Subgroups — #26124210

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245791	C₂	1	1	yes	maximal, characteristic
2	#20243121	C₂	1	1	yes	maximal, characteristic
3	#20243099	C₂	1	2	yes	maximal
4	C₂×#15688884	C₂	1	2	yes	maximal
5	C₄₇×#2905	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093983	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	#15688992	C₄	1	2	yes	complemented
10	#15688988	C₄	1	2	yes	complemented
11	#15688981	—	2	2	no	—
12	C₄×Dic₄₇⋊C₄	—	2	2	no	—
13	#15688890	—	2	2	no	—
14	#15688884	—	2	4	no	—
15	#15688884	—	2	4	no	—
16	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
17	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#424	—	2	4	no	—
19	C₄₇×#423	—	2	2	no	—
20	C₂³×Dic₄₇	—	2	1	no	—
21	#13093983	—	2	2	no	—
22	#13093983	—	2	2	no	—
23	#13093983	—	2	2	no	—
24	#13093983	—	2	2	no	—
25	#13093983	—	2	2	no	—
26	#13093983	Q₈	1	4	yes	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₄₇⋊C₄	—	2	4	no	—
30	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
31	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
32	C₂×C₄×Dic₄₇	—	2	1	no	—
33	C₂×C₄×Dic₄₇	—	2	1	no	—
34	C₂×C₄×Dic₄₇	—	2	1	no	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	1	no	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
40	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
41	C₄₇⋊#95	—	2	4	no	—
42	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₄₇	—	2	2	no	—
44	C₂×M₁₆×C₄₇	—	2	2	no	—
45	C₂×M₁₆×C₄₇	—	2	2	no	—
46	C₂×M₁₆×C₄₇	—	2	2	no	—
47	C₂×M₁₆×C₄₇	—	2	2	no	—
48	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
49	C₂²×C₈×C₄₇	—	2	2	no	—
50	#117×C₄₇	—	2	2	no	—
51	#117×C₄₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
54	C₂×C₄²×C₄₇	D₄	1	2	yes	—
55	C₄⋊C₈×C₄₇	—	2	8	no	—
56	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	2	yes	—
68	C₂²×Dic₄₇	—	2	1	no	—
69	C₂²×Dic₄₇	—	2	1	no	—
70	C₂²×Dic₄₇	—	2	2	no	—
71	C₂²×Dic₄₇	—	2	2	no	—
72	C₂²×Dic₄₇	—	2	2	no	—
73	C₂²×Dic₄₇	—	4	4	no	—
74	C₂²×Dic₄₇	—	4	2	no	—
75	M₁₆×C₄₇	—	2	4	no	—
76	M₁₆×C₄₇	—	2	4	no	—
77	M₁₆×C₄₇	—	4	2	no	—
78	M₁₆×C₄₇	—	4	2	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	4	2	no	—
81	C₂×C₈×C₄₇	—	4	2	no	—
82	C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₂²⋊C₄×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₂²⋊Dic₄₇	—	2	4	no	—
87	C₂²⋊Dic₄₇	—	4	2	no	—
88	C₂²⋊Dic₄₇	—	4	2	no	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	C₄⋊Dic₄₇	—	2	4	no	—
91	C₄⋊Dic₄₇	—	4	2	no	—
92	Dic₄₇⋊C₄	—	4	8	no	—
93	C₄×Dic₄₇	—	2	1	no	—
94	C₄×Dic₄₇	—	2	1	no	—
95	C₄×Dic₄₇	—	2	2	no	—
96	C₄×Dic₄₇	—	2	2	no	—
97	C₄×Dic₄₇	—	2	4	no	—
98	C₄×Dic₄₇	—	4	1	no	—
99	C₄×Dic₄₇	—	4	2	no	—
100	C₄×Dic₄₇	—	4	1	no	—
101	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
102	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
103	C₂³×C₄₇	—	4	2	no	—
104	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂³×C₄₇	#121	1	2	yes	—
106	C₂³×C₄₇	#121	1	1	yes	characteristic
107	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₄₇	#121	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₄₇	#123	1	2	yes	—
122	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#125	1	2	yes	—
