Subgroups — #16807287

Order 3328.

508 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	#13131253	C₂	1	2	yes	maximal, complemented
2	#13131221	C₂	1	2	yes	maximal, complemented
3	#13131219	C₂	1	2	yes	maximal, complemented
4	#13131217	C₂	1	2	yes	maximal, complemented
5	#13131203	C₂	1	2	yes	maximal, complemented
6	#13131197	C₂	1	1	yes	maximal, characteristic, complemented
7	#13131046	C₂	1	2	yes	maximal, complemented
8	C₄×#88106	C₂	1	1	yes	maximal, characteristic, complemented
9	#13123805	C₂	1	1	yes	maximal, characteristic
10	#11719184	C₄	1	1	yes	characteristic
11	C₂×#88106	C₂²	1	2	yes	—
12	C₂×#88098	C₂²	1	2	yes	—
13	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
14	C₁₃×#432	C₄	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	#11718499	C₄	1	1	yes	characteristic
17	#11718495	C₂²	1	1	yes	characteristic
18	#11718492	C₄	1	1	yes	characteristic
19	#11718491	C₂²	1	1	yes	characteristic
20	#11718391	C₄	1	2	yes	—
21	#11718386	C₂²	1	2	yes	—
22	#11718370	C₄	1	2	yes	—
23	#11718369	C₂²	1	2	yes	—
24	C₂²⋊#6092	C₂²	1	1	yes	characteristic
25	C₂²⋊#6092	C₂²	1	1	yes	characteristic
26	#11718260	C₂²	1	2	yes	—
27	#11718260	C₂²	1	2	yes	—
28	#11718259	C₂²	1	2	yes	—
29	#11718259	C₂²	1	2	yes	—
30	#11718258	C₂²	1	2	yes	—
31	#11718257	C₂²	1	2	yes	—
32	#11718231	C₂²	1	2	yes	—
33	#11718231	C₂²	1	2	yes	—
34	C₄⋊#6092	C₂²	1	2	yes	—
35	C₄×#6092	C₂²	1	1	yes	characteristic
36	C₄×#6092	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	#88184	C₈	1	1	yes	characteristic
39	#88183	C₈	1	2	yes	—
40	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
41	#88178	C₈	1	1	yes	characteristic
42	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
43	C₂²×#1055	C₂³	1	1	yes	characteristic
44	C₂×#6094	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₃	C₂³	1	2	yes	—
46	#88130	C₈	1	2	yes	—
47	#88129	C₈	1	1	yes	characteristic
48	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
49	#88126	C₂×C₄	1	2	yes	—
50	#88124	C₈	1	1	yes	characteristic
51	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
52	#88106	—	2	4	no	—
53	C₂×#6092	C₂³	1	2	yes	—
54	C₂×#6092	C₂³	1	2	yes	—
55	C₂×#6092	C₂³	1	2	yes	—
56	C₂×#6092	C₂³	1	2	yes	—
57	#88098	—	2	4	no	—
58	D₁₃⋊C₁₆	—	2	4	no	—
59	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
62	C₁₃×#97	C₂×C₄	1	2	yes	—
63	#88060	C₂×C₄	1	2	yes	—
64	#88060	C₂×C₄	1	2	yes	—
65	#88055	C₂×C₄	1	2	yes	—
66	#88054	C₂×C₄	1	2	yes	—
67	#88054	C₂×C₄	1	2	yes	—
68	C₈×Dic₁₃	C₂³	1	2	yes	—
69	C₄×#1055	C₂×C₄	1	1	yes	characteristic
70	C₄×#1055	C₂×C₄	1	1	yes	characteristic
71	C₄×#1055	C₂³	1	1	yes	characteristic
72	C₄×#1055	C₂³	1	1	yes	characteristic
73	#33481	—	13	1	no	maximal, Sylow
74	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
78	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
80	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
88	#6103	C₂×C₈	1	2	yes	—
89	#6103	C₂×C₈	1	2	yes	—
90	#6103	C₂×C₈	1	2	yes	—
91	#6103	C₂×C₈	1	2	yes	—
92	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
93	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
94	#6101	C₂×C₈	1	2	yes	—
95	#6101	C₂×C₈	1	2	yes	—
96	#6101	C₂×C₈	1	2	yes	—
97	#6101	C₂×C₈	1	2	yes	—
