Subgroups — #11790501

Order 1152.

445 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63646	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#818	C₂	1	1	yes	maximal, characteristic
3	C₃×#5101	C₂	1	1	yes	maximal, characteristic
4	#10591139	C₂	1	2	yes	maximal
5	#10589167	C₂	1	2	yes	maximal
6	#70625	—	3	1	no	maximal
7	#69894	—	3	1	no	maximal
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3748	C₂²	1	2	yes	—
10	C₃×#926	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×#208	C₂²	1	1	yes	characteristic
13	#63646	C₄	1	4	yes	complemented
14	C₃×#818	—	2	1	no	—
15	C₃×#818	—	2	1	no	—
16	C₃×#815	—	2	1	no	—
17	C₃×C₈×Dic₃	—	2	1	no	—
18	#63576	—	2	2	no	—
19	#63574	—	2	2	no	—
20	C₂×C₃×#107	—	3	1	no	—
21	C₃×#405	—	3	1	no	—
22	C₂×#819	—	3	1	no	—
23	C₂×#819	—	3	1	no	—
24	C₂×#819	—	6	1	no	—
25	C₂×#818	D₃	1	1	yes	characteristic
26	#5101	D₃	1	1	yes	characteristic
27	#4995	—	3	2	no	—
28	#4964	—	3	2	no	—
29	#4554	—	3	2	no	—
30	#4530	—	3	2	no	—
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃×Dic₃	—	2	2	no	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
35	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
36	C₂×C₃²×C₈	Q₈	1	2	yes	—
37	#3748	—	2	2	no	—
38	#3748	C₂×C₄	1	4	yes	—
39	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
40	C₃×#217	—	2	1	no	—
41	C₃×C₄⋊Dic₃	—	2	2	no	—
42	C₃×C₄⋊Dic₃	—	2	2	no	—
43	C₃×C₄⋊Dic₃	D₄	1	1	yes	characteristic
44	C₃×C₄⋊Dic₃	D₄	1	1	yes	characteristic
45	C₃×C₄×Dic₃	—	2	1	no	—
46	C₃×#208	—	2	1	no	—
47	C₃×#208	Q₈	1	2	yes	—
48	C₂×C₃×C₃⋊C₈	—	2	1	no	—
49	#1952	—	9	1	no	Sylow
50	C₂×C₃×M₁₆	—	3	1	no	—
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
54	C₃×#117	—	3	1	no	—
55	C₂×C₃×C₄⋊C₄	—	3	2	no	—
56	#926	C₂×D₃	1	1	yes	characteristic
57	C₂×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
58	C₂×#208	—	3	2	no	—
59	C₂×#208	—	3	2	no	—
60	C₂×#208	—	6	2	no	—
61	C₂×#208	C₂×D₃	1	1	yes	characteristic
62	C₃×#107	—	6	1	no	—
63	C₃×#107	—	6	1	no	—
64	C₃×#97	—	6	1	no	—
65	C₃×C₄×C₈	—	6	1	no	—
66	#819	—	3	4	no	—
67	#819	—	3	4	no	—
68	#819	—	6	4	no	—
69	#818	—	2	1	no	—
70	#818	—	2	1	no	—
71	#815	—	2	1	no	—
72	C₈×Dic₃	—	2	1	no	—
73	#808	—	6	2	no	—
74	#808	—	6	2	no	—
75	#807	—	6	2	no	—
76	#807	—	6	2	no	—
77	C₂³×C₃²	#41	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₃²×C₄	#41	1	1	yes	characteristic
84	C₂×C₃×Dic₃	—	2	1	no	—
85	C₂×C₃×Dic₃	—	2	1	no	—
86	C₂×C₃×Dic₃	—	4	2	no	—
87	C₂×C₃×Dic₃	—	8	2	no	—
88	C₃²×C₈	C₄⋊C₄	1	2	yes	—
89	C₃²×C₈	C₄⋊C₄	1	2	yes	—
90	C₃⋊(C₃⋊C₈)	—	2	4	no	—
91	C₃×C₃⋊C₈	—	2	2	no	—
92	C₂×#108	—	9	1	no	—
93	#428	—	9	2	no	—
94	C₂×#107	—	3	1	no	—
95	#405	—	3	1	no	—
96	#343	—	9	2	no	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂²×C₃×C₄	—	6	2	no	—
99	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂²×Dic₃	—	2	2	no	—
102	C₃×M₁₆	—	3	2	no	—
103	C₃×M₁₆	—	6	1	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	2	2	no	—
108	C₂×C₃×C₈	—	6	1	no	—
109	C₂×C₃×C₈	Dic₆	1	2	yes	—
110	C₂×C₃×C₈	Dic₆	1	2	yes	—
111	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
