Subgroups — #16807132

Order 3328.

710 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719162	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131217	C₂	1	4	yes	maximal, complemented
4	#13131214	C₂	1	4	yes	maximal, complemented
5	#13131173	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131066	C₂	1	1	yes	maximal, characteristic, complemented
7	#13131065	C₂	1	1	yes	maximal, characteristic, complemented
8	#13131047	C₂	1	1	yes	maximal, characteristic, complemented
9	#13123533	C₂	1	1	yes	maximal, characteristic
10	#11719533	C₄	1	1	yes	characteristic
11	C₂×#88202	C₂²	1	1	yes	characteristic
12	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
13	#11719162	—	2	4	no	—
14	C₂×#88106	C₂²	1	1	yes	characteristic
15	C₂×#88106	C₂²	1	1	yes	characteristic
16	C₂×#88106	C₂²	1	2	yes	complemented
17	C₂²×#6092	C₂²	1	1	yes	characteristic
18	C₂²×#6092	C₂²	1	1	yes	characteristic
19	C₂²×#6092	C₂²	1	2	yes	complemented
20	C₂×D₁₃⋊C₁₆	—	2	4	no	—
21	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
22	C₂×D₁₃⋊C₁₆	C₂²	1	2	yes	—
23	C₁₃×#408	C₄	1	1	yes	characteristic
24	#11718787	C₄	1	1	yes	characteristic
25	#11718783	C₂²	1	1	yes	characteristic
26	#11718763	C₄	1	1	yes	characteristic
27	#11718389	C₄	1	4	yes	—
28	#11718386	C₂²	1	4	yes	—
29	C₂²⋊#6092	C₂²	1	4	yes	complemented
30	#11718260	C₂²	1	4	yes	complemented
31	#11718259	C₂²	1	4	yes	—
32	#11718258	C₂²	1	2	yes	complemented
33	#11718257	C₂²	1	2	yes	complemented
34	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	#88202	—	2	2	no	—
39	#88202	—	2	2	no	—
40	#88200	C₈	1	2	yes	—
41	#88192	C₈	1	2	yes	—
42	#88191	C₈	1	2	yes	—
43	#88189	C₈	1	2	yes	—
44	C₄×C₂²⋊D₁₃	C₂×C₄	1	4	yes	—
45	#88180	C₂×C₄	1	1	yes	characteristic
46	C₂×#6099	C₂×C₄	1	1	yes	characteristic
47	C₂×#6099	C₂×C₄	1	2	yes	—
48	C₂²×#1055	C₂³	1	2	yes	—
49	C₂²×#1055	C₂³	1	1	yes	characteristic
50	C₂×C₈×D₁₃	—	2	2	no	—
51	C₂×C₈×D₁₃	—	2	2	no	—
52	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₁₃	C₂³	1	2	yes	—
54	#88106	—	2	2	no	—
55	#88106	—	2	2	no	—
56	#88106	—	2	4	no	—
57	C₂×#6092	—	2	2	no	—
58	C₂×#6092	—	2	4	no	—
59	C₂×#6092	D₄	1	4	yes	complemented
60	C₂×#6092	C₂³	1	2	yes	—
61	C₂×#6092	C₂³	1	2	yes	—
62	C₂×#6092	C₂³	1	2	yes	—
63	C₂×#6092	C₂³	1	1	yes	characteristic
64	C₂×#6092	C₂³	1	1	yes	characteristic
65	D₁₃⋊C₁₆	—	2	4	no	—
66	D₁₃⋊C₁₆	—	2	4	no	—
67	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
68	#88060	C₂×C₄	1	4	yes	—
69	#88055	C₂×C₄	1	2	yes	—
70	#88054	C₂×C₄	1	4	yes	—
71	C₈×Dic₁₃	C₂³	1	2	yes	—
72	#33432	—	13	1	no	maximal, Sylow
73	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
74	C₁₃×#41	—	2	2	no	—
75	C₁₃×#41	—	2	2	no	—
76	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
77	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
78	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₁₃	C₂×C₈	1	4	yes	—
82	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₁₃	—	2	2	no	—
84	C₂×C₄×D₁₃	—	2	2	no	—
85	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
87	C₁₃×M₁₆	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
92	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
93	#6103	C₂×C₈	1	4	yes	—
94	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
95	#6101	C₂×C₈	1	4	yes	—
96	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
97	#6099	—	2	2	no	—
