Subgroups — #15881928

Order 3328.

622 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₂₆	C₂	1	2	yes	maximal
2	C₂×#11718340	C₂	1	1	yes	maximal, characteristic
3	#13115611	C₂	1	2	yes	maximal
4	#13115608	C₂	1	4	yes	maximal
5	#13114975	C₂	1	2	yes	maximal
6	#13114974	C₂	1	1	yes	maximal, characteristic
7	#13114741	C₂	1	2	yes	maximal
8	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#108	C₂²	1	2	yes	—
11	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88062	C₂²	1	4	yes	—
14	#11718779	C₂²	1	4	yes	—
15	M₁₆×Dic₁₃	C₂²	1	4	yes	—
16	C₂×#88055	C₂²	1	2	yes	—
17	C₂×#88054	C₂²	1	2	yes	—
18	#11718585	C₂²	1	2	yes	—
19	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
20	#11718362	—	2	4	no	—
21	#11718340	—	2	2	no	—
22	#11718340	C₄	1	4	yes	—
23	#11718340	C₄	1	4	yes	—
24	C₈×Dic₂₆	—	2	2	no	—
25	C₁₃×#341	C₂²	1	1	yes	characteristic
26	C₁₃×#341	C₂²	1	1	yes	characteristic
27	#11718150	C₂²	1	4	yes	—
28	#11718144	C₂²	1	2	yes	—
29	#11718144	C₂²	1	2	yes	—
30	C₂²×Dic₂₆	Q₈	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
32	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
37	C₂×#6101	Q₈	1	2	yes	—
38	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
39	C₂×#6099	C₂³	1	4	yes	—
40	C₂²×#1055	C₂³	1	2	yes	—
41	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
42	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
43	C₄×Dic₂₆	D₄	1	2	yes	—
44	C₄×Dic₂₆	D₄	1	2	yes	—
45	C₁₃×#108	—	2	4	no	—
46	C₁₃×#97	—	2	2	no	—
47	C₄×C₈×C₁₃	—	2	2	no	—
48	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
50	#88062	—	2	8	no	—
51	#88055	—	2	4	no	—
52	#88055	C₂×C₄	1	8	yes	—
53	#88054	—	2	4	no	—
54	#88054	—	2	4	no	—
55	#88054	C₂×C₄	1	8	yes	—
56	C₈×Dic₁₃	—	2	4	no	—
57	C₄⋊#1055	—	2	2	no	—
58	C₄⋊#1055	—	2	2	no	—
59	C₄⋊#1055	C₂×C₄	1	2	yes	—
60	C₄⋊#1055	C₂×C₄	1	2	yes	—
61	#22747	—	13	1	no	maximal, Sylow
62	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
66	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
67	C₂×Dic₂₆	—	4	2	no	—
68	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
69	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
70	C₁₃×M₁₆	—	2	2	no	—
71	C₁₃×M₁₆	—	2	2	no	—
72	C₁₃×M₁₆	—	2	2	no	—
73	C₁₃×M₁₆	#41	1	4	yes	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
85	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
86	#6101	C₄⋊C₄	1	4	yes	—
87	#6101	C₄⋊C₄	1	4	yes	—
88	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
89	C₄×Dic₁₃	C₂×D₄	1	4	yes	—
90	#6099	—	2	4	no	—
91	#6099	—	2	4	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	—	2	4	no	—
94	C₂×#1055	C₂²×C₄	1	4	yes	—
95	Q₈×M₁₆	—	13	2	no	—
96	C₂×#446	—	13	1	no	—
97	#1950	—	13	4	no	—
98	#1950	—	13	2	no	—
99	#1838	—	13	2	no	—
100	#1837	—	13	1	no	—
101	#1778	—	13	2	no	—
102	C₄×#108	D₁₃	1	1	yes	characteristic
103	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	4	1	no	—
106	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
113	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
114	C₂×Dic₁₃	—	4	2	no	—
115	C₂×Dic₁₃	—	4	2	no	—
116	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
118	Dic₂₆	—	2	4	no	—
119	Dic₂₆	—	4	2	no	—
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	—	2	2	no	—
122	C₈×C₁₃	—	2	4	no	—
123	C₈×C₁₃	—	4	2	no	—
124	#1055	—	2	4	no	—
125	#1055	—	4	4	no	—
126	C₂×C₄×Q₈	—	13	1	no	—
127	#446	—	13	4	no	—
128	#446	—	13	4	no	—
129	#446	—	26	2	no	—
130	#446	—	26	4	no	—
131	C₈×Q₈	—	26	2	no	—
132	C₂×#108	—	13	4	no	—
133	C₂×#108	C₂×D₁₃	1	2	yes	—
134	#423	—	13	4	no	—
135	#423	—	13	2	no	—
136	C₂×C₄⋊C₈	—	13	2	no	—
137	C₂×C₄⋊C₈	—	13	1	no	—
138	C₄×M₁₆	—	13	4	no	—
139	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
140	C₂×#97	—	13	2	no	—
141	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
142	#341	—	13	4	no	—
143	#341	—	13	2	no	—
144	#341	—	13	2	no	—
145	#341	C₂×D₁₃	1	1	yes	characteristic
146	#341	C₂×D₁₃	1	1	yes	characteristic
147	C₂²×C₁₃	—	2	1	no	—
