Subgroups — C₂²×C₁₃×#562 #15749028

Order 3328.

8102 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#562	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₁₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₃×#124	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₃×#123	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₃×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
6	C₂²×C₁₃×#121	C₂	1	4	yes	maximal, complemented
7	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₂³×D₄×C₁₃	C₂²	1	4	yes	—
9	C₂³×D₄×C₁₃	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₁₃	C₂²	1	4	yes	—
11	C₁₃×#562	C₂²	1	256	yes	direct product factor
12	C₂×C₁₃×#125	C₂²	1	24	yes	complemented
13	C₂×C₁₃×#124	C₂²	1	48	yes	complemented
14	C₂×C₁₃×#123	C₂²	1	96	yes	complemented
15	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	96	yes	complemented
16	C₂×C₁₃×#121	C₂²	1	96	yes	complemented
17	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
18	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
19	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
22	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
24	C₂⁵×C₁₃	C₂³	1	4	yes	—
25	C₂²×Q₈×C₁₃	—	4	8	no	—
26	C₂²×Q₈×C₁₃	C₂³	1	24	yes	—
27	C₂²×D₄×C₁₃	—	4	16	no	—
28	C₂²×D₄×C₁₃	—	4	8	no	—
29	C₂²×D₄×C₁₃	C₂³	1	48	yes	—
30	C₂²×D₄×C₁₃	C₂³	1	24	yes	—
31	C₂³×C₄×C₁₃	—	4	16	no	—
32	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	48	yes	—
37	C₁₃×#125	C₂³	1	64	yes	complemented
38	C₁₃×#124	C₂³	1	128	yes	complemented
39	C₁₃×#123	C₂³	1	256	yes	complemented
40	C₁₃×C₂²⋊Q₈	C₂³	1	256	yes	complemented
41	C₁₃×#121	C₂³	1	256	yes	complemented
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	48	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	48	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	—
45	C₂×C₁₃×C₂²⋊C₄	C₂³	1	48	yes	—
46	C₂×C₁₃×C₂²⋊C₄	C₂³	1	48	yes	—
47	C₂×C₁₃×C₂²⋊C₄	C₂³	1	48	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
49	C₂²×#562	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂⁴×C₁₃	—	2	16	no	—
51	C₂⁴×C₁₃	—	2	16	no	—
52	C₂⁴×C₁₃	—	2	16	no	—
53	C₂⁴×C₁₃	C₂⁴	1	24	yes	—
54	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
55	C₂×Q₈×C₁₃	—	4	48	no	—
56	C₂×Q₈×C₁₃	C₂⁴	1	32	yes	—
57	C₂×D₄×C₁₃	—	4	96	no	—
58	C₂×D₄×C₁₃	—	4	48	no	—
59	C₂×D₄×C₁₃	C₂⁴	1	64	yes	—
60	C₂×D₄×C₁₃	C₂⁴	1	32	yes	—
61	C₂²×C₄×C₁₃	—	2	16	no	—
62	C₂²×C₄×C₁₃	—	2	16	no	—
63	C₂²×C₄×C₁₃	—	2	8	no	—
64	C₂²×C₄×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	—	4	96	no	—
66	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	64	yes	—
71	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	—
72	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	—
73	C₁₃×C₄⋊C₄	C₂⁴	1	32	yes	—
74	C₁₃×C₂²⋊C₄	C₂⁴	1	64	yes	—
75	C₁₃×C₂²⋊C₄	C₂⁴	1	64	yes	—
76	C₁₃×C₂²⋊C₄	C₂⁴	1	64	yes	—
77	C₄²×C₁₃	C₂⁴	1	16	yes	—
78	C₂×#562	C₂×C₁₃	1	48	yes	direct product factor
79	C₂²×#125	C₂×C₁₃	1	1	yes	characteristic, complemented
80	C₂²×#124	C₂×C₁₃	1	2	yes	complemented
81	C₂²×#123	C₂×C₁₃	1	4	yes	complemented
82	C₂²×C₂²⋊Q₈	C₂×C₁₃	1	4	yes	complemented
83	C₂²×#121	C₂×C₁₃	1	4	yes	complemented
84	C₂³×C₁₃	—	2	48	no	—
85	C₂³×C₁₃	—	2	48	no	—
86	C₂³×C₁₃	—	2	48	no	—
87	C₂³×C₁₃	—	4	64	no	—
