Subgroups — #16023030

Order 3328.

3006 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	C₂×#11719532	C₂	1	2	yes	maximal
2	#13117757	C₂	1	4	yes	maximal, complemented
3	#13117746	C₂	1	12	yes	maximal, complemented
4	#13117459	C₂	1	6	yes	maximal, complemented
5	#13117243	C₂	1	3	yes	maximal, complemented
6	C₂×#11718597	C₂	1	3	yes	maximal, complemented
7	C₂×C₄×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×#41	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×#41	C₄	1	16	yes	complemented
10	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₁₃	C₂²	1	6	yes	—
12	C₂×C₁₃×#117	C₂²	1	3	yes	—
13	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
14	#11719532	—	2	16	no	—
15	C₂×Q₈×Dic₁₃	C₂²	1	4	yes	—
16	C₂×D₄×Dic₁₃	C₂²	1	12	yes	—
17	C₂×#88180	C₂²	1	6	yes	—
18	C₂²×C₄⋊Dic₁₃	C₂²	1	6	yes	—
19	#11718873	C₂²	1	12	yes	—
20	C₂×#88071	C₂²	1	12	yes	—
21	#11718646	C₂²	1	24	yes	—
22	#11718611	C₂²	1	24	yes	—
23	#11718597	C₂²	1	6	yes	—
24	#11718597	C₂²	1	3	yes	—
25	#11718597	C₂²	1	12	yes	—
26	#11718597	C₂²	1	1	yes	characteristic
27	#11718596	C₂²	1	6	yes	—
28	C₄×C₄⋊Dic₁₃	C₂²	1	12	yes	—
29	C₂×C₁₃×#41	C₂×C₄	1	8	yes	—
30	C₂×C₁₃×#41	D₄	1	8	yes	—
31	C₂×C₁₃×#41	Q₈	1	8	yes	—
32	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×D₄×C₁₃	C₂³	1	3	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
36	C₂³×Dic₁₃	C₂³	1	12	yes	—
37	C₄×Q₈×C₁₃	C₂×C₄	1	16	yes	complemented
38	C₄×D₄×C₁₃	C₂×C₄	1	48	yes	complemented
39	C₁₃×#117	C₂×C₄	1	24	yes	complemented
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
41	C₂×C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	24	yes	complemented
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
46	C₂×C₂²⋊Dic₁₃	C₂³	1	12	yes	—
47	Q₈×Dic₁₃	—	2	16	no	—
48	D₄×Dic₁₃	—	2	48	no	—
49	#88180	—	2	24	no	—
50	C₂×C₄⋊Dic₁₃	—	2	24	no	—
51	C₂×C₄⋊Dic₁₃	C₂³	1	6	yes	—
52	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
53	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
54	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
55	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
56	#88071	C₂³	1	4	yes	—
57	#88071	C₂³	1	12	yes	—
58	#88071	C₂³	1	24	yes	—
59	#34184	—	13	1	no	maximal, Sylow
60	C₂⁴×C₁₃	C₂⁴	1	3	yes	—
61	C₁₃×#41	C₄⋊C₄	1	64	yes	complemented
62	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
63	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
64	C₂×Q₈×C₁₃	C₂×Q₈	1	4	yes	—
65	C₂×D₄×C₁₃	C₂²×C₄	1	12	yes	—
66	C₂×D₄×C₁₃	C₂×D₄	1	12	yes	—
67	C₂×D₄×C₁₃	C₂×Q₈	1	12	yes	—
68	C₂²×C₄×C₁₃	—	4	12	no	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	24	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
72	C₂²×C₄×C₁₃	C₂×Q₈	1	12	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
75	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₁₃	—	2	24	no	—
78	C₂²×Dic₁₃	—	4	24	no	—
79	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
80	C₂²×Dic₁₃	C₂⁴	1	4	yes	—
81	C₁₃×C₄⋊C₄	C₂²×C₄	1	24	yes	—
82	C₁₃×C₂²⋊C₄	C₂²×C₄	1	24	yes	—
83	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
84	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
85	C₄²×C₁₃	C₂²×C₄	1	24	yes	—
86	C₂²⋊Dic₁₃	—	2	24	no	—
87	C₄⋊Dic₁₃	—	2	12	no	—
88	C₄⋊Dic₁₃	—	2	24	no	—
89	C₄⋊Dic₁₃	—	2	4	no	—
90	C₄⋊Dic₁₃	—	4	24	no	—
91	C₄×Dic₁₃	—	2	24	no	—
92	C₂×#520	—	13	2	no	—
93	C₂×C₄×#41	D₁₃	1	1	yes	characteristic
94	#2351	—	13	4	no	—
