Subgroups — #16807117

Order 3328.

528 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131212	C₂	1	4	yes	maximal, complemented
2	#13131208	C₂	1	4	yes	maximal, complemented
3	C₂×#11718233	C₂	1	2	yes	maximal, complemented
4	#13131074	C₂	1	2	yes	maximal
5	#13131059	C₂	1	2	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	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
12	#11718783	C₄	1	1	yes	characteristic
13	#11718783	C₂²	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	#11718369	C₄	1	2	yes	—
16	#11718369	C₂²	1	2	yes	—
17	C₄×#6094	C₂²	1	2	yes	—
18	C₄×C₈×D₁₃	C₄	1	2	yes	—
19	#11718263	C₂²	1	8	yes	—
20	#11718262	C₂²	1	8	yes	—
21	#11718233	C₂²	1	4	yes	—
22	#11718233	C₂²	1	4	yes	—
23	C₂×#6127	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₂²⋊D₁₃	—	2	2	no	—
28	#88180	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂³	1	1	yes	characteristic
31	#88161	—	2	4	no	—
32	#88161	—	2	2	no	—
33	#88161	—	2	2	no	—
34	C₂×#6094	C₂×C₄	1	2	yes	—
35	C₂×#6094	C₂³	1	2	yes	—
36	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₁₃	C₂×C₄	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	—	2	4	no	—
43	#88106	C₂³	1	8	yes	—
44	C₂×#6092	—	2	4	no	—
45	C₁₃×#97	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
47	#88055	C₂×C₄	1	2	yes	—
48	#88055	C₂³	1	2	yes	—
49	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
50	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
51	C₄×#1055	C₂×C₄	1	1	yes	characteristic
52	C₄×#1055	C₂×C₄	1	1	yes	characteristic
53	C₄×#1055	C₂³	1	1	yes	characteristic
54	C₄×#1055	C₂³	1	1	yes	characteristic
55	#33428	—	13	1	no	maximal, Sylow
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₁₃	M₁₆	1	2	yes	—
59	#6127	—	2	2	no	—
60	#6127	—	2	2	no	—
61	C₂×D₅₂	M₁₆	1	1	yes	characteristic
62	C₂×C₄×D₁₃	—	2	2	no	—
63	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
64	C₂×Dic₂₆	M₁₆	1	1	yes	characteristic
65	C₁₃×M₁₆	—	2	2	no	—
66	C₁₃×M₁₆	—	2	2	no	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₁₃	—	2	1	no	—
70	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²⋊Dic₁₃	—	2	1	no	—
73	#6103	—	2	2	no	—
74	C₄⋊Dic₁₃	—	2	1	no	—
75	#6101	—	2	2	no	—
76	C₄×Dic₁₃	—	2	2	no	—
77	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
78	C₂×#1055	M₁₆	1	4	yes	—
79	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
80	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
81	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
82	C₂×#1055	#41	1	2	yes	—
83	C₂×#1055	#41	1	2	yes	—
84	C₂×#1055	C₂⁴	1	1	yes	characteristic
85	#6094	—	2	4	no	—
86	#6094	—	2	4	no	—
87	C₈×D₁₃	—	2	2	no	—
88	C₈×D₁₃	—	2	2	no	—
89	C₈×D₁₃	—	2	4	no	—
90	#6092	—	2	8	no	—
91	C₄×#131	—	13	1	no	—
92	#2166	—	13	4	no	—
93	#2165	—	13	4	no	—
94	C₂×#364	—	13	2	no	—
95	#2115	—	13	2	no	—
96	#2110	—	13	2	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	—	4	2	no	—
102	C₂×C₄×C₁₃	#117	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	—	2	4	no	—
107	C₂×Dic₁₃	—	2	2	no	—
108	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
109	D₅₂	—	2	2	no	—
110	C₄×D₁₃	—	2	2	no	—
111	C₄×D₁₃	—	2	2	no	—
112	C₄×D₁₃	—	4	4	no	—
113	Dic₂₆	—	2	2	no	—
114	C₈×C₁₃	—	2	2	no	—
115	C₈×C₁₃	—	2	2	no	—
116	#1055	#117	1	1	yes	characteristic
117	#1055	#117	1	1	yes	characteristic
118	#1055	C₂×M₁₆	1	2	yes	—
119	#1055	C₂×M₁₆	1	2	yes	—
120	#1055	C₂×#41	1	2	yes	—
121	C₂×#131	—	13	1	no	—
122	C₂×#131	—	13	1	no	—
123	C₄×#41	—	13	1	no	—
124	C₂×M₃₂	—	13	4	no	—
125	#405	—	13	1	no	—
126	#405	—	13	1	no	—
127	#405	—	13	2	no	—
128	#405	—	13	2	no	—
129	C₄×M₁₆	—	13	2	no	—
130	C₄×M₁₆	#261	1	1	yes	characteristic
131	C₂×C₄×C₈	—	13	1	no	—
132	C₂×C₄×C₈	—	13	2	no	—
133	#364	—	13	4	no	—
134	#364	—	13	4	no	—
135	#349	—	13	8	no	—
136	#347	—	13	8	no	—
137	C₂²×C₁₃	—	4	1	no	—
138	C₂²×C₁₃	#432	1	2	yes	—
139	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
