Subgroups — C₂²×C₁₃×#497 #15746973

Order 3328.

3350 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	C₂³×C₁₃×Q₁₆	C₂	1	2	yes	maximal, complemented
2	C₂³×C₁₃×SD₁₆	C₂	1	2	yes	maximal
3	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₁₃×#124	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#497	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×Q₁₆	—	2	16	no	—
11	C₂²×C₁₃×Q₁₆	C₂²	1	24	yes	complemented
12	C₂²×C₁₃×SD₁₆	—	2	16	no	—
13	C₂²×C₁₃×SD₁₆	C₂²	1	24	yes	—
14	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊Q₈	—	2	6	no	—
16	C₂×C₁₃×C₄⋊Q₈	C₂²	1	12	yes	complemented
17	C₂×C₁₃×#124	—	2	6	no	—
18	C₂×C₁₃×#124	C₂²	1	12	yes	—
19	C₂²×C₁₃×C₄⋊C₄	—	2	2	no	—
20	C₂²×C₁₃×C₂²⋊C₄	—	2	2	no	—
21	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×#497	C₂²	1	64	yes	direct product factor
23	C₂×C₁₃×#97	C₂²	1	12	yes	complemented
24	C₂⁵×C₁₃	—	2	1	no	—
25	C₂²×Q₈×C₁₃	—	2	3	no	—
26	C₂²×Q₈×C₁₃	—	2	6	no	—
27	C₂²×Q₈×C₁₃	—	2	8	no	—
28	C₂²×Q₈×C₁₃	—	2	8	no	—
29	C₂²×Q₈×C₁₃	—	2	8	no	—
30	C₂²×Q₈×C₁₃	C₂³	1	6	yes	—
31	C₂²×Q₈×C₁₃	C₂³	1	12	yes	—
32	C₂²×D₄×C₁₃	—	2	8	no	—
33	C₂²×D₄×C₁₃	—	2	3	no	—
34	C₂²×D₄×C₁₃	C₂³	1	6	yes	—
35	C₂³×C₄×C₁₃	—	2	1	no	—
36	C₂³×C₄×C₁₃	—	2	2	no	—
37	C₂³×C₄×C₁₃	D₄	1	2	yes	—
38	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₁₃×Q₁₆	—	2	96	no	—
40	C₂×C₁₃×Q₁₆	C₂³	1	32	yes	complemented
41	C₂×C₁₃×SD₁₆	—	2	96	no	—
42	C₂×C₁₃×SD₁₆	C₂³	1	32	yes	—
43	C₂²×C₈×C₁₃	D₄	1	16	yes	—
44	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
45	C₁₃×C₄⋊Q₈	—	2	24	no	—
46	C₁₃×C₄⋊Q₈	C₂³	1	16	yes	complemented
47	C₁₃×#124	—	2	24	no	—
48	C₁₃×#124	C₂³	1	16	yes	—
49	C₂×C₁₃×C₄⋊C₄	—	2	24	no	—
50	C₂×C₁₃×C₂²⋊C₄	—	2	24	no	—
51	C₂×C₄²×C₁₃	D₄	1	6	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
53	C₁₃×#97	C₂³	1	16	yes	complemented
54	C₂²×#497	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁴×C₁₃	—	2	6	no	—
56	C₂⁴×C₁₃	—	4	4	no	—
57	C₂⁴×C₁₃	—	4	8	no	—
58	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	—	2	6	no	—
60	C₂×Q₈×C₁₃	—	2	12	no	—
61	C₂×Q₈×C₁₃	—	2	24	no	—
62	C₂×Q₈×C₁₃	—	2	24	no	—
63	C₂×Q₈×C₁₃	—	2	24	no	—
64	C₂×Q₈×C₁₃	—	4	12	no	—
65	C₂×Q₈×C₁₃	—	4	24	no	—
66	C₂×Q₈×C₁₃	C₂⁴	1	4	yes	—
67	C₂×Q₈×C₁₃	C₂⁴	1	8	yes	—
68	C₂×D₄×C₁₃	—	2	24	no	—
69	C₂×D₄×C₁₃	—	2	6	no	—
70	C₂×D₄×C₁₃	—	4	12	no	—
71	C₂×D₄×C₁₃	C₂⁴	1	4	yes	—
72	C₂²×C₄×C₁₃	—	2	8	no	—
