Subgroups — #15833627

Order 3328.

726 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123548	C₂	1	2	yes	maximal, complemented
2	#13113666	C₂	1	8	yes	maximal, complemented
3	#13113595	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113287	C₂	1	2	yes	maximal, complemented
5	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#532	C₂²	1	2	yes	—
8	C₂×#88202	—	2	2	no	—
9	C₁₃×#500	C₄	1	1	yes	characteristic
10	C₁₃×#493	C₄	1	1	yes	characteristic
11	C₁₃×#486	C₄	1	2	yes	—
12	C₁₃×#466	C₄	1	4	yes	—
13	C₄×C₁₃×D₈	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	#11718895	—	2	2	no	—
16	#11718673	—	2	2	no	—
17	#11718673	C₂²	1	4	yes	—
18	D₄×#1055	—	2	2	no	—
19	D₄×#1055	C₂²	1	4	yes	—
20	#11718585	C₂²	1	2	yes	—
21	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
22	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
23	D₄⋊#1055	C₂²	1	8	yes	—
24	#11718050	C₂²	1	4	yes	—
25	#11718046	C₂²	1	4	yes	—
26	C₂×C₁₃×#41	—	2	2	no	—
27	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
30	C₁₃×#127	C₂×C₄	1	2	yes	—
31	C₁₃×#124	—	2	2	no	—
32	C₁₃×#121	—	2	2	no	—
33	C₁₃×#121	C₂×C₄	1	4	yes	—
34	C₄×D₄×C₁₃	—	2	2	no	—
35	C₄×D₄×C₁₃	C₂³	1	4	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	#88202	—	4	8	no	—
38	C₂×#6099	—	2	2	no	—
39	C₂×#6099	—	2	4	no	—
40	C₂²×#1055	—	2	1	no	—
41	C₂²×#1055	—	2	4	no	—
42	C₂²×#1055	D₄	1	1	yes	characteristic
43	C₂²×#1055	D₄	1	1	yes	characteristic
44	C₁₃×#107	C₂×C₄	1	4	yes	—
45	C₁₃×#102	C₂×C₄	1	8	yes	—
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	C₂³	1	2	yes	—
49	C₂²⋊#1055	—	2	4	no	—
50	C₂²⋊#1055	—	2	4	no	—
51	C₄⋊#1055	—	2	2	no	—
52	C₄⋊#1055	C₂³	1	4	yes	—
53	C₄×#1055	—	2	1	no	—
54	C₄×#1055	D₄	1	2	yes	—
55	C₄×#1055	C₂³	1	2	yes	—
56	C₄×#1055	C₂³	1	2	yes	—
57	#17975	—	13	1	no	maximal, Sylow
58	C₁₃×#41	—	4	4	no	—
59	C₁₃×#41	—	4	4	no	—
60	C₂×Q₈×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	2	4	no	—
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	2	2	no	—
64	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
69	C₁₃×D₈	—	4	4	no	—
70	C₂×C₈×C₁₃	—	4	2	no	—
71	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
73	C₁₃×C₄⋊C₄	—	2	2	no	—
74	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
75	C₁₃×C₂²⋊C₄	—	2	4	no	—
76	C₁₃×C₂²⋊C₄	—	2	4	no	—
77	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₃	#41	1	1	yes	characteristic
80	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
81	#6099	—	4	4	no	—
82	#6099	—	4	8	no	—
83	C₂×#1055	—	2	2	no	—
84	C₂×#1055	—	2	4	no	—
85	C₂×#1055	—	2	2	no	—
86	C₂×#1055	—	4	2	no	—
87	C₂×#1055	—	4	8	no	—
88	C₂×#1055	D₈	1	4	yes	complemented
89	C₂×#1055	C₂×D₄	1	2	yes	—
90	C₂×#1055	C₂×D₄	1	2	yes	—
91	#3169	D₁₃	1	1	yes	characteristic
92	#2931	—	13	2	no	—
93	#1581	—	13	8	no	—
94	#1567	—	13	1	no	—
95	#1492	—	13	2	no	—
96	C₂×#322	—	13	1	no	—
97	C₂³×C₁₃	—	2	4	no	—
98	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
99	Q₈×C₁₃	—	4	4	no	—
100	D₄×C₁₃	—	2	8	no	—
101	D₄×C₁₃	—	4	8	no	—
102	D₄×C₁₃	—	4	4	no	—
103	D₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	4	4	no	—
109	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₁₃	#135	1	2	yes	—
114	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₈×C₁₃	—	2	2	no	—
118	C₈×C₁₃	—	2	2	no	—
119	#1055	—	2	2	no	—
120	#1055	—	4	2	no	—
121	#1055	—	4	4	no	—
122	#1055	C₂×D₈	1	4	yes	—
123	C₂×#131	—	26	2	no	—
124	#532	C₂×D₁₃	1	2	yes	—
125	#500	Dic₁₃	1	1	yes	characteristic
126	#493	Dic₁₃	1	1	yes	characteristic
127	#486	Dic₁₃	1	2	yes	—
128	#466	Dic₁₃	1	4	yes	—
129	C₄×D₈	C₂×D₁₃	1	4	yes	—
130	#436	—	13	4	no	—
131	#436	—	26	2	no	—
132	C₈×D₄	—	13	4	no	—
133	C₈×D₄	—	26	2	no	—
134	#423	—	13	2	no	—
135	#408	—	26	2	no	—
136	C₂×C₄×C₈	—	13	1	no	—
137	C₂×C₄×C₈	—	13	1	no	—
138	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
139	#335	—	13	4	no	—
140	D₄⋊C₈	—	13	8	no	—
141	#322	—	13	4	no	—
142	C₂²×C₁₃	—	2	1	no	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	—	4	4	no	—
145	C₂²×C₁₃	—	4	4	no	—
