Subgroups — #16806759

Order 3328.

716 conjugacy classes of subgroups.

The subgroup lattice has 260 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	#13131059	C₂	1	2	yes	maximal, complemented
3	#13131058	C₂	1	4	yes	maximal, complemented
4	#13131053	C₂	1	4	yes	maximal, complemented
5	C₂×#11718232	C₂	1	2	yes	maximal, complemented
6	#13123265	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	C₄	1	16	yes	complemented
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×#88043	C₂²	1	1	yes	characteristic
17	#11718368	C₄	1	2	yes	—
18	D₁₃×#97	C₂²	1	2	yes	—
19	#11718350	C₄	1	2	yes	—
20	#11718350	C₂²	1	2	yes	—
21	#11718262	C₂²	1	8	yes	—
22	#11718232	C₂²	1	4	yes	—
23	#11718232	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	—	2	4	no	—
33	#88161	C₂×C₄	1	8	yes	—
34	C₂×#6094	C₂×C₄	1	2	yes	—
35	C₂×#6094	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₁₃	C₂³	1	2	yes	—
38	C₄×D₅₂	—	2	1	no	—
39	#88126	C₂×C₄	1	2	yes	—
40	C₄²×D₁₃	C₂×C₄	1	2	yes	—
41	C₄×Dic₂₆	—	2	1	no	—
42	#88106	C₂×C₄	1	8	yes	complemented
43	#88106	C₂³	1	8	yes	—
44	C₂×#6092	C₂×C₄	1	8	yes	complemented
45	#88098	C₂×C₄	1	16	yes	complemented
46	D₁₃⋊C₁₆	C₂×C₄	1	16	yes	complemented
47	C₁₃×#97	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
49	#88055	C₂×C₄	1	2	yes	—
50	#88055	C₂³	1	2	yes	—
51	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
52	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
53	#88043	C₂×C₄	1	1	yes	characteristic
54	#88043	C₂×C₄	1	1	yes	characteristic
55	#88043	C₂³	1	1	yes	characteristic
56	#88043	C₂³	1	1	yes	characteristic
57	#33329	—	13	1	no	maximal, Sylow
58	C₂²×C₄×C₁₃	—	2	1	no	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₃	M₁₆	1	2	yes	—
61	#6127	C₄²	1	4	yes	—
62	#6127	M₁₆	1	4	yes	—
63	C₂×D₅₂	M₁₆	1	1	yes	characteristic
64	C₂×C₄×D₁₃	—	2	2	no	—
65	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
66	C₂×Dic₂₆	M₁₆	1	1	yes	characteristic
67	C₁₃×M₁₆	—	2	2	no	—
68	C₁₃×M₁₆	C₄²	1	4	yes	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²⋊Dic₁₃	—	2	1	no	—
75	#6103	—	2	2	no	—
76	C₄⋊Dic₁₃	—	2	1	no	—
77	#6101	—	2	2	no	—
78	C₄×Dic₁₃	—	2	2	no	—
79	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
80	C₂×#1055	—	2	2	no	—
81	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
82	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
83	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
84	C₂×#1055	C₂²×C₄	1	4	yes	—
85	C₂×#1055	C₂⁴	1	1	yes	characteristic
86	#6094	—	2	4	no	—
87	#6094	C₄²	1	8	yes	—
88	C₈×D₁₃	—	2	4	no	—
89	C₈×D₁₃	C₂²×C₄	1	8	yes	—
90	#6092	C₂²×C₄	1	16	yes	—
91	#3407	—	13	2	no	—
92	#2875	—	13	1	no	—
93	#2110	—	13	4	no	—
94	#2110	—	13	2	no	—
95	C₂×#347	—	13	2	no	—
96	C₂×#347	—	13	4	no	—
97	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂²×D₁₃	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	8	1	no	—
102	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
106	C₂²⋊D₁₃	#97	1	8	yes	—
107	C₂×Dic₁₃	—	2	2	no	—
108	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
109	D₅₂	#97	1	4	yes	—
110	C₄×D₁₃	—	8	2	no	—
111	C₄×D₁₃	C₂×C₄²	1	4	yes	—
112	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
113	Dic₂₆	#97	1	4	yes	—
114	C₈×C₁₃	—	2	2	no	—
115	C₈×C₁₃	C₂×C₄²	1	4	yes	—
116	#1055	—	2	1	no	—
117	#1055	—	2	1	no	—
118	#1055	C₂×C₄²	1	2	yes	—
119	#1055	C₂³×C₄	1	2	yes	—
120	C₂×#131	—	13	1	no	—
121	C₂×#131	—	13	1	no	—
122	C₄×#41	—	13	1	no	—
123	#504	—	13	16	no	—
124	C₂×M₃₂	—	13	8	no	—
125	C₂×M₃₂	—	13	4	no	—
126	#405	—	13	1	no	—
127	#405	—	13	1	no	—
128	#405	—	13	2	no	—
129	#405	—	13	2	no	—
130	C₄×M₁₆	—	13	2	no	—
131	C₄×M₁₆	#261	1	1	yes	characteristic
132	C₂×#97	—	13	1	no	—
133	C₂×#97	—	13	2	no	—
134	#347	—	13	4	no	—
135	#347	—	13	4	no	—
136	#347	—	13	8	no	—
137	C₂²×C₁₃	—	4	1	no	—
138	C₂²×C₁₃	C₂×#97	1	2	yes	—
139	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
140	C₂×D₁₃	—	4	2	no	—
141	C₂×D₁₃	C₂×#97	1	4	yes	—
142	C₄×C₁₃	—	4	1	no	—
143	C₄×C₁₃	—	4	1	no	—
