Subgroups — #26124213

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	#20251952	C₂	1	1	yes	maximal, characteristic
2	#20243121	C₂	1	1	yes	maximal, characteristic
3	#20243097	C₂	1	2	yes	maximal
4	C₂×#15688883	C₂	1	2	yes	maximal
5	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	2	yes	—
8	#15689150	—	2	4	no	—
9	#15689147	—	2	2	no	—
10	#15688890	—	2	2	no	—
11	#15688883	—	2	4	no	—
12	#15688882	—	2	4	no	—
13	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#384	C₄	1	2	yes	complemented
17	C₄₇×#382	C₄	1	2	yes	complemented
18	C₄₇×#380	—	2	2	no	—
19	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	D₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	—	2	2	no	—
27	#13093983	—	2	2	no	—
28	#13093983	—	2	2	no	—
29	#13093983	—	2	2	no	—
30	#13093983	Q₈	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
35	C₂³×C₄×C₄₇	—	2	1	no	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	#117×C₄₇	—	2	2	no	—
38	#117×C₄₇	—	2	2	no	—
39	#117×C₄₇	—	2	2	no	—
40	#117×C₄₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
46	C₂×C₄²×C₄₇	—	2	1	no	—
47	C₂×C₄²×C₄₇	—	2	1	no	—
48	C₂×C₄²×C₄₇	—	2	1	no	—
49	C₂×C₄²×C₄₇	—	2	1	no	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	D₄	1	2	yes	—
53	C₂×C₄²×C₄₇	D₄	1	2	yes	—
54	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
55	#95×C₄₇	—	2	4	no	—
56	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	1	no	—
58	C₂²×C₄×C₄₇	—	2	1	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	4	4	no	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₂²×C₄×C₄₇	#41	1	2	yes	—
74	C₂²×Dic₄₇	—	4	2	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	—	4	8	no	—
78	C₂²⋊C₄×C₄₇	—	2	4	no	—
79	C₂²⋊C₄×C₄₇	—	4	2	no	—
80	C₄²×C₄₇	—	2	1	no	—
81	C₄²×C₄₇	—	2	1	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	2	2	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	4	1	no	—
87	C₄²×C₄₇	—	4	1	no	—
88	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
89	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
90	C₂²⋊Dic₄₇	—	2	4	no	—
91	C₂²⋊Dic₄₇	—	4	2	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	C₄×Dic₄₇	—	2	4	no	—
95	C₄×Dic₄₇	—	4	2	no	—
96	#10625329	—	2	4	no	—
97	#10625329	—	2	4	no	—
98	#10625329	—	4	2	no	—
99	#10625329	—	4	2	no	—
100	C₂×C₄₇⋊C₈	—	2	4	no	—
101	C₂×C₄₇⋊C₈	—	4	2	no	—
102	C₂×C₄₇⋊C₈	—	4	2	no	—
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	2	yes	—
108	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#121	1	2	yes	—
126	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#123	1	2	yes	—
131	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
132	C₂×C₄×C₄₇	#125	1	2	yes	—
133	C₂×C₄×C₄₇	#125	1	2	yes	—
134	C₂×C₄×C₄₇	#126	1	2	yes	—
135	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
137	C₂×Dic₄₇	—	4	4	no	—
138	C₂×Dic₄₇	—	8	4	no	—
139	C₄₇⋊C₈	—	4	4	no	—
140	#11212	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	4	no	—
142	C₂²×C₄₇	—	2	2	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	—	2	4	no	—
152	C₄×C₄₇	—	4	2	no	—
153	C₄×C₄₇	—	4	4	no	—
154	C₄×C₄₇	—	8	2	no	—
155	C₄×C₄₇	#387	1	2	yes	—
156	C₄×C₄₇	#389	1	2	yes	—
157	C₄×C₄₇	#394	1	1	yes	characteristic
158	C₄×C₄₇	#395	1	1	yes	characteristic
159	C₄×C₄₇	#398	1	1	yes	characteristic
160	C₄×C₄₇	#400	1	1	yes	characteristic
161	Dic₄₇	—	8	4	no	—
162	#2905	—	47	1	no	—
163	#2285	D₄₇	1	1	yes	characteristic
164	#1740	—	47	1	no	—
165	#1733	—	47	2	no	—
166	C₂×#339	—	47	2	no	—
167	C₂×C₄₇	—	2	2	no	—
168	C₂×C₄₇	—	2	1	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	—	47	2	no	—
174	C₂×#117	—	47	2	no	—
175	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
176	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
177	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
178	#424	—	94	4	no	—
179	#423	—	94	2	no	—
180	#384	Dic₄₇	1	2	yes	complemented
181	#382	Dic₄₇	1	2	yes	complemented
182	#380	—	2	2	no	—
183	C₄×C₄⋊C₄	—	2	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	—	2	1	no	—
189	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