95	#2349	—	13	12	no	—
96	#2317	—	13	6	no	—
97	#2285	—	13	3	no	—
98	C₂×#382	—	13	3	no	—
99	C₂³×C₁₃	—	2	12	no	—
100	C₂³×C₁₃	C₂³×C₄	1	6	yes	—
101	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
102	C₂³×C₁₃	C₂²×Q₈	1	6	yes	—
103	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
104	Q₈×C₁₃	C₂×C₄⋊C₄	1	16	yes	—
105	D₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
106	C₂×C₄×C₁₃	—	2	12	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	4	24	no	—
109	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
110	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
111	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
114	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
115	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
118	C₂×Dic₁₃	—	2	4	no	—
119	C₂×Dic₁₃	—	2	12	no	—
120	C₂×Dic₁₃	—	2	8	no	—
121	C₂×Dic₁₃	—	2	24	no	—
122	C₂×Dic₁₃	—	8	24	no	—
123	C₂²×#41	C₂×D₁₃	1	1	yes	characteristic
124	#520	—	26	16	no	—
125	C₄×#41	Dic₁₃	1	16	yes	complemented
126	C₂×C₄×Q₈	—	13	4	no	—
127	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
128	C₂×C₄×D₄	—	13	12	no	—
129	C₂×C₄×D₄	C₂×D₁₃	1	6	yes	—
130	C₂×#117	—	13	6	no	—
131	C₂×#117	C₂×D₁₃	1	3	yes	—
132	C₂²×C₄⋊C₄	—	13	6	no	—
133	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
134	#389	—	13	24	no	—
135	#385	—	13	24	no	—
136	#384	—	13	6	no	—
137	#382	—	13	6	no	—
138	#382	—	13	12	no	—
139	#382	—	13	3	no	—
140	#382	—	13	1	no	—
141	#380	—	13	12	no	—
142	C₄×C₄⋊C₄	—	13	12	no	—
143	C₂×#95	—	13	12	no	—
144	C₂²×C₁₃	—	2	12	no	—
145	C₂²×C₁₃	—	2	12	no	—
146	C₂²×C₁₃	—	2	12	no	—
147	C₂²×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
148	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
149	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
150	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
151	C₂²×C₁₃	#585	1	4	yes	—
152	C₄×C₁₃	—	2	4	no	—
153	C₄×C₁₃	—	2	12	no	—
154	C₄×C₁₃	C₂²×C₄⋊C₄	1	4	yes	—
155	C₄×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
156	Dic₁₃	—	4	4	no	—
157	Dic₁₃	—	4	12	no	—
158	C₂×#41	Dic₂₆	1	8	yes	—
159	C₂×#41	D₅₂	1	8	yes	—
160	C₂×#41	C₂×Dic₁₃	1	8	yes	—
161	C₂²×Q₈	C₂²×D₁₃	1	1	yes	characteristic
162	C₂²×D₄	C₂²×D₁₃	1	3	yes	—
163	C₂³×C₄	—	13	12	no	—
164	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
165	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
166	C₄×Q₈	—	26	16	no	—
167	C₄×Q₈	C₂×Dic₁₃	1	16	yes	complemented
168	C₄×D₄	—	26	48	no	—
169	C₄×D₄	C₂×Dic₁₃	1	48	yes	complemented
170	#117	—	26	24	no	—
171	#117	C₂×Dic₁₃	1	24	yes	complemented
172	C₂×C₄⋊C₄	—	13	6	no	—
173	C₂×C₄⋊C₄	—	13	12	no	—
174	C₂×C₄⋊C₄	—	13	2	no	—
175	C₂×C₄⋊C₄	—	13	24	no	—
176	C₂×C₄⋊C₄	—	26	24	no	—
177	C₂×C₄⋊C₄	C₂²×D₁₃	1	6	yes	—
178	C₂×C₂²⋊C₄	—	13	12	no	—
179	C₂×C₂²⋊C₄	C₂²×D₁₃	1	6	yes	—
180	C₂×C₄²	—	13	12	no	—
181	C₂×C₄²	C₂×Dic₁₃	1	24	yes	complemented
182	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
183	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
184	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
185	#95	—	13	12	no	—
186	#95	—	13	24	no	—
187	#95	—	13	4	no	—
188	C₂×C₁₃	—	2	12	no	—
189	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₁₃	#3428	1	2	yes	—
191	C₂×C₁₃	#3500	1	2	yes	—
192	C₂×C₁₃	#3504	1	2	yes	—
193	C₂⁴	C₂³×D₁₃	1	3	yes	—
194	#41	C₄⋊Dic₁₃	1	64	yes	complemented
195	C₂×Q₈	C₂×Dic₂₆	1	4	yes	—
196	C₂×Q₈	C₂×D₅₂	1	4	yes	—