125	C₂×C₄×C₄₇	#125	1	2	yes	—
126	C₂×C₄×C₄₇	#126	1	2	yes	—
127	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
129	C₂×Dic₄₇	—	2	4	no	—
130	C₂×Dic₄₇	—	2	2	no	—
131	C₂×Dic₄₇	—	2	2	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	—	4	4	no	—
136	C₂×Dic₄₇	—	4	2	no	—
137	C₂×Dic₄₇	—	4	4	no	—
138	C₂×Dic₄₇	—	8	4	no	—
139	C₈×C₄₇	—	4	4	no	—
140	#11212	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	4	no	—
143	C₂²×C₄₇	—	4	1	no	—
144	C₂²×C₄₇	—	4	1	no	—
145	C₂²×C₄₇	—	4	2	no	—
146	C₂²×C₄₇	#380	1	1	yes	characteristic
147	C₂²×C₄₇	#386	1	1	yes	characteristic
148	C₂²×C₄₇	#391	1	1	yes	characteristic
149	C₂²×C₄₇	#396	1	2	yes	—
150	C₂²×C₄₇	#398	1	2	yes	—
151	C₄×C₄₇	—	4	4	no	—
152	C₄×C₄₇	#387	1	2	yes	—
153	C₄×C₄₇	#389	1	2	yes	—
154	C₄×C₄₇	#394	1	1	yes	characteristic
155	C₄×C₄₇	#395	1	1	yes	characteristic
156	C₄×C₄₇	#398	1	1	yes	characteristic
157	C₄×C₄₇	#400	1	1	yes	characteristic
158	Dic₄₇	—	2	4	no	—
159	Dic₄₇	—	4	2	no	—
160	Dic₄₇	—	8	4	no	—
161	Dic₄₇	—	8	2	no	—
162	#2905	D₄₇	1	1	yes	characteristic
163	#2285	—	47	1	no	—
164	#1740	—	47	1	no	—
165	#1733	—	47	2	no	—
166	C₂×#339	—	47	2	no	—
167	C₂×C₄₇	—	2	1	no	—
168	C₂×C₄₇	—	2	2	no	—
169	C₂×C₄₇	—	2	1	no	—
170	C₂×C₄₇	—	2	2	no	—
171	C₂×C₄₇	#1441	1	1	yes	characteristic
172	C₂×C₄₇	#1884	1	2	yes	—
173	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
174	C₂×#117	—	47	1	no	—
175	C₂×#117	—	47	2	no	—
176	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
177	C₂²×C₄²	—	47	1	no	—
178	#424	—	2	4	no	—
179	#423	—	2	2	no	—
180	#384	—	47	2	no	—
181	#382	—	47	2	no	—
182	#380	—	94	2	no	—
183	C₄×C₄⋊C₄	—	94	2	no	—
184	#342	—	94	2	no	—
185	#339	—	94	4	no	—
186	#339	—	94	4	no	—
187	C₄₇	#11212	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	94	1	no	—
189	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	C₂²⋊D₄₇	1	4	yes	—
196	C₂²×C₈	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	47	4	no	—
200	#117	—	94	2	no	—
201	#117	—	94	2	no	—
202	#117	—	94	2	no	—
203	#117	—	94	2	no	—
204	#117	—	94	2	no	—
205	C₂×C₄⋊C₄	—	47	2	no	—
206	C₂×C₄⋊C₄	—	94	2	no	—
207	C₂×C₄⋊C₄	—	94	4	no	—
208	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
209	C₂×C₂²⋊C₄	—	94	2	no	—
210	C₂×C₂²⋊C₄	—	94	1	no	—
211	C₂×C₂²⋊C₄	D₁₈₈	1	2	yes	—
212	C₂×C₄²	—	47	1	no	—
213	C₂×C₄²	—	47	1	no	—
214	C₂×C₄²	—	47	2	no	—
215	C₂×C₄²	—	47	2	no	—
216	C₂×C₄²	—	94	1	no	—
217	C₂×C₄²	—	94	1	no	—
218	C₂×C₄²	—	94	1	no	—
219	C₂×C₄²	—	94	2	no	—
220	C₂×C₄²	—	94	1	no	—
221	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
222	C₄⋊C₈	—	2	8	no	—
223	#95	—	94	4	no	—
224	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
225	C₂²×C₄	—	94	1	no	—
226	C₂²×C₄	—	94	1	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	—	94	2	no	—
229	C₂²×C₄	—	94	2	no	—
230	C₂²×C₄	—	188	4	no	—
231	C₂²×C₄	—	188	2	no	—
232	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
233	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
234	C₂²×C₄	#10625350	1	2	yes	—
235	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	#10625352	1	1	yes	characteristic
237	C₂²×C₄	#10625352	1	2	yes	—
238	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
240	C₂²×C₄	#10625354	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
