Subgroups — #26218539

Order 12032.

822 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251978	C₂	1	1	yes	maximal, characteristic
2	#20251951	C₂	1	1	yes	maximal, characteristic
3	#20251806	C₂	1	1	yes	maximal, characteristic
4	#20243988	C₂	1	8	yes	maximal, complemented
5	#20243115	C₂	1	1	yes	maximal, characteristic
6	C₂×#15688883	C₂	1	1	yes	maximal, characteristic
7	C₂×#15688883	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#1728	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10625329	C₂²	1	1	yes	characteristic
10	C₂²×#10625329	C₂²	1	1	yes	characteristic
11	C₂×#13093983	C₂²	1	1	yes	characteristic
12	C₂×#13093983	C₂²	1	1	yes	characteristic
13	#15689165	C₄	1	4	yes	complemented
14	#15689165	C₄	1	4	yes	complemented
15	#15689161	C₂²	1	4	yes	—
16	#15689151	—	2	2	no	—
17	#15689149	C₂²	1	4	yes	—
18	#15689146	—	2	2	no	—
19	#15689081	—	2	2	no	—
20	#15689078	C₂²	1	4	yes	—
21	M₁₆×Dic₄₇	—	2	2	no	—
22	#15688888	C₂²	1	4	yes	—
23	#15688883	C₂²	1	4	yes	—
24	#15688883	C₂²	1	4	yes	—
25	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
26	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#336	C₂²	1	4	yes	—
29	C₂×#10625329	—	2	1	no	—
30	C₂×#10625329	—	2	2	no	—
31	C₂×#10625329	—	2	1	no	—
32	C₂×#10625329	—	2	2	no	—
33	C₂×#10625329	—	2	2	no	—
34	C₂×#10625329	Q₈	1	4	yes	—
35	C₂×#10625329	C₂³	1	2	yes	—
36	C₂×#10625329	C₂³	1	2	yes	—
37	C₂²×C₄₇⋊C₈	—	2	1	no	—
38	C₂²×C₄₇⋊C₈	—	2	1	no	—
39	#13093983	—	2	1	no	—
40	#13093983	—	2	1	no	—
41	#13093983	—	2	1	no	—
42	#13093983	—	2	1	no	—
43	#13093983	C₂×C₄	1	2	yes	—
44	#13093983	C₂×C₄	1	2	yes	—
45	#13093983	C₂³	1	2	yes	—
46	#13093983	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
50	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
51	C₂×C₄×Dic₄₇	—	2	1	no	—
52	C₂×C₄×Dic₄₇	—	2	1	no	—
53	#13093958	C₂×C₄	1	8	yes	complemented
54	#13093956	C₂×C₄	1	8	yes	complemented
55	Dic₄₇⋊C₈	—	2	8	no	—
56	#13093934	—	2	4	no	—
57	C₈×Dic₄₇	—	2	4	no	—
58	C₂³×C₄×C₄₇	—	2	1	no	—
59	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₄₇	—	2	2	no	—
61	C₂×M₁₆×C₄₇	—	2	2	no	—
62	C₂×M₁₆×C₄₇	—	2	2	no	—
63	C₂×M₁₆×C₄₇	—	2	2	no	—
64	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
66	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
67	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
68	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₄₇	—	2	2	no	—
71	C₂×C₄²×C₄₇	—	2	2	no	—
72	C₂×C₄²×C₄₇	—	2	1	no	—
73	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
74	C₂⁴×C₄₇	#41	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	—	2	2	no	—
76	C₂²×C₄×C₄₇	—	4	4	no	—
77	C₂²×C₄×C₄₇	—	4	2	no	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₄₇	#41	1	2	yes	—
84	C₂²×C₄×C₄₇	#41	1	2	yes	—
85	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
86	C₂²×C₄×C₄₇	#41	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
88	C₂²×Dic₄₇	—	2	2	no	—
