Subgroups — #16016331

Order 3328.

652 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127972	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718556	C₂	1	2	yes	maximal, complemented
3	#13116721	C₂	1	4	yes	maximal, complemented
4	#13116716	C₂	1	4	yes	maximal, complemented
5	#13116563	C₂	1	2	yes	maximal, complemented
6	M₃₂×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88161	—	2	2	no	—
9	C₂×#88158	C₂²	1	2	yes	—
10	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#108	C₂²	1	2	yes	—
12	C₂×#88065	—	2	2	no	—
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88052	C₂²	1	1	yes	characteristic
15	#11718570	C₄	1	4	yes	complemented
16	#11718569	C₄	1	4	yes	complemented
17	#11718565	C₂²	1	4	yes	—
18	#11718556	—	2	2	no	—
19	#11718556	—	2	2	no	—
20	#11718556	C₂²	1	4	yes	complemented
21	#11718556	C₂²	1	4	yes	complemented
22	C₁₃×#368	C₂²	1	4	yes	complemented
23	#11718094	C₂²	1	2	yes	complemented
24	C₁₆×Dic₁₃	C₂²	1	2	yes	complemented
25	#11718089	C₂²	1	8	yes	complemented
26	C₂×#6127	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
30	C₂²×#1055	D₄	1	1	yes	characteristic
31	#88161	—	2	4	no	—
32	#88161	—	2	4	no	—
33	#88161	—	4	2	no	—
34	#88161	—	4	2	no	—
35	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
36	#88158	—	2	2	no	—
37	#88158	—	2	2	no	—
38	#88158	C₂×C₄	1	4	yes	—
39	#88158	C₂³	1	4	yes	—
40	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
41	C₂×#6095	—	2	2	no	—
42	C₂×#6094	—	4	2	no	—
43	C₂×C₈×D₁₃	—	4	2	no	—
44	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
45	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
46	C₁₃×#108	—	2	2	no	—
47	C₁₃×#108	C₂³	1	4	yes	—
48	#88065	—	2	4	no	—
49	#88065	—	2	4	no	—
50	C₈×Dic₁₃	D₄	1	1	yes	characteristic
51	C₈×Dic₁₃	D₄	1	1	yes	characteristic
52	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
53	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
54	#88052	—	2	2	no	—
55	C₂×#6090	C₂³	1	2	yes	—
56	Dic₁₀₄	C₂×C₄	1	4	yes	complemented
57	#88040	C₂×C₄	1	8	yes	complemented
58	D₂₀₈	C₂×C₄	1	4	yes	complemented
59	#33463	—	13	1	no	maximal, Sylow
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃	—	4	2	no	—
62	C₂²×Dic₁₃	—	2	1	no	—
63	#6127	—	2	4	no	—
64	#6127	—	2	4	no	—
65	#6127	—	4	2	no	—
66	#6127	—	4	2	no	—
67	C₂×D₅₂	—	2	2	no	—
68	C₂×C₄×D₁₃	—	4	2	no	—
69	C₂×Dic₂₆	—	2	2	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	—	4	2	no	—
72	C₂×C₈×C₁₃	—	4	2	no	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	#41	1	2	yes	—
76	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₄×Dic₁₃	—	2	1	no	—
78	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
79	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×#1055	D₈	1	2	yes	—
81	C₂×#1055	D₈	1	2	yes	—
82	C₂×#1055	C₂×D₄	1	1	yes	characteristic
83	C₂×#1055	C₂×D₄	1	1	yes	characteristic
84	Dic₅₂	—	2	2	no	—
85	Dic₅₂	C₂²×C₄	1	4	yes	—
86	D₁₀₄	—	2	2	no	—
87	D₁₀₄	C₂²×C₄	1	4	yes	—
