Subgroups — #16807017

Order 3328.

684 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131805	C₂	1	2	yes	maximal
2	#13131178	C₂	1	4	yes	maximal, complemented
3	#13131177	C₂	1	2	yes	maximal, complemented
4	C₂×#11718233	C₂	1	2	yes	maximal, complemented
5	D₁₃⋊#364	C₂	1	4	yes	maximal, complemented
6	C₄×#88161	C₂	1	1	yes	maximal, characteristic
7	C₂×#88161	C₄	1	1	yes	characteristic
8	C₂×#88161	C₂²	1	1	yes	characteristic
9	C₄×#6127	C₄	1	1	yes	characteristic
10	C₂×#88106	C₂²	1	4	yes	—
11	#11719096	—	2	8	no	—
12	D₁₃⋊M₃₂	C₂²	1	8	yes	—
13	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
14	#11718783	C₄	1	1	yes	characteristic
15	#11718783	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	#11718369	C₄	1	2	yes	—
18	#11718369	C₂²	1	2	yes	—
19	C₄×#6094	C₄	1	2	yes	—
20	C₄×C₈×D₁₃	C₂²	1	2	yes	—
21	#11718261	C₂²	1	8	yes	—
22	#11718233	C₂²	1	4	yes	—
23	#11718233	C₂²	1	4	yes	—
24	C₂×#6127	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₂²⋊D₁₃	—	2	2	no	—
29	#88180	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1055	C₂³	1	1	yes	characteristic
32	#88161	C₈	1	8	yes	—
33	#88161	D₄	1	4	yes	—
34	#88161	Q₈	1	4	yes	—
35	C₂×#6094	C₂×C₄	1	2	yes	—
36	C₂×#6094	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₃	C₂³	1	2	yes	—
39	C₄×D₅₂	—	2	1	no	—
40	#88126	C₂×C₄	1	2	yes	—
41	C₄²×D₁₃	C₂×C₄	1	2	yes	—
42	C₄×Dic₂₆	—	2	1	no	—
43	#88106	—	2	4	no	—
44	#88106	C₂³	1	8	yes	—
45	C₂×#6092	—	2	4	no	—
46	#88098	—	2	8	no	—
47	D₁₃⋊C₁₆	—	2	8	no	—
48	C₁₃×#97	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
50	#88055	C₂×C₄	1	2	yes	—
51	#88055	C₂×C₄	1	2	yes	—
52	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
53	C₈×Dic₁₃	C₂³	1	2	yes	—
54	C₄×#1055	C₂×C₄	1	1	yes	characteristic
55	C₄×#1055	C₂×C₄	1	1	yes	characteristic
56	C₄×#1055	C₂³	1	1	yes	characteristic
57	C₄×#1055	C₂³	1	1	yes	characteristic
58	#33401	—	13	1	no	maximal, Sylow
59	C₂²×C₄×C₁₃	—	2	1	no	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₃	M₁₆	1	2	yes	—
62	#6127	C₄⋊C₄	1	4	yes	—
63	#6127	C₂×C₈	1	4	yes	—
64	C₂×D₅₂	M₁₆	1	1	yes	characteristic
65	C₂×C₄×D₁₃	—	2	2	no	—
66	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
67	C₂×Dic₂₆	M₁₆	1	1	yes	characteristic
68	C₁₃×M₁₆	C₄⋊C₄	1	4	yes	—
69	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₁₃	—	2	1	no	—
73	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²⋊Dic₁₃	—	2	1	no	—
76	#6103	—	2	2	no	—
77	C₄⋊Dic₁₃	—	2	1	no	—
78	#6101	—	2	2	no	—
79	C₄×Dic₁₃	—	2	2	no	—
80	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
81	C₂×#1055	C₂×C₈	1	4	yes	—
82	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
83	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
84	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
85	C₂×#1055	C₂×D₄	1	2	yes	—
86	C₂×#1055	C₂×Q₈	1	2	yes	—
