—
272	C₂²×C₄	—	47	2	no	—
273	C₂²×C₄	—	188	2	no	—
274	C₂²×C₄	—	188	1	no	—
275	C₂²×C₄	—	188	1	no	—
276	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
277	Q₁₆	—	94	2	no	—
278	Q₁₆	—	94	2	no	—
279	Q₁₆	—	188	2	no	—
280	M₁₆	—	94	2	no	—
281	M₁₆	—	94	4	no	—
282	C₂×C₈	—	47	2	no	—
283	C₂×C₈	—	94	2	no	—
284	C₂×C₈	—	94	2	no	—
285	C₂×C₈	#10625350	1	2	yes	—
286	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
287	C₂×C₈	#10625354	1	2	yes	—
288	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	94	2	no	—
294	C₄⋊C₄	—	188	2	no	—
295	C₄⋊C₄	—	188	2	no	—
296	C₄⋊C₄	—	188	2	no	—
297	C₄⋊C₄	—	188	1	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	—	188	2	no	—
301	C₂²⋊C₄	—	188	2	no	—
302	C₂²⋊C₄	—	188	2	no	—
303	C₄²	—	4	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	188	2	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	94	2	no	—
309	C₂³	—	47	2	no	—
310	C₂³	D₈×D₄₇	1	1	yes	characteristic
311	Q₈	—	94	2	no	—
312	Q₈	—	376	2	no	—
313	Q₈	—	188	2	no	—
314	Q₈	—	188	2	no	—
315	D₄	—	94	4	no	—
316	D₄	—	94	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	47	2	no	—
322	C₂×C₄	—	94	2	no	—
323	C₂×C₄	—	94	2	no	—
324	C₂×C₄	—	376	2	no	—
325	C₂×C₄	—	188	2	no	—
326	C₂×C₄	—	188	2	no	—
327	C₂×C₄	—	752	2	no	—
328	C₂×C₄	—	752	1	no	—
329	C₂×C₄	#13094009	1	2	yes	—
330	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
331	C₂×C₄	#13094056	1	2	yes	—
332	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
333	C₈	—	94	2	no	—
334	C₈	#13094009	1	2	yes	—
335	C₈	C₂×#10625350	1	1	yes	characteristic
336	C₈	C₂×#10625354	1	1	yes	characteristic
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	94	2	no	—
340	C₂²	—	94	2	no	—
341	C₂²	—	94	2	no	—
342	C₂²	#15689438	1	2	yes	—
343	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
344	C₄	—	8	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	1	no	—
348	C₄	—	94	2	no	—
349	C₄	—	376	2	no	—
350	C₄	—	376	2	no	—
351	C₄	#15689438	1	2	yes	—
352	C₄	C₂×#13094009	1	1	yes	characteristic
353	C₄	C₂×#13094056	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	188	2	no	—
356	C₂	C₂×#15689438	1	1	yes	characteristic
357	C₂	#20256531	1	1	yes	characteristic
358	C₂	#20256538	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2158	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#10625350	D₄
hypercenter	#2158	D₄₇
hypocenter	C₄₇	#33651
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #20245599	C₂	semidirect
4. #20245598	C₂	semidirect
5. #20245592	C₂	semidirect
6. #20245587	C₂	semidirect
7. #20245177	C₂	semidirect
8. #20245107	C₂	semidirect
9. C₂×#15688928	C₂	semidirect
11. C₄₇×#2158	C₂	semidirect
182. #2158	D₄₇	semidirect
220. C₄₇	#33651	semidirect

Maximal subgroups

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

subgroup	index
#20253044	2
#20251985	2
#20245599	2
#20245598	2
#20245592	2
#20245587	2
#20245177	2
#20245107	2
C₂×#15688928	2
#20245001	2
C₄₇×#2158	2
#33651	47