88	#6095	—	2	4	no	—
89	#6095	—	2	4	no	—
90	#6094	—	4	4	no	—
91	#6094	—	4	4	no	—
92	C₈×D₁₃	—	4	4	no	—
93	C₈×D₁₃	—	4	4	no	—
94	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
95	#6090	—	2	2	no	—
96	#3414	—	13	1	no	—
97	C₂×#443	—	13	2	no	—
98	#2179	—	13	4	no	—
99	#2178	—	13	4	no	—
100	#2161	D₁₃	1	1	yes	characteristic, complemented
101	#2143	—	13	2	no	—
102	C₄×M₃₂	—	13	1	no	—
103	C₂³×C₁₃	#135	1	1	yes	characteristic
104	C₂²×D₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₂²⋊D₁₃	—	4	4	no	—
110	C₂²⋊D₁₃	—	4	4	no	—
111	C₂×Dic₁₃	—	2	1	no	—
112	C₂×Dic₁₃	—	2	1	no	—
113	C₂×Dic₁₃	—	4	2	no	—
114	C₂×Dic₁₃	—	4	2	no	—
115	D₅₂	—	2	4	no	—
116	D₅₂	—	4	2	no	—
117	C₄×D₁₃	—	4	4	no	—
118	C₄×D₁₃	—	4	4	no	—
119	Dic₂₆	—	2	4	no	—
120	Dic₂₆	—	4	2	no	—
121	C₈×C₁₃	—	4	4	no	—
122	C₈×C₁₃	C₄×D₄	1	2	yes	—
123	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
125	#1055	C₂×D₈	1	1	yes	characteristic
126	#1055	C₂×D₈	1	2	yes	—
127	#1055	C₂×D₈	1	1	yes	characteristic
128	C₂×#135	—	13	2	no	—
129	C₂×#131	—	26	2	no	—
130	#510	—	13	4	no	—
131	#509	—	13	4	no	—
132	#508	—	13	4	no	—
133	C₂×M₃₂	—	13	1	no	—
134	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
135	#443	—	13	4	no	—
136	#443	—	13	4	no	—
137	#443	—	26	2	no	—
138	#443	—	26	2	no	—
139	C₂×#108	C₂×D₁₃	1	2	yes	—
140	C₂×#104	—	26	2	no	—
141	C₂×C₄×C₈	—	13	1	no	—
142	#368	C₂×D₁₃	1	4	yes	complemented
143	#359	—	13	8	no	—
144	#346	—	13	2	no	—
145	C₄×C₁₆	—	13	2	no	—
146	C₂²×C₁₃	—	4	1	no	—
147	C₂²×C₁₃	C₄×D₈	1	2	yes	—
148	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
149	C₂×D₁₃	—	4	4	no	—
150	C₂×D₁₃	—	8	4	no	—
151	C₄×C₁₃	C₄×D₈	1	2	yes	—
152	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
154	Dic₁₃	—	2	2	no	—
155	Dic₁₃	—	4	1	no	—
156	Dic₁₃	—	4	4	no	—
157	C₂×#41	—	26	2	no	—
158	#135	—	13	4	no	—
159	#135	—	13	4	no	—
160	#135	—	26	2	no	—
161	#135	—	26	2	no	—
162	C₂×Q₁₆	—	13	2	no	—
163	C₂×SD₁₆	—	26	2	no	—
164	C₂×D₈	—	13	2	no	—
165	#131	—	26	4	no	—
166	#131	—	26	4	no	—
167	#131	—	52	2	no	—
168	#131	—	52	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	52	2	no	—
171	C₂²×C₈	—	13	1	no	—
172	C₂²×C₈	—	52	2	no	—
173	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
174	C₂×C₄²	—	13	1	no	—
175	Q₃₂	—	13	4	no	—
176	SD₃₂	—	13	8	no	—
177	D₁₆	—	13	4	no	—
178	M₃₂	—	26	2	no	—
179	M₃₂	C₄×D₁₃	1	4	yes	—
180	C₂×C₁₆	—	13	2	no	—
181	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
182	#108	—	2	2	no	—
183	#108	C₂²×D₁₃	1	4	yes	—
184	#104	—	26	4	no	—
185	#104	—	26	4	no	—
186	C₄×C₈	—	13	1	no	—
187	C₄×C₈	—	13	1	no	—
188	C₄×C₈	—	13	1	no	—
189	C₄×C₈	—	13	1	no	—
190	C₂×C₁₃	—	2	1	no	—
191	C₂×C₁₃	—	2	2	no	—
192	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
193	D₁₃	—	8	4	no	—
194	#41	—	26	4	no	—
195	#41	—	26	4	no	—
196	#41	—	52	2	no	—
197	#41	—	52	2	no	—
198	C₂×Q₈	—	26	2	no	—
199	C₂×D₄	—	26	2	no	—
200	C₂×D₄	—	52	2	no	—
201	C₂²×C₄	—	26	1	no	—
202	C₂²×C₄	—	52	2	no	—