148	C₂²×C₁₃	—	2	1	no	—
149	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₁₃	#557	1	2	yes	—
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	—	2	1	no	—
153	C₄×C₁₃	—	2	1	no	—
154	C₄×C₁₃	#430	1	2	yes	—
155	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
156	C₄×C₁₃	#519	1	1	yes	characteristic
157	C₄×C₁₃	#548	1	2	yes	—
158	Dic₁₃	—	2	4	no	—
159	Dic₁₃	—	2	4	no	—
160	C₂²×Q₈	—	13	1	no	—
161	C₂×M₁₆	—	13	4	no	—
162	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
163	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
164	C₂²×C₈	—	13	2	no	—
165	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
167	C₄×Q₈	—	13	2	no	—
168	C₄×Q₈	—	13	2	no	—
169	C₄×Q₈	—	13	2	no	—
170	C₄×Q₈	—	13	2	no	—
171	C₂×C₄⋊C₄	—	13	2	no	—
172	C₂×C₄⋊C₄	—	13	1	no	—
173	C₂×C₄²	—	13	2	no	—
174	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
175	#108	—	2	4	no	—
176	#108	—	26	8	no	—
177	C₄⋊C₈	—	13	8	no	—
178	C₄⋊C₈	—	13	2	no	—
179	C₄⋊C₈	—	13	2	no	—
180	C₄⋊C₈	—	26	4	no	—
181	C₄⋊C₈	—	26	4	no	—
182	C₄⋊C₈	—	26	2	no	—
183	C₄⋊C₈	—	26	2	no	—
184	#97	—	2	2	no	—
185	#97	—	13	8	no	—
186	#97	—	26	4	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	26	4	no	—
189	C₄×C₈	C₄×D₁₃	1	2	yes	—
190	C₄×C₈	C₄×D₁₃	1	2	yes	—
191	C₂×C₁₃	—	2	2	no	—
192	C₂×C₁₃	#2351	1	1	yes	characteristic
193	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
194	C₂×C₁₃	#2916	1	1	yes	characteristic
195	C₂×Q₈	—	13	2	no	—
196	C₂×Q₈	—	13	2	no	—
197	C₂×Q₈	—	52	2	no	—
198	C₂²×C₄	—	13	2	no	—
199	C₂²×C₄	—	13	2	no	—
200	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
201	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
203	M₁₆	—	2	2	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	26	4	no	—
207	M₁₆	—	26	4	no	—
208	M₁₆	#6127	1	4	yes	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	13	4	no	—
214	C₂×C₈	—	26	4	no	—
215	C₂×C₈	—	26	4	no	—
216	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
217	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
218	C₄⋊C₄	—	13	4	no	—
219	C₄⋊C₄	—	13	4	no	—
220	C₄⋊C₄	—	13	2	no	—
221	C₄⋊C₄	—	13	2	no	—
222	C₄²	—	13	4	no	—
223	C₄²	—	13	4	no	—
224	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
225	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
226	C₄²	D₄×D₁₃	1	1	yes	characteristic
227	C₄²	D₄×D₁₃	1	1	yes	characteristic
228	C₁₃	#22747	1	1	yes	characteristic, complemented, Sylow
229	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
230	Q₈	—	26	4	no	—
231	Q₈	—	52	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	13	4	no	—
235	C₂×C₄	—	13	4	no	—
236	C₂×C₄	—	52	2	no	—
237	C₂×C₄	—	52	2	no	—
238	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
239	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
240	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
241	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
243	C₂×C₄	C₂×#6127	1	2	yes	—
244	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
245	C₂×C₄	#88257	1	2	yes	—
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	4	no	—
249	C₈	—	4	2	no	—
250	C₈	—	26	4	no	—
251	C₈	—	52	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
255	C₂²	#11719272	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	26	4	no	—
260	C₄	—	26	4	no	—
261	C₄	#11718554	1	2	yes	—
262	C₄	C₄×#6127	1	1	yes	characteristic
263	C₄	#11719221	1	1	yes	characteristic
264	C₄	#11719257	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	#13117756	1	1	yes	characteristic
267	C₂	C₂×#11718554	1	1	yes	characteristic
268	C₂	#13123444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₃	C₂²×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#108	D₁₃
hypocenter	C₁₃	#22747
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
228. C₁₃	#22747	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₂₆	2
C₂×#11718340	2
#13115611	2
#13115608	2
#13114975	2
#13114974	2
#13114741	2
C₄×C₁₃×#108	2
#22747	13