Subgroups — #16726620

Order 3328.

2564 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719524	C₂	1	1	yes	maximal, characteristic
2	C₂×#11719524	C₂	1	1	yes	maximal, characteristic
3	#13123445	C₂	1	16	yes	maximal, complemented
4	C₂×#11718785	C₂	1	12	yes	maximal, complemented
5	C₂×C₁₃×#568	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×#41	C₂²	1	1	yes	characteristic
7	C₁₃×#568	—	2	4	no	—
8	C₁₃×#568	C₂²	1	8	yes	—
9	C₂×C₁₃×#131	—	2	4	no	—
10	C₂²×C₁₃×M₁₆	C₂²	1	6	yes	—
11	#11719524	—	2	4	no	—
12	#11719524	C₄	1	8	yes	—
13	#11719524	C₄	1	8	yes	—
14	#11719524	C₂²	1	8	yes	—
15	C₂×#88202	—	2	4	no	—
16	C₂×#88202	C₂²	1	8	yes	—
17	C₂²×#6099	C₂²	1	6	yes	—
18	C₂²×#6099	C₂²	1	6	yes	—
19	#11718785	C₂²	1	16	yes	—
20	#11718785	C₂²	1	48	yes	complemented
21	C₂×#88062	C₂²	1	48	yes	complemented
22	C₂×C₁₃×#41	C₂×C₄	1	4	yes	—
23	C₂×C₁₃×#41	C₂×C₄	1	4	yes	—
24	C₂×C₁₃×#41	D₄	1	4	yes	—
25	C₂×C₁₃×#41	D₄	1	4	yes	—
26	C₂×C₁₃×#41	Q₈	1	4	yes	—
27	C₂×C₁₃×#41	C₂³	1	4	yes	—
28	C₂²×Q₈×C₁₃	Q₈	1	1	yes	characteristic
29	C₂²×D₄×C₁₃	Q₈	1	3	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
31	C₁₃×#131	—	2	32	no	—
32	C₂×C₁₃×M₁₆	—	2	6	no	—
33	C₂×C₁₃×M₁₆	—	2	12	no	—
34	C₂×C₁₃×M₁₆	C₂³	1	24	yes	—
35	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₃	—	2	6	no	—
37	#88202	—	2	32	no	—
38	#88202	C₂×C₄	1	64	yes	—
39	C₂×#6099	—	2	6	no	—
40	C₂×#6099	—	2	12	no	—
41	C₂×#6099	C₂×C₄	1	24	yes	—
42	C₂×#6099	C₂×C₄	1	24	yes	—
43	C₂×#6099	C₂³	1	4	yes	—
44	C₂×#6099	C₂³	1	24	yes	—
45	C₂×#6099	C₂³	1	4	yes	—
46	C₂×#6099	C₂³	1	12	yes	—
47	C₂²×#1055	—	2	6	no	—
48	C₂²×#1055	C₂³	1	12	yes	—
49	#88062	C₂³	1	64	yes	—
50	#60113	—	13	1	no	maximal, Sylow
51	C₂⁴×C₁₃	C₂×Q₈	1	3	yes	—
52	C₁₃×#41	C₄⋊C₄	1	16	yes	—
53	C₁₃×#41	C₄⋊C₄	1	16	yes	—
54	C₁₃×#41	C₂²×C₄	1	16	yes	—
55	C₁₃×#41	C₂×D₄	1	16	yes	—
56	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
57	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
58	C₂×Q₈×C₁₃	C₂×Q₈	1	4	yes	—
59	C₂×D₄×C₁₃	C₄⋊C₄	1	12	yes	—
60	C₂×D₄×C₁₃	C₄⋊C₄	1	12	yes	—
61	C₂×D₄×C₁₃	C₂×Q₈	1	12	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	6	yes	—
66	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
67	C₂²×C₄×C₁₃	C₂×Q₈	1	6	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	—	2	24	no	—
72	C₁₃×M₁₆	C₂⁴	1	8	yes	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	24	no	—
75	#6099	—	2	4	no	—
76	#6099	—	2	24	no	—
77	#6099	C₂²×C₄	1	8	yes	—
78	#6099	C₂²×C₄	1	8	yes	—
79	#6099	C₂²×C₄	1	48	yes	—
80	#6099	C₂⁴	1	8	yes	—
81	C₂×#1055	—	2	4	no	—
82	C₂×#1055	—	2	24	no	—
83	C₂×#1055	C₂²×C₄	1	48	yes	—
84	C₂×#1055	C₂⁴	1	8	yes	—
85	C₂×#568	—	13	1	no	—
86	C₂×#568	—	13	1	no	—
87	C₂×#568	D₁₃	1	1	yes	characteristic
88	#2916	—	13	16	no	—
89	C₂×#430	—	13	12	no	—