203	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
204	Q₁₆	—	13	4	no	—
205	Q₁₆	—	26	2	no	—
206	SD₁₆	—	26	4	no	—
207	SD₁₆	—	26	4	no	—
208	D₈	—	13	4	no	—
209	D₈	—	26	2	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	52	4	no	—
213	M₁₆	—	52	4	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	13	2	no	—
216	C₂×C₈	—	13	2	no	—
217	C₂×C₈	—	13	1	no	—
218	C₂×C₈	—	13	1	no	—
219	C₂×C₈	—	52	4	no	—
220	C₂×C₈	—	52	4	no	—
221	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
222	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
223	C₂×C₈	#6131	1	2	yes	—
224	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
225	C₄²	—	13	1	no	—
226	C₄²	—	13	1	no	—
227	C₄²	—	26	1	no	—
228	C₁₆	—	26	2	no	—
229	C₁₆	C₂×C₄×D₁₃	1	4	yes	—
230	C₁₃	#33463	1	1	yes	characteristic, complemented, Sylow
231	C₂³	—	52	2	no	—
232	C₂³	#88173	1	1	yes	characteristic
233	Q₈	—	26	4	no	—
234	Q₈	—	52	2	no	—
235	D₄	—	26	4	no	—
236	D₄	—	52	4	no	—
237	D₄	—	52	4	no	—
238	D₄	—	52	2	no	—
239	C₂×C₄	—	26	1	no	—
240	C₂×C₄	—	26	1	no	—
241	C₂×C₄	—	52	2	no	—
242	C₂×C₄	—	52	2	no	—
243	C₂×C₄	—	52	4	no	—
244	C₂×C₄	—	52	4	no	—
245	C₂×C₄	#88147	1	2	yes	—
246	C₂×C₄	D₈×D₁₃	1	2	yes	—
247	C₂×C₄	#88173	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
249	C₈	—	4	4	no	—
250	C₈	—	13	1	no	—
251	C₈	—	13	2	no	—
252	C₈	—	13	1	no	—
253	C₈	#88147	1	2	yes	—
254	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
255	C₈	C₂×#6131	1	1	yes	characteristic
256	C₂²	—	4	1	no	—
257	C₂²	—	52	4	no	—
258	C₂²	—	104	4	no	—
259	C₂²	#11718533	1	2	yes	—
260	C₂²	C₂×#88173	1	1	yes	characteristic
261	C₄	—	26	2	no	—
262	C₄	—	52	1	no	—
263	C₄	—	52	4	no	—
264	C₄	#11718533	1	2	yes	—
265	C₄	C₂×#88147	1	1	yes	characteristic
266	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	2	no	—
269	C₂	—	104	4	no	—
270	C₂	C₂×#11718533	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2161	C₂
socle	C₂×C₁₃	C₂×C₄×D₈
prime radical	#13127972	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2161	D₁₃
hypocenter	C₁₃	#33463
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

Complements

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

N	H ≅ G/N	type
2. C₂×#11718556	C₂	semidirect
3. #13116721	C₂	semidirect
4. #13116716	C₂	semidirect
5. #13116563	C₂	semidirect
6. M₃₂×Dic₁₃	C₂	semidirect
7. C₁₃×#2161	C₂	semidirect
15. #11718570	C₄	semidirect
16. #11718569	C₄	semidirect
20. #11718556	C₂²	semidirect
21. #11718556	C₂²	semidirect
22. C₁₃×#368	C₂²	semidirect
23. #11718094	C₂²	semidirect
24. C₁₆×Dic₁₃	C₂²	semidirect
25. #11718089	C₂²	semidirect
56. Dic₁₀₄	C₂×C₄	semidirect
57. #88040	C₂×C₄	semidirect
58. D₂₀₈	C₂×C₄	semidirect
100. #2161	D₁₃	semidirect
142. #368	C₂×D₁₃	semidirect
230. C₁₃	#33463	semidirect

Maximal subgroups

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

subgroup	index
#13127972	2
C₂×#11718556	2
#13116721	2
#13116716	2
#13116563	2
M₃₂×Dic₁₃	2
C₁₃×#2161	2
#33463	13