Subgroups — #26124376

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689161	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688895	C₂	1	2	yes	maximal
3	C₂×#15688888	C₂	1	1	yes	maximal, characteristic
4	#20243097	C₂	1	2	yes	maximal
5	C₂×C₄₇×#424	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	2	yes	—
8	#15689161	—	2	4	no	—
9	C₂×#13093958	—	2	1	no	—
10	C₂×#13093956	—	2	1	no	—
11	#15688895	—	2	4	no	—
12	#15688888	—	2	2	no	—
13	#15688882	—	2	4	no	—
14	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#424	—	2	1	no	—
18	C₄₇×#424	—	2	1	no	—
19	C₄₇×#424	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₄₇	—	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	—	2	2	no	—
26	C₂×#10625329	Q₈	1	4	yes	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	#13093983	—	2	2	no	—
29	#13093983	—	2	2	no	—
30	#13093983	—	2	2	no	—
31	#13093983	D₄	1	4	yes	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
34	C₂×C₄×Dic₄₇	—	2	2	no	—
35	#13093958	—	2	4	no	—
36	#13093956	—	2	4	no	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₄₇	—	2	1	no	—
39	C₂×M₁₆×C₄₇	—	2	2	no	—
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	—	2	2	no	—
44	C₂²×C₈×C₄₇	—	2	1	no	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₄₇	D₄	1	2	yes	—
47	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
49	#117×C₄₇	—	2	1	no	—
50	#117×C₄₇	—	2	1	no	—
51	#117×C₄₇	C₂×C₄	1	2	yes	—
52	#117×C₄₇	D₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₄₇	Q₈	1	2	yes	—
54	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
55	C₂×C₄²×C₄₇	—	2	1	no	—
56	C₄⋊C₈×C₄₇	—	2	4	no	—
57	C₄⋊C₈×C₄₇	—	2	4	no	—
58	C₂⁴×C₄₇	#41	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	2	yes	—
68	C₂²×C₄×C₄₇	#41	1	2	yes	—
69	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	4	2	no	—
71	M₁₆×C₄₇	—	2	4	no	—
72	M₁₆×C₄₇	—	4	1	no	—
73	M₁₆×C₄₇	—	4	1	no	—
74	C₂×C₈×C₄₇	—	2	4	no	—
75	C₂×C₈×C₄₇	—	2	1	no	—
76	C₂×C₈×C₄₇	—	2	1	no	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	2	1	no	—
79	C₂×C₈×C₄₇	—	2	1	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	4	2	no	—
82	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
84	C₄⋊C₄×C₄₇	—	2	2	no	—
85	C₄⋊C₄×C₄₇	C₄⋊C₄	1	4	yes	—
86	C₂²⋊C₄×C₄₇	—	2	2	no	—
87	C₂²⋊C₄×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	2	2	no	—
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	#117	1	1	yes	characteristic
105	C₂³×C₄₇	#121	1	1	yes	characteristic
106	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₄₇	#123	1	2	yes	—
108	C₂³×C₄₇	#125	1	2	yes	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	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	2	yes	—
117	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#121	1	2	yes	—
119	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#125	1	2	yes	—
126	C₂×C₄×C₄₇	#126	1	2	yes	—
127	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₂×Dic₄₇	—	8	4	no	—
131	C₈×C₄₇	—	2	2	no	—
132	C₈×C₄₇	—	2	2	no	—
133	C₈×C₄₇	—	4	2	no	—
134	C₄₇⋊C₈	—	4	4	no	—
135	#11241	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	4	2	no	—
140	C₂²×C₄₇	—	4	2	no	—
141	C₂²×C₄₇	#380	1	1	yes	characteristic
142	C₂²×C₄₇	#387	1	2	yes	—
143	C₂²×C₄₇	#394	1	1	yes	characteristic
144	C₂²×C₄₇	#398	1	1	yes	characteristic
145	C₂²×C₄₇	#398	1	2	yes	—
146	C₄×C₄₇	—	4	4	no	—
147	C₄×C₄₇	#386	1	1	yes	characteristic
148	C₄×C₄₇	#389	1	2	yes	—
149	C₄×C₄₇	#391	1	1	yes	characteristic
150	C₄×C₄₇	#395	1	1	yes	characteristic
151	C₄×C₄₇	#396	1	2	yes	—
152	C₄×C₄₇	#400	1	1	yes	characteristic
153	Dic₄₇	—	8	4	no	—
154	C₂×#427	—	47	1	no	—
155	C₂×#424	D₄₇	1	1	yes	characteristic
156	C₂×#344	—	47	2	no	—
157	C₂×#341	—	47	1	no	—
158	#1733	—	47	2	no	—
159	C₂×C₄₇	—	2	1	no	—
160	C₂×C₄₇	—	2	2	no	—
161	C₂×C₄₇	—	2	1	no	—
162	C₂×C₄₇	—	2	2	no	—
163	C₂×C₄₇	#1441	1	1	yes	characteristic
164	C₂×C₄₇	#1814	1	2	yes	—
165	C₂²×M₁₆	—	47	2	no	—
166	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
167	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
168	C₂×#117	—	47	2	no	—
169	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
170	#427	—	94	4	no	—
171	C₂×#107	—	94	1	no	—
172	C₂×#106	—	94	1	no	—
173	#424	—	2	1	no	—
174	#424	—	2	1	no	—
175	#424	Dic₄₇	1	4	yes	complemented
176	C₂×C₄⋊C₈	—	2	2	no	—
