Subgroups — #16021063

Order 3328.

466 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	C₂×#11718565	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117154	C₂	1	2	yes	maximal, complemented
3	#13117153	C₂	1	2	yes	maximal, complemented
4	#13117149	C₂	1	2	yes	maximal, complemented
5	#13117148	C₂	1	2	yes	maximal, complemented
6	#13117143	C₂	1	4	yes	maximal, complemented
7	C₂×#11718091	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	—	2	1	no	—
10	C₂×#88158	—	2	1	no	—
11	C₁₃×#508	—	2	1	no	—
12	C₁₃×#508	—	2	1	no	—
13	C₁₃×#508	C₂²	1	2	yes	—
14	C₁₃×#508	C₂²	1	2	yes	complemented
15	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₁₃×SD₃₂	—	2	1	no	—
17	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic, complemented
20	#11718565	—	2	1	no	—
21	#11718565	—	2	1	no	—
22	#11718565	C₂²	1	2	yes	—
23	#11718565	C₂²	1	2	yes	complemented
24	C₂×#88040	—	2	1	no	—
25	C₂×D₂₀₈	C₂²	1	1	yes	characteristic, complemented
26	#11718114	C₂²	1	4	yes	complemented
27	#11718113	C₂²	1	4	yes	complemented
28	#11718112	C₂²	1	4	yes	complemented
29	#11718111	C₂²	1	4	yes	complemented
30	#11718091	C₂²	1	2	yes	complemented
31	#11718091	C₂²	1	2	yes	complemented
32	C₂×#88034	C₂²	1	2	yes	complemented
33	C₂×C₁₃×#41	—	4	1	no	—
34	C₂×#6127	—	4	1	no	—
35	C₁₃×#135	—	2	2	no	—
36	C₁₃×#135	—	2	2	no	—
37	C₁₃×#135	—	4	1	no	—
38	C₁₃×#135	—	4	1	no	—
39	C₂×C₁₃×Q₁₆	—	2	1	no	—
40	C₂×C₁₃×SD₁₆	—	4	1	no	—
41	C₂×C₁₃×D₈	—	2	1	no	—
42	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂×Dic₅₂	—	2	1	no	—
44	#88158	—	2	2	no	—
45	#88158	—	2	2	no	—
46	#88158	—	4	1	no	—
47	#88158	—	4	1	no	—
48	C₂×D₁₀₄	—	2	1	no	—
49	C₂×#6095	—	4	1	no	—
50	C₁₃×Q₃₂	—	2	1	no	—
51	C₁₃×Q₃₂	C₂³	1	2	yes	—
52	C₁₃×SD₃₂	—	2	2	no	—
53	C₁₃×SD₃₂	—	2	2	no	—
54	C₁₃×D₁₆	—	2	1	no	—
55	C₁₃×D₁₆	C₂³	1	2	yes	—
56	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
57	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
58	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
62	Dic₁₀₄	—	2	1	no	—
63	Dic₁₀₄	C₂³	1	2	yes	—
64	#88040	—	2	2	no	—
65	#88040	—	2	2	no	—
66	D₂₀₈	—	2	1	no	—
67	D₂₀₈	C₂³	1	2	yes	—
68	#88034	C₂³	1	4	yes	—
69	#33981	—	13	1	no	maximal, Sylow
70	C₁₃×#41	—	4	2	no	—
71	C₁₃×#41	—	4	2	no	—
72	C₁₃×#41	—	8	1	no	—
73	C₁₃×#41	—	8	1	no	—
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂×D₄×C₁₃	—	4	1	no	—
76	C₂×D₄×C₁₃	—	8	1	no	—
77	C₂²×C₄×C₁₃	—	8	1	no	—
78	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
79	C₂×C₂²⋊D₁₃	—	8	1	no	—
80	#6127	—	4	2	no	—
81	#6127	—	4	2	no	—
82	#6127	—	8	1	no	—
83	#6127	—	8	1	no	—
84	C₂×D₅₂	—	4	1	no	—
85	C₂×C₄×D₁₃	—	8	1	no	—
86	C₂×Dic₂₆	—	4	1	no	—
87	C₁₃×Q₁₆	—	2	2	no	—
88	C₁₃×Q₁₆	—	4	1	no	—
89	C₁₃×SD₁₆	—	4	2	no	—
90	C₁₃×SD₁₆	—	4	2	no	—
91	C₁₃×D₈	—	2	2	no	—
92	C₁₃×D₈	—	4	1	no	—
93	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
94	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
95	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
99	Dic₅₂	—	2	2	no	—
100	Dic₅₂	—	4	1	no	—
101	D₁₀₄	—	2	2	no	—
102	D₁₀₄	—	4	1	no	—
103	#6095	—	4	2	no	—
104	#6095	—	4	2	no	—
105	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
106	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
107	C₁₃×C₁₆	C₂×D₄	1	1	yes	characteristic
108	C₁₃×C₁₆	C₂⁴	1	1	yes	characteristic
109	C₂×#508	—	13	1	no	—
110	C₂×#508	D₁₃	1	1	yes	characteristic, complemented
111	#2264	—	13	2	no	—
112	#2264	—	13	2	no	—
113	#2263	—	13	2	no	—
114	#2263	—	13	2	no	—
115	#2262	—	13	4	no	—
