Subgroups — C₂²×C₁₃×#557 #15749048

Order 3328.

7414 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#557	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₁₃×C₄⋊Q₈	C₂	1	3	yes	maximal
3	C₂²×C₁₃×#125	C₂	1	6	yes	maximal
4	C₂²×C₄×Q₈×C₁₃	C₂	1	2	yes	maximal
5	C₂²×C₄×Q₈×C₁₃	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃	C₂²	1	3	yes	—
8	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#557	C₂²	1	256	yes	direct product factor
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	72	yes	—
11	C₂×C₁₃×#125	C₂²	1	144	yes	—
12	C₂×C₄×Q₈×C₁₃	C₂²	1	48	yes	—
13	C₂×C₄×Q₈×C₁₃	C₂²	1	72	yes	—
14	C₂×C₄×Q₈×C₁₃	C₂²	1	24	yes	—
15	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂²×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
17	C₂²×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
18	C₂²×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
19	C₂²×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
20	C₂²×C₄²×C₁₃	C₂²	1	6	yes	—
21	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
22	C₂²×Q₈×C₁₃	—	2	24	no	—
23	C₂²×Q₈×C₁₃	Q₈	1	16	yes	complemented
24	C₂²×Q₈×C₁₃	C₂³	1	36	yes	—
25	C₂²×Q₈×C₁₃	C₂³	1	12	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₁₃×C₄⋊Q₈	C₂³	1	192	yes	—
32	C₁₃×#125	C₂³	1	384	yes	—
33	C₄×Q₈×C₁₃	C₂³	1	128	yes	—
34	C₄×Q₈×C₁₃	C₂³	1	192	yes	—
35	C₄×Q₈×C₁₃	C₂³	1	64	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂³	1	72	yes	—
38	C₂×C₁₃×C₄⋊C₄	C₂³	1	36	yes	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	72	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	72	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	72	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	36	yes	—
43	C₂²×#557	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
45	C₂×Q₈×C₁₃	—	2	144	no	—
46	C₂×Q₈×C₁₃	C₂×Q₈	1	96	yes	complemented
47	C₂×Q₈×C₁₃	C₂⁴	1	48	yes	—
48	C₂×Q₈×C₁₃	C₂⁴	1	16	yes	—
49	C₂²×C₄×C₁₃	—	2	24	no	—
50	C₂²×C₄×C₁₃	—	2	8	no	—
51	C₂²×C₄×C₁₃	—	2	12	no	—
52	C₂²×C₄×C₁₃	C₂×Q₈	1	24	yes	—
53	C₂²×C₄×C₁₃	#41	1	8	yes	—
54	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
55	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
56	C₂²×C₄×C₁₃	C₂⁴	1	36	yes	—
57	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
59	C₁₃×C₄⋊C₄	C₂⁴	1	16	yes	—
60	C₁₃×C₄⋊C₄	C₂⁴	1	96	yes	—
61	C₁₃×C₄⋊C₄	C₂⁴	1	48	yes	—
62	C₁₃×C₄⋊C₄	C₂⁴	1	96	yes	—
63	C₁₃×C₄⋊C₄	C₂⁴	1	96	yes	—
64	C₄²×C₁₃	C₂⁴	1	96	yes	—
65	C₄²×C₁₃	C₂⁴	1	48	yes	—
66	C₂×#557	C₂×C₁₃	1	48	yes	direct product factor
67	C₂²×C₄⋊Q₈	C₂×C₁₃	1	3	yes	—
68	C₂²×#125	C₂×C₁₃	1	6	yes	—
69	C₂²×C₄×Q₈	C₂×C₁₃	1	3	yes	—
