Subgroups — #16021148

Order 3328.

3598 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×Dic₁₃	C₂	1	2	yes	maximal
2	#13117201	C₂	1	4	yes	maximal, complemented
3	#13117197	C₂	1	12	yes	maximal, complemented
4	#117×Dic₁₃	C₂	1	6	yes	maximal, complemented
5	C₄×#88180	C₂	1	3	yes	maximal, complemented
6	C₂×#11718593	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	#41×Dic₁₃	C₄	1	32	yes	complemented
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	C₄×C₂²⋊Dic₁₃	C₂²	1	12	yes	—
20	C₂×#88071	C₂²	1	12	yes	—
21	C₄⋊C₄×Dic₁₃	C₂²	1	24	yes	—
22	C₂²⋊C₄×Dic₁₃	C₂²	1	24	yes	—
23	C₄×C₄⋊Dic₁₃	C₂²	1	12	yes	—
24	#11718593	C₂²	1	12	yes	—
25	#11718593	C₂²	1	1	yes	characteristic
26	#11718593	C₂²	1	3	yes	—
27	#11718593	C₂²	1	6	yes	—
28	C₄²×Dic₁₃	C₂²	1	6	yes	—
29	C₂×C₁₃×#41	C₂×C₄	1	16	yes	—
30	C₂×C₁₃×#41	C₂×C₄	1	8	yes	—
31	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×D₄×C₁₃	C₂³	1	3	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
35	C₂³×Dic₁₃	C₂³	1	12	yes	—
36	C₄×Q₈×C₁₃	C₂×C₄	1	16	yes	complemented
37	C₄×D₄×C₁₃	C₂×C₄	1	48	yes	complemented
38	C₁₃×#117	C₂×C₄	1	24	yes	complemented
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
40	C₂×C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	24	yes	complemented
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
45	C₂×C₂²⋊Dic₁₃	C₂³	1	12	yes	—
46	Q₈×Dic₁₃	C₂×C₄	1	32	yes	complemented
47	D₄×Dic₁₃	C₂×C₄	1	96	yes	complemented
48	#88180	C₂×C₄	1	48	yes	complemented
49	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
50	C₂×C₄×Dic₁₃	C₂×C₄	1	48	yes	complemented
51	C₂×C₄×Dic₁₃	C₂³	1	24	yes	—
52	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
53	C₂×C₄×Dic₁₃	C₂³	1	6	yes	—
54	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
55	#88071	C₂³	1	4	yes	—
56	#88071	C₂³	1	12	yes	—
57	#88071	C₂³	1	24	yes	—
58	#33999	—	13	1	no	maximal, Sylow
59	C₂⁴×C₁₃	C₂⁴	1	3	yes	—
60	C₁₃×#41	C₄²	1	64	yes	complemented
61	C₂×Q₈×C₁₃	C₂²×C₄	1	8	yes	—
62	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
63	C₂×D₄×C₁₃	C₂²×C₄	1	24	yes	—
64	C₂×D₄×C₁₃	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₁₃	—	4	12	no	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	24	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	24	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
71	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₁₃	—	4	24	no	—
74	C₂²×Dic₁₃	C₂²×C₄	1	48	yes	—
75	C₂²×Dic₁₃	C₂⁴	1	4	yes	—
76	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
77	C₁₃×C₄⋊C₄	C₂²×C₄	1	24	yes	—
78	C₁₃×C₂²⋊C₄	C₂²×C₄	1	24	yes	—
79	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
80	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
81	C₄²×C₁₃	C₂²×C₄	1	24	yes	—
82	C₂²⋊Dic₁₃	C₂²×C₄	1	48	yes	—
83	C₄⋊Dic₁₃	C₂²×C₄	1	48	yes	—
84	C₄×Dic₁₃	—	4	24	no	—
85	C₄×Dic₁₃	C₂²×C₄	1	8	yes	—
86	C₄×Dic₁₃	C₂²×C₄	1	24	yes	—
87	C₄×Dic₁₃	C₂²×C₄	1	48	yes	—
88	C₂×C₄×#41	—	13	2	no	—
89	C₂×C₄×#41	D₁₃	1	1	yes	characteristic
90	#2276	—	13	4	no	—
91	#2275	—	13	12	no	—
92	C₄×#117	—	13	3	no	—
93	C₄×#117	—	13	6	no	—
94	C₂×#376	—	13	3	no	—