116	C₂×M₆₄	—	13	1	no	—
117	C₂³×C₁₃	—	8	1	no	—
118	C₂³×C₁₃	D₁₆	1	1	yes	characteristic
119	C₂²×D₁₃	—	8	1	no	—
120	Q₈×C₁₃	—	4	2	no	—
121	Q₈×C₁₃	—	8	1	no	—
122	D₄×C₁₃	—	4	2	no	—
123	D₄×C₁₃	—	8	2	no	—
124	D₄×C₁₃	—	8	2	no	—
125	D₄×C₁₃	—	8	1	no	—
126	C₂×C₄×C₁₃	—	8	1	no	—
127	C₂×C₄×C₁₃	—	8	2	no	—
128	C₂×C₄×C₁₃	—	8	2	no	—
129	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₃	D₁₆	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂²⋊D₁₃	—	8	2	no	—
136	C₂²⋊D₁₃	—	8	2	no	—
137	C₂×Dic₁₃	—	8	1	no	—
138	D₅₂	—	4	2	no	—
139	D₅₂	—	8	1	no	—
140	C₄×D₁₃	—	8	2	no	—
141	C₄×D₁₃	—	8	2	no	—
142	Dic₂₆	—	4	2	no	—
143	Dic₂₆	—	8	1	no	—
144	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
146	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×#135	—	2	1	no	—
149	C₂×#135	—	26	1	no	—
150	#508	—	2	1	no	—
151	#508	—	2	1	no	—
152	#508	—	13	2	no	—
153	#508	—	13	2	no	—
154	#508	—	26	1	no	—
155	#508	—	26	1	no	—
156	#508	C₂×D₁₃	1	2	yes	—
157	#508	C₂×D₁₃	1	2	yes	complemented
158	C₂×Q₃₂	—	13	1	no	—
159	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic, complemented
160	C₂×SD₃₂	—	2	1	no	—
161	C₂×SD₃₂	—	26	1	no	—
162	C₂×D₁₆	—	13	1	no	—
163	C₂×D₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
164	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
165	Q₆₄	—	13	4	no	—
166	SD₆₄	—	13	4	no	—
167	SD₆₄	—	13	4	no	—
168	D₃₂	—	13	4	no	—
169	M₆₄	—	13	2	no	—
170	M₆₄	—	13	2	no	—
171	C₂×C₃₂	—	13	2	no	—
172	C₂²×C₁₃	—	4	1	no	—
173	C₂²×C₁₃	—	8	2	no	—
174	C₂²×C₁₃	—	16	2	no	—
175	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
176	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
177	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
178	C₂×D₁₃	—	8	2	no	—
179	C₂×D₁₃	—	16	2	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	C₂×D₁₆	1	1	yes	characteristic
182	C₄×C₁₃	C₂×D₁₆	1	1	yes	characteristic
183	C₄×C₁₃	C₂×D₁₆	1	1	yes	characteristic
184	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
185	Dic₁₃	—	8	2	no	—
186	C₂×#41	—	4	1	no	—
187	C₂×#41	—	52	1	no	—
188	#135	—	2	2	no	—
189	#135	—	2	2	no	—
190	#135	—	4	1	no	—
191	#135	—	4	1	no	—
192	#135	—	26	2	no	—
193	#135	—	26	2	no	—
194	#135	—	52	1	no	—
195	#135	—	52	1	no	—
196	C₂×Q₁₆	—	2	1	no	—
197	C₂×Q₁₆	—	26	1	no	—
198	C₂×SD₁₆	—	4	1	no	—
199	C₂×SD₁₆	—	52	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	26	1	no	—
202	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
203	Q₃₂	—	2	1	no	—
204	Q₃₂	—	13	2	no	—
205	Q₃₂	—	26	1	no	—
206	Q₃₂	C₂²×D₁₃	1	2	yes	—
207	SD₃₂	—	2	2	no	—
208	SD₃₂	—	2	2	no	—
209	SD₃₂	—	26	2	no	—
210	SD₃₂	—	26	2	no	—
211	D₁₆	—	2	1	no	—
212	D₁₆	—	13	2	no	—
213	D₁₆	—	26	1	no	—
214	D₁₆	C₂²×D₁₃	1	2	yes	—
215	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
216	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
217	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
218	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
219	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
220	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
221	C₃₂	—	13	4	no	—
222	C₂×C₁₃	—	2	1	no	—
223	C₂×C₁₃	—	2	1	no	—
224	C₂×C₁₃	—	2	1	no	—
225	C₂×C₁₃	—	16	2	no	—
226	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
227	D₁₃	—	16	2	no	—
228	#41	—	4	2	no	—
229	#41	—	4	2	no	—
230	#41	—	8	1	no	—
231	#41	—	8	1	no	—
232	#41	—	52	2	no	—
233	#41	—	52	2	no	—
234	#41	—	104	1	no	—
235	#41	—	104	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	52	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	8	1	no	—