87	C₂×#1055	C₂⁴	1	1	yes	characteristic
88	#6094	C₄⋊C₄	1	8	yes	—
89	#6094	C₂×C₈	1	8	yes	—
90	C₈×D₁₃	C₂×C₈	1	8	yes	—
91	C₈×D₁₃	C₂×D₄	1	4	yes	—
92	C₈×D₁₃	C₂×Q₈	1	4	yes	—
93	#6092	—	2	8	no	—
94	#3407	—	13	2	no	—
95	C₄×#131	—	13	1	no	—
96	#2153	—	13	2	no	—
97	#2153	—	13	4	no	—
98	C₂×#364	—	13	4	no	—
99	C₂×#364	—	13	2	no	—
100	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₁₃	C₂×M₁₆	1	2	yes	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	C₂²⋊D₁₃	C₄⋊C₈	1	8	yes	—
110	C₂×Dic₁₃	—	2	2	no	—
111	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
112	D₅₂	C₄⋊C₈	1	4	yes	—
113	C₄×D₁₃	—	4	4	no	—
114	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
115	C₄×D₁₃	C₂²×C₈	1	4	yes	—
116	Dic₂₆	C₄⋊C₈	1	4	yes	—
117	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
118	C₈×C₁₃	C₂²×C₈	1	4	yes	—
119	#1055	C₂×C₄⋊C₄	1	2	yes	—
120	#1055	C₂²×C₈	1	2	yes	—
121	#1055	C₂²×C₈	1	2	yes	—
122	#1055	C₂²×D₄	1	1	yes	characteristic
123	#1055	C₂²×Q₈	1	1	yes	characteristic
124	C₂×#131	—	13	1	no	—
125	C₂×#131	—	13	1	no	—
126	C₄×#41	—	13	1	no	—
127	#504	—	26	8	no	—
128	C₂×M₃₂	—	13	8	no	—
129	C₂×M₃₂	—	13	4	no	—
130	#405	—	13	1	no	—
131	#405	—	13	1	no	—
132	#405	—	13	2	no	—
133	#405	—	13	2	no	—
134	C₄×M₁₆	—	13	2	no	—
135	C₄×M₁₆	#261	1	1	yes	characteristic
136	C₂×C₄×C₈	—	13	1	no	—
137	C₂×C₄×C₈	—	13	2	no	—
138	#364	—	13	4	no	—
139	#364	—	13	4	no	—
140	#364	—	13	8	no	—
141	C₂²×C₁₃	—	4	1	no	—
142	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
143	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
144	C₂×D₁₃	—	4	2	no	—
145	C₂×D₁₃	C₂×C₄⋊C₈	1	4	yes	—
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	4	1	no	—
148	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
149	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
151	Dic₁₃	—	4	2	no	—
152	Dic₁₃	C₂×C₄⋊C₈	1	4	yes	—
153	C₂×#41	—	13	1	no	—
154	#131	—	13	8	no	—
155	#131	—	13	4	no	—
156	#131	—	13	4	no	—
157	C₂×M₁₆	—	13	2	no	—
158	C₂×M₁₆	—	13	2	no	—
159	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
160	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
161	C₂²×C₈	—	13	1	no	—
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	—	13	2	no	—
165	C₄×Q₈	—	26	1	no	—
166	C₄×D₄	—	26	1	no	—
167	C₄×D₄	—	26	2	no	—
168	#117	—	13	1	no	—
169	#117	—	13	2	no	—
170	C₂×C₄²	—	13	2	no	—
171	C₂×C₄²	C₂×#261	1	1	yes	characteristic
172	M₃₂	—	13	8	no	—
173	M₃₂	—	26	8	no	—
174	M₃₂	—	26	4	no	—
175	C₂×C₁₆	—	26	8	no	—
176	C₂×C₁₆	—	26	4	no	—
177	#97	—	13	2	no	—
178	#97	—	13	2	no	—
179	#97	C₂×#261	1	2	yes	—
180	C₄×C₈	—	13	2	no	—
181	C₄×C₈	—	13	2	no	—
182	C₄×C₈	—	13	1	no	—
183	C₄×C₈	—	13	1	no	—
184	C₄×C₈	—	13	1	no	—
185	C₄×C₈	—	13	1	no	—
186	C₄×C₈	C₂×#261	1	2	yes	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	—	2	2	no	—
