Subgroups — #16807220

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131616	C₂	1	2	yes	maximal, complemented
2	#13131239	C₂	1	4	yes	maximal, complemented
3	#13131237	C₂	1	4	yes	maximal, complemented
4	D₁₃⋊#368	C₂	1	1	yes	maximal, characteristic, complemented
5	#13131152	C₂	1	2	yes	maximal, complemented
6	#13131044	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718580	C₂	1	1	yes	maximal, characteristic
8	#11719625	—	2	2	no	—
9	C₂×#88173	C₂²	1	2	yes	—
10	#11719155	—	2	2	no	—
11	#11719112	—	2	4	no	—
12	#11719111	C₂²	1	8	yes	complemented
13	D₁₃⋊#108	C₂²	1	2	yes	—
14	#11719089	C₂²	1	1	yes	characteristic
15	D₁₃⋊M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×Q₃₂	C₄	1	1	yes	characteristic, complemented
17	C₂×#88068	C₄	1	2	yes	complemented
18	#11718580	C₄	1	2	yes	complemented
19	#11718580	C₄	1	2	yes	complemented
20	#11718580	C₂²	1	2	yes	—
21	#11718580	C₂²	1	2	yes	—
22	C₂×D₂₀₈	C₄	1	1	yes	characteristic, complemented
23	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
24	#11718281	C₂²	1	8	yes	complemented
25	#11718224	C₂²	1	2	yes	complemented
26	#11718223	C₂²	1	2	yes	complemented
27	#11718216	C₂²	1	2	yes	complemented
28	C₁₆×#261	C₂²	1	2	yes	complemented
29	C₂×#6131	—	2	2	no	—
30	#88240	—	2	8	no	—
31	#88240	—	4	4	no	—
32	#88237	D₄	1	1	yes	characteristic
33	D₁₃⋊M₁₆	—	2	2	no	—
34	D₁₃⋊M₁₆	—	4	4	no	—
35	D₁₃⋊M₁₆	D₄	1	1	yes	characteristic
36	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
37	C₂×#6106	—	2	2	no	—
38	#88173	—	2	2	no	—
39	#88173	—	2	2	no	—
40	#88173	C₂×C₄	1	4	yes	—
41	#88173	C₂³	1	4	yes	—
42	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
44	#88118	—	2	8	no	—
45	#88106	—	2	1	no	—
46	#88104	C₂³	1	4	yes	—
47	#88103	—	2	2	no	—
48	#88100	D₄	1	2	yes	—
49	C₈×#261	C₂³	1	2	yes	—
50	#88098	—	2	1	no	—
51	D₁₃⋊C₁₆	C₂³	1	2	yes	—
52	C₁₃×Q₃₂	C₂×C₄	1	2	yes	—
53	C₁₃×Q₃₂	C₂×C₄	1	2	yes	—
54	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
55	#88068	C₂×C₄	1	4	yes	—
56	#88068	C₂×C₄	1	4	yes	—
57	C₂×#6090	C₂×C₄	1	1	yes	characteristic
58	D₂₀₈	C₂×C₄	1	2	yes	—
59	D₂₀₈	C₂×C₄	1	2	yes	—
60	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic
61	C₁₆×D₁₃	C₂×C₄	1	1	yes	characteristic
62	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
63	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
64	#33465	—	13	1	no	maximal, Sylow
65	C₂×Q₈×C₁₃	—	2	2	no	—
66	#6131	—	2	4	no	—
67	#6131	—	2	4	no	—
68	#6131	—	4	2	no	—
69	#6131	—	4	2	no	—
70	C₂×D₅₂	—	2	2	no	—
71	C₂×D₅₂	—	4	2	no	—
72	C₂×C₄×D₁₃	—	4	2	no	—
73	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²⋊#261	—	2	1	no	—
75	#6122	—	4	2	no	—
76	#6122	D₈	1	2	yes	—
77	C₄⋊#261	—	2	1	no	—
78	C₄×#261	C₂×D₄	1	2	yes	—
79	#6118	—	2	4	no	—
80	#6118	—	4	8	no	—
81	#6118	D₈	1	2	yes	—
82	D₁₃⋊C₈	—	4	2	no	—
83	D₁₃⋊C₈	—	4	8	no	—
84	D₁₃⋊C₈	C₂×D₄	1	2	yes	—
85	C₁₃×Q₁₆	—	2	2	no	—
86	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	#6106	—	2	4	no	—
89	#6106	—	2	4	no	—
90	C₂×#1055	#41	1	1	yes	characteristic
91	D₁₀₄	—	2	2	no	—
92	D₁₀₄	C₂²×C₄	1	4	yes	—
93	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