242	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
243	M₁₆	—	2	4	no	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	4	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	94	4	no	—
252	C₄⋊C₄	—	94	4	no	—
253	C₄⋊C₄	—	188	8	no	—
254	C₄⋊C₄	—	188	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	94	4	no	—
257	C₂²⋊C₄	—	188	2	no	—
258	C₂²⋊C₄	—	188	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	47	2	no	—
262	C₄²	—	47	2	no	—
263	C₄²	—	94	1	no	—
264	C₄²	—	94	1	no	—
265	C₄²	—	94	2	no	—
266	C₄²	—	94	2	no	—
267	C₄²	—	94	4	no	—
268	C₄²	—	188	1	no	—
269	C₄²	—	188	2	no	—
270	C₄²	—	188	1	no	—
271	C₂³	—	4	2	no	—
272	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
273	C₂³	#13093998	1	2	yes	—
274	C₂³	#13093999	1	1	yes	characteristic
275	C₂³	#13094004	1	1	yes	characteristic
276	C₂³	#13094004	1	2	yes	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	94	4	no	—
280	C₂×C₄	—	94	2	no	—
281	C₂×C₄	—	94	2	no	—
282	C₂×C₄	—	376	4	no	—
283	C₂×C₄	—	188	4	no	—
284	C₂×C₄	—	188	4	no	—
285	C₂×C₄	—	188	4	no	—
286	C₂×C₄	—	188	2	no	—
287	C₂×C₄	—	188	4	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
290	C₂×C₄	#13093982	1	1	yes	characteristic
291	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
292	C₂×C₄	#13093989	1	2	yes	—
293	C₂×C₄	#13093993	1	1	yes	characteristic
294	C₂×C₄	#13093995	1	1	yes	characteristic
295	C₂×C₄	#13093996	1	2	yes	—
296	C₂×C₄	#13094000	1	2	yes	—
297	C₂×C₄	#13094000	1	1	yes	characteristic
298	C₂×C₄	#13094002	1	2	yes	—
299	C₂×C₄	#13094005	1	2	yes	—
300	C₂×C₄	#13094007	1	1	yes	characteristic
301	C₂×C₄	#13094011	1	1	yes	characteristic
302	C₂×C₄	#13094014	1	2	yes	—
303	C₂×C₄	#13094014	1	2	yes	—
304	C₂×C₄	#13094019	1	2	yes	—
305	C₂×C₄	#13094022	1	2	yes	—
306	C₂×C₄	#13094022	1	1	yes	characteristic
307	C₈	—	4	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	2	no	—
313	C₂²	#15688980	1	1	yes	characteristic
314	C₂²	#15689002	1	1	yes	characteristic
315	C₂²	#15689028	1	1	yes	characteristic
316	C₂²	#15689045	1	2	yes	—
317	C₂²	#15689053	1	2	yes	—
318	C₄	—	4	4	no	—
319	C₄	—	94	4	no	—
320	C₄	—	376	4	no	—
321	C₄	—	376	2	no	—
322	C₄	—	188	2	no	—
323	C₄	#15689004	1	2	yes	—
324	C₄	#15689018	1	2	yes	—
325	C₄	#15689037	1	1	yes	characteristic
326	C₄	#15689042	1	1	yes	characteristic
327	C₄	#15689054	1	1	yes	characteristic
328	C₄	#15689061	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	2	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	2	no	—
333	C₂	#20241638	1	1	yes	characteristic
334	C₂	#20243771	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093993
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2905	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2905	D₄₇
hypocenter	C₄₇	#11212
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2905

Complements

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

N	H ≅ G/N	type
9. #15688992	C₄	semidirect
10. #15688988	C₄	semidirect
187. C₄₇	#11212	semidirect

Maximal subgroups

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

subgroup	index
#20245791	2
#20243121	2
#20243099	2
C₂×#15688884	2
C₄₇×#2905	2
#11212	47