Subgroups — #16807030

Order 3328.

674 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719092	C₂	1	2	yes	maximal
2	#13131188	C₂	1	8	yes	maximal, complemented
3	#13131187	C₂	1	4	yes	maximal, complemented
4	C₂×D₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂×#88237	—	2	2	no	—
6	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
7	#11719092	—	2	2	no	—
8	#11719092	—	2	2	no	—
9	#11719092	C₂²	1	4	yes	—
10	#11719092	C₂²	1	4	yes	—
11	C₂×#88102	C₂²	1	4	yes	—
12	C₂×#88101	—	2	2	no	—
13	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic, complemented
14	C₂×#88052	C₄	1	1	yes	characteristic, complemented
15	D₁₃×M₃₂	C₄	1	4	yes	complemented
16	D₁₃×M₃₂	C₂²	1	4	yes	—
17	C₂×#88038	C₄	1	2	yes	complemented
18	C₂×C₁₆×D₁₃	C₂²	1	2	yes	—
19	#11718222	C₂²	1	8	yes	complemented
20	#11718221	C₂²	1	8	yes	complemented
21	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
22	C₂×C₂²⋊#261	—	4	2	no	—
23	#88237	—	2	4	no	—
24	#88237	—	2	4	no	—
25	#88237	—	4	2	no	—
26	#88237	—	4	2	no	—
27	C₂×C₄⋊#261	—	2	4	no	—
28	C₂×C₄×#261	—	4	2	no	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	Q₈	1	1	yes	characteristic
31	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₁₃	D₄	1	2	yes	—
34	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
35	C₂×C₈×D₁₃	Q₈	1	2	yes	—
36	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
37	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₈×D₁₃	C₂³	1	2	yes	—
39	#88102	—	2	4	no	—
40	#88102	C₂³	1	8	yes	—
41	#88101	—	2	4	no	—
42	#88101	—	2	4	no	—
43	C₁₃×M₃₂	C₂×C₄	1	4	yes	complemented
44	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
45	#88052	C₂×C₄	1	4	yes	complemented
46	C₂×#6090	C₂×C₄	1	2	yes	complemented
47	#88038	C₂×C₄	1	8	yes	complemented
48	C₁₆×D₁₃	C₂×C₄	1	4	yes	complemented
49	C₁₆×D₁₃	C₂³	1	4	yes	—
50	#33409	—	13	1	no	maximal, Sylow
51	C₂³×D₁₃	D₈	1	1	yes	characteristic
52	C₂²×#261	—	4	4	no	—
53	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	Q₁₆	1	1	yes	characteristic
55	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×D₁₃	D₈	1	2	yes	—
58	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
59	C₂×C₄×D₁₃	Q₁₆	1	1	yes	characteristic
60	C₂×C₄×D₁₃	Q₁₆	1	2	yes	—
61	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²⋊#261	—	4	4	no	—
64	C₂²⋊#261	—	4	4	no	—
65	C₄⋊#261	—	2	8	no	—
66	C₄⋊#261	—	4	4	no	—
67	C₄×#261	—	4	4	no	—
68	C₄×#261	—	4	4	no	—
69	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
74	C₂×#1055	C₄⋊C₄	1	2	yes	—
75	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
76	C₂×#1055	C₂×Q₈	1	2	yes	—
77	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
78	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
79	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
80	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₈×D₁₃	C₂²×C₄	1	2	yes	—
82	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
83	C₈×D₁₃	C₂×D₄	1	2	yes	—
84	C₈×D₁₃	C₂×Q₈	1	1	yes	characteristic
85	C₈×D₁₃	C₂×Q₈	1	2	yes	—
86	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
87	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
88	#6090	C₂²×C₄	1	4	yes	—
