Subgroups — #15778326

Order 3328.

358 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×#11718097	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718096	C₂	1	1	yes	maximal, characteristic
3	C₄×#88058	C₂	1	2	yes	maximal
4	#13112928	C₂	1	2	yes	maximal
5	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88058	C₂²	1	2	yes	—
10	C₂×#88058	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₁₃	—	2	2	no	—
12	C₁₃×#346	C₄	1	4	yes	complemented
13	#11718143	—	2	2	no	—
14	#11718097	C₄	1	4	yes	complemented
15	#11718096	C₄	1	4	yes	complemented
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃	—	2	2	no	—
20	C₂×C₄⋊Dic₁₃	—	2	2	no	—
21	C₂×C₄×Dic₁₃	—	4	2	no	—
22	C₂×C₁₃×C₁₆	—	2	1	no	—
23	C₂×C₁₃×C₁₆	—	2	1	no	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₃	Q₈	1	2	yes	—
30	#88071	—	4	2	no	—
31	#88058	—	2	2	no	—
32	#88058	—	2	2	no	—
33	#88058	C₂×C₄	1	4	yes	—
34	#88058	C₂×C₄	1	4	yes	—
35	#12674	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	D₈	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₁₃	—	4	2	no	—
39	C₂²×Dic₁₃	—	4	2	no	—
40	C₂×C₈×C₁₃	—	2	1	no	—
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
48	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
49	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₄²×C₁₃	Q₁₆	1	2	yes	—
51	C₄⋊Dic₁₃	—	2	4	no	—
52	C₄⋊Dic₁₃	—	2	4	no	—
53	C₄⋊Dic₁₃	—	4	2	no	—
54	C₄⋊Dic₁₃	—	4	2	no	—
55	C₄×Dic₁₃	—	8	4	no	—
56	C₁₃×C₁₆	—	2	2	no	—
57	C₁₃×C₁₆	C₄²	1	4	yes	complemented
58	C₂×#367	—	13	1	no	—
59	C₂×#366	—	13	1	no	—
60	C₂×#346	D₁₃	1	1	yes	characteristic
61	C₄×#107	—	13	2	no	—
62	#1380	—	13	2	no	—
63	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	#107	1	4	yes	—
67	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
73	C₂×Dic₁₃	—	4	4	no	—
74	C₂×Dic₁₃	—	4	4	no	—
75	C₂×Dic₁₃	—	8	2	no	—
76	C₂×Dic₁₃	—	8	2	no	—
77	C₂×Dic₁₃	—	8	2	no	—
78	C₂×Dic₁₃	—	8	2	no	—
79	C₈×C₁₃	—	2	2	no	—
80	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
81	C₈×C₁₃	#117	1	1	yes	characteristic
82	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
83	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
84	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
85	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
86	C₂×#107	—	13	2	no	—
87	C₂×#107	—	13	2	no	—
88	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
89	C₄×C₄⋊C₄	—	26	2	no	—
90	#367	—	13	4	no	—
91	#366	—	13	4	no	—
92	#346	Dic₁₃	1	4	yes	complemented
93	#340	—	26	2	no	—
94	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
95	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
96	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
97	C₂²×C₁₃	#509	1	2	yes	—
98	C₂²×C₁₃	#510	1	2	yes	—
99	C₄×C₁₃	—	2	4	no	—
100	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₁₃	#427	1	1	yes	characteristic
102	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
103	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
104	Dic₁₃	—	8	4	no	—
105	Dic₁₃	—	8	4	no	—
106	C₂²×C₈	D₅₂	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	26	2	no	—
109	C₂×C₄⋊C₄	—	26	2	no	—
110	C₂×C₄²	—	52	2	no	—
111	C₂×C₄²	D₅₂	1	1	yes	characteristic
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
115	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
116	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
117	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
118	#107	—	13	4	no	—
119	#107	—	13	4	no	—
120	#107	—	26	2	no	—
121	#107	—	26	2	no	—
122	C₄×C₈	Dic₂₆	1	2	yes	—
123	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
124	#95	—	52	2	no	—
125	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
126	C₂×C₁₃	#2159	1	2	yes	—
127	C₂×C₁₃	#2175	1	2	yes	—
128	C₂×C₁₃	#2176	1	1	yes	characteristic
129	C₂×C₁₃	#2177	1	1	yes	characteristic
130	C₂²×C₄	—	52	2	no	—
131	C₂²×C₄	—	52	2	no	—
132	C₂²×C₄	D₁₀₄	1	2	yes	—
133	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
137	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
138	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
140	C₂×C₈	#6127	1	1	yes	characteristic
141	C₂×C₈	#6127	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
143	C₄⋊C₄	—	26	4	no	—
144	C₄⋊C₄	—	26	4	no	—
145	C₄⋊C₄	—	52	2	no	—
146	C₄⋊C₄	—	52	2	no	—
147	C₄²	—	104	4	no	—
148	C₄²	Dic₅₂	1	2	yes	—
149	C₄²	C₄⋊Dic₁₃	1	2	yes	—
150	C₁₆	—	2	2	no	—
151	C₁₆	C₄×Dic₁₃	1	4	yes	complemented
152	C₁₃	#12674	1	1	yes	characteristic, complemented, Sylow
153	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	52	4	no	—
157	C₂×C₄	—	52	4	no	—
158	C₂×C₄	—	104	2	no	—
159	C₂×C₄	—	104	2	no	—
160	C₂×C₄	—	104	2	no	—
161	C₂×C₄	—	104	2	no	—
162	C₂×C₄	#88058	1	4	yes	—
163	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
164	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
165	C₂×C₄	#88158	1	1	yes	characteristic
166	C₂×C₄	#88158	1	1	yes	characteristic
167	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	C₄×Dic₂₆	1	1	yes	characteristic
171	C₈	C₄×Dic₂₆	1	1	yes	characteristic
172	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
173	C₈	#88180	1	1	yes	characteristic
174	C₂²	C₄×D₁₀₄	1	1	yes	characteristic
175	C₂²	C₄×D₁₀₄	1	1	yes	characteristic
176	C₂²	#11718569	1	2	yes	—
177	C₂²	#11718570	1	2	yes	—
178	C₂²	C₂×#88058	1	1	yes	characteristic
179	C₄	—	2	4	no	—
180	C₄	—	104	4	no	—
181	C₄	—	104	4	no	—
182	C₄	C₄×Dic₅₂	1	1	yes	characteristic
183	C₄	C₄×Dic₅₂	1	1	yes	characteristic
184	C₄	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
185	C₄	#11718767	1	1	yes	characteristic
186	C₂	C₄×#88058	1	1	yes	characteristic
187	C₂	#13116624	1	2	yes	—
188	C₂	#13116702	1	2	yes	—
189	C₂	#13116708	1	1	yes	characteristic
190	C₂	#13116713	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#346	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#12674
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
12. C₁₃×#346	C₄	semidirect
14. #11718097	C₄	semidirect
15. #11718096	C₄	semidirect
57. C₁₃×C₁₆	C₄²	semidirect
92. #346	Dic₁₃	semidirect
151. C₁₆	C₄×Dic₁₃	semidirect
152. C₁₃	#12674	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718097	2
C₂×#11718096	2
C₄×#88058	2
#13112928	2
C₂×C₁₃×#346	2
#12674	13