73	C₂²×C₄×C₁₃	—	2	6	no	—
74	C₂²×C₄×C₁₃	—	2	12	no	—
75	C₂²×C₄×C₁₃	—	4	4	no	—
76	C₂²×C₄×C₁₃	—	4	8	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
81	C₁₃×Q₁₆	—	2	64	no	—
82	C₁₃×SD₁₆	—	2	64	no	—
83	C₂×C₈×C₁₃	C₂×D₄	1	48	yes	—
84	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
85	C₁₃×C₄⋊C₄	—	2	32	no	—
86	C₁₃×C₂²⋊C₄	—	2	32	no	—
87	C₄²×C₁₃	C₂×D₄	1	12	yes	—
88	C₄²×C₁₃	C₂⁴	1	4	yes	—
89	C₂³×Q₁₆	C₂×C₁₃	1	2	yes	complemented
90	C₂³×SD₁₆	C₂×C₁₃	1	2	yes	—
91	C₂²×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
92	C₂²×#124	C₂×C₁₃	1	1	yes	characteristic
93	C₂×#497	C₂×C₁₃	1	24	yes	direct product factor
94	C₂²×#97	C₂×C₁₃	1	1	yes	characteristic, complemented
95	C₂³×C₁₃	—	2	4	no	—
96	C₂³×C₁₃	—	4	12	no	—
97	C₂³×C₁₃	—	4	24	no	—
98	C₂³×C₁₃	—	8	16	no	—
99	C₂³×C₁₃	#127	1	4	yes	—
100	C₂³×C₁₃	#137	1	8	yes	—
101	C₂³×C₁₃	C₂²×D₄	1	3	yes	—
102	Q₈×C₁₃	—	2	8	no	—
103	Q₈×C₁₃	—	2	8	no	—
104	Q₈×C₁₃	—	2	8	no	—
105	Q₈×C₁₃	—	4	12	no	—
106	Q₈×C₁₃	—	4	24	no	—
107	D₄×C₁₃	—	2	8	no	—
108	D₄×C₁₃	—	4	12	no	—
109	C₂×C₄×C₁₃	—	2	24	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	8	no	—
112	C₂×C₄×C₁₃	—	4	12	no	—
113	C₂×C₄×C₁₃	—	4	24	no	—
114	C₂×C₄×C₁₃	#127	1	12	yes	—
115	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
116	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
117	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
118	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
119	C₈×C₁₃	C₂²×D₄	1	16	yes	—
120	C₂³×Q₈	C₂²×C₁₃	1	1	yes	characteristic
121	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
122	C₂³×D₄	C₂²×C₁₃	1	1	yes	characteristic
123	C₂²×Q₁₆	—	2	16	no	—
124	C₂²×Q₁₆	C₂²×C₁₃	1	24	yes	complemented
125	C₂²×SD₁₆	—	2	16	no	—
126	C₂²×SD₁₆	C₂²×C₁₃	1	24	yes	—
127	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
128	C₂×C₄⋊Q₈	—	2	6	no	—
129	C₂×C₄⋊Q₈	C₂²×C₁₃	1	12	yes	complemented
130	C₂×#124	—	2	6	no	—
131	C₂×#124	C₂²×C₁₃	1	12	yes	—
132	C₂²×C₄⋊C₄	—	2	2	no	—
133	C₂²×C₂²⋊C₄	—	2	2	no	—
134	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
135	#497	C₂²×C₁₃	1	64	yes	direct product factor
136	C₂×#97	C₂²×C₁₃	1	12	yes	complemented
137	C₂²×C₁₃	—	4	4	no	—
138	C₂²×C₁₃	—	4	8	no	—
139	C₂²×C₁₃	—	8	24	no	—
140	C₂²×C₁₃	#497	1	16	yes	direct product factor
141	C₂²×C₁₃	C₂×#127	1	6	yes	—
142	C₂²×C₁₃	C₂×#137	1	12	yes	—
