Subgroups — #15881858

Order 3328.

822 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₁₃	C₂	1	2	yes	maximal
2	#13115600	C₂	1	6	yes	maximal
3	#13114964	C₂	1	3	yes	maximal
4	C₄×#88057	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×Q₈×Dic₁₃	—	2	2	no	—
8	C₈×Q₈×C₁₃	C₄	1	8	yes	complemented
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
11	C₂×#88057	C₂²	1	6	yes	—
12	#11718646	—	2	6	no	—
13	#11718596	C₂²	1	6	yes	—
14	C₄×C₄⋊Dic₁₃	C₂²	1	6	yes	—
15	#11718143	C₂²	1	3	yes	—
16	#11718143	C₂²	1	6	yes	—
17	#11718143	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
21	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
22	C₄×Q₈×C₁₃	Q₈	1	4	yes	—
23	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
24	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
25	Q₈×Dic₁₃	—	4	8	no	—
26	C₂×C₄⋊Dic₁₃	—	2	6	no	—
27	C₂×C₄⋊Dic₁₃	—	2	6	no	—
28	C₂×C₄⋊Dic₁₃	C₂³	1	6	yes	—
29	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
30	C₂×C₄×Dic₁₃	—	2	6	no	—
31	C₁₃×C₄⋊C₈	C₂×C₄	1	12	yes	—
32	C₄×C₈×C₁₃	C₂×C₄	1	12	yes	—
33	#88071	—	2	6	no	—
34	#88057	—	2	12	no	—
35	#22738	—	13	1	no	maximal, Sylow
36	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
37	C₂×Q₈×C₁₃	SD₁₆	1	8	yes	—
38	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
39	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
41	C₂²×Dic₁₃	—	2	2	no	—
42	C₂²×Dic₁₃	—	2	6	no	—
43	C₂×C₈×C₁₃	—	2	6	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	6	yes	—
47	C₁₃×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
48	C₄²×C₁₃	C₂²×C₄	1	6	yes	—
49	C₄²×C₁₃	C₂×Q₈	1	6	yes	—
50	C₄⋊Dic₁₃	—	2	2	no	—
51	C₄⋊Dic₁₃	—	2	6	no	—
52	C₄⋊Dic₁₃	—	2	6	no	—
53	C₄⋊Dic₁₃	—	4	12	no	—
54	C₄⋊Dic₁₃	—	8	6	no	—
55	C₄⋊Dic₁₃	#41	1	4	yes	—
56	C₄×Dic₁₃	—	4	12	no	—
57	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
58	Q₈×C₄⋊C₄	—	13	2	no	—
59	#1948	—	13	6	no	—
60	#1834	—	13	3	no	—
61	C₄×#106	—	13	3	no	—
62	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
63	Q₈×C₁₃	#106	1	16	yes	complemented
64	C₂×C₄×C₁₃	—	2	3	no	—
65	C₂×C₄×C₁₃	—	2	3	no	—
66	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
67	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
68	C₂×C₄×C₁₃	C₂×SD₁₆	1	12	yes	—
69	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
72	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
73	C₂×Dic₁₃	—	2	4	no	—
74	C₂×Dic₁₃	—	4	2	no	—
75	C₂×Dic₁₃	—	4	2	no	—
76	C₂×Dic₁₃	—	4	6	no	—
77	C₂×Dic₁₃	—	4	6	no	—
78	C₂×Dic₁₃	—	4	6	no	—
79	C₈×C₁₃	—	2	6	no	—
80	C₈×C₁₃	—	2	2	no	—
81	C₂×C₄×Q₈	—	26	2	no	—
82	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
83	C₈×Q₈	Dic₁₃	1	8	yes	complemented
84	C₂×#106	—	13	6	no	—
85	C₂×C₄⋊C₈	C₂×D₁₃	1	3	yes	—
86	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
87	#389	—	26	6	no	—
88	#384	—	13	6	no	—
89	C₄×C₄⋊C₄	—	13	6	no	—
90	#340	—	13	3	no	—
91	#340	—	13	6	no	—
92	#340	—	13	1	no	—
93	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₁₃	#548	1	2	yes	—
95	C₂²×C₁₃	C₂²×SD₁₆	1	2	yes	—
96	C₂²×C₁₃	#576	1	1	yes	characteristic
97	C₂²×C₁₃	#578	1	1	yes	characteristic
