Subgroups — #15867727

Order 3328.

854 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	C₂×#11718586	C₂	1	1	yes	maximal, characteristic
2	#13117234	C₂	1	1	yes	maximal, characteristic
3	#13115554	C₂	1	4	yes	maximal, complemented
4	#13115019	C₂	1	4	yes	maximal
5	#13114696	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×M₁₆	—	2	4	no	—
10	C₂×C₁₃×#97	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
12	#11718873	C₄	1	2	yes	—
13	#11718872	C₄	1	2	yes	—
14	C₄×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊#1055	C₂²	1	2	yes	—
16	C₂×#88054	C₂²	1	8	yes	—
17	#11718598	C₄	1	2	yes	—
18	#11718597	C₄	1	2	yes	—
19	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
20	#11718593	C₂²	1	2	yes	—
21	#11718586	—	2	4	no	—
22	C₂×C₄⋊#1055	C₂²	1	2	yes	—
23	C₂×C₄×#1055	C₂²	1	2	yes	—
24	#11718142	C₂²	1	8	yes	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₃×M₁₆	—	2	4	no	—
29	C₂×C₁₃×M₁₆	—	2	8	no	—
30	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	4	yes	—
34	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₁₃	C₂³	1	4	yes	—
40	C₂²×#1055	C₂³	1	4	yes	—
41	C₁₃×#97	—	2	4	no	—
42	C₄×C₈×C₁₃	—	2	4	no	—
43	#88071	C₂×C₄	1	4	yes	—
44	#88071	C₂×C₄	1	4	yes	—
45	#88071	C₂×C₄	1	4	yes	—
46	#88071	C₂×C₄	1	4	yes	—
47	C₂²⋊#1055	—	2	4	no	—
48	#88054	—	2	16	no	—
49	C₄⋊#1055	—	2	4	no	—
50	C₄×#1055	—	2	4	no	—
51	#21354	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	4	no	—
54	C₂²×C₄×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	—	4	2	no	—
56	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₂²×C₄	1	8	yes	—
67	C₁₃×M₁₆	—	2	16	no	—
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	#41	1	8	yes	—
71	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
72	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
73	C₄²×C₁₃	C₂×D₄	1	2	yes	—
74	C₄²×C₁₃	C₂×D₄	1	2	yes	—
75	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
76	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
77	C₂²⋊Dic₁₃	—	4	4	no	—
78	C₄⋊Dic₁₃	—	2	8	no	—
79	C₄×Dic₁₃	—	2	8	no	—
80	C₂×#1055	—	2	4	no	—
81	C₂×#1055	—	2	8	no	—
82	C₂×#431	—	13	1	no	—
83	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
84	#2283	—	13	1	no	—
85	#1939	—	13	4	no	—
86	#1851	—	13	4	no	—
87	#1768	—	13	4	no	—
88	C₂³×C₁₃	—	2	2	no	—
89	C₂³×C₁₃	—	2	2	no	—
90	C₂³×C₁₃	—	2	2	no	—
91	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	4	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	8	no	—
100	C₂×C₄×C₁₃	—	4	4	no	—
101	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₃	#131	1	4	yes	—
107	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
110	C₂×Dic₁₃	—	2	8	no	—
111	C₂×Dic₁₃	—	2	8	no	—
112	C₂×Dic₁₃	—	2	8	no	—
113	C₈×C₁₃	—	2	8	no	—
114	#1055	—	4	4	no	—
115	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
116	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
117	#431	—	26	4	no	—
118	C₂×C₄⋊C₈	—	13	8	no	—
119	C₂×C₄⋊C₈	—	13	2	no	—
120	C₂×C₂²⋊C₈	—	13	2	no	—
121	C₄×M₁₆	—	2	4	no	—
122	C₂×#97	C₂×D₁₃	1	2	yes	—
123	C₂×C₄×C₈	—	13	2	no	—
124	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
125	#383	—	13	2	no	—
126	#382	—	13	2	no	—
127	#381	—	13	2	no	—
128	#380	—	13	2	no	—
129	C₄×C₄⋊C₄	—	13	2	no	—
130	C₄×C₂²⋊C₄	—	13	2	no	—
131	#376	—	13	2	no	—
132	#336	—	13	8	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	2	2	no	—
136	C₂²×C₁₃	—	4	4	no	—
137	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₁₃	C₄×#41	1	2	yes	—
139	C₂²×C₁₃	C₂×#131	1	2	yes	—