140	C₂×D₁₃	—	2	2	no	—
141	C₂×D₁₃	—	4	2	no	—
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	—	4	1	no	—
144	C₄×C₁₃	#432	1	2	yes	—
145	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
146	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
147	Dic₁₃	—	2	2	no	—
148	Dic₁₃	—	4	2	no	—
149	C₂×#41	—	13	1	no	—
150	#131	—	26	4	no	—
151	#131	—	26	2	no	—
152	#131	—	26	2	no	—
153	C₂×M₁₆	—	13	2	no	—
154	C₂×M₁₆	—	13	2	no	—
155	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
156	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
157	C₂²×C₈	—	13	1	no	—
158	C₂²×C₈	—	13	1	no	—
159	C₂²×C₈	—	13	2	no	—
160	C₂²×C₈	—	13	2	no	—
161	C₄×Q₈	—	26	1	no	—
162	C₄×D₄	—	26	1	no	—
163	C₄×D₄	—	26	2	no	—
164	#117	—	13	1	no	—
165	#117	—	13	2	no	—
166	C₂×C₄²	—	13	2	no	—
167	C₂×C₄²	C₂×#261	1	1	yes	characteristic
168	M₃₂	—	13	8	no	—
169	M₃₂	—	26	4	no	—
170	C₂×C₁₆	—	26	4	no	—
171	#97	—	13	2	no	—
172	#97	—	13	2	no	—
173	#97	C₂×#261	1	2	yes	—
174	C₄×C₈	—	13	2	no	—
175	C₄×C₈	—	13	2	no	—
176	C₄×C₈	—	13	1	no	—
177	C₄×C₈	—	13	1	no	—
178	C₄×C₈	—	13	1	no	—
179	C₄×C₈	—	13	1	no	—
180	C₄×C₈	C₂×#261	1	2	yes	—
181	C₂×C₁₃	—	2	1	no	—
182	C₂×C₁₃	—	2	2	no	—
183	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
184	D₁₃	—	4	2	no	—
185	#41	—	26	2	no	—
186	#41	—	26	2	no	—
187	C₂×Q₈	—	13	1	no	—
188	C₂×D₄	—	13	2	no	—
189	C₂×D₄	—	13	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	13	2	no	—
192	C₂²×C₄	—	26	2	no	—
193	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
194	M₁₆	—	2	2	no	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	26	4	no	—
197	M₁₆	—	26	4	no	—
198	C₂×C₈	—	13	4	no	—
199	C₂×C₈	—	13	2	no	—
200	C₂×C₈	—	13	2	no	—
201	C₂×C₈	—	13	1	no	—
202	C₂×C₈	—	13	1	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	—	26	2	no	—
208	C₂×C₈	C₂²×#261	1	2	yes	—
209	C₂×C₈	C₂²×#261	1	2	yes	—
210	C₄⋊C₄	—	26	2	no	—
211	C₄⋊C₄	—	26	1	no	—
212	C₂²⋊C₄	—	26	2	no	—
213	C₂²⋊C₄	—	26	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	13	2	no	—
216	C₄²	—	26	2	no	—
217	C₄²	C₂²×#261	1	1	yes	characteristic
218	C₄²	C₂²×#261	1	1	yes	characteristic
219	C₁₆	—	26	8	no	—
220	C₁₃	#33428	1	1	yes	characteristic, complemented, Sylow
221	C₂³	—	13	2	no	—
222	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
223	Q₈	—	26	2	no	—
224	D₄	—	26	4	no	—
225	D₄	—	26	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	13	2	no	—
230	C₂×C₄	—	26	2	no	—
231	C₂×C₄	—	26	2	no	—
232	C₂×C₄	—	26	2	no	—
233	C₂×C₄	—	52	4	no	—
234	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
235	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
236	C₂×C₄	#88237	1	2	yes	—
237	C₂×C₄	C₂³×#261	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	—	13	2	no	—
241	C₈	—	13	2	no	—
242	C₈	—	13	2	no	—
243	C₈	—	13	1	no	—
244	C₈	—	13	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	26	2	no	—
247	C₂²	—	52	2	no	—
248	C₂²	#11719055	1	2	yes	—
249	C₂²	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	4	1	no	—
252	C₄	—	26	2	no	—
253	C₄	—	52	2	no	—
254	C₄	#11719055	1	2	yes	—
255	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
256	C₄	C₂×#88237	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	2	no	—
259	C₂	—	52	2	no	—
260	C₂	C₂×#11719055	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88237
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₄×C₁₃×M₁₆	C₄
socle	C₂×C₁₃	C₂×#432
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	C₄×M₁₆	#261
hypocenter	C₁₃	#33428
LCS	G ▷ C₂×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
1. #13131212	C₂	semidirect
2. #13131208	C₂	semidirect
3. C₂×#11718233	C₂	semidirect
5. #13131059	C₂	semidirect
220. C₁₃	#33428	semidirect

Maximal subgroups

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

subgroup	index
#13131212	2
#13131208	2
C₂×#11718233	2
#13131074	2
#13131059	2
C₄×#88161	2
#33428	13