143	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
144	C₄×C₁₃	—	2	8	no	—
145	C₄×C₁₃	—	4	4	no	—
146	C₄×C₁₃	—	4	8	no	—
147	C₄×C₁₃	C₂×#127	1	6	yes	—
148	C₄×C₁₃	C₂³×D₄	1	2	yes	—
149	C₂⁵	—	2	1	no	—
150	C₂²×Q₈	—	2	3	no	—
151	C₂²×Q₈	—	2	6	no	—
152	C₂²×Q₈	—	2	8	no	—
153	C₂²×Q₈	—	2	8	no	—
154	C₂²×Q₈	—	2	8	no	—
155	C₂²×Q₈	C₂³×C₁₃	1	6	yes	—
156	C₂²×Q₈	C₂³×C₁₃	1	12	yes	—
157	C₂²×D₄	—	2	8	no	—
158	C₂²×D₄	—	2	3	no	—
159	C₂²×D₄	C₂³×C₁₃	1	6	yes	—
160	C₂³×C₄	—	2	1	no	—
161	C₂³×C₄	—	2	2	no	—
162	C₂³×C₄	D₄×C₁₃	1	2	yes	—
163	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
164	C₂×Q₁₆	—	2	96	no	—
165	C₂×Q₁₆	C₂³×C₁₃	1	32	yes	complemented
166	C₂×SD₁₆	—	2	96	no	—
167	C₂×SD₁₆	C₂³×C₁₃	1	32	yes	—
168	C₂²×C₈	D₄×C₁₃	1	16	yes	—
169	C₂²×C₈	C₂³×C₁₃	1	12	yes	—
170	C₄⋊Q₈	—	2	24	no	—
171	C₄⋊Q₈	C₂³×C₁₃	1	16	yes	complemented
172	#124	—	2	24	no	—
173	#124	C₂³×C₁₃	1	16	yes	—
174	C₂×C₄⋊C₄	—	2	24	no	—
175	C₂×C₂²⋊C₄	—	2	24	no	—
176	C₂×C₄²	D₄×C₁₃	1	6	yes	—
177	C₂×C₄²	C₂³×C₁₃	1	6	yes	—
178	#97	C₂³×C₁₃	1	16	yes	complemented
179	C₂×C₁₃	—	8	4	no	—
180	C₂×C₁₃	C₂×#497	1	12	yes	direct product factor
181	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
182	C₂×C₁₃	C₂²×#137	1	2	yes	—
183	C₂⁴	—	2	6	no	—
184	C₂⁴	—	4	4	no	—
185	C₂⁴	—	4	8	no	—
186	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
187	C₂×Q₈	—	2	6	no	—
188	C₂×Q₈	—	2	12	no	—
189	C₂×Q₈	—	2	24	no	—
190	C₂×Q₈	—	2	24	no	—
191	C₂×Q₈	—	2	24	no	—
192	C₂×Q₈	—	4	12	no	—
193	C₂×Q₈	—	4	24	no	—
194	C₂×Q₈	C₂⁴×C₁₃	1	4	yes	—
195	C₂×Q₈	C₂⁴×C₁₃	1	8	yes	—
196	C₂×D₄	—	2	24	no	—
197	C₂×D₄	—	2	6	no	—
198	C₂×D₄	—	4	12	no	—
199	C₂×D₄	C₂⁴×C₁₃	1	4	yes	—
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	2	6	no	—
202	C₂²×C₄	—	2	12	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	—	4	8	no	—
205	C₂²×C₄	C₂×D₄×C₁₃	1	12	yes	—
206	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
207	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
208	C₂²×C₄	C₂⁴×C₁₃	1	3	yes	—
209	Q₁₆	—	2	64	no	—
210	SD₁₆	—	2	64	no	—
211	C₂×C₈	C₂×D₄×C₁₃	1	48	yes	—
212	C₂×C₈	C₂⁴×C₁₃	1	8	yes	—
213	C₄⋊C₄	—	2	32	no	—
214	C₂²⋊C₄	—	2	32	no	—
215	C₄²	C₂×D₄×C₁₃	1	12	yes	—
216	C₄²	C₂⁴×C₁₃	1	4	yes	—
217	C₁₃	C₂²×#497	1	1	yes	characteristic, direct product factor, Sylow
218	C₂³	—	2	4	no	—
219	C₂³	—	4	12	no	—
220	C₂³	—	4	24	no	—
221	C₂³	—	8	16	no	—