98	#6099	—	2	4	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	2	2	no	—
101	C₂×#1055	C₂²×C₄	1	2	yes	—
102	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
103	C₂×#1055	C₂×D₄	1	2	yes	—
104	C₂×#1055	#41	1	2	yes	—
105	C₂×#1055	C₂⁴	1	1	yes	characteristic
106	C₈×D₁₃	—	2	4	no	—
107	C₈×D₁₃	—	2	2	no	—
108	C₈×D₁₃	—	2	2	no	—
109	#6092	—	2	2	no	—
110	#6092	—	2	2	no	—
111	#6092	—	2	2	no	—
112	#6092	C₂×D₄	1	2	yes	—
113	#6092	C₂×D₄	1	2	yes	—
114	C₂×#504	—	13	1	no	—
115	C₂³×C₁₆	—	13	1	no	—
116	#2929	—	13	1	no	—
117	#2168	—	13	4	no	—
118	D₄×C₁₆	—	13	4	no	—
119	#2151	—	13	1	no	—
120	#2113	—	13	1	no	—
121	#2112	—	13	1	no	—
122	#2108	—	13	1	no	—
123	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
124	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
125	C₂²×D₁₃	—	2	2	no	—
126	C₂²×D₁₃	—	2	4	no	—
127	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
128	Q₈×C₁₃	—	2	2	no	—
129	D₄×C₁₃	—	2	4	no	—
130	D₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	#131	1	2	yes	—
137	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
138	C₂²⋊D₁₃	—	2	8	no	—
139	C₂×Dic₁₃	—	2	2	no	—
140	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
141	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
142	C₄×D₁₃	—	2	2	no	—
143	C₄×D₁₃	—	2	2	no	—
144	C₄×D₁₃	—	2	4	no	—
145	C₈×C₁₃	—	2	2	no	—
146	C₈×C₁₃	—	2	2	no	—
147	#1055	—	2	2	no	—
148	#1055	C₄×D₄	1	2	yes	—
149	#1055	C₂²×D₄	1	1	yes	characteristic
150	#1055	C₂×#41	1	1	yes	characteristic
151	C₂×#131	—	13	1	no	—
152	C₂³×C₈	—	13	1	no	—
153	#525	—	13	1	no	—
154	#504	—	26	4	no	—
155	C₂×M₃₂	—	13	1	no	—
156	C₂×M₃₂	—	13	1	no	—
157	C₂×M₃₂	—	13	2	no	—
158	C₂²×C₁₆	—	13	1	no	—
159	C₂²×C₁₆	—	13	1	no	—
160	C₂²×C₁₆	—	13	2	no	—
161	C₂²×C₁₆	—	13	2	no	—
162	C₂²×C₁₆	—	13	2	no	—
163	C₂²×C₁₆	—	26	4	no	—
164	#435	—	13	4	no	—
165	C₈×D₄	—	13	4	no	—
166	#424	—	13	1	no	—
167	#408	#261	1	1	yes	characteristic
168	#407	—	13	1	no	—
169	#405	—	13	1	no	—
170	C₄⋊C₁₆	—	13	4	no	—
171	C₂²⋊C₁₆	—	13	4	no	—
172	C₂²⋊C₁₆	—	13	4	no	—
173	#346	—	13	2	no	—
174	C₄×C₁₆	—	13	2	no	—
175	C₂²×C₁₃	—	2	2	no	—
176	C₂²×C₁₃	—	2	4	no	—
177	C₂²×C₁₃	—	2	2	no	—
178	C₂²×C₁₃	C₈×D₄	1	2	yes	—
179	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
180	C₂×D₁₃	—	2	8	no	—
181	C₂×D₁₃	—	2	2	no	—
182	C₂×D₁₃	—	2	2	no	—
183	C₂×D₁₃	#504	1	4	yes	complemented
184	C₄×C₁₃	—	2	2	no	—
185	C₄×C₁₃	C₈×D₄	1	2	yes	—
186	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
188	Dic₁₃	—	2	4	no	—
189	Dic₁₃	—	2	2	no	—
190	C₂×#41	C₂×#261	1	1	yes	characteristic
191	C₂³×C₄	—	13	1	no	—
192	#131	—	26	2	no	—
193	#131	—	26	2	no	—
194	C₂×M₁₆	—	13	1	no	—
195	C₂×M₁₆	—	13	2	no	—
196	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
197	C₂²×C₈	—	13	1	no	—
198	C₂²×C₈	—	13	2	no	—
199	C₂²×C₈	—	13	2	no	—
200	C₂²×C₈	—	13	2	no	—
201	C₂²×C₈	—	26	2	no	—
202	C₂²×C₈	—	26	2	no	—
203	C₂²×C₈	C₂×#261	1	1	yes	characteristic
204	#123	—	13	2	no	—
205	C₂²⋊Q₈	—	13	2	no	—
206	#121	—	13	2	no	—
207	#120	—	13	2	no	—
208	C₄×D₄	—	13	4	no	—
209	#117	—	13	1	no	—
210	M₃₂	—	26	4	no	—
211	M₃₂	—	26	2	no	—
212	M₃₂	—	26	2	no	—
213	C₂×C₁₆	—	13	4	no	—
214	C₂×C₁₆	—	13	1	no	—
215	C₂×C₁₆	—	13	1	no	—
216	C₂×C₁₆	—	13	2	no	—
217	C₂×C₁₆	—	13	2	no	—
218	C₂×C₁₆	—	13	2	no	—
219	C₂×C₁₆	—	26	4	no	—
220	C₂×C₁₆	—	26	4	no	—
221	C₂×C₁₆	—	26	4	no	—
222	C₂×C₁₆	—	26	2	no	—
223	C₄⋊C₈	—	13	4	no	—
224	C₂²⋊C₈	—	13	4	no	—
225	C₂²⋊C₈	C₂×#261	1	4	yes	—