88	C₂³×C₁₃	H(5,𝔽₂)	1	8	yes	—
89	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
90	C₂³×C₁₃	C₂⁵	1	16	yes	—
91	C₂³×C₁₃	C₂⁵	1	3	yes	—
92	Q₈×C₁₃	—	4	32	no	—
93	D₄×C₁₃	—	4	64	no	—
94	D₄×C₁₃	—	4	32	no	—
95	C₂×C₄×C₁₃	—	2	48	no	—
96	C₂×C₄×C₁₃	—	2	48	no	—
97	C₂×C₄×C₁₃	—	2	24	no	—
98	C₂×C₄×C₁₃	—	2	12	no	—
99	C₂×C₄×C₁₃	—	4	64	no	—
100	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
101	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
102	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
103	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
104	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
105	C₂³×D₄	C₂²×C₁₃	1	4	yes	—
106	C₂³×D₄	C₂²×C₁₃	1	2	yes	—
107	C₂⁴×C₄	C₂²×C₁₃	1	4	yes	—
108	#562	C₂²×C₁₃	1	256	yes	direct product factor
109	C₂×#125	C₂²×C₁₃	1	24	yes	complemented
110	C₂×#124	C₂²×C₁₃	1	48	yes	complemented
111	C₂×#123	C₂²×C₁₃	1	96	yes	complemented
112	C₂×C₂²⋊Q₈	C₂²×C₁₃	1	96	yes	complemented
113	C₂×#121	C₂²×C₁₃	1	96	yes	complemented
114	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
115	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
116	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
117	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
118	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
119	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	—
120	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
121	C₂²×C₁₃	—	2	16	no	—
122	C₂²×C₁₃	—	2	16	no	—
123	C₂²×C₁₃	—	2	16	no	—
124	C₂²×C₁₃	—	4	96	no	—
125	C₂²×C₁₃	#562	1	16	yes	direct product factor
126	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	12	yes	—
127	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	6	yes	—
128	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
129	C₄×C₁₃	—	2	16	no	—
130	C₄×C₁₃	—	2	16	no	—
131	C₄×C₁₃	—	2	8	no	—
132	C₄×C₁₃	—	2	4	no	—
133	C₂⁵	C₂³×C₁₃	1	4	yes	—
134	C₂²×Q₈	—	4	8	no	—
135	C₂²×Q₈	C₂³×C₁₃	1	24	yes	—
136	C₂²×D₄	—	4	16	no	—
137	C₂²×D₄	—	4	8	no	—
138	C₂²×D₄	C₂³×C₁₃	1	48	yes	—
139	C₂²×D₄	C₂³×C₁₃	1	24	yes	—
140	C₂³×C₄	—	4	16	no	—
141	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
142	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
143	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
144	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
145	C₂³×C₄	C₂³×C₁₃	1	48	yes	—
146	#125	C₂³×C₁₃	1	64	yes	complemented
147	#124	C₂³×C₁₃	1	128	yes	complemented
148	#123	C₂³×C₁₃	1	256	yes	complemented
149	C₂²⋊Q₈	C₂³×C₁₃	1	256	yes	complemented
150	#121	C₂³×C₁₃	1	256	yes	complemented
151	C₂×C₄⋊C₄	C₂³×C₁₃	1	48	yes	—
152	C₂×C₄⋊C₄	C₂³×C₁₃	1	48	yes	—
153	C₂×C₄⋊C₄	C₂³×C₁₃	1	24	yes	—
154	C₂×C₂²⋊C₄	C₂³×C₁₃	1	48	yes	—
155	C₂×C₂²⋊C₄	C₂³×C₁₃	1	48	yes	—
156	C₂×C₂²⋊C₄	C₂³×C₁₃	1	48	yes	—
157	C₂×C₄²	C₂³×C₁₃	1	12	yes	—
158	C₂×C₁₃	—	4	16	no	—
159	C₂×C₁₃	C₂×#562	1	12	yes	direct product factor
160	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	2	yes	—
161	C₂×C₁₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
162	C₂⁴	—	2	16	no	—
163	C₂⁴	—	2	16	no	—
164	C₂⁴	—	2	16	no	—
165	C₂⁴	C₂⁴×C₁₃	1	24	yes	—
166	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
167	C₂×Q₈	—	4	48	no	—
168	C₂×Q₈	C₂⁴×C₁₃	1	32	yes	—
169	C₂×D₄	—	4	96	no	—