95	C₈×D₁₃	#41	1	1	yes	characteristic
96	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
97	#6092	—	2	2	no	—
98	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
99	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
100	#6090	C₂²×C₄	1	1	yes	characteristic
101	#6090	C₂²×C₄	1	1	yes	characteristic
102	C₂×#508	—	13	1	no	—
103	#2957	—	13	2	no	—
104	#2179	—	13	4	no	—
105	#2178	—	13	4	no	—
106	C₂×#368	—	13	1	no	—
107	#2143	—	13	2	no	—
108	#2108	—	13	1	no	—
109	C₂²×D₁₃	—	4	2	no	—
110	C₂²×D₁₃	#135	1	1	yes	characteristic
111	C₂×#261	—	2	2	no	—
112	Q₈×C₁₃	—	2	4	no	—
113	Q₈×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
116	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
117	D₅₂	—	2	4	no	—
118	D₅₂	—	4	4	no	—
119	D₅₂	—	4	4	no	—
120	D₅₂	—	4	2	no	—
121	C₄×D₁₃	—	4	4	no	—
122	C₄×D₁₃	—	4	4	no	—
123	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
124	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
125	C₄×D₁₃	#135	1	1	yes	characteristic
126	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
127	#1057	—	4	4	no	—
128	#1057	C₂×D₈	1	4	yes	—
129	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
130	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
131	#1055	C₄×D₄	1	1	yes	characteristic
132	#1055	C₂×#41	1	1	yes	characteristic
133	C₂×#135	—	13	2	no	—
134	#568	—	26	2	no	—
135	#508	—	13	2	no	—
136	#508	—	13	2	no	—
137	#508	—	13	2	no	—
138	#508	—	13	2	no	—
139	C₂×Q₃₂	#261	1	1	yes	characteristic, complemented
140	C₂×SD₃₂	—	13	2	no	—
141	C₂×D₁₆	—	13	1	no	—
142	C₂×M₃₂	—	13	1	no	—
143	C₂²×C₁₆	—	13	1	no	—
144	#444	—	26	4	no	—
145	#443	—	13	8	no	—
146	C₂×#108	—	13	2	no	—
147	#421	—	26	2	no	—
148	#405	—	13	1	no	—
149	#368	—	13	2	no	—
150	#368	—	13	2	no	—
151	#359	—	13	8	no	—
152	#346	—	13	2	no	—
153	C₄×C₁₆	—	13	2	no	—
154	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
155	C₂×D₁₃	—	4	1	no	—
156	C₂×D₁₃	—	4	4	no	—
157	C₂×D₁₃	—	16	2	no	—
158	C₂×D₁₃	C₄×D₈	1	1	yes	characteristic
159	C₂×D₁₃	C₂×#135	1	1	yes	characteristic
160	#261	—	4	2	no	—
161	C₄×C₁₃	—	4	4	no	—
162	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
163	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
164	Dic₁₃	C₄×D₈	1	1	yes	characteristic
165	Dic₁₃	C₂²×D₈	1	1	yes	characteristic
166	C₂×#41	—	26	2	no	—
167	#135	—	13	4	no	—
168	#135	—	13	4	no	—
169	#135	—	26	2	no	—
170	#135	—	26	2	no	—
171	C₂×Q₁₆	C₂×#261	1	2	yes	—
172	C₂×SD₁₆	—	26	2	no	—
173	C₂×D₈	—	13	2	no	—
174	#131	—	26	8	no	—
175	#131	—	52	4	no	—
176	C₂×M₁₆	—	13	1	no	—
177	C₂×M₁₆	—	26	2	no	—
178	C₂×M₁₆	—	52	4	no	—
179	C₂²×C₈	—	13	1	no	—
180	#117	—	13	1	no	—
181	Q₃₂	C₂×#261	1	2	yes	—
182	Q₃₂	C₂×#261	1	2	yes	—
183	SD₃₂	—	13	4	no	—
184	SD₃₂	—	13	4	no	—
185	D₁₆	—	13	2	no	—
186	D₁₆	—	13	2	no	—
187	M₃₂	—	26	1	no	—
188	M₃₂	—	26	1	no	—
189	C₂×C₁₆	—	13	1	no	—
190	C₂×C₁₆	—	13	1	no	—
191	C₂×C₁₆	—	13	1	no	—
192	C₂×C₁₆	—	13	1	no	—
193	C₂×C₁₆	—	13	1	no	—
194	C₂×C₁₆	—	13	2	no	—
195	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
196	#108	—	13	4	no	—
197	#108	—	26	2	no	—
198	#104	—	26	8	no	—
199	#97	—	13	2	no	—