70	C₂²×C₄×Q₈	C₂×C₁₃	1	1	yes	characteristic
71	C₂²×C₄×Q₈	C₂×C₁₃	1	2	yes	—
72	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
73	C₂³×C₁₃	C₂×#41	1	4	yes	—
74	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
75	C₂³×C₁₃	C₂⁵	1	3	yes	—
76	Q₈×C₁₃	—	2	96	no	—
77	Q₈×C₁₃	C₂²×Q₈	1	64	yes	complemented
78	C₂×C₄×C₁₃	—	2	72	no	—
79	C₂×C₄×C₁₃	—	2	24	no	—
80	C₂×C₄×C₁₃	—	2	36	no	—
81	C₂×C₄×C₁₃	C₂²×Q₈	1	72	yes	—
82	C₂×C₄×C₁₃	C₂×#41	1	24	yes	—
83	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
84	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
85	C₂×C₄×C₁₃	C₂⁵	1	24	yes	—
86	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
87	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
88	C₂³×Q₈	C₂²×C₁₃	1	3	yes	—
89	C₂³×Q₈	C₂²×C₁₃	1	1	yes	characteristic
90	#557	C₂²×C₁₃	1	256	yes	direct product factor
91	C₂×C₄⋊Q₈	C₂²×C₁₃	1	72	yes	—
92	C₂×#125	C₂²×C₁₃	1	144	yes	—
93	C₂×C₄×Q₈	C₂²×C₁₃	1	48	yes	—
94	C₂×C₄×Q₈	C₂²×C₁₃	1	72	yes	—
95	C₂×C₄×Q₈	C₂²×C₁₃	1	24	yes	—
96	C₂²×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
97	C₂²×C₄⋊C₄	C₂²×C₁₃	1	3	yes	—
98	C₂²×C₄⋊C₄	C₂²×C₁₃	1	6	yes	—
99	C₂²×C₄⋊C₄	C₂²×C₁₃	1	6	yes	—
100	C₂²×C₄⋊C₄	C₂²×C₁₃	1	6	yes	—
101	C₂²×C₄²	C₂²×C₁₃	1	6	yes	—
102	C₂²×C₄²	C₂²×C₁₃	1	3	yes	—
103	C₂²×C₁₃	#557	1	16	yes	direct product factor
104	C₂²×C₁₃	C₂³×Q₈	1	6	yes	—
105	C₂²×C₁₃	C₂²×#41	1	6	yes	—
106	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	6	yes	—
107	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
108	C₄×C₁₃	—	2	24	no	—
109	C₄×C₁₃	—	2	8	no	—
110	C₄×C₁₃	—	2	12	no	—
111	C₄×C₁₃	C₂³×Q₈	1	24	yes	—
112	C₄×C₁₃	C₂²×#41	1	8	yes	—
113	C₂²×Q₈	—	2	24	no	—
114	C₂²×Q₈	Q₈×C₁₃	1	16	yes	complemented
115	C₂²×Q₈	C₂³×C₁₃	1	36	yes	—
116	C₂²×Q₈	C₂³×C₁₃	1	12	yes	—
117	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
118	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
119	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
120	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
121	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
122	C₄⋊Q₈	C₂³×C₁₃	1	192	yes	—
123	#125	C₂³×C₁₃	1	384	yes	—
124	C₄×Q₈	C₂³×C₁₃	1	128	yes	—
125	C₄×Q₈	C₂³×C₁₃	1	192	yes	—
126	C₄×Q₈	C₂³×C₁₃	1	64	yes	—
127	C₂×C₄⋊C₄	C₂³×C₁₃	1	12	yes	—
128	C₂×C₄⋊C₄	C₂³×C₁₃	1	72	yes	—
129	C₂×C₄⋊C₄	C₂³×C₁₃	1	36	yes	—
130	C₂×C₄⋊C₄	C₂³×C₁₃	1	72	yes	—
131	C₂×C₄⋊C₄	C₂³×C₁₃	1	72	yes	—
132	C₂×C₄²	C₂³×C₁₃	1	72	yes	—
133	C₂×C₄²	C₂³×C₁₃	1	36	yes	—
134	C₂×C₁₃	C₂×#557	1	12	yes	direct product factor
135	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
136	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
