Subgroups — #16016311

Order 3328.

674 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718573	C₂	1	1	yes	maximal, characteristic
2	#13123706	C₂	1	2	yes	maximal, complemented
3	#13116712	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116708	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116706	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116691	C₂	1	2	yes	maximal, complemented
7	C₄×#88067	C₂	1	2	yes	maximal, complemented
8	D₁₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116545	C₂	1	2	yes	maximal, complemented
10	#13116434	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2177	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88166	C₂²	1	2	yes	—
13	#11719251	—	2	2	no	—
14	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×D₈	C₂²	1	2	yes	—
16	D₈×Dic₁₃	C₂²	1	2	yes	—
17	C₂×#88067	C₂²	1	2	yes	—
18	#11718679	—	2	2	no	—
19	#11718573	C₄	1	8	yes	complemented
20	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
21	C₂×#88038	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
23	D₁₃×#107	C₂²	1	2	yes	—
24	#11718534	C₂²	1	2	yes	—
25	#11718432	—	2	2	no	—
26	#11718418	—	2	2	no	—
27	C₄×Dic₅₂	C₂²	1	2	yes	—
28	#11718350	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	1	yes	characteristic
30	C₁₃×#357	C₂²	1	2	yes	—
31	C₁₃×#346	C₂²	1	1	yes	characteristic
32	#11718154	C₂²	1	2	yes	—
33	#11718097	C₂²	1	1	yes	characteristic
34	#11718095	C₂²	1	2	yes	—
35	#11718094	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
37	#11718084	C₂²	1	2	yes	—
38	#11718082	C₂²	1	2	yes	—
39	C₄×#6090	C₂²	1	1	yes	characteristic
40	C₂×#6129	—	2	2	no	—
41	C₂×C₁₃×D₈	C₂³	1	2	yes	—
42	C₄×D₄×C₁₃	—	2	2	no	—
43	D₄×Dic₁₃	—	2	2	no	—
44	C₂×#6105	—	2	2	no	—
45	C₄×C₂²⋊D₁₃	—	4	2	no	—
46	C₂×#6101	—	4	2	no	—
47	#88166	—	2	4	no	—
48	#88166	C₂×C₄	1	8	yes	complemented
49	C₂×Dic₅₂	C₂³	1	2	yes	—
50	C₂×#6094	D₄	1	1	yes	characteristic
51	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×D₁₃	—	2	2	no	—
53	#88141	—	2	2	no	—
54	#88135	—	4	2	no	—
55	#88131	—	4	2	no	—
56	#88126	D₄	1	1	yes	characteristic
57	C₄×Dic₂₆	—	2	2	no	—
58	C₁₃×D₁₆	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#102	—	2	2	no	—
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	#88072	—	2	2	no	—
65	#88067	C₂×C₄	1	8	yes	complemented
66	#88058	C₂³	1	2	yes	—
67	#88056	—	2	2	no	—
68	#88055	D₄	1	1	yes	characteristic
69	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
70	C₂×#6090	C₂³	1	1	yes	characteristic
71	C₂×#6090	C₂³	1	1	yes	characteristic
72	#88050	—	2	2	no	—
73	#88047	—	2	2	no	—
74	#88043	D₄	1	1	yes	characteristic
75	Dic₁₀₄	C₂×C₄	1	4	yes	complemented
76	#88038	—	2	2	no	—
77	C₁₆×D₁₃	C₂×C₄	1	4	yes	complemented
78	#33460	—	13	1	no	maximal, Sylow
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	—	4	2	no	—
81	C₂×C₂²⋊D₁₃	—	4	2	no	—
82	C₂²×Dic₁₃	—	4	2	no	—
83	C₂²×Dic₁₃	—	4	2	no	—
84	#6129	—	2	8	no	—
85	#6129	—	4	4	no	—
86	C₂×C₄×D₁₃	—	4	2	no	—
87	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×Dic₂₆	—	2	2	no	—
89	C₁₃×D₈	—	2	2	no	—
90	C₁₃×D₈	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₂²⋊C₄	—	4	2	no	—
95	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂²⋊Dic₁₃	—	4	2	no	—
97	#6105	—	2	8	no	—
98	#6103	D₈	1	2	yes	—
99	C₄⋊Dic₁₃	—	2	2	no	—
100	#6101	—	4	2	no	—
101	#6101	—	4	2	no	—
102	#6101	—	4	2	no	—
103	#6101	D₈	1	2	yes	—
104	C₄×Dic₁₃	—	4	2	no	—