189	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
190	D₁₃	—	2	4	no	—
191	#41	—	13	4	no	—
192	#41	—	13	4	no	—
193	C₂×Q₈	—	13	1	no	—
194	C₂×D₄	—	13	2	no	—
195	C₂×D₄	—	13	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	13	2	no	—
198	C₂²×C₄	—	26	2	no	—
199	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
200	M₁₆	—	13	8	no	—
201	M₁₆	—	13	8	no	—
202	M₁₆	D₁₃⋊C₈	1	4	yes	—
203	M₁₆	C₄⋊#261	1	4	yes	—
204	C₂×C₈	—	13	4	no	—
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	13	2	no	—
207	C₂×C₈	—	13	1	no	—
208	C₂×C₈	—	13	1	no	—
209	C₂×C₈	—	13	1	no	—
210	C₂×C₈	—	13	1	no	—
211	C₂×C₈	—	13	8	no	—
212	C₂×C₈	—	13	4	no	—
213	C₂×C₈	—	13	4	no	—
214	C₂×C₈	C₂²×#261	1	2	yes	—
215	C₂×C₈	C₂²×#261	1	2	yes	—
216	C₄⋊C₄	—	26	2	no	—
217	C₄⋊C₄	—	26	1	no	—
218	C₂²⋊C₄	—	26	2	no	—
219	C₂²⋊C₄	—	26	1	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	13	2	no	—
222	C₄²	—	26	2	no	—
223	C₄²	C₂²×#261	1	1	yes	characteristic
224	C₄²	C₂²×#261	1	1	yes	characteristic
225	C₁₆	—	26	8	no	—
226	C₁₃	#33401	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	13	2	no	—
228	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
229	Q₈	—	13	4	no	—
230	D₄	—	13	8	no	—
231	D₄	—	13	4	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	13	2	no	—
236	C₂×C₄	—	13	4	no	—
237	C₂×C₄	—	13	4	no	—
238	C₂×C₄	—	26	2	no	—
239	C₂×C₄	—	52	4	no	—
240	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
241	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
242	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
243	C₂×C₄	C₂³×#261	1	1	yes	characteristic
244	C₈	—	13	2	no	—
245	C₈	—	13	2	no	—
246	C₈	—	13	1	no	—
247	C₈	—	13	2	no	—
248	C₈	—	13	1	no	—
249	C₈	C₂×D₁₃⋊C₈	1	4	yes	—
250	C₈	C₂×C₄⋊#261	1	4	yes	—
251	C₂²	—	4	1	no	—
252	C₂²	—	13	4	no	—
253	C₂²	—	52	2	no	—
254	C₂²	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
255	C₂²	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	13	4	no	—
259	C₄	—	52	2	no	—
260	C₄	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
261	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
262	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
263	C₂	—	2	1	no	—
264	C₂	—	2	2	no	—
265	C₂	—	26	4	no	—
266	C₂	C₂×D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄⋊#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₄×C₁₃×M₁₆	C₄
socle	C₂×C₁₃	C₂²×C₄⋊C₈
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	C₄×M₁₆	#261
hypocenter	C₁₃	#33401
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13131178	C₂	semidirect
3. #13131177	C₂	semidirect
4. C₂×#11718233	C₂	semidirect
5. D₁₃⋊#364	C₂	semidirect
226. C₁₃	#33401	semidirect

Maximal subgroups

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

subgroup	index
#13131805	2
#13131178	2
#13131177	2
C₂×#11718233	2
D₁₃⋊#364	2
C₄×#88161	2
#33401	13