146	C₂²×C₁₃	—	4	4	no	—
147	C₂²×C₁₃	C₄×D₈	1	2	yes	—
148	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	4	4	no	—
152	C₄×C₁₃	C₄×D₈	1	2	yes	—
153	C₄×C₁₃	#444	1	2	yes	—
154	C₄×C₁₃	#568	1	2	yes	—
155	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
156	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
157	C₂×#41	—	2	2	no	—
158	C₂×D₈	C₂×Dic₁₃	1	4	yes	—
159	#131	—	52	8	no	—
160	C₂×M₁₆	—	26	4	no	—
161	C₂×M₁₆	—	26	2	no	—
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	1	no	—
164	C₂²×C₈	—	26	4	no	—
165	C₂²×C₈	—	26	1	no	—
166	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
167	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
168	#127	C₂×Dic₁₃	1	2	yes	—
169	#124	—	2	2	no	—
170	#121	—	2	2	no	—
171	#121	C₂×Dic₁₃	1	4	yes	—
172	C₄×D₄	—	2	2	no	—
173	C₄×D₄	C₂²×D₁₃	1	4	yes	—
174	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
175	#107	C₂×Dic₁₃	1	4	yes	—
176	C₄⋊C₈	—	13	4	no	—
177	C₄⋊C₈	—	26	2	no	—
178	#102	C₂×Dic₁₃	1	8	yes	—
179	C₂²⋊C₈	—	26	4	no	—
180	C₂²⋊C₈	—	26	4	no	—
181	C₄×C₈	—	13	2	no	—
182	C₄×C₈	—	13	2	no	—
183	C₄×C₈	—	13	2	no	—
184	C₄×C₈	—	26	1	no	—
185	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
186	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₁₃	1	2	yes	—
188	C₂×C₁₃	—	2	2	no	—
189	C₂×C₁₃	—	8	4	no	—
190	C₂×C₁₃	#2932	1	1	yes	characteristic
191	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
192	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
193	#41	—	4	4	no	—
194	#41	—	4	4	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	C₂²×Dic₁₃	1	4	yes	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
202	C₂²×C₄	#6129	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
204	D₈	—	4	4	no	—
205	M₁₆	—	52	8	no	—
206	M₁₆	—	52	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	13	4	no	—
209	C₂×C₈	—	13	2	no	—
210	C₂×C₈	—	13	2	no	—
211	C₂×C₈	—	26	2	no	—
212	C₂×C₈	—	26	4	no	—
213	C₂×C₈	—	26	2	no	—
214	C₂×C₈	—	52	8	no	—
215	C₂×C₈	—	52	2	no	—
216	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
217	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
220	C₂²⋊C₄	—	2	4	no	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₄²	D₄×D₁₃	1	1	yes	characteristic
223	C₄²	#6129	1	1	yes	characteristic
224	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
225	C₄²	C₂³×D₁₃	1	1	yes	characteristic
226	C₁₃	#17975	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	2	4	no	—
228	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	D₈×D₁₃	1	2	yes	—
240	C₂×C₄	#88166	1	2	yes	—
241	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
242	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
243	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂×#6129	1	1	yes	characteristic
246	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	13	4	no	—
250	C₈	—	26	2	no	—
251	C₈	—	52	2	no	—
252	C₈	—	52	4	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	—	4	4	no	—
258	C₂²	D₈×Dic₁₃	1	2	yes	—
259	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	4	no	—
263	C₄	D₈×Dic₁₃	1	2	yes	—
264	C₄	#11718858	1	2	yes	—
265	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
266	C₄	C₂×#88166	1	1	yes	characteristic
267	C₄	#11719524	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	8	4	no	—
270	C₂	#13123554	1	1	yes	characteristic
271	C₂	C₂×D₈×Dic₁₃	1	1	yes	characteristic
272	C₂	C₂×#11718858	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting, prime radical	C₁₃×#3169	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#3169	D₁₃
hypocenter	C₁₃	#17975
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #13123548	C₂	semidirect
2. #13113666	C₂	semidirect
3. #13113595	C₂	semidirect
4. #13113287	C₂	semidirect
5. C₂×#11718046	C₂	semidirect
88. C₂×#1055	D₈	semidirect
226. C₁₃	#17975	semidirect

Maximal subgroups

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

subgroup	index
#13123548	2
#13113666	2
#13113595	2
#13113287	2
C₂×#11718046	2
C₁₃×#3169	2
#17975	13