98	C₄×C₁₃	—	2	6	no	—
99	C₄×C₁₃	C₂×#106	1	12	yes	—
100	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
101	C₄×C₁₃	#519	1	1	yes	characteristic
102	C₄×C₁₃	#557	1	2	yes	—
103	Dic₁₃	—	4	4	no	—
104	Dic₁₃	—	8	6	no	—
105	C₂²×Q₈	D₅₂	1	1	yes	characteristic
106	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
108	C₄×Q₈	—	52	8	no	—
109	C₄×Q₈	Dic₂₆	1	4	yes	—
110	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
111	C₂×C₄⋊C₄	—	13	2	no	—
112	C₂×C₄⋊C₄	—	13	6	no	—
113	C₂×C₄⋊C₄	—	26	6	no	—
114	C₂×C₄⋊C₄	—	26	6	no	—
115	C₂×C₄⋊C₄	D₅₂	1	3	yes	—
116	C₂×C₄²	—	26	6	no	—
117	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
118	#106	—	26	12	no	—
119	C₄⋊C₈	C₂×Dic₁₃	1	12	yes	—
120	C₄×C₈	C₂×Dic₁₃	1	12	yes	—
121	#95	—	26	6	no	—
122	C₂×C₁₃	#2351	1	1	yes	characteristic
123	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
124	C₂×C₁₃	#2913	1	1	yes	characteristic
125	C₂×C₁₃	#3359	1	2	yes	—
126	C₂×C₁₃	#3362	1	2	yes	—
127	C₂×Q₈	#6095	1	8	yes	—
128	C₂×Q₈	C₄⋊Dic₁₃	1	4	yes	—
129	C₂²×C₄	—	26	2	no	—
130	C₂²×C₄	—	26	6	no	—
131	C₂²×C₄	C₂×D₅₂	1	3	yes	—
132	C₂²×C₄	C₂×D₅₂	1	3	yes	—
133	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
137	C₂×C₈	C₂²×Dic₁₃	1	6	yes	—
138	C₄⋊C₄	—	13	4	no	—
139	C₄⋊C₄	—	26	2	no	—
140	C₄⋊C₄	—	26	6	no	—
141	C₄⋊C₄	—	26	6	no	—
142	C₄⋊C₄	—	52	12	no	—
143	C₄⋊C₄	—	104	6	no	—
144	C₄⋊C₄	C₄⋊Dic₁₃	1	12	yes	—
145	C₄²	—	52	12	no	—
146	C₄²	C₂×Dic₂₆	1	6	yes	—
147	C₄²	C₂²×Dic₁₃	1	6	yes	—
148	C₁₃	#22738	1	1	yes	characteristic, complemented, Sylow
149	C₂³	C₂²×D₅₂	1	1	yes	characteristic
150	Q₈	#88057	1	16	yes	complemented
151	C₂×C₄	—	2	3	no	—
152	C₂×C₄	—	2	3	no	—
153	C₂×C₄	—	26	4	no	—
154	C₂×C₄	—	52	2	no	—
155	C₂×C₄	—	52	2	no	—
156	C₂×C₄	—	52	6	no	—
157	C₂×C₄	—	52	6	no	—
158	C₂×C₄	—	52	6	no	—
159	C₂×C₄	C₂×#6095	1	12	yes	—
160	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
161	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
162	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
163	C₂×C₄	#88254	1	2	yes	—
164	C₂×C₄	#41×D₁₃	1	2	yes	—
165	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	2	6	no	—
168	C₂²	#11719281	1	2	yes	—
169	C₂²	C₂²×#6095	1	2	yes	—
170	C₂²	#11719457	1	1	yes	characteristic
171	C₂²	#11719458	1	1	yes	characteristic
172	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
173	C₄	—	2	6	no	—
174	C₄	—	52	4	no	—
175	C₄	—	104	6	no	—
176	C₄	C₂×#88057	1	12	yes	—
177	C₄	#11719274	1	2	yes	—
178	C₄	#11719517	1	1	yes	characteristic
179	C₄	#41×Dic₁₃	1	1	yes	characteristic
180	C₂	#13117757	1	1	yes	characteristic
181	C₂	C₂²×#88057	1	1	yes	characteristic
182	C₂	#13123436	1	1	yes	characteristic
183	C₂	#13127479	1	2	yes	—
184	C₂	#13127508	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	#22738
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
8. C₈×Q₈×C₁₃	C₄	semidirect
63. Q₈×C₁₃	#106	semidirect
83. C₈×Q₈	Dic₁₃	semidirect
148. C₁₃	#22738	semidirect
150. Q₈	#88057	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₁₃	2
#13115600	2
#13114964	2
C₄×#88057	2
C₂×C₈×Q₈×C₁₃	2
#22738	13