140	C₂²×C₁₃	#568	1	2	yes	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	—	2	2	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	#424	1	4	yes	—
146	C₄×C₁₃	#525	1	2	yes	—
147	C₄×C₁₃	#533	1	2	yes	—
148	Dic₁₃	—	4	8	no	—
149	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
150	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
151	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	—	2	8	no	—
154	C₂²×C₈	—	13	4	no	—
155	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
156	C₂×C₄⋊C₄	—	13	4	no	—
157	C₂×C₂²⋊C₄	—	13	4	no	—
158	C₂×C₄²	—	13	4	no	—
159	C₂×C₄²	Dic₂₆	1	4	yes	—
160	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
161	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
162	C₂×C₄²	C₂²⋊D₁₃	1	4	yes	—
163	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
164	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
165	C₄⋊C₈	—	26	16	no	—
166	C₄⋊C₈	—	26	4	no	—
167	C₂²⋊C₈	—	26	4	no	—
168	#97	—	2	4	no	—
169	C₄×C₈	—	2	4	no	—
170	C₄×C₈	—	26	4	no	—
171	#95	—	13	4	no	—
172	#95	—	13	4	no	—
173	#95	—	13	4	no	—
174	#95	—	13	4	no	—
175	C₂×C₁₃	—	4	2	no	—
176	C₂×C₁₃	#2296	1	1	yes	characteristic
177	C₂×C₁₃	C₂×#424	1	1	yes	characteristic
178	C₂×C₁₃	#2905	1	1	yes	characteristic
179	C₂×C₁₃	#2969	1	4	yes	—
180	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	13	8	no	—
185	C₂²×C₄	#6101	1	4	yes	—
186	C₂²×C₄	#6101	1	4	yes	—
187	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
188	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
194	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
195	M₁₆	—	2	16	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	26	8	no	—
199	C₂×C₈	—	26	4	no	—
200	C₂×C₈	#6127	1	8	yes	—
201	C₄⋊C₄	—	26	8	no	—
202	C₂²⋊C₄	—	52	4	no	—
203	C₄²	—	26	8	no	—
204	C₄²	#6101	1	4	yes	—
205	C₄²	#6101	1	4	yes	—
206	C₄²	C₂×Dic₂₆	1	2	yes	—
207	C₄²	C₂×Dic₂₆	1	2	yes	—
208	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
209	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
210	C₁₃	#21354	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×#6101	1	2	yes	—
215	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	26	8	no	—
225	C₂×C₄	—	26	8	no	—
226	C₂×C₄	—	26	8	no	—
227	C₂×C₄	#88161	1	4	yes	—
228	C₂×C₄	C₂×#6101	1	2	yes	—
229	C₂×C₄	C₂×#6101	1	2	yes	—
230	C₂×C₄	C₂×#6101	1	2	yes	—
231	C₂×C₄	C₂×#6101	1	2	yes	—
232	C₂×C₄	C₂×#6101	1	2	yes	—
233	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
234	C₂×C₄	C₂×#6127	1	4	yes	—
235	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
236	C₈	—	2	8	no	—
237	C₈	—	52	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	4	no	—
242	C₂²	C₄×#6127	1	2	yes	—
243	C₂²	C₂×#88161	1	2	yes	—
244	C₂²	#11719450	1	2	yes	—
245	C₂²	C₂²×#6101	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	52	8	no	—
251	C₄	#11718780	1	4	yes	—
252	C₄	#11719175	1	2	yes	—
253	C₄	#11719496	1	2	yes	—
254	C₂	—	4	2	no	—
255	C₂	#13117319	1	1	yes	characteristic
256	C₂	C₂×#11718780	1	1	yes	characteristic
257	C₂	#13123399	1	1	yes	characteristic
258	C₂	#13123840	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator, socle	C₂³×C₁₃	C₂³×C₄
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	#21354
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13115554	C₂	semidirect
5. #13114696	C₂	semidirect
210. C₁₃	#21354	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718586	2
#13117234	2
#13115554	2
#13115019	2
#13114696	2
C₂×C₄×C₁₃×M₁₆	2
#21354	13