197	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
198	C₂×D₄	C₂×Dic₂₆	1	12	yes	—
199	C₂×D₄	C₂×D₅₂	1	12	yes	—
200	C₂×D₄	C₂²×Dic₁₃	1	12	yes	—
201	C₂²×C₄	—	4	12	no	—
202	C₂²×C₄	—	13	12	no	—
203	C₂²×C₄	—	13	4	no	—
204	C₂²×C₄	—	26	24	no	—
205	C₂²×C₄	—	52	24	no	—
206	C₂²×C₄	C₂×Dic₂₆	1	12	yes	—
207	C₂²×C₄	C₂×D₅₂	1	12	yes	—
208	C₂²×C₄	C₂²×Dic₁₃	1	24	yes	—
209	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
210	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
211	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
212	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
213	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
214	C₄⋊C₄	—	26	12	no	—
215	C₄⋊C₄	—	26	24	no	—
216	C₄⋊C₄	—	26	4	no	—
217	C₄⋊C₄	—	52	24	no	—
218	C₄⋊C₄	C₂²×Dic₁₃	1	24	yes	—
219	C₂²⋊C₄	—	26	24	no	—
220	C₂²⋊C₄	C₂²×Dic₁₃	1	24	yes	—
221	C₄²	—	26	24	no	—
222	C₄²	C₂²×Dic₁₃	1	4	yes	—
223	C₄²	C₂²×Dic₁₃	1	12	yes	—
224	C₄²	C₂²×Dic₁₃	1	24	yes	—
225	C₁₃	#34184	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	2	12	no	—
227	C₂³	C₂²×Dic₂₆	1	6	yes	—
228	C₂³	C₂²×D₅₂	1	6	yes	—
229	C₂³	C₂³×Dic₁₃	1	6	yes	—
230	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
231	Q₈	C₂×C₄⋊Dic₁₃	1	16	yes	—
232	D₄	C₂×C₄⋊Dic₁₃	1	48	yes	—
233	C₂×C₄	—	2	12	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	24	no	—
236	C₂×C₄	—	26	4	no	—
237	C₂×C₄	—	26	12	no	—
238	C₂×C₄	—	26	8	no	—
239	C₂×C₄	—	26	24	no	—
240	C₂×C₄	—	104	24	no	—
241	C₂×C₄	C₂×C₄⋊Dic₁₃	1	48	yes	—
242	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
243	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
244	C₂×C₄	C₂²×D₅₂	1	6	yes	—
245	C₂×C₄	C₂²×D₅₂	1	2	yes	—
246	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
247	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
248	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
249	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
250	C₂²	—	2	12	no	—
251	C₂²	—	2	12	no	—
252	C₂²	—	2	12	no	—
253	C₂²	C₂²×C₄⋊Dic₁₃	1	12	yes	—
254	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
255	C₂²	C₂³×D₅₂	1	1	yes	characteristic
256	C₂²	#11719648	1	4	yes	—
257	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	12	no	—
260	C₄	—	52	4	no	—
261	C₄	—	52	12	no	—
262	C₄	C₂²×C₄⋊Dic₁₃	1	4	yes	—
263	C₄	C₂²×C₄⋊Dic₁₃	1	12	yes	—
264	C₂	—	2	12	no	—
265	C₂	C₂³×C₄⋊Dic₁₃	1	1	yes	characteristic
266	C₂	#13128039	1	2	yes	—
267	C₂	#13128629	1	2	yes	—
268	C₂	#13128662	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Fitting	C₂×C₄×C₁₃×#41	C₂
socle, cosocle	C₂³×C₁₃	C₂⁵
prime radical	C₂²×C₁₃×#41	C₂²
hypercenter	C₂×C₄×#41	D₁₃
hypocenter	C₁₃	#34184
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
2. #13117757	C₂	semidirect
3. #13117746	C₂	semidirect
4. #13117459	C₂	semidirect
5. #13117243	C₂	semidirect
6. C₂×#11718597	C₂	semidirect
9. C₄×C₁₃×#41	C₄	semidirect
37. C₄×Q₈×C₁₃	C₂×C₄	semidirect
38. C₄×D₄×C₁₃	C₂×C₄	semidirect
39. C₁₃×#117	C₂×C₄	semidirect
42. C₂×C₄²×C₁₃	C₂×C₄	semidirect
61. C₁₃×#41	C₄⋊C₄	semidirect
125. C₄×#41	Dic₁₃	semidirect
167. C₄×Q₈	C₂×Dic₁₃	semidirect
169. C₄×D₄	C₂×Dic₁₃	semidirect
171. #117	C₂×Dic₁₃	semidirect
181. C₂×C₄²	C₂×Dic₁₃	semidirect
194. #41	C₄⋊Dic₁₃	semidirect
225. C₁₃	#34184	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719532	2
#13117757	2
#13117746	2
#13117459	2
#13117243	2
C₂×#11718597	2
C₂×C₄×C₁₃×#41	2
#34184	13