240	C₂×D₄	—	52	1	no	—
241	C₂×D₄	—	104	1	no	—
242	C₂²×C₄	—	8	1	no	—
243	C₂²×C₄	—	104	1	no	—
244	C₂²×C₄	#6104	1	1	yes	characteristic
245	Q₁₆	—	2	2	no	—
246	Q₁₆	—	4	1	no	—
247	Q₁₆	—	26	2	no	—
248	Q₁₆	—	52	1	no	—
249	SD₁₆	—	4	2	no	—
250	SD₁₆	—	4	2	no	—
251	SD₁₆	—	52	2	no	—
252	SD₁₆	—	52	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	4	1	no	—
255	D₈	—	26	2	no	—
256	D₈	—	52	1	no	—
257	C₂×C₈	#6104	1	1	yes	characteristic
258	C₂×C₈	#6104	1	1	yes	characteristic
259	C₂×C₈	#6104	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
263	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
264	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
265	C₁₆	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
266	C₁₆	C₂³×D₁₃	1	1	yes	characteristic
267	C₁₃	#33981	1	1	yes	characteristic, complemented, Sylow
268	C₂³	—	8	1	no	—
269	C₂³	—	104	1	no	—
270	C₂³	#88066	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	8	1	no	—
273	Q₈	—	52	2	no	—
274	Q₈	—	104	1	no	—
275	D₄	—	4	2	no	—
276	D₄	—	8	2	no	—
277	D₄	—	8	2	no	—
278	D₄	—	8	1	no	—
279	D₄	—	52	2	no	—
280	D₄	—	104	2	no	—
281	D₄	—	104	2	no	—
282	D₄	—	104	1	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	104	1	no	—
287	C₂×C₄	—	104	2	no	—
288	C₂×C₄	—	104	2	no	—
289	C₂×C₄	#88066	1	1	yes	characteristic
290	C₂×C₄	#88066	1	1	yes	characteristic
291	C₂×C₄	#88066	1	1	yes	characteristic
292	C₂×C₄	C₂×#6104	1	1	yes	characteristic
293	C₂×C₄	C₂×#6104	1	1	yes	characteristic
294	C₂×C₄	C₂×#6104	1	1	yes	characteristic
295	C₈	C₂×#6104	1	1	yes	characteristic
296	C₈	C₂×#6104	1	1	yes	characteristic
297	C₈	C₂×#6104	1	1	yes	characteristic
298	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
299	C₂²	—	4	1	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	16	2	no	—
302	C₂²	—	104	2	no	—
303	C₂²	—	208	2	no	—
304	C₂²	C₂×#88066	1	1	yes	characteristic
305	C₂²	C₂×#88066	1	1	yes	characteristic
306	C₂²	C₂×#88066	1	1	yes	characteristic
307	C₄	—	8	2	no	—
308	C₄	—	104	2	no	—
309	C₄	C₂×#88066	1	1	yes	characteristic
310	C₄	C₂×#88066	1	1	yes	characteristic
311	C₄	C₂×#88066	1	1	yes	characteristic
312	C₄	C₂²×#6104	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	16	2	no	—
317	C₂	—	208	2	no	—
318	C₂	C₂²×#88066	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88066
commutator, cosocle	C₁₃×C₁₆	C₂⁴
Frattini	C₁₆	C₂³×D₁₃
Fitting	C₂×C₁₃×#508	C₂
socle	C₂×C₁₃	C₂²×D₁₆
hypercenter	C₂×#508	D₁₃
hypocenter	C₁₃	#33981
LCS	G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#11718565	C₂	semidirect
2. #13117154	C₂	semidirect
3. #13117153	C₂	semidirect
4. #13117149	C₂	semidirect
5. #13117148	C₂	semidirect
6. #13117143	C₂	semidirect
7. C₂×#11718091	C₂	semidirect
8. C₂×C₁₃×#508	C₂	semidirect
14. C₁₃×#508	C₂²	semidirect
15. C₂×C₁₃×Q₃₂	C₂²	semidirect
17. C₂×C₁₃×D₁₆	C₂²	semidirect
19. C₂×Dic₁₀₄	C₂²	semidirect
23. #11718565	C₂²	semidirect
25. C₂×D₂₀₈	C₂²	semidirect
26. #11718114	C₂²	semidirect
27. #11718113	C₂²	semidirect
28. #11718112	C₂²	semidirect
29. #11718111	C₂²	semidirect
30. #11718091	C₂²	semidirect
31. #11718091	C₂²	semidirect
32. C₂×#88034	C₂²	semidirect
110. C₂×#508	D₁₃	semidirect
157. #508	C₂×D₁₃	semidirect
159. C₂×Q₃₂	C₂×D₁₃	semidirect
163. C₂×D₁₆	C₂×D₁₃	semidirect
267. C₁₃	#33981	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718565	2
#13117154	2
#13117153	2
#13117149	2
#13117148	2
#13117143	2
C₂×#11718091	2
C₂×C₁₃×#508	2
#33981	13