200	C₄×C₈	—	13	2	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
203	D₁₃	—	2	1	no	—
204	D₁₃	—	2	1	no	—
205	D₁₃	—	8	4	no	—
206	#41	—	26	4	no	—
207	#41	—	26	4	no	—
208	#41	—	52	2	no	—
209	#41	—	52	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×D₄	—	26	2	no	—
212	C₂×D₄	—	52	2	no	—
213	C₂²×C₄	—	13	1	no	—
214	C₂²×C₄	—	52	2	no	—
215	Q₁₆	—	2	2	no	—
216	Q₁₆	C₂²×#261	1	4	yes	—
217	SD₁₆	—	26	4	no	—
218	SD₁₆	—	26	4	no	—
219	D₈	—	13	4	no	—
220	D₈	—	26	2	no	—
221	M₁₆	—	13	2	no	—
222	M₁₆	—	13	2	no	—
223	M₁₆	—	26	4	no	—
224	M₁₆	—	52	8	no	—
225	M₁₆	—	52	2	no	—
226	C₂×C₈	—	13	2	no	—
227	C₂×C₈	—	13	1	no	—
228	C₂×C₈	—	13	1	no	—
229	C₂×C₈	—	13	1	no	—
230	C₂×C₈	—	13	1	no	—
231	C₂×C₈	—	13	1	no	—
232	C₂×C₈	—	52	8	no	—
233	C₂×C₈	—	52	2	no	—
234	C₂×C₈	C₂²×#261	1	1	yes	characteristic
235	C₄⋊C₄	—	26	1	no	—
236	C₂²⋊C₄	—	26	1	no	—
237	C₄²	—	13	2	no	—
238	C₁₆	—	13	1	no	—
239	C₁₆	—	13	1	no	—
240	C₁₆	—	26	2	no	—
241	C₁₆	C₂²×#261	1	1	yes	characteristic
242	C₁₆	C₂²×#261	1	1	yes	characteristic
243	C₁₃	#33465	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	13	1	no	—
245	C₂³	—	52	2	no	—
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	26	4	no	—
249	D₄	—	52	4	no	—
250	D₄	—	52	4	no	—
251	D₄	—	52	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	13	1	no	—
254	C₂×C₄	—	13	1	no	—
255	C₂×C₄	—	13	1	no	—
256	C₂×C₄	—	13	1	no	—
257	C₂×C₄	—	13	1	no	—
258	C₂×C₄	—	26	2	no	—
259	C₂×C₄	—	52	4	no	—
260	C₂×C₄	—	52	4	no	—
261	C₂×C₄	D₄×#261	1	1	yes	characteristic
262	C₈	—	13	4	no	—
263	C₈	—	13	1	no	—
264	C₈	—	13	1	no	—
265	C₈	—	52	4	no	—
266	C₈	D₄×#261	1	1	yes	characteristic
267	C₈	C₂³×#261	1	1	yes	characteristic
268	C₂²	—	13	1	no	—
269	C₂²	—	13	1	no	—
270	C₂²	—	52	1	no	—
271	C₂²	—	52	4	no	—
272	C₂²	—	208	2	no	—
273	C₂²	D₈×#261	1	1	yes	characteristic
274	C₄	—	4	4	no	—
275	C₄	—	13	1	no	—
276	C₄	—	13	1	no	—
277	C₄	—	52	2	no	—
278	C₄	D₈×#261	1	1	yes	characteristic
279	C₄	C₂×D₄×#261	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	26	1	no	—
282	C₂	—	26	1	no	—
283	C₂	—	104	4	no	—
284	C₂	C₂×D₈×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#261
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₈	C₂³×#261
Fitting	C₂×C₁₃×Q₃₂	C₄
socle	C₂×C₁₃	C₂×C₄×D₈
prime radical	C₂×#11718580	C₂
cosocle	C₈×D₁₃	C₂⁴
hypercenter	C₂×Q₃₂	#261
hypocenter	C₁₃	#33465
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
1. #13131616	C₂	semidirect
2. #13131239	C₂	semidirect
3. #13131237	C₂	semidirect
4. D₁₃⋊#368	C₂	semidirect
5. #13131152	C₂	semidirect
6. #13131044	C₂	semidirect
12. #11719111	C₂²	semidirect
16. C₂×C₁₃×Q₃₂	C₄	semidirect
17. C₂×#88068	C₄	semidirect
18. #11718580	C₄	semidirect
19. #11718580	C₄	semidirect
22. C₂×D₂₀₈	C₄	semidirect
24. #11718281	C₂²	semidirect
25. #11718224	C₂²	semidirect
26. #11718223	C₂²	semidirect
27. #11718216	C₂²	semidirect
28. C₁₆×#261	C₂²	semidirect
139. C₂×Q₃₂	#261	semidirect
243. C₁₃	#33465	semidirect

Maximal subgroups

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

subgroup	index
#13131616	2
#13131239	2
#13131237	2
D₁₃⋊#368	2
#13131152	2
#13131044	2
C₂×#11718580	2
#33465	13