90	C₂³×C₁₃	—	8	3	no	—
91	C₂³×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
92	C₂³×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
93	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
94	C₂³×C₁₃	C₂²×Q₈	1	6	yes	—
95	Q₈×C₁₃	C₂×C₄⋊C₄	1	16	yes	—
96	D₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
97	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
98	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
101	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
102	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
106	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
109	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
110	C₈×C₁₃	—	2	8	no	—
111	#1055	—	2	8	no	—
112	#1055	C₂³×C₄	1	16	yes	—
113	C₂²×#41	C₂×D₁₃	1	1	yes	characteristic
114	#568	—	2	4	no	—
115	#568	—	13	8	no	—
116	#568	—	13	8	no	—
117	#568	—	13	8	no	—
118	#568	—	26	4	no	—
119	#568	C₂×D₁₃	1	8	yes	—
120	C₂×#131	—	2	4	no	—
121	C₂×#131	—	13	8	no	—
122	C₂×#131	—	26	4	no	—
123	C₂²×M₁₆	—	13	6	no	—
124	C₂²×M₁₆	—	13	6	no	—
125	C₂²×M₁₆	C₂×D₁₃	1	6	yes	—
126	#430	—	13	16	no	—
127	#430	—	13	48	no	—
128	C₂×#108	—	13	48	no	—
129	C₂²×C₁₃	—	4	1	no	—
130	C₂²×C₁₃	—	4	6	no	—
131	C₂²×C₁₃	—	4	6	no	—
132	C₂²×C₁₃	—	4	6	no	—
133	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
136	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
137	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
140	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
141	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
142	C₂×#41	Dic₂₆	1	4	yes	—
143	C₂×#41	C₄×D₁₃	1	4	yes	—
144	C₂×#41	C₄×D₁₃	1	4	yes	—
145	C₂×#41	C₂²⋊D₁₃	1	4	yes	—
146	C₂×#41	C₂²⋊D₁₃	1	4	yes	—
147	C₂×#41	C₂²×D₁₃	1	4	yes	—
148	C₂²×Q₈	Dic₂₆	1	1	yes	characteristic
149	C₂²×D₄	Dic₂₆	1	3	yes	—
150	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
151	#131	—	2	32	no	—
152	#131	—	13	64	no	—
153	#131	—	26	32	no	—
154	C₂×M₁₆	—	2	6	no	—
155	C₂×M₁₆	—	2	12	no	—
156	C₂×M₁₆	—	13	24	no	—
157	C₂×M₁₆	—	13	24	no	—
158	C₂×M₁₆	—	13	4	no	—
159	C₂×M₁₆	—	13	12	no	—
160	C₂×M₁₆	—	13	4	no	—
161	C₂×M₁₆	—	13	24	no	—
162	C₂×M₁₆	—	26	6	no	—
163	C₂×M₁₆	—	26	12	no	—
164	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
165	C₂×M₁₆	C₂²×D₁₃	1	24	yes	—
166	C₂²×C₈	—	2	6	no	—
167	C₂²×C₈	—	13	12	no	—
168	C₂²×C₈	—	26	6	no	—
169	#108	—	13	64	no	—
170	C₂×C₁₃	—	2	1	no	—
171	C₂×C₁₃	—	2	1	no	—
172	C₂×C₁₃	—	2	1	no	—
173	C₂×C₁₃	—	2	12	no	—
174	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
175	C₂⁴	C₂×Dic₂₆	1	3	yes	—
176	#41	#6101	1	16	yes	—
177	#41	#6101	1	16	yes	—
178	#41	C₂×C₄×D₁₃	1	16	yes	—
179	#41	C₂×C₂²⋊D₁₃	1	16	yes	—
180	C₂×Q₈	#6101	1	4	yes	—
181	C₂×Q₈	#6101	1	4	yes	—
182	C₂×Q₈	C₂×Dic₂₆	1	4	yes	—
183	C₂×D₄	#6101	1	12	yes	—