190	C₂×M₁₆	—	47	4	no	—
191	C₂×M₁₆	—	94	2	no	—
192	C₂×M₁₆	—	94	2	no	—
193	C₂×M₁₆	—	94	2	no	—
194	C₂×M₁₆	—	94	2	no	—
195	C₂×M₁₆	—	94	2	no	—
196	C₂²×C₈	—	94	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	47	4	no	—
202	#117	—	94	2	no	—
203	#117	—	94	2	no	—
204	#117	—	94	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	94	2	no	—
207	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
208	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₂²⋊C₄	—	94	2	no	—
211	C₂×C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	—	94	2	no	—
217	C₂×C₄²	Dic₉₄	1	2	yes	—
218	C₂×C₄²	D₁₈₈	1	2	yes	—
219	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
220	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
221	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
222	C₄⋊C₈	—	94	8	no	—
223	#95	—	2	4	no	—
224	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	188	2	no	—
232	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
233	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
235	C₂²×C₄	#10625350	1	1	yes	characteristic
236	C₂²×C₄	#10625350	1	1	yes	characteristic
237	C₂²×C₄	D₄×D₄₇	1	2	yes	—
238	C₂²×C₄	#10625352	1	2	yes	—
239	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
240	C₂²×C₄	#10625354	1	2	yes	—
241	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
243	M₁₆	—	94	4	no	—
244	M₁₆	—	94	4	no	—
245	M₁₆	—	188	2	no	—
246	M₁₆	—	188	2	no	—
247	C₂×C₈	—	94	4	no	—
248	C₂×C₈	—	188	2	no	—
249	C₂×C₈	—	188	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	8	no	—
253	C₄⋊C₄	—	94	4	no	—
254	C₄⋊C₄	—	188	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	94	4	no	—
258	C₂²⋊C₄	—	188	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	4	1	no	—
266	C₄²	—	4	1	no	—
267	C₄²	—	94	4	no	—
268	C₄²	—	188	2	no	—
269	C₄²	C₄⋊Dic₄₇	1	2	yes	—
270	C₄²	C₄⋊Dic₄₇	1	2	yes	—
271	C₂³	—	4	2	no	—
272	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
273	C₂³	#13093996	1	2	yes	—
274	C₂³	#13093998	1	1	yes	characteristic
275	C₂³	#13094000	1	2	yes	—
276	C₂³	#13094003	1	1	yes	characteristic
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	376	4	no	—
288	C₂×C₄	—	188	4	no	—
289	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
290	C₂×C₄	#13093983	1	1	yes	characteristic
291	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
292	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
293	C₂×C₄	#13093992	1	1	yes	characteristic
294	C₂×C₄	#13093998	1	1	yes	characteristic
295	C₂×C₄	#13093999	1	2	yes	—
296	C₂×C₄	#13094002	1	2	yes	—
297	C₂×C₄	#13094003	1	1	yes	characteristic
298	C₂×C₄	#13094004	1	2	yes	—
299	C₂×C₄	#13094005	1	1	yes	characteristic
300	C₂×C₄	#13094009	1	2	yes	—
301	C₂×C₄	#13094013	1	1	yes	characteristic
302	C₂×C₄	#13094014	1	2	yes	—
303	C₂×C₄	#13094015	1	2	yes	—
304	C₂×C₄	#13094019	1	2	yes	—
305	C₂×C₄	#13094022	1	1	yes	characteristic
306	C₂×C₄	#13094023	1	2	yes	—
307	C₈	—	188	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	#15688983	1	1	yes	characteristic
314	C₂²	#15689003	1	1	yes	characteristic
315	C₂²	#15689030	1	1	yes	characteristic
316	C₂²	#15689045	1	2	yes	—
317	C₂²	#15689056	1	2	yes	—
318	C₄	—	2	4	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	8	2	no	—
322	C₄	—	376	4	no	—
323	C₄	#15689009	1	2	yes	—
324	C₄	#15689022	1	2	yes	—
325	C₄	#15689040	1	1	yes	characteristic
326	C₄	#15689043	1	1	yes	characteristic
327	C₄	#15689053	1	1	yes	characteristic
328	C₄	#15689062	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	#20241641	1	1	yes	characteristic
334	C₂	#20243772	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093992
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2285	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2285	D₄₇
hypocenter	C₄₇	#11212
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
16. C₄₇×#384	C₄	semidirect
17. C₄₇×#382	C₄	semidirect
180. #384	Dic₄₇	semidirect
181. #382	Dic₄₇	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
#20251952	2
#20243121	2
#20243097	2
C₂×#15688883	2
C₄₇×#2285	2
#11212	47