105	C₄×Dic₁₃	—	4	2	no	—
106	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×#1055	C₂×D₄	1	1	yes	characteristic
108	C₂×#1055	C₂×D₄	1	1	yes	characteristic
109	Dic₅₂	—	2	2	no	—
110	Dic₅₂	C₂²×C₄	1	4	yes	—
111	#6094	—	2	2	no	—
112	C₈×D₁₃	C₂²×C₄	1	2	yes	—
113	C₈×D₁₃	#41	1	2	yes	—
114	C₁₃×C₁₆	—	2	1	no	—
115	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
116	#6090	—	2	1	no	—
117	#6090	C₂²×C₄	1	2	yes	—
118	C₂×#508	—	13	1	no	—
119	#2951	—	13	2	no	—
120	#2177	—	13	1	no	—
121	#2177	D₁₃	1	1	yes	characteristic, complemented
122	#2176	—	13	1	no	—
123	#2176	—	13	1	no	—
124	C₄×SD₃₂	—	13	2	no	—
125	C₄×SD₃₂	—	13	2	no	—
126	C₂×#367	—	13	1	no	—
127	#2139	—	13	2	no	—
128	#2108	—	13	1	no	—
129	C₂³×C₁₃	—	4	2	no	—
130	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
131	D₄×C₁₃	—	2	4	no	—
132	D₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	16	2	no	—
135	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₁₃	—	4	8	no	—
138	C₂×Dic₁₃	—	4	2	no	—
139	C₂×Dic₁₃	—	4	8	no	—
140	C₂×Dic₁₃	—	4	2	no	—
141	C₂×Dic₁₃	—	16	2	no	—
142	C₂×Dic₁₃	C₂×D₈	1	2	yes	—
143	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
144	C₄×D₁₃	—	16	2	no	—
145	C₄×D₁₃	C₄×D₄	1	2	yes	—
146	C₄×D₁₃	#135	1	2	yes	—
147	Dic₂₆	—	2	4	no	—
148	Dic₂₆	—	4	2	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
151	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
152	#1055	—	2	1	no	—
153	#1055	C₄×D₄	1	2	yes	—
154	C₂×#135	—	13	2	no	—
155	#520	—	26	2	no	—
156	#508	—	13	8	no	—
157	C₂×Q₃₂	—	13	1	no	—
158	C₂×SD₃₂	—	13	2	no	—
159	C₂×D₁₆	C₂×D₁₃	1	1	yes	characteristic
160	C₂×M₃₂	—	13	1	no	—
161	C₂²×C₁₆	—	13	1	no	—
162	#440	—	26	2	no	—
163	#440	—	26	2	no	—
164	C₄×Q₁₆	—	13	2	no	—
165	C₄×Q₁₆	—	13	2	no	—
166	C₄×D₈	—	13	2	no	—
167	C₄×D₈	C₂×D₁₃	1	2	yes	—
168	C₂×#107	—	13	2	no	—
169	#417	—	26	2	no	—
170	#405	—	13	1	no	—
171	#367	—	13	1	no	—
172	#367	—	13	2	no	—
173	#367	C₂×D₁₃	1	1	yes	characteristic
174	#358	—	13	2	no	—
175	#358	—	13	2	no	—
176	#357	—	13	2	no	—
177	#357	C₂×D₁₃	1	2	yes	—
178	#346	—	13	1	no	—
179	#346	C₂×D₁₃	1	1	yes	characteristic
180	C₄×C₁₆	—	13	1	no	—
181	C₄×C₁₆	—	13	1	no	—
182	C₂²×C₁₃	—	4	4	no	—
183	C₂²×C₁₃	—	8	2	no	—
184	C₂²×C₁₃	—	8	2	no	—
185	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
186	C₂×D₁₃	—	2	2	no	—
187	C₂×D₁₃	C₄×D₈	1	2	yes	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	8	2	no	—
190	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
191	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
192	Dic₁₃	—	2	2	no	—
193	Dic₁₃	—	4	4	no	—
194	Dic₁₃	—	8	2	no	—
195	Dic₁₃	C₄×D₈	1	2	yes	—
196	C₂×#41	—	26	2	no	—
197	#135	—	13	8	no	—
198	#135	—	26	4	no	—
199	C₂×Q₁₆	—	13	2	no	—
200	C₂×SD₁₆	—	26	2	no	—
201	C₂×D₈	C₂²×D₁₃	1	2	yes	—
202	C₂×M₁₆	—	13	1	no	—
203	C₂²×C₈	—	13	1	no	—
204	C₄×Q₈	—	26	2	no	—
205	C₄×Q₈	—	26	2	no	—
206	C₄×D₄	—	2	2	no	—
207	C₄×D₄	—	26	2	no	—
208	C₄×D₄	—	52	2	no	—
209	C₄×D₄	—	52	2	no	—
210	#117	—	13	1	no	—
211	#117	—	52	2	no	—
212	C₂×C₄⋊C₄	—	26	2	no	—
213	C₂×C₄⋊C₄	—	52	2	no	—
214	Q₃₂	—	13	4	no	—
215	SD₃₂	—	13	8	no	—
216	D₁₆	C₄×D₁₃	1	4	yes	complemented
217	M₃₂	—	26	2	no	—
218	C₂×C₁₆	—	13	1	no	—
219	C₂×C₁₆	—	13	4	no	—
220	C₂×C₁₆	—	13	1	no	—
221	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
222	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
223	#107	—	13	2	no	—
224	#107	—	26	2	no	—
225	#107	C₂²×D₁₃	1	2	yes	—
226	#103	—	26	2	no	—