177	#344	—	94	4	no	—
178	#341	—	94	2	no	—
179	#339	—	94	4	no	—
180	C₄₇	#11241	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	47	4	no	—
186	C₂×M₁₆	—	94	2	no	—
187	C₂×M₁₆	—	94	2	no	—
188	C₂×M₁₆	—	94	2	no	—
189	C₂×M₁₆	—	94	2	no	—
190	C₂×M₁₆	—	94	2	no	—
191	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	1	no	—
195	C₂²×C₈	—	94	2	no	—
196	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
197	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
198	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
199	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
200	#117	—	2	1	no	—
201	#117	—	2	1	no	—
202	#117	—	47	4	no	—
203	#117	—	94	2	no	—
204	#117	—	94	2	no	—
205	#117	—	94	2	no	—
206	#117	D₁₈₈	1	2	yes	—
207	#117	C₂×Dic₄₇	1	2	yes	—
208	C₂×C₄⋊C₄	—	94	2	no	—
209	C₂×C₄⋊C₄	Dic₉₄	1	2	yes	—
210	C₂×C₂²⋊C₄	—	2	1	no	—
211	C₂×C₂²⋊C₄	—	94	2	no	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	—	94	2	no	—
214	#107	—	94	4	no	—
215	#106	—	94	4	no	—
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	—	2	4	no	—
218	C₂⁴	#10625350	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	188	2	no	—
221	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
223	C₂²×C₄	#10625350	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₄₇	1	2	yes	—
225	C₂²×C₄	#10625352	1	2	yes	—
226	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
227	C₂²×C₄	#10625354	1	2	yes	—
228	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
231	M₁₆	—	2	4	no	—
232	M₁₆	—	4	1	no	—
233	M₁₆	—	4	1	no	—
234	M₁₆	—	94	4	no	—
235	M₁₆	—	94	4	no	—
236	M₁₆	—	188	2	no	—
237	M₁₆	—	188	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	94	4	no	—
247	C₂×C₈	—	188	2	no	—
248	C₂×C₈	—	188	2	no	—
249	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
250	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	94	4	no	—
253	C₄⋊C₄	—	188	2	no	—
254	C₄⋊C₄	C₄⋊Dic₄₇	1	4	yes	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	2	2	no	—
257	C₂²⋊C₄	—	94	4	no	—
258	C₂²⋊C₄	—	188	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	94	4	no	—
262	C₄²	—	188	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	#13093983	1	1	yes	characteristic
265	C₂³	#13093998	1	1	yes	characteristic
266	C₂³	#13094003	1	1	yes	characteristic
267	C₂³	#13094009	1	2	yes	—
268	C₂³	#13094014	1	2	yes	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	376	4	no	—
272	C₂×C₄	—	188	4	no	—
273	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
275	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
276	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
277	C₂×C₄	#13093992	1	1	yes	characteristic
278	C₂×C₄	#13093996	1	2	yes	—
279	C₂×C₄	#13093998	1	1	yes	characteristic
280	C₂×C₄	#13093999	1	2	yes	—
281	C₂×C₄	#13094000	1	2	yes	—
282	C₂×C₄	#13094002	1	2	yes	—
283	C₂×C₄	#13094003	1	1	yes	characteristic
284	C₂×C₄	#13094004	1	2	yes	—
285	C₂×C₄	#13094005	1	1	yes	characteristic
286	C₂×C₄	#13094013	1	1	yes	characteristic
287	C₂×C₄	#13094015	1	2	yes	—
288	C₂×C₄	#13094019	1	2	yes	—
289	C₂×C₄	#13094022	1	1	yes	characteristic
290	C₂×C₄	#13094023	1	2	yes	—
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	188	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	#15688983	1	1	yes	characteristic
301	C₂²	#15689009	1	2	yes	—
302	C₂²	#15689040	1	1	yes	characteristic
303	C₂²	#15689053	1	1	yes	characteristic
304	C₂²	#15689056	1	2	yes	—
305	C₄	—	4	4	no	—
306	C₄	—	376	4	no	—
307	C₄	#15689003	1	1	yes	characteristic
308	C₄	#15689022	1	2	yes	—
309	C₄	#15689030	1	1	yes	characteristic
310	C₄	#15689043	1	1	yes	characteristic
311	C₄	#15689045	1	2	yes	—
312	C₄	#15689062	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	#20241641	1	1	yes	characteristic
318	C₂	#20243442	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093998
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#424	C₂
socle	C₂²×C₄₇	#380
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#424	D₄₇
hypocenter	C₄₇	#11241
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
19. C₄₇×#424	C₄	semidirect
175. #424	Dic₄₇	semidirect
180. C₄₇	#11241	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689161	2
C₂×#15688895	2
C₂×#15688888	2
#20243097	2
C₂×C₄₇×#424	2
#11241	47