89	C₂²×Dic₄₇	—	2	2	no	—
90	M₁₆×C₄₇	—	2	4	no	—
91	M₁₆×C₄₇	—	2	4	no	—
92	M₁₆×C₄₇	—	2	4	no	—
93	M₁₆×C₄₇	—	2	4	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
100	C₄²×C₄₇	—	2	2	no	—
101	C₄²×C₄₇	—	2	2	no	—
102	C₄²×C₄₇	—	4	2	no	—
103	C₂²⋊Dic₄₇	—	2	2	no	—
104	C₂²⋊Dic₄₇	—	2	2	no	—
105	C₂²⋊Dic₄₇	—	2	2	no	—
106	C₂²⋊Dic₄₇	—	2	2	no	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	2	2	no	—
109	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
110	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
111	C₄×Dic₄₇	—	2	2	no	—
112	C₄×Dic₄₇	—	2	2	no	—
113	C₄×Dic₄₇	—	2	2	no	—
114	C₄×Dic₄₇	—	2	2	no	—
115	#10625329	—	2	4	no	—
116	#10625329	—	2	2	no	—
117	#10625329	—	2	2	no	—
118	#10625329	—	2	2	no	—
119	#10625329	—	4	2	no	—
120	#10625329	C₂×Q₈	1	4	yes	—
121	C₂×C₄₇⋊C₈	—	2	4	no	—
122	C₂×C₄₇⋊C₈	—	2	2	no	—
123	C₂×C₄₇⋊C₈	—	2	2	no	—
124	C₂×C₄₇⋊C₈	—	4	2	no	—
125	C₂³×C₄₇	—	4	2	no	—
126	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
127	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
128	C₂³×C₄₇	#125	1	2	yes	—
129	C₂³×C₄₇	#126	1	2	yes	—
130	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₄₇	—	2	4	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	—	4	8	no	—
134	C₂×C₄×C₄₇	—	4	2	no	—
135	C₂×C₄×C₄₇	#117	1	2	yes	—
136	C₂×C₄×C₄₇	#117	1	2	yes	—
137	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
140	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
141	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
142	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₄₇	#123	1	2	yes	—
144	C₂×C₄×C₄₇	#123	1	2	yes	—
145	C₂×C₄×C₄₇	#125	1	2	yes	—
146	C₂×C₄×C₄₇	#126	1	2	yes	—
147	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
153	C₂×Dic₄₇	—	2	4	no	—
154	C₂×Dic₄₇	—	2	4	no	—
155	C₂×Dic₄₇	—	8	2	no	—
156	C₂×Dic₄₇	—	8	2	no	—
157	C₈×C₄₇	—	2	4	no	—
158	C₈×C₄₇	—	2	4	no	—
159	C₄₇⋊C₈	—	2	4	no	—
160	C₄₇⋊C₈	—	4	2	no	—
161	#21042	—	47	1	no	maximal, Sylow
162	C₂²×C₄₇	—	2	2	no	—
163	C₂²×C₄₇	—	2	2	no	—
164	C₂²×C₄₇	—	2	2	no	—
165	C₂²×C₄₇	—	4	2	no	—
166	C₂²×C₄₇	—	4	2	no	—
167	C₂²×C₄₇	#387	1	2	yes	—
168	C₂²×C₄₇	#387	1	2	yes	—
169	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₄₇	C₂×#125	1	1	yes	characteristic
171	C₂²×C₄₇	C₂×#126	1	1	yes	characteristic
172	C₄×C₄₇	—	4	4	no	—
173	C₄×C₄₇	#387	1	2	yes	—
174	C₄×C₄₇	#387	1	2	yes	—
175	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
176	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
177	C₄×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
178	C₄×C₄₇	C₂×#123	1	1	yes	characteristic
179	Dic₄₇	—	4	4	no	—
180	Dic₄₇	—	4	4	no	—
181	#2911	—	47	1	no	—
182	#2905	—	47	1	no	—
183	#2873	—	47	1	no	—
184	#1922	—	47	8	no	—
185	#1738	—	47	1	no	—
186	C₂×#339	—	47	1	no	—
187	C₂×#339	—	47	1	no	—
188	#1728	D₄₇	1	1	yes	characteristic
189	C₂×C₄₇	—	2	1	no	—
190	C₂×C₄₇	—	2	1	no	—
191	C₂×C₄₇	—	2	2	no	—
192	C₂×C₄₇	—	2	2	no	—