98	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	2	2	no	—
101	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
102	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
103	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
104	C₂×#1055	C₂⁴	1	1	yes	characteristic
105	#6094	—	2	4	no	—
106	C₈×D₁₃	—	2	2	no	—
107	C₈×D₁₃	—	2	2	no	—
108	#6092	—	2	2	no	—
109	#6092	—	2	2	no	—
110	#6092	—	2	2	no	—
111	#6092	#41	1	4	yes	—
112	#2966	—	13	1	no	—
113	#2183	—	13	2	no	—
114	#2169	—	13	2	no	—
115	#2168	—	13	2	no	—
116	#2168	—	13	2	no	—
117	#2164	—	13	2	no	—
118	#2163	—	13	1	no	—
119	#2108	—	13	2	no	—
120	C₄×M₃₂	—	13	1	no	—
121	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
122	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	4	1	no	—
126	C₂×C₄×C₁₃	—	4	1	no	—
127	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
133	C₂²⋊D₁₃	—	4	2	no	—
134	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
135	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
136	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
137	D₅₂	—	4	1	no	—
138	C₄×D₁₃	—	2	2	no	—
139	C₄×D₁₃	—	2	2	no	—
140	Dic₂₆	—	4	1	no	—
141	C₈×C₁₃	—	2	2	no	—
142	C₈×C₁₃	—	2	2	no	—
143	#1055	—	2	1	no	—
144	#1055	—	2	1	no	—
145	#1055	#131	1	2	yes	—
146	#1055	C₂×#41	1	2	yes	—
147	#529	—	13	1	no	—
148	C₂×M₃₂	—	13	2	no	—
149	C₂×M₃₂	—	13	2	no	—
150	C₂²×C₁₆	—	13	2	no	—
151	#446	—	13	1	no	—
152	C₈×Q₈	—	13	1	no	—
153	#436	—	13	1	no	—
154	#435	—	13	2	no	—
155	C₈×D₄	—	13	1	no	—
156	C₈×D₄	—	13	2	no	—
157	#433	—	13	2	no	—
158	#432	#261	1	1	yes	characteristic
159	#405	—	13	2	no	—
160	C₂×C₄×C₈	—	13	1	no	—
161	C₄⋊C₁₆	—	13	2	no	—
162	C₄⋊C₁₆	—	13	2	no	—
163	C₄⋊C₁₆	—	13	2	no	—
164	C₂²⋊C₁₆	—	13	1	no	—
165	C₂²⋊C₁₆	—	13	1	no	—
166	C₂²⋊C₁₆	—	13	2	no	—
167	C₂²⋊C₁₆	—	13	2	no	—
168	#346	—	13	2	no	—
169	#346	—	13	2	no	—
170	#346	—	13	2	no	—
171	C₄×C₁₆	—	13	1	no	—
172	C₄×C₁₆	—	13	1	no	—
173	C₄×C₁₆	—	13	2	no	—
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
178	C₂×D₁₃	—	2	2	no	—
179	C₂×D₁₃	—	2	2	no	—
180	C₂×D₁₃	—	2	2	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
187	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
188	Dic₁₃	—	2	2	no	—
189	Dic₁₃	—	2	2	no	—
190	Dic₁₃	—	2	2	no	—
191	C₂×M₁₆	—	13	2	no	—
192	C₂²×C₈	—	13	1	no	—
193	C₂²×C₈	—	13	1	no	—
194	C₂²×C₈	—	13	2	no	—
195	#126	—	13	2	no	—
196	#125	—	13	1	no	—
197	#124	—	13	1	no	—
198	#123	—	13	2	no	—
199	C₂²⋊Q₈	—	13	1	no	—
200	#121	—	13	1	no	—
201	C₄×Q₈	—	13	1	no	—
202	C₄×D₄	—	13	2	no	—
203	C₄×D₄	—	13	1	no	—
204	#117	—	13	2	no	—
205	C₂×C₄²	C₂×#261	1	1	yes	characteristic
206	M₃₂	—	26	4	no	—
207	M₃₂	—	26	4	no	—
208	C₂×C₁₆	—	13	2	no	—
209	C₂×C₁₆	—	13	2	no	—
210	C₂×C₁₆	—	13	2	no	—
211	C₂×C₁₆	—	13	2	no	—
212	C₂×C₁₆	—	26	4	no	—
213	C₄⋊C₈	—	13	2	no	—
214	C₄⋊C₈	—	13	2	no	—
215	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
216	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
217	C₂²⋊C₈	—	13	2	no	—
218	C₂²⋊C₈	—	13	2	no	—
219	C₂²⋊C₈	C₂×#261	1	2	yes	—
220	#97	—	13	2	no	—
221	#97	C₂×#261	1	2	yes	—
222	C₄×C₈	—	13	2	no	—
223	C₄×C₈	—	13	1	no	—
224	C₄×C₈	—	13	1	no	—
225	C₄×C₈	—	13	1	no	—
226	C₄×C₈	—	13	1	no	—
227	C₂×C₁₃	—	4	1	no	—