95	C₂³×C₁₃	—	2	12	no	—
96	C₂³×C₁₃	C₂³×C₄	1	12	yes	—
97	C₂³×C₁₃	C₂³×C₄	1	6	yes	—
98	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
99	Q₈×C₁₃	C₂×C₄²	1	16	yes	—
100	D₄×C₁₃	C₂×C₄²	1	48	yes	—
101	C₂×C₄×C₁₃	—	2	12	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	4	24	no	—
104	C₂×C₄×C₁₃	C₂×C₄²	1	48	yes	—
105	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
107	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
108	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
109	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
111	C₂×Dic₁₃	—	2	24	no	—
112	C₂×Dic₁₃	—	2	8	no	—
113	C₂×Dic₁₃	—	4	48	no	—
114	C₂×Dic₁₃	C₂³×C₄	1	8	yes	—
115	C₂×Dic₁₃	C₂³×C₄	1	24	yes	—
116	C₂²×#41	C₂×D₁₃	1	1	yes	characteristic
117	C₄×#41	—	13	32	no	—
118	C₄×#41	Dic₁₃	1	16	yes	complemented
119	C₂×C₄×Q₈	—	13	4	no	—
120	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
121	C₂×C₄×D₄	—	13	12	no	—
122	C₂×C₄×D₄	C₂×D₁₃	1	6	yes	—
123	C₂×#117	—	13	6	no	—
124	C₂×#117	C₂×D₁₃	1	3	yes	—
125	C₂²×C₄²	—	13	6	no	—
126	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
127	C₄×C₄⋊C₄	—	13	24	no	—
128	C₄×C₄⋊C₄	—	13	12	no	—
129	C₄×C₂²⋊C₄	—	13	12	no	—
130	C₄×C₂²⋊C₄	—	13	24	no	—
131	#376	—	13	1	no	—
132	#376	—	13	3	no	—
133	#376	—	13	6	no	—
134	#376	—	13	12	no	—
135	C₂×#95	—	13	12	no	—
136	C₄³	—	13	6	no	—
137	C₂²×C₁₃	—	2	24	no	—
138	C₂²×C₁₃	—	2	12	no	—
139	C₂²×C₁₃	C₂²×C₄²	1	12	yes	—
140	C₂²×C₁₃	C₂⁴×C₄	1	2	yes	—
141	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
142	C₂²×C₁₃	#585	1	4	yes	—
143	C₄×C₁₃	—	2	4	no	—
144	C₄×C₁₃	—	2	12	no	—
145	C₄×C₁₃	C₂²×C₄²	1	4	yes	—
146	C₄×C₁₃	C₂²×C₄²	1	12	yes	—
147	Dic₁₃	—	2	8	no	—
148	Dic₁₃	—	2	24	no	—
149	C₂×#41	C₄×D₁₃	1	16	yes	—
150	C₂×#41	C₂×Dic₁₃	1	8	yes	—
151	C₂²×Q₈	C₂²×D₁₃	1	1	yes	characteristic
152	C₂²×D₄	C₂²×D₁₃	1	3	yes	—
153	C₂³×C₄	—	13	12	no	—
154	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
155	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
156	C₄×Q₈	—	13	32	no	—
157	C₄×Q₈	C₂×Dic₁₃	1	16	yes	complemented
158	C₄×D₄	—	13	96	no	—
159	C₄×D₄	C₂×Dic₁₃	1	48	yes	complemented
160	#117	—	13	48	no	—
161	#117	C₂×Dic₁₃	1	24	yes	complemented
162	C₂×C₄⋊C₄	—	13	12	no	—
163	C₂×C₄⋊C₄	C₂²×D₁₃	1	6	yes	—
164	C₂×C₂²⋊C₄	—	13	12	no	—
165	C₂×C₂²⋊C₄	C₂²×D₁₃	1	6	yes	—
166	C₂×C₄²	—	13	48	no	—
167	C₂×C₄²	—	13	12	no	—
168	C₂×C₄²	—	13	6	no	—
169	C₂×C₄²	—	13	2	no	—
170	C₂×C₄²	—	13	24	no	—
171	C₂×C₄²	C₂×Dic₁₃	1	24	yes	complemented
172	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
173	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
174	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
175	#95	—	13	4	no	—
176	#95	—	13	24	no	—
177	#95	—	13	12	no	—
178	C₂×C₁₃	—	2	12	no	—
179	C₂×C₁₃	C₂³×C₄²	1	1	yes	characteristic
180	C₂×C₁₃	#3428	1	4	yes	—
181	C₂×C₁₃	#3428	1	2	yes	—
182	C₂⁴	C₂³×D₁₃	1	3	yes	—
183	#41	C₄×Dic₁₃	1	64	yes	complemented
184	C₂×Q₈	C₂×C₄×D₁₃	1	8	yes	—
185	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
186	C₂×D₄	C₂×C₄×D₁₃	1	24	yes	—
187	C₂×D₄	C₂²×Dic₁₃	1	12	yes	—
188	C₂²×C₄	—	4	12	no	—