89	C₂²×M₃₂	—	13	1	no	—
90	C₂×#427	—	13	2	no	—
91	#2159	—	13	8	no	—
92	#2158	—	13	4	no	—
93	C₂³×C₁₃	#107	1	1	yes	characteristic
94	C₂²×D₁₃	—	4	2	no	—
95	C₂²×D₁₃	#107	1	2	yes	—
96	C₂²×D₁₃	#107	1	1	yes	characteristic
97	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
98	C₂²×D₁₃	C₂×D₈	1	2	yes	—
99	C₂×#261	—	4	8	no	—
100	C₂×#261	—	8	8	no	—
101	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#107	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×Dic₁₃	#107	1	2	yes	—
106	C₂×Dic₁₃	#107	1	1	yes	characteristic
107	C₂×Dic₁₃	C₂×Q₁₆	1	2	yes	—
108	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
109	C₄×D₁₃	#107	1	2	yes	—
110	C₄×D₁₃	#107	1	2	yes	—
111	C₄×D₁₃	#107	1	2	yes	—
112	C₄×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
115	C₄×D₁₃	C₂×D₈	1	2	yes	—
116	C₄×D₁₃	C₂×Q₁₆	1	2	yes	—
117	C₄×D₁₃	C₂×Q₁₆	1	1	yes	characteristic
118	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
123	#1055	C₂×C₄⋊C₄	1	2	yes	—
124	#1055	C₂²×Q₈	1	1	yes	characteristic
125	C₂³×C₈	—	13	1	no	—
126	C₂×#117	—	26	2	no	—
127	C₂×M₃₂	—	13	1	no	—
128	C₂×M₃₂	—	13	4	no	—
129	C₂×M₃₂	—	13	4	no	—
130	C₂×M₃₂	—	13	2	no	—
131	C₂×M₃₂	#261	1	1	yes	characteristic, complemented
132	C₂²×C₁₆	—	13	2	no	—
133	#427	—	13	4	no	—
134	#427	—	13	4	no	—
135	#427	—	26	2	no	—
136	#427	—	26	2	no	—
137	C₂×#107	—	13	4	no	—
138	C₂×#106	—	26	2	no	—
139	#367	—	13	8	no	—
140	#366	—	13	8	no	—
141	C₂²×C₁₃	—	4	1	no	—
142	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
143	C₂²×C₁₃	C₂×#107	1	2	yes	—
144	C₂×D₁₃	—	2	4	no	—
145	C₂×D₁₃	—	2	2	no	—
146	C₂×D₁₃	—	4	1	no	—
147	C₂×D₁₃	—	4	2	no	—
148	C₂×D₁₃	C₂×#107	1	1	yes	characteristic
149	C₂×D₁₃	C₂×#107	1	2	yes	—
150	C₂×D₁₃	C₂²×D₈	1	1	yes	characteristic
151	#261	—	8	8	no	—
152	C₄×C₁₃	C₂×#107	1	2	yes	—
153	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
154	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	Dic₁₃	C₂×#107	1	1	yes	characteristic
156	Dic₁₃	C₂×#107	1	2	yes	—
157	Dic₁₃	C₂²×Q₁₆	1	1	yes	characteristic
158	C₂³×C₄	—	13	1	no	—
159	C₂²×C₈	—	13	1	no	—
160	C₂²×C₈	—	13	2	no	—
161	C₂²×C₈	—	13	1	no	—
162	C₂²×C₈	—	13	2	no	—
163	C₂²×C₈	—	13	1	no	—
164	C₂²×C₈	—	13	2	no	—
165	C₂²×C₈	—	13	1	no	—
166	C₂²×C₈	—	13	2	no	—
167	C₂²×C₈	—	13	1	no	—
168	C₂²×C₈	C₂×#261	1	1	yes	characteristic
169	#117	—	26	4	no	—
170	#117	—	26	4	no	—
171	#117	—	52	2	no	—
172	#117	—	52	2	no	—
173	C₂×C₄⋊C₄	—	26	4	no	—
174	C₂×C₂²⋊C₄	—	52	2	no	—
175	C₂×C₄²	—	52	2	no	—
176	M₃₂	—	13	4	no	—
177	M₃₂	—	13	8	no	—
178	M₃₂	C₂×#261	1	4	yes	complemented
179	C₂×C₁₆	—	13	2	no	—
180	C₂×C₁₆	—	13	4	no	—
181	C₂×C₁₆	—	13	4	no	—
182	C₂×C₁₆	C₂×#261	1	2	yes	complemented
183	#107	—	13	8	no	—
184	#107	—	26	4	no	—
185	#106	—	26	4	no	—
186	#106	—	26	4	no	—
187	C₂×C₁₃	—	2	2	no	—
188	C₂×C₁₃	—	2	1	no	—
189	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
190	D₁₃	—	2	2	no	—
191	D₁₃	—	2	1	no	—
192	D₁₃	#3414	1	2	yes	—
193	C₂⁴	—	13	1	no	—
194	C₂²×C₄	—	13	1	no	—
195	C₂²×C₄	—	13	2	no	—
196	C₂²×C₄	—	13	1	no	—
197	C₂²×C₄	—	13	1	no	—
198	C₂²×C₄	—	13	2	no	—
199	C₂²×C₄	—	13	2	no	—
200	C₂²×C₄	—	13	1	no	—
201	C₂²×C₄	—	13	2	no	—
202	C₂²×C₄	—	13	1	no	—