222	C₂³	C₁₃×#127	1	4	yes	—
223	C₂³	C₁₃×#137	1	8	yes	—
224	C₂³	C₂²×D₄×C₁₃	1	3	yes	—
225	Q₈	—	2	8	no	—
226	Q₈	—	2	8	no	—
227	Q₈	—	2	8	no	—
228	Q₈	—	4	12	no	—
229	Q₈	—	4	24	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	12	no	—
232	C₂×C₄	—	2	24	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	4	12	no	—
236	C₂×C₄	—	4	24	no	—
237	C₂×C₄	C₁₃×#127	1	12	yes	—
238	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
239	C₂×C₄	C₂²×D₄×C₁₃	1	3	yes	—
240	C₂×C₄	C₂²×D₄×C₁₃	1	12	yes	—
241	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
242	C₈	C₂²×D₄×C₁₃	1	16	yes	—
243	C₂²	—	4	8	no	—
244	C₂²	—	4	4	no	—
245	C₂²	—	8	24	no	—
246	C₂²	C₁₃×#497	1	16	yes	direct product factor
247	C₂²	C₂×C₁₃×#127	1	6	yes	—
248	C₂²	C₂×C₁₃×#137	1	12	yes	—
249	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
250	C₄	—	2	8	no	—
251	C₄	—	4	4	no	—
252	C₄	—	4	8	no	—
253	C₄	C₂×C₁₃×#127	1	6	yes	—
254	C₄	C₂³×D₄×C₁₃	1	2	yes	—
255	C₂	—	8	4	no	—
256	C₂	C₂×C₁₃×#497	1	12	yes	direct product factor
257	C₂	C₂²×C₁₃×#127	1	1	yes	characteristic
258	C₂	C₂²×C₁₃×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
prime radical	C₂³×D₄×C₁₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₁₃×Q₁₆	C₂	semidirect
3. C₂²×C₁₃×C₄⋊Q₈	C₂	semidirect
5. C₂×C₁₃×#497	C₂	direct (N × H)
6. C₂²×C₁₃×#97	C₂	semidirect
11. C₂²×C₁₃×Q₁₆	C₂²	semidirect
16. C₂×C₁₃×C₄⋊Q₈	C₂²	semidirect
22. C₁₃×#497	C₂²	direct (N × H)
23. C₂×C₁₃×#97	C₂²	semidirect
40. C₂×C₁₃×Q₁₆	C₂³	semidirect
46. C₁₃×C₄⋊Q₈	C₂³	semidirect
53. C₁₃×#97	C₂³	semidirect
54. C₂²×#497	C₁₃	direct (N × H)
89. C₂³×Q₁₆	C₂×C₁₃	semidirect
91. C₂²×C₄⋊Q₈	C₂×C₁₃	semidirect
93. C₂×#497	C₂×C₁₃	direct (N × H)
94. C₂²×#97	C₂×C₁₃	semidirect
124. C₂²×Q₁₆	C₂²×C₁₃	semidirect
129. C₂×C₄⋊Q₈	C₂²×C₁₃	semidirect
135. #497	C₂²×C₁₃	direct (N × H)
136. C₂×#97	C₂²×C₁₃	semidirect
140. C₂²×C₁₃	#497	direct (N × H)
165. C₂×Q₁₆	C₂³×C₁₃	semidirect
171. C₄⋊Q₈	C₂³×C₁₃	semidirect
178. #97	C₂³×C₁₃	semidirect
180. C₂×C₁₃	C₂×#497	direct (N × H)
217. C₁₃	C₂²×#497	direct (N × H)
246. C₂²	C₁₃×#497	direct (N × H)
256. C₂	C₂×C₁₃×#497	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×Q₁₆	2
C₂³×C₁₃×SD₁₆	2
C₂²×C₁₃×C₄⋊Q₈	2
C₂²×C₁₃×#124	2
C₂×C₁₃×#497	2
C₂²×C₁₃×#97	2
C₂²×#497	13

Subgroups — C22×C13×#497 #15746973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#497 #15746973