189	C₂²×C₄	—	13	4	no	—
190	C₂²×C₄	—	13	12	no	—
191	C₂²×C₄	—	13	48	no	—
192	C₂²×C₄	—	52	24	no	—
193	C₂²×C₄	C₂×C₄×D₁₃	1	24	yes	—
194	C₂²×C₄	C₂²×Dic₁₃	1	24	yes	—
195	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
196	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
197	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
198	C₂²×C₄	C₂³×D₁₃	1	3	yes	—
199	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
200	C₄⋊C₄	—	13	48	no	—
201	C₄⋊C₄	C₂²×Dic₁₃	1	24	yes	—
202	C₂²⋊C₄	—	13	48	no	—
203	C₂²⋊C₄	C₂²×Dic₁₃	1	24	yes	—
204	C₄²	—	13	8	no	—
205	C₄²	—	13	24	no	—
206	C₄²	—	13	48	no	—
207	C₄²	—	52	24	no	—
208	C₄²	C₂²×Dic₁₃	1	4	yes	—
209	C₄²	C₂²×Dic₁₃	1	12	yes	—
210	C₄²	C₂²×Dic₁₃	1	24	yes	—
211	C₁₃	#33999	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	2	12	no	—
213	C₂³	C₂²×C₄×D₁₃	1	12	yes	—
214	C₂³	C₂³×Dic₁₃	1	6	yes	—
215	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
216	Q₈	C₂×C₄×Dic₁₃	1	16	yes	—
217	D₄	C₂×C₄×Dic₁₃	1	48	yes	—
218	C₂×C₄	—	2	12	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	24	no	—
221	C₂×C₄	—	13	8	no	—
222	C₂×C₄	—	13	24	no	—
223	C₂×C₄	—	26	24	no	—
224	C₂×C₄	—	26	8	no	—
225	C₂×C₄	—	52	48	no	—
226	C₂×C₄	C₂×C₄×Dic₁₃	1	48	yes	—
227	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
228	C₂×C₄	C₂²×C₄×D₁₃	1	12	yes	—
229	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
230	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
231	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
232	C₂×C₄	C₂³×Dic₁₃	1	6	yes	—
233	C₂²	—	2	24	no	—
234	C₂²	—	2	12	no	—
235	C₂²	C₂²×C₄×Dic₁₃	1	12	yes	—
236	C₂²	C₂³×C₄×D₁₃	1	2	yes	—
237	C₂²	#11719648	1	4	yes	—
238	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	2	12	no	—
241	C₄	—	26	24	no	—
242	C₄	—	26	8	no	—
243	C₄	C₂²×C₄×Dic₁₃	1	4	yes	—
244	C₄	C₂²×C₄×Dic₁₃	1	12	yes	—
245	C₂	—	2	12	no	—
246	C₂	C₂³×C₄×Dic₁₃	1	1	yes	characteristic
247	C₂	#13128035	1	4	yes	—
248	C₂	#13128039	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator	C₂×C₁₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂×C₄×C₁₃×#41	C₂
socle, cosocle	C₂³×C₁₃	C₂⁵
prime radical	C₂²×C₁₃×#41	C₂²
hypercenter	C₂×C₄×#41	D₁₃
hypocenter	C₁₃	#33999
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
2. #13117201	C₂	semidirect
3. #13117197	C₂	semidirect
4. #117×Dic₁₃	C₂	semidirect
5. C₄×#88180	C₂	semidirect
6. C₂×#11718593	C₂	semidirect
9. C₄×C₁₃×#41	C₄	semidirect
14. #41×Dic₁₃	C₄	semidirect
36. C₄×Q₈×C₁₃	C₂×C₄	semidirect
37. C₄×D₄×C₁₃	C₂×C₄	semidirect
38. C₁₃×#117	C₂×C₄	semidirect
41. C₂×C₄²×C₁₃	C₂×C₄	semidirect
46. Q₈×Dic₁₃	C₂×C₄	semidirect
47. D₄×Dic₁₃	C₂×C₄	semidirect
48. #88180	C₂×C₄	semidirect
50. C₂×C₄×Dic₁₃	C₂×C₄	semidirect
60. C₁₃×#41	C₄²	semidirect
118. C₄×#41	Dic₁₃	semidirect
157. C₄×Q₈	C₂×Dic₁₃	semidirect
159. C₄×D₄	C₂×Dic₁₃	semidirect
161. #117	C₂×Dic₁₃	semidirect
171. C₂×C₄²	C₂×Dic₁₃	semidirect
183. #41	C₄×Dic₁₃	semidirect
211. C₁₃	#33999	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×Dic₁₃	2
#13117201	2
#13117197	2
#117×Dic₁₃	2
C₄×#88180	2
C₂×#11718593	2
C₂×C₄×C₁₃×#41	2
#33999	13