170	C₂×D₄	—	4	48	no	—
171	C₂×D₄	C₂⁴×C₁₃	1	64	yes	—
172	C₂×D₄	C₂⁴×C₁₃	1	32	yes	—
173	C₂²×C₄	—	2	16	no	—
174	C₂²×C₄	—	2	16	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	4	96	no	—
178	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
179	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
180	C₂²×C₄	C₂⁴×C₁₃	1	24	yes	—
181	C₂²×C₄	C₂⁴×C₁₃	1	24	yes	—
182	C₂²×C₄	C₂⁴×C₁₃	1	64	yes	—
183	C₄⋊C₄	C₂⁴×C₁₃	1	64	yes	—
184	C₄⋊C₄	C₂⁴×C₁₃	1	64	yes	—
185	C₄⋊C₄	C₂⁴×C₁₃	1	32	yes	—
186	C₂²⋊C₄	C₂⁴×C₁₃	1	64	yes	—
187	C₂²⋊C₄	C₂⁴×C₁₃	1	64	yes	—
188	C₂²⋊C₄	C₂⁴×C₁₃	1	64	yes	—
189	C₄²	C₂⁴×C₁₃	1	16	yes	—
190	C₁₃	C₂²×#562	1	1	yes	characteristic, direct product factor, Sylow
191	C₂³	—	2	48	no	—
192	C₂³	—	2	48	no	—
193	C₂³	—	2	48	no	—
194	C₂³	—	4	64	no	—
195	C₂³	C₁₃×H(5,𝔽₂)	1	8	yes	—
196	C₂³	C₁₃×2₋¹⁺⁴	1	4	yes	—
197	C₂³	C₂⁵×C₁₃	1	16	yes	—
198	C₂³	C₂⁵×C₁₃	1	3	yes	—
199	Q₈	—	4	32	no	—
200	D₄	—	4	64	no	—
201	D₄	—	4	32	no	—
202	C₂×C₄	—	2	48	no	—
203	C₂×C₄	—	2	48	no	—
204	C₂×C₄	—	2	24	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	—	4	64	no	—
207	C₂×C₄	C₂⁵×C₁₃	1	4	yes	—
208	C₂×C₄	C₂⁵×C₁₃	1	8	yes	—
209	C₂×C₄	C₂⁵×C₁₃	1	16	yes	—
210	C₂×C₄	C₂⁵×C₁₃	1	16	yes	—
211	C₂²	—	2	16	no	—
212	C₂²	—	2	16	no	—
213	C₂²	—	2	16	no	—
214	C₂²	—	4	96	no	—
215	C₂²	C₁₃×#562	1	16	yes	direct product factor
216	C₂²	C₂×C₁₃×H(5,𝔽₂)	1	12	yes	—
217	C₂²	C₂×C₁₃×2₋¹⁺⁴	1	6	yes	—
218	C₂²	C₂⁶×C₁₃	1	1	yes	characteristic
219	C₄	—	2	16	no	—
220	C₄	—	2	16	no	—
221	C₄	—	2	8	no	—
222	C₄	—	2	4	no	—
223	C₂	—	4	16	no	—
224	C₂	C₂×C₁₃×#562	1	12	yes	direct product factor
225	C₂	C₂²×C₁₃×H(5,𝔽₂)	1	2	yes	—
226	C₂	C₂²×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#562	C₂	direct (N × H)
2. C₂²×C₁₃×#125	C₂	semidirect
3. C₂²×C₁₃×#124	C₂	semidirect
4. C₂²×C₁₃×#123	C₂	semidirect
5. C₂²×C₁₃×C₂²⋊Q₈	C₂	semidirect
6. C₂²×C₁₃×#121	C₂	semidirect
11. C₁₃×#562	C₂²	direct (N × H)
12. C₂×C₁₃×#125	C₂²	semidirect
13. C₂×C₁₃×#124	C₂²	semidirect
14. C₂×C₁₃×#123	C₂²	semidirect
15. C₂×C₁₃×C₂²⋊Q₈	C₂²	semidirect
16. C₂×C₁₃×#121	C₂²	semidirect
37. C₁₃×#125	C₂³	semidirect
38. C₁₃×#124	C₂³	semidirect
39. C₁₃×#123	C₂³	semidirect
40. C₁₃×C₂²⋊Q₈	C₂³	semidirect
41. C₁₃×#121	C₂³	semidirect
49. C₂²×#562	C₁₃	direct (N × H)
78. C₂×#562	C₂×C₁₃	direct (N × H)
79. C₂²×#125	C₂×C₁₃	semidirect
80. C₂²×#124	C₂×C₁₃	semidirect
81. C₂²×#123	C₂×C₁₃	semidirect
82. C₂²×C₂²⋊Q₈	C₂×C₁₃	semidirect
83. C₂²×#121	C₂×C₁₃	semidirect
108. #562	C₂²×C₁₃	direct (N × H)
109. C₂×#125	C₂²×C₁₃	semidirect
110. C₂×#124	C₂²×C₁₃	semidirect
111. C₂×#123	C₂²×C₁₃	semidirect
112. C₂×C₂²⋊Q₈	C₂²×C₁₃	semidirect
113. C₂×#121	C₂²×C₁₃	semidirect
125. C₂²×C₁₃	#562	direct (N × H)
146. #125	C₂³×C₁₃	semidirect
147. #124	C₂³×C₁₃	semidirect
148. #123	C₂³×C₁₃	semidirect
149. C₂²⋊Q₈	C₂³×C₁₃	semidirect
150. #121	C₂³×C₁₃	semidirect
159. C₂×C₁₃	C₂×#562	direct (N × H)
190. C₁₃	C₂²×#562	direct (N × H)
215. C₂²	C₁₃×#562	direct (N × H)
224. C₂	C₂×C₁₃×#562	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#562	2
C₂²×C₁₃×#125	2
C₂²×C₁₃×#124	2
C₂²×C₁₃×#123	2
C₂²×C₁₃×C₂²⋊Q₈	2
C₂²×C₁₃×#121	2
C₂²×#562	13

Subgroups — C22×C13×#562 #15749028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#562 #15749028