228	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
229	C₂×C₁₃	#3407	1	1	yes	characteristic
230	C₂×C₁₃	#3407	1	1	yes	characteristic
231	D₁₃	—	4	2	no	—
232	C₂×Q₈	—	13	1	no	—
233	C₂×D₄	—	13	2	no	—
234	C₂×D₄	—	13	1	no	—
235	C₂²×C₄	—	13	2	no	—
236	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
237	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
238	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
239	M₁₆	—	26	4	no	—
240	C₂×C₈	—	13	1	no	—
241	C₂×C₈	—	13	1	no	—
242	C₂×C₈	—	13	1	no	—
243	C₂×C₈	—	13	1	no	—
244	C₂×C₈	—	26	2	no	—
245	C₂×C₈	—	26	2	no	—
246	C₂×C₈	—	26	2	no	—
247	C₂×C₈	—	26	2	no	—
248	C₂×C₈	C₂²×#261	1	2	yes	—
249	C₂×C₈	C₂²×#261	1	2	yes	—
250	C₄⋊C₄	—	13	2	no	—
251	C₄⋊C₄	—	13	2	no	—
252	C₄⋊C₄	—	13	1	no	—
253	C₄⋊C₄	—	13	2	no	—
254	C₄⋊C₄	—	13	1	no	—
255	C₄⋊C₄	—	13	2	no	—
256	C₂²⋊C₄	—	13	2	no	—
257	C₂²⋊C₄	—	13	2	no	—
258	C₂²⋊C₄	—	13	2	no	—
259	C₂²⋊C₄	—	13	2	no	—
260	C₂²⋊C₄	—	13	2	no	—
261	C₄²	—	13	2	no	—
262	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
263	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
264	C₄²	C₂²×#261	1	1	yes	characteristic
265	C₄²	C₂²×#261	1	1	yes	characteristic
266	C₁₆	—	13	4	no	—
267	C₁₆	—	26	2	no	—
268	C₁₆	—	26	2	no	—
269	C₁₆	—	26	2	no	—
270	C₁₃	#33481	1	1	yes	characteristic, complemented, Sylow
271	C₂³	—	13	2	no	—
272	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
273	Q₈	—	52	1	no	—
274	D₄	—	52	1	no	—
275	D₄	—	52	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	13	2	no	—
281	C₂×C₄	—	13	2	no	—
282	C₂×C₄	—	13	2	no	—
283	C₂×C₄	—	26	2	no	—
284	C₂×C₄	—	26	2	no	—
285	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
286	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
287	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
288	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
289	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
290	C₂×C₄	C₂³×#261	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	13	2	no	—
294	C₈	—	13	2	no	—
295	C₈	—	26	1	no	—
296	C₈	—	26	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	26	2	no	—
301	C₂²	—	26	2	no	—
302	C₂²	—	26	2	no	—
303	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	26	2	no	—
310	C₄	—	26	2	no	—
311	C₄	—	26	2	no	—
312	C₄	D₁₃⋊#131	1	1	yes	characteristic
313	C₄	#41×#261	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	52	2	no	—
316	C₂	#13131640	1	1	yes	characteristic
317	C₂	#13131805	1	1	yes	characteristic
318	C₂	#13131805	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#432	C₄
prime radical	#88184	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#432	#261
hypocenter	C₁₃	#33481
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
1. #13131253	C₂	semidirect
2. #13131221	C₂	semidirect
3. #13131219	C₂	semidirect
4. #13131217	C₂	semidirect
5. #13131203	C₂	semidirect
6. #13131197	C₂	semidirect
7. #13131046	C₂	semidirect
8. C₄×#88106	C₂	semidirect
270. C₁₃	#33481	semidirect

Maximal subgroups

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

subgroup	index
#13131253	2
#13131221	2
#13131219	2
#13131217	2
#13131203	2
#13131197	2
#13131046	2
C₄×#88106	2
#13123805	2
#33481	13