226	#97	—	13	2	no	—
227	C₄×C₈	—	13	2	no	—
228	C₂×C₁₃	—	2	2	no	—
229	C₂×C₁₃	—	4	2	no	—
230	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
231	C₂×C₁₃	C₂×#504	1	2	yes	—
232	D₁₃	—	2	4	no	—
233	C₂⁴	—	13	1	no	—
234	#41	—	2	2	no	—
235	#41	—	2	2	no	—
236	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
237	C₂×D₄	—	13	4	no	—
238	C₂×D₄	D₁₃⋊C₈	1	2	yes	—
239	C₂×D₄	D₁₃⋊C₈	1	1	yes	characteristic
240	C₂²×C₄	—	13	1	no	—
241	C₂²×C₄	—	13	2	no	—
242	C₂²×C₄	—	13	2	no	—
243	C₂²×C₄	—	26	2	no	—
244	C₂²×C₄	—	26	2	no	—
245	C₂²×C₄	C₂²×#261	1	2	yes	—
246	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
247	M₁₆	—	2	2	no	—
248	M₁₆	—	26	4	no	—
249	M₁₆	—	26	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	13	2	no	—
252	C₂×C₈	—	13	2	no	—
253	C₂×C₈	—	13	1	no	—
254	C₂×C₈	—	13	1	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	—	26	2	no	—
258	C₂×C₈	—	26	4	no	—
259	C₂×C₈	—	26	2	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	C₂²×#261	1	2	yes	—
262	C₂×C₈	C₂²×#261	1	2	yes	—
263	C₄⋊C₄	—	13	2	no	—
264	C₄⋊C₄	—	13	4	no	—
265	C₂²⋊C₄	—	13	2	no	—
266	C₂²⋊C₄	—	13	4	no	—
267	C₂²⋊C₄	—	13	4	no	—
268	C₄²	—	13	2	no	—
269	C₁₆	—	13	2	no	—
270	C₁₆	—	13	2	no	—
271	C₁₆	—	26	2	no	—
272	C₁₆	—	26	2	no	—
273	C₁₆	—	26	2	no	—
274	C₁₃	#33432	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	13	2	no	—
276	C₂³	—	26	4	no	—
277	C₂³	—	26	2	no	—
278	C₂³	C₂×D₁₃⋊C₈	1	2	yes	—
279	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	D₄	—	2	4	no	—
282	D₄	—	2	2	no	—
283	D₄	—	26	8	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	13	4	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	26	2	no	—
289	C₂×C₄	—	26	4	no	—
290	C₂×C₄	—	26	2	no	—
291	C₂×C₄	—	26	2	no	—
292	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
293	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
294	C₂×C₄	#88238	1	2	yes	—
295	C₂×C₄	D₄×#261	1	2	yes	—
296	C₂×C₄	C₂³×#261	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	—	13	1	no	—
300	C₈	—	13	1	no	—
301	C₈	—	13	2	no	—
302	C₈	—	26	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	13	4	no	—
307	C₂²	—	26	8	no	—
308	C₂²	—	26	2	no	—
309	C₂²	—	26	2	no	—
310	C₂²	#11719063	1	2	yes	—
311	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	26	2	no	—
314	C₄	—	26	4	no	—
315	C₄	#11719063	1	2	yes	—
316	C₄	C₂×#88238	1	1	yes	characteristic
317	C₄	C₂×D₄×#261	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	26	4	no	—
321	C₂	C₂×#11719063	1	1	yes	characteristic
322	C₂	D₁₃⋊#504	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#408	C₄
prime radical	#11719533	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#408	#261
hypocenter	C₁₃	#33432
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#11719162	C₂	semidirect
2. C₂²×D₁₃⋊C₁₆	C₂	semidirect
3. #13131217	C₂	semidirect
4. #13131214	C₂	semidirect
5. #13131173	C₂	semidirect
6. #13131066	C₂	semidirect
7. #13131065	C₂	semidirect
8. #13131047	C₂	semidirect
16. C₂×#88106	C₂²	semidirect
19. C₂²×#6092	C₂²	semidirect
29. C₂²⋊#6092	C₂²	semidirect
30. #11718260	C₂²	semidirect
32. #11718258	C₂²	semidirect
33. #11718257	C₂²	semidirect
59. C₂×#6092	D₄	semidirect
183. C₂×D₁₃	#504	semidirect
274. C₁₃	#33432	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719162	2
C₂²×D₁₃⋊C₁₆	2
#13131217	2
#13131214	2
#13131173	2
#13131066	2
#13131065	2
#13131047	2
#13123533	2
#33432	13