193	C₂×C₄₇	C₂×#387	1	1	yes	characteristic
194	C₂×C₄₇	#2956	1	2	yes	—
195	C₂²×M₁₆	—	47	1	no	—
196	C₂²×M₁₆	—	47	1	no	—
197	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
198	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
199	C₂×#117	—	47	1	no	—
200	C₂×#117	—	47	1	no	—
201	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
202	#428	—	47	4	no	—
203	#428	—	47	4	no	—
204	#427	—	47	4	no	—
205	#424	—	47	4	no	—
206	#424	—	94	2	no	—
207	#423	—	94	2	no	—
208	#405	—	47	4	no	—
209	#405	—	94	2	no	—
210	C₄×M₁₆	—	94	2	no	—
211	#341	—	47	4	no	—
212	#339	—	47	4	no	—
213	#339	—	47	4	no	—
214	#336	C₂×D₄₇	1	4	yes	—
215	C₄₇	#21042	1	1	yes	characteristic, complemented, Sylow
216	C₂³×C₄	—	2	1	no	—
217	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
218	C₂×M₁₆	—	2	2	no	—
219	C₂×M₁₆	—	2	2	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	—	47	4	no	—
223	C₂×M₁₆	—	47	2	no	—
224	C₂×M₁₆	—	47	2	no	—
225	C₂×M₁₆	—	94	1	no	—
226	C₂×M₁₆	—	94	2	no	—
227	C₂×M₁₆	—	94	1	no	—
228	C₂×M₁₆	—	94	2	no	—
229	C₂×M₁₆	—	94	2	no	—
230	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
231	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
232	C₂×M₁₆	D₁₈₈	1	2	yes	—
233	C₂×M₁₆	D₁₈₈	1	2	yes	—
234	C₂²×C₈	—	94	1	no	—
235	C₂²×C₈	—	94	1	no	—
236	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
237	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
238	#117	—	47	2	no	—
239	#117	—	47	2	no	—
240	#117	—	47	2	no	—
241	#117	—	47	2	no	—
242	#117	—	94	1	no	—
243	#117	—	94	1	no	—
244	#117	—	94	1	no	—
245	#117	—	94	1	no	—
246	C₂×C₄⋊C₄	—	47	2	no	—
247	C₂×C₄⋊C₄	—	47	2	no	—
248	C₂×C₂²⋊C₄	—	94	1	no	—
249	C₂×C₂²⋊C₄	—	94	1	no	—
250	C₂×C₄²	—	2	2	no	—
251	C₂×C₄²	—	2	2	no	—
252	C₂×C₄²	—	2	1	no	—
253	C₂×C₄²	—	94	1	no	—
254	C₂×C₄²	—	94	1	no	—
255	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
256	#107	—	47	8	no	—
257	#106	—	47	8	no	—
258	C₄⋊C₈	—	94	8	no	—
259	#97	—	94	4	no	—
260	C₄×C₈	—	94	4	no	—
261	C₂⁴	#10625350	1	1	yes	characteristic
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	4	4	no	—
264	C₂²×C₄	—	4	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	#10625352	1	2	yes	—
273	C₂²×C₄	#10625352	1	2	yes	—
274	C₂²×C₄	#10625352	1	2	yes	—
275	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
276	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
277	M₁₆	—	2	4	no	—
278	M₁₆	—	2	4	no	—
279	M₁₆	—	2	4	no	—
280	M₁₆	—	2	4	no	—
281	M₁₆	—	47	4	no	—
282	M₁₆	—	94	4	no	—
283	M₁₆	—	94	2	no	—
284	M₁₆	—	94	2	no	—
285	M₁₆	—	94	2	no	—
286	M₁₆	—	188	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	94	4	no	—
292	C₂×C₈	—	94	2	no	—
293	C₂×C₈	—	94	2	no	—
294	C₂×C₈	—	188	2	no	—
295	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
296	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
297	C₄⋊C₄	—	47	4	no	—
298	C₄⋊C₄	—	47	4	no	—
299	C₄⋊C₄	—	94	2	no	—
300	C₄⋊C₄	—	94	2	no	—