112	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
113	C₂×C₃×C₈	#67	1	1	yes	characteristic
114	C₂×C₃×C₈	#67	1	1	yes	characteristic
115	C₂×C₃×C₈	#67	1	1	yes	characteristic
116	C₂×C₃×C₈	#67	1	1	yes	characteristic
117	C₃×C₄⋊C₄	—	3	1	no	—
118	C₃×C₄⋊C₄	—	3	1	no	—
119	C₃×C₄⋊C₄	—	6	2	no	—
120	C₃×C₄⋊C₄	—	6	2	no	—
121	C₃×C₂²⋊C₄	—	6	1	no	—
122	C₃×C₄²	—	6	1	no	—
123	#217	—	2	1	no	—
124	C₄⋊Dic₃	—	2	2	no	—
125	C₄⋊Dic₃	—	2	2	no	—
126	C₄⋊Dic₃	D₁₂	1	1	yes	characteristic
127	C₄⋊Dic₃	D₁₂	1	1	yes	characteristic
128	C₄×Dic₃	—	2	1	no	—
129	#208	—	2	1	no	—
130	#208	—	3	4	no	—
131	#208	—	3	4	no	—
132	#208	—	6	4	no	—
133	#208	—	6	2	no	—
134	#208	—	6	2	no	—
135	#208	—	12	2	no	—
136	#208	Dic₆	1	2	yes	—
137	C₂×C₃⋊C₈	—	2	1	no	—
138	C₂×C₃⋊C₈	—	6	2	no	—
139	C₂×C₃⋊C₈	—	6	2	no	—
140	C₂×C₃⋊C₈	—	12	2	no	—
141	C₂²×C₃²	—	2	1	no	—
142	C₂²×C₃²	—	2	1	no	—
143	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
144	C₂²×C₃²	#121	1	1	yes	characteristic
145	C₂²×C₃²	#125	1	1	yes	characteristic
146	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
148	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
149	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
150	C₃×Dic₃	—	4	1	no	—
151	C₃×Dic₃	—	4	1	no	—
152	C₃×Dic₃	—	8	2	no	—
153	C₂×M₁₆	—	3	1	no	—
154	C₂×M₁₆	—	9	2	no	—
155	C₂²×C₈	D₃²	1	1	yes	characteristic
156	#117	—	3	1	no	—
157	C₂×C₄⋊C₄	—	3	2	no	—
158	#108	—	9	4	no	—
159	#107	—	6	1	no	—
160	#107	—	6	1	no	—
161	#107	—	18	2	no	—
162	#106	—	18	2	no	—
163	#97	—	6	1	no	—
164	C₄×C₈	—	6	1	no	—
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	#235	1	1	yes	characteristic
167	C₂³×C₃	#237	1	1	yes	characteristic
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	6	1	no	—
175	C₂×C₃×C₄	—	6	1	no	—
176	C₂×C₃×C₄	—	12	2	no	—
177	C₂×C₃×C₄	—	24	2	no	—
178	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
184	C₂×C₃×C₄	#235	1	1	yes	characteristic
185	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
187	C₂×C₃×C₄	#239	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
190	C₂×Dic₃	—	2	1	no	—
191	C₂×Dic₃	—	2	1	no	—
192	C₂×Dic₃	—	4	2	no	—
193	C₂×Dic₃	—	8	2	no	—
194	C₃×C₈	—	2	2	no	—
195	C₃×C₈	—	2	2	no	—
196	C₃×C₈	—	6	2	no	—
197	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
198	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
199	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
200	C₃×C₈	Dic₃⋊C₄	1	2	yes	—
201	C₃⋊C₈	—	2	2	no	—
202	C₃⋊C₈	—	6	4	no	—
203	C₃⋊C₈	—	6	4	no	—
204	C₃⋊C₈	—	12	4	no	—
205	C₂×C₃²	—	2	1	no	—
206	C₂×C₃²	—	2	1	no	—
207	C₂×C₃²	#389	1	1	yes	characteristic
208	C₂×C₃²	#472	1	1	yes	characteristic
209	C₂×C₃²	#473	1	1	yes	characteristic
210	C₂²×C₄	—	6	2	no	—
211	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
212	M₁₆	—	3	2	no	—
213	M₁₆	—	6	1	no	—
214	M₁₆	—	9	4	no	—
215	M₁₆	—	18	2	no	—
216	C₂×C₈	—	6	1	no	—
217	C₂×C₈	—	18	2	no	—
218	C₂×C₈	#624	1	2	yes	—
219	C₂×C₈	#625	1	1	yes	characteristic
220	C₂×C₈	#625	1	1	yes	characteristic
221	C₂×C₈	#627	1	2	yes	—
222	C₄⋊C₄	—	3	1	no	—
223	C₄⋊C₄	—	3	1	no	—
224	C₄⋊C₄	—	6	2	no	—
225	C₄⋊C₄	—	6	2	no	—