137	C₂×C₁₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
138	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
139	C₂×Q₈	—	2	144	no	—
140	C₂×Q₈	C₂×Q₈×C₁₃	1	96	yes	complemented
141	C₂×Q₈	C₂⁴×C₁₃	1	48	yes	—
142	C₂×Q₈	C₂⁴×C₁₃	1	16	yes	—
143	C₂²×C₄	—	2	24	no	—
144	C₂²×C₄	—	2	12	no	—
145	C₂²×C₄	—	2	8	no	—
146	C₂²×C₄	C₂×Q₈×C₁₃	1	24	yes	—
147	C₂²×C₄	C₁₃×#41	1	8	yes	—
148	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
149	C₂²×C₄	C₂⁴×C₁₃	1	18	yes	—
150	C₂²×C₄	C₂⁴×C₁₃	1	36	yes	—
151	C₂²×C₄	C₂⁴×C₁₃	1	18	yes	—
152	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
153	C₄⋊C₄	C₂⁴×C₁₃	1	16	yes	—
154	C₄⋊C₄	C₂⁴×C₁₃	1	96	yes	—
155	C₄⋊C₄	C₂⁴×C₁₃	1	48	yes	—
156	C₄⋊C₄	C₂⁴×C₁₃	1	96	yes	—
157	C₄⋊C₄	C₂⁴×C₁₃	1	96	yes	—
158	C₄²	C₂⁴×C₁₃	1	96	yes	—
159	C₄²	C₂⁴×C₁₃	1	48	yes	—
160	C₁₃	C₂²×#557	1	1	yes	characteristic, direct product factor, Sylow
161	C₂³	C₂²×Q₈×C₁₃	1	4	yes	—
162	C₂³	C₂×C₁₃×#41	1	4	yes	—
163	C₂³	C₁₃×2₋¹⁺⁴	1	4	yes	—
164	C₂³	C₂⁵×C₁₃	1	3	yes	—
165	Q₈	—	2	96	no	—
166	Q₈	C₂²×Q₈×C₁₃	1	64	yes	complemented
167	C₂×C₄	—	2	72	no	—
168	C₂×C₄	—	2	24	no	—
169	C₂×C₄	—	2	36	no	—
170	C₂×C₄	C₂²×Q₈×C₁₃	1	72	yes	—
171	C₂×C₄	C₂×C₁₃×#41	1	24	yes	—
172	C₂×C₄	C₂⁵×C₁₃	1	8	yes	—
173	C₂×C₄	C₂⁵×C₁₃	1	12	yes	—
174	C₂×C₄	C₂⁵×C₁₃	1	24	yes	—
175	C₂×C₄	C₂⁵×C₁₃	1	12	yes	—
176	C₂×C₄	C₂⁵×C₁₃	1	4	yes	—
177	C₂²	C₁₃×#557	1	16	yes	direct product factor
178	C₂²	C₂³×Q₈×C₁₃	1	6	yes	—
179	C₂²	C₂²×C₁₃×#41	1	6	yes	—
180	C₂²	C₂×C₁₃×2₋¹⁺⁴	1	6	yes	—
181	C₂²	C₂⁶×C₁₃	1	1	yes	characteristic
182	C₄	—	2	24	no	—
183	C₄	—	2	8	no	—
184	C₄	—	2	12	no	—
185	C₄	C₂³×Q₈×C₁₃	1	24	yes	—
186	C₄	C₂²×C₁₃×#41	1	8	yes	—
187	C₂	C₂×C₁₃×#557	1	12	yes	direct product factor
188	C₂	C₂⁴×Q₈×C₁₃	1	1	yes	characteristic
189	C₂	C₂³×C₁₃×#41	1	1	yes	characteristic
190	C₂	C₂²×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	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×#557	C₂	direct (N × H)
9. C₁₃×#557	C₂²	direct (N × H)
23. C₂²×Q₈×C₁₃	Q₈	semidirect
43. C₂²×#557	C₁₃	direct (N × H)
46. C₂×Q₈×C₁₃	C₂×Q₈	semidirect
66. C₂×#557	C₂×C₁₃	direct (N × H)
77. Q₈×C₁₃	C₂²×Q₈	semidirect
90. #557	C₂²×C₁₃	direct (N × H)
103. C₂²×C₁₃	#557	direct (N × H)
114. C₂²×Q₈	Q₈×C₁₃	semidirect
134. C₂×C₁₃	C₂×#557	direct (N × H)
140. C₂×Q₈	C₂×Q₈×C₁₃	semidirect
160. C₁₃	C₂²×#557	direct (N × H)
166. Q₈	C₂²×Q₈×C₁₃	semidirect
177. C₂²	C₁₃×#557	direct (N × H)
187. C₂	C₂×C₁₃×#557	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×#557	2
C₂²×C₁₃×C₄⋊Q₈	2
C₂²×C₁₃×#125	2
C₂²×C₄×Q₈×C₁₃	2
C₂²×#557	13

Subgroups — C22×C13×#557 #15749048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#557 #15749048