301	C₂²⋊C₄	—	94	2	no	—
302	C₂²⋊C₄	—	94	2	no	—
303	C₂²⋊C₄	—	94	2	no	—
304	C₂²⋊C₄	—	94	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	2	2	no	—
307	C₄²	—	4	2	no	—
308	C₄²	—	94	2	no	—
309	C₄²	—	94	2	no	—
310	C₄²	—	94	2	no	—
311	C₄²	—	94	2	no	—
312	C₂³	—	4	2	no	—
313	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
314	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
315	C₂³	#13094014	1	2	yes	—
316	C₂³	#13094017	1	2	yes	—
317	C₂³	C₂×#10625350	1	1	yes	characteristic
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	8	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	94	4	no	—
323	C₂×C₄	—	94	4	no	—
324	C₂×C₄	—	376	2	no	—
325	C₂×C₄	—	376	2	no	—
326	C₂×C₄	#13093980	1	2	yes	—
327	C₂×C₄	#13093980	1	2	yes	—
328	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
329	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
330	C₂×C₄	#13093989	1	2	yes	—
331	C₂×C₄	#13093989	1	2	yes	—
332	C₂×C₄	#13094004	1	2	yes	—
333	C₂×C₄	#13094004	1	2	yes	—
334	C₂×C₄	#13094008	1	2	yes	—
335	C₂×C₄	#13094008	1	2	yes	—
336	C₂×C₄	#13094014	1	2	yes	—
337	C₂×C₄	#13094017	1	2	yes	—
338	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
340	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
341	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
342	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
343	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
344	C₈	—	2	4	no	—
345	C₈	—	2	4	no	—
346	C₈	—	94	4	no	—
347	C₈	—	188	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	—	4	2	no	—
352	C₂²	—	4	2	no	—
353	C₂²	#15689008	1	2	yes	—
354	C₂²	#15689008	1	2	yes	—
355	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
356	C₂²	C₂×#13094014	1	1	yes	characteristic
357	C₂²	C₂×#13094017	1	1	yes	characteristic
358	C₄	—	4	4	no	—
359	C₄	—	188	4	no	—
360	C₄	—	188	4	no	—
361	C₄	#15689008	1	2	yes	—
362	C₄	#15689008	1	2	yes	—
363	C₄	C₂×#13093980	1	1	yes	characteristic
364	C₄	C₂×#13093989	1	1	yes	characteristic
365	C₄	C₂×#13094004	1	1	yes	characteristic
366	C₄	C₂×#13094008	1	1	yes	characteristic
367	C₂	—	2	1	no	—
368	C₂	—	2	2	no	—
369	C₂	—	2	2	no	—
370	C₂	—	2	1	no	—
371	C₂	C₂×#15689008	1	1	yes	characteristic
372	C₂	#20252299	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1728	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1728	D₄₇
hypocenter	C₄₇	#21042
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ #1728

Complements

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

N	H ≅ G/N	type
4. #20243988	C₂	semidirect
13. #15689165	C₄	semidirect
14. #15689165	C₄	semidirect
53. #13093958	C₂×C₄	semidirect
54. #13093956	C₂×C₄	semidirect
215. C₄₇	#21042	semidirect

Maximal subgroups

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

subgroup	index
#20251978	2
#20251951	2
#20251806	2
#20243988	2
#20243115	2
C₂×#15688883	2
C₄₇×#1728	2
#21042	47