203	C₂²×C₄	—	52	4	no	—
204	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	13	1	no	—
207	C₂×C₈	—	13	2	no	—
208	C₂×C₈	—	13	1	no	—
209	C₂×C₈	—	13	1	no	—
210	C₂×C₈	—	13	2	no	—
211	C₂×C₈	—	13	2	no	—
212	C₂×C₈	—	13	2	no	—
213	C₂×C₈	—	13	2	no	—
214	C₂×C₈	—	13	2	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	13	1	no	—
217	C₂×C₈	—	13	1	no	—
218	C₂×C₈	—	13	2	no	—
219	C₂×C₈	C₄⋊#261	1	2	yes	—
220	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
221	C₂×C₈	C₂²×#261	1	1	yes	characteristic
222	C₂×C₈	C₂²×#261	1	2	yes	—
223	C₄⋊C₄	—	26	8	no	—
224	C₄⋊C₄	—	52	4	no	—
225	C₂²⋊C₄	—	52	4	no	—
226	C₂²⋊C₄	—	52	4	no	—
227	C₄²	—	52	4	no	—
228	C₄²	—	52	4	no	—
229	C₁₆	—	13	4	no	—
230	C₁₆	C₂²×#261	1	4	yes	—
231	C₁₃	#33409	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	13	1	no	—
233	C₂³	—	13	2	no	—
234	C₂³	—	13	2	no	—
235	C₂³	—	13	1	no	—
236	C₂³	—	52	2	no	—
237	C₂³	#88102	1	1	yes	characteristic
238	C₂×C₄	—	13	2	no	—
239	C₂×C₄	—	13	1	no	—
240	C₂×C₄	—	13	2	no	—
241	C₂×C₄	—	13	1	no	—
242	C₂×C₄	—	13	2	no	—
243	C₂×C₄	—	13	2	no	—
244	C₂×C₄	—	13	2	no	—
245	C₂×C₄	—	13	2	no	—
246	C₂×C₄	—	13	1	no	—
247	C₂×C₄	—	13	2	no	—
248	C₂×C₄	—	13	1	no	—
249	C₂×C₄	—	13	1	no	—
250	C₂×C₄	—	13	1	no	—
251	C₂×C₄	—	13	2	no	—
252	C₂×C₄	—	52	8	no	—
253	C₂×C₄	—	104	8	no	—
254	C₂×C₄	#88102	1	1	yes	characteristic
255	C₂×C₄	#88102	1	2	yes	—
256	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
257	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
258	C₈	—	13	1	no	—
259	C₈	—	13	2	no	—
260	C₈	—	13	1	no	—
261	C₈	C₂×C₄⋊#261	1	1	yes	characteristic
262	C₈	C₂×C₄⋊#261	1	2	yes	—
263	C₈	C₂³×#261	1	1	yes	characteristic
264	C₂²	—	4	1	no	—
265	C₂²	—	13	1	no	—
266	C₂²	—	13	1	no	—
267	C₂²	—	13	2	no	—
268	C₂²	—	26	4	no	—
269	C₂²	—	26	2	no	—
270	C₂²	—	52	2	no	—
271	C₂²	—	52	1	no	—
272	C₂²	C₂×#88102	1	1	yes	characteristic
273	C₂²	C₂×#88102	1	2	yes	—
274	C₄	—	13	1	no	—
275	C₄	—	13	2	no	—
276	C₄	—	13	1	no	—
277	C₄	—	104	8	no	—
278	C₄	C₂×#88102	1	2	yes	—
279	C₄	C₂×#88102	1	1	yes	characteristic
280	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	13	2	no	—
284	C₂	—	26	2	no	—
285	C₂	—	26	1	no	—
286	C₂	C₂²×#88102	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88102
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₈	C₂³×#261
Fitting	C₂×C₁₃×M₃₂	C₄
socle	C₂×C₁₃	C₂²×#107
prime radical	C₂³×D₁₃	D₈
cosocle	C₈×D₁₃	C₂⁴
hypercenter	C₂×M₃₂	#261
hypocenter	C₁₃	#33409
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #13131188	C₂	semidirect
3. #13131187	C₂	semidirect
13. C₂×C₁₃×M₃₂	C₄	semidirect
14. C₂×#88052	C₄	semidirect
15. D₁₃×M₃₂	C₄	semidirect
17. C₂×#88038	C₄	semidirect
19. #11718222	C₂²	semidirect
20. #11718221	C₂²	semidirect
43. C₁₃×M₃₂	C₂×C₄	semidirect
44. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
45. #88052	C₂×C₄	semidirect
46. C₂×#6090	C₂×C₄	semidirect
47. #88038	C₂×C₄	semidirect
48. C₁₆×D₁₃	C₂×C₄	semidirect
131. C₂×M₃₂	#261	semidirect
178. M₃₂	C₂×#261	semidirect
182. C₂×C₁₆	C₂×#261	semidirect
231. C₁₃	#33409	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719092	2
#13131188	2
#13131187	2
C₂×D₁₃×M₃₂	2
#33409	13