227	#103	—	26	2	no	—
228	#102	—	2	2	no	—
229	#102	—	26	2	no	—
230	#97	—	13	1	no	—
231	#97	—	13	1	no	—
232	C₄×C₈	—	13	1	no	—
233	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
234	C₂×C₁₃	—	8	4	no	—
235	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
236	C₂×C₁₃	#3416	1	2	yes	—
237	D₁₃	—	2	2	no	—
238	#41	—	26	8	no	—
239	#41	—	52	4	no	—
240	C₂×Q₈	—	26	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	52	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	13	1	no	—
245	C₂²×C₄	—	52	2	no	—
246	C₂²×C₄	—	52	2	no	—
247	C₂²×C₄	—	52	2	no	—
248	Q₁₆	—	13	4	no	—
249	Q₁₆	—	26	2	no	—
250	SD₁₆	—	26	8	no	—
251	D₈	—	2	2	no	—
252	D₈	C₂×C₄×D₁₃	1	4	yes	—
253	M₁₆	—	26	2	no	—
254	C₂×C₈	—	13	1	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	13	2	no	—
257	C₂×C₈	—	13	1	no	—
258	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	13	2	no	—
262	C₄⋊C₄	—	26	2	no	—
263	C₄⋊C₄	—	52	2	no	—
264	C₄⋊C₄	—	52	2	no	—
265	C₄⋊C₄	—	52	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	13	2	no	—
268	C₂²⋊C₄	—	52	2	no	—
269	C₄²	—	13	1	no	—
270	C₄²	—	52	2	no	—
271	C₄²	—	52	2	no	—
272	C₄²	D₄×D₁₃	1	1	yes	characteristic
273	C₁₆	—	2	1	no	—
274	C₁₆	—	13	2	no	—
275	C₁₆	—	26	1	no	—
276	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
277	C₁₃	#33460	1	1	yes	characteristic, complemented, Sylow
278	C₂³	—	4	2	no	—
279	C₂³	—	13	1	no	—
280	Q₈	—	26	4	no	—
281	Q₈	—	52	2	no	—
282	D₄	—	2	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	52	8	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	13	1	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	13	2	no	—
289	C₂×C₄	—	13	2	no	—
290	C₂×C₄	—	16	2	no	—
291	C₂×C₄	—	52	8	no	—
292	C₂×C₄	—	52	2	no	—
293	C₂×C₄	—	52	2	no	—
294	C₂×C₄	—	208	2	no	—
295	C₂×C₄	—	208	2	no	—
296	C₂×C₄	D₈×D₁₃	1	2	yes	—
297	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	13	2	no	—
300	C₈	—	26	1	no	—
301	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
302	C₈	#41×D₁₃	1	1	yes	characteristic
303	C₂²	—	4	4	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	13	2	no	—
307	C₂²	—	26	2	no	—
308	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	13	2	no	—
312	C₄	—	26	2	no	—
313	C₄	—	52	4	no	—
314	C₄	—	104	2	no	—
315	C₄	C₄×D₄×D₁₃	1	1	yes	characteristic
316	C₄	D₁₃×#135	1	1	yes	characteristic
317	C₂	—	8	4	no	—
318	C₂	—	26	2	no	—
319	C₂	C₄×D₈×D₁₃	1	1	yes	characteristic
320	C₂	#13127982	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2177	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#11718573	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2177	D₁₃
hypocenter	C₁₃	#33460
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
2. #13123706	C₂	semidirect
3. #13116712	C₂	semidirect
4. #13116708	C₂	semidirect
5. #13116706	C₂	semidirect
6. #13116691	C₂	semidirect
7. C₄×#88067	C₂	semidirect
8. D₁₃×#367	C₂	semidirect
9. #13116545	C₂	semidirect
10. #13116434	C₂	semidirect
11. C₁₃×#2177	C₂	semidirect
19. #11718573	C₄	semidirect
48. #88166	C₂×C₄	semidirect
58. C₁₃×D₁₆	C₂×C₄	semidirect
65. #88067	C₂×C₄	semidirect
75. Dic₁₀₄	C₂×C₄	semidirect
77. C₁₆×D₁₃	C₂×C₄	semidirect
121. #2177	D₁₃	semidirect
216. D₁₆	C₄×D₁₃	semidirect
277. C₁₃	#33460	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718573	2
#13123706	2
#13116712	2
#13116708	2
#13116706	2
#13116691	2
C₄×#88067	2
D₁₃×#367	2
#13116545	2
#13116434	2
C₁₃×#2177	2
#33460	13