144	C₄×C₁₃	C₂×#97	1	2	yes	—
145	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
146	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
147	Dic₁₃	—	4	2	no	—
148	Dic₁₃	C₂×#97	1	4	yes	—
149	C₂×#41	—	13	1	no	—
150	#131	—	13	8	no	—
151	#131	—	26	4	no	—
152	C₂×M₁₆	—	13	2	no	—
153	C₂×M₁₆	—	13	2	no	—
154	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
155	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
156	C₂²×C₈	—	13	1	no	—
157	C₂²×C₈	—	13	2	no	—
158	C₂²×C₈	—	13	1	no	—
159	C₂²×C₈	—	13	2	no	—
160	C₄×Q₈	—	26	1	no	—
161	C₄×D₄	—	26	1	no	—
162	C₄×D₄	—	26	2	no	—
163	#117	—	13	1	no	—
164	#117	—	13	2	no	—
165	C₂×C₄²	—	13	2	no	—
166	C₂×C₄²	C₂×#261	1	1	yes	characteristic
167	M₃₂	—	13	16	no	—
168	M₃₂	—	13	8	no	—
169	M₃₂	—	13	8	no	—
170	C₂×C₁₆	—	13	16	no	—
171	C₂×C₁₆	—	13	8	no	—
172	#97	—	13	2	no	—
173	#97	—	13	2	no	—
174	#97	—	13	1	no	—
175	#97	—	13	1	no	—
176	#97	—	13	1	no	—
177	#97	—	13	1	no	—
178	#97	C₂×#261	1	2	yes	—
179	C₄×C₈	—	13	2	no	—
180	C₄×C₈	—	13	2	no	—
181	C₄×C₈	C₂×#261	1	2	yes	—
182	C₂×C₁₃	—	2	1	no	—
183	C₂×C₁₃	—	2	2	no	—
184	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
185	D₁₃	—	2	4	no	—
186	#41	—	13	4	no	—
187	#41	—	13	4	no	—
188	C₂×Q₈	—	13	1	no	—
189	C₂×D₄	—	13	2	no	—
190	C₂×D₄	—	13	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	13	2	no	—
193	C₂²×C₄	—	26	2	no	—
194	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
195	M₁₆	—	2	2	no	—
196	M₁₆	—	13	8	no	—
197	M₁₆	—	26	4	no	—
198	M₁₆	C₄×#261	1	4	yes	—
199	C₂×C₈	—	13	4	no	—
200	C₂×C₈	—	13	1	no	—
201	C₂×C₈	—	13	1	no	—
202	C₂×C₈	—	13	8	no	—
203	C₂×C₈	—	13	1	no	—
204	C₂×C₈	—	13	1	no	—
205	C₂×C₈	—	26	4	no	—
206	C₂×C₈	—	26	2	no	—
207	C₂×C₈	C₂²×#261	1	2	yes	—
208	C₂×C₈	C₂²×#261	1	2	yes	—
209	C₄⋊C₄	—	26	2	no	—
210	C₄⋊C₄	—	26	1	no	—
211	C₂²⋊C₄	—	26	2	no	—
212	C₂²⋊C₄	—	26	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	13	2	no	—
215	C₄²	—	26	2	no	—
216	C₄²	C₂²×#261	1	1	yes	characteristic
217	C₄²	C₂²×#261	1	1	yes	characteristic
218	C₁₆	—	13	16	no	—
219	C₁₃	#33329	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	13	2	no	—
221	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
222	Q₈	—	13	4	no	—
223	D₄	—	13	8	no	—
224	D₄	—	13	4	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	8	1	no	—
228	C₂×C₄	—	13	2	no	—
229	C₂×C₄	—	13	4	no	—
230	C₂×C₄	—	13	4	no	—
231	C₂×C₄	—	26	2	no	—
232	C₂×C₄	—	104	2	no	—
233	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
234	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
235	C₂×C₄	C₂×C₄×#261	1	2	yes	—
236	C₂×C₄	C₂³×#261	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	13	2	no	—
239	C₈	—	13	2	no	—
240	C₈	—	26	1	no	—
241	C₈	—	26	1	no	—
242	C₈	C₂×C₄×#261	1	4	yes	—
243	C₂²	—	4	1	no	—
244	C₂²	—	13	4	no	—
245	C₂²	—	52	2	no	—
246	C₂²	#11719047	1	2	yes	—
247	C₂²	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
248	C₄	—	4	1	no	—
249	C₄	—	4	1	no	—
250	C₄	—	13	4	no	—
251	C₄	—	52	2	no	—
252	C₄	#11719047	1	2	yes	—
253	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
254	C₄	C₂²×C₄×#261	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	2	no	—
257	C₂	—	26	4	no	—
258	C₂	C₂×#11719047	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×#261
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#261
Fitting	C₄×C₁₃×M₁₆	C₄
socle	C₂×C₁₃	C₂²×#97
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	C₄×M₁₆	#261
hypocenter	C₁₃	#33329
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13131059	C₂	semidirect
3. #13131058	C₂	semidirect
4. #13131053	C₂	semidirect
5. C₂×#11718232	C₂	semidirect
11. #11719096	C₄	semidirect
42. #88106	C₂×C₄	semidirect
44. C₂×#6092	C₂×C₄	semidirect
45. #88098	C₂×C₄	semidirect
46. D₁₃⋊C₁₆	C₂×C₄	semidirect
219. C₁₃	#33329	semidirect

Maximal subgroups

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

subgroup	index
#13131805	2
#13131059	2
#13131058	2
#13131053	2
C₂×#11718232	2
#13123265	2
#33329	13