226	C₂²⋊C₄	—	6	1	no	—
227	C₄²	—	6	1	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	4	1	no	—
236	C₂²×C₃	—	4	1	no	—
237	C₂²×C₃	C₄×Dic₆	1	1	yes	characteristic
238	C₂²×C₃	#870	1	1	yes	characteristic
239	C₂²×C₃	#887	1	1	yes	characteristic
240	C₂²×C₃	#890	1	1	yes	characteristic
241	C₂²×C₃	#930	1	1	yes	characteristic
242	C₂²×C₃	#938	1	1	yes	characteristic
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	12	1	no	—
248	C₃×C₄	—	12	1	no	—
249	C₃×C₄	—	24	2	no	—
250	C₃×C₄	#869	1	1	yes	characteristic
251	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
252	C₃×C₄	#888	1	1	yes	characteristic
253	C₃×C₄	#893	1	1	yes	characteristic
254	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
255	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
256	C₃×C₄	#924	1	1	yes	characteristic
257	C₃×C₄	#946	1	1	yes	characteristic
258	Dic₃	—	4	1	no	—
259	Dic₃	—	4	1	no	—
260	Dic₃	—	8	2	no	—
261	C₃²	#1952	1	1	yes	characteristic, complemented, Sylow
262	C₂³	#3795	1	1	yes	characteristic
263	C₂×C₄	—	6	1	no	—
264	C₂×C₄	—	6	1	no	—
265	C₂×C₄	—	12	2	no	—
266	C₂×C₄	—	24	2	no	—
267	C₂×C₄	D₃×Dic₆	1	1	yes	characteristic
268	C₂×C₄	#3799	1	1	yes	characteristic
269	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
270	C₂×C₄	C₂×#624	1	1	yes	characteristic
271	C₂×C₄	C₂×#625	1	1	yes	characteristic
272	C₂×C₄	C₂×#627	1	1	yes	characteristic
273	C₈	—	6	2	no	—
274	C₈	—	18	4	no	—
275	C₈	#3724	1	2	yes	—
276	C₈	#3724	1	2	yes	—
277	C₂×C₃	—	2	1	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	—	2	1	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	2	1	no	—
282	C₂×C₃	—	2	1	no	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	4	1	no	—
285	C₂×C₃	—	4	1	no	—
286	C₂×C₃	#4928	1	1	yes	characteristic
287	C₂×C₃	#4972	1	1	yes	characteristic
288	C₂×C₃	#5142	1	1	yes	characteristic
289	C₂×C₃	#5143	1	1	yes	characteristic
290	C₂×C₃	#5160	1	1	yes	characteristic
291	C₂×C₃	#5192	1	1	yes	characteristic
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	#63844	1	1	yes	characteristic
295	C₂²	#63869	1	1	yes	characteristic
296	C₂²	#63908	1	1	yes	characteristic
297	C₄	—	12	1	no	—
298	C₄	—	12	1	no	—
299	C₄	—	24	2	no	—
300	C₄	#63856	1	1	yes	characteristic
301	C₄	#63894	1	1	yes	characteristic
302	C₄	#63918	1	1	yes	characteristic
303	C₄	C₂×#3724	1	1	yes	characteristic
304	C₃	—	2	1	no	—
305	C₃	#69894	1	1	yes	characteristic, complemented
306	C₃	#70625	1	1	yes	characteristic, complemented
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	#10590926	1	1	yes	characteristic
310	C₂	#10590930	1	1	yes	characteristic
311	C₂	#10591212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#63908
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	#121
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1952
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
13. #63646	C₄	semidirect
261. C₃²	#1952	semidirect
305. C₃	#69894	semidirect
306. C₃	#70625	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63646	2
C₂×C₃×#818	2
C₃×#5101	2
#10591139	2
#10589167	2
#70625	3
#69894	3