184	C₂×D₄	#6101	1	12	yes	—
185	C₂×D₄	C₂×Dic₂₆	1	12	yes	—
186	C₂²×C₄	C₂×Dic₂₆	1	3	yes	—
187	C₂²×C₄	C₂×Dic₂₆	1	6	yes	—
188	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
189	C₂²×C₄	C₂×C₄×D₁₃	1	6	yes	—
190	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
192	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
193	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
194	M₁₆	—	2	4	no	—
195	M₁₆	—	2	24	no	—
196	M₁₆	—	13	8	no	—
197	M₁₆	—	13	8	no	—
198	M₁₆	—	13	48	no	—
199	M₁₆	—	13	8	no	—
200	M₁₆	—	26	4	no	—
201	M₁₆	—	26	24	no	—
202	M₁₆	C₂³×D₁₃	1	8	yes	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	24	no	—
205	C₂×C₈	—	13	48	no	—
206	C₂×C₈	—	13	8	no	—
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	—	26	24	no	—
209	C₁₃	#60113	1	1	yes	characteristic, complemented, Sylow
210	C₂³	—	8	3	no	—
211	C₂³	C₂×#6101	1	6	yes	—
212	C₂³	C₂×#6101	1	6	yes	—
213	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
214	C₂³	C₂²×Dic₂₆	1	6	yes	—
215	Q₈	C₂×#6101	1	16	yes	—
216	D₄	C₂×#6101	1	48	yes	—
217	C₂×C₄	C₂×#6101	1	6	yes	—
218	C₂×C₄	C₂×#6101	1	6	yes	—
219	C₂×C₄	C₂×#6101	1	12	yes	—
220	C₂×C₄	C₂×#6101	1	12	yes	—
221	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
222	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
223	C₂×C₄	C₂²×C₄×D₁₃	1	12	yes	—
224	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
226	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₂²⋊D₁₃	1	12	yes	—
228	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
229	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
230	C₈	—	2	8	no	—
231	C₈	—	13	16	no	—
232	C₈	—	26	8	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	6	no	—
235	C₂²	—	4	6	no	—
236	C₂²	—	4	6	no	—
237	C₂²	C₂²×#6101	1	1	yes	characteristic
238	C₂²	C₂²×#6101	1	1	yes	characteristic
239	C₂²	C₂²×#6101	1	12	yes	—
240	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
241	C₄	C₂²×#6101	1	1	yes	characteristic
242	C₄	C₂²×#6101	1	1	yes	characteristic
243	C₄	C₂²×#6101	1	12	yes	—
244	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
245	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	1	no	—
248	C₂	—	2	1	no	—
249	C₂	—	2	12	no	—
250	C₂	C₂³×#6101	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂³×C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#568	C₂
socle	C₂×C₁₃	C₂³×C₄⋊C₄
prime radical	C₂²×C₁₃×#41	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂×#568	D₁₃
hypocenter	C₁₃	#60113
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
3. #13123445	C₂	semidirect
4. C₂×#11718785	C₂	semidirect
20. #11718785	C₂²	semidirect
21. C₂×#88062	C₂²	semidirect
209. C₁₃	#60113	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719524	2
#13123445	2
C₂×#11718785	2
C₂×C₁₃×#568	2
#60113	13