Subgroups — #16794918

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131751	C₂	1	2	yes	maximal
2	#13130387	C₂	1	6	yes	maximal, complemented
3	#13130383	C₂	1	3	yes	maximal, complemented
4	C₂×#11718253	C₂	1	3	yes	maximal, complemented
5	#131×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
7	#41×Dic₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88202	C₄	1	1	yes	characteristic
9	Q₈×#1057	—	2	2	no	—
10	D₄×#1057	—	2	6	no	—
11	#11719119	C₂²	1	6	yes	—
12	C₂×C₄⋊#1057	C₂²	1	6	yes	—
13	#11718783	C₄	1	3	yes	—
14	M₁₆×Dic₁₃	C₂²	1	3	yes	—
15	#11718762	C₄	1	3	yes	—
16	C₂×C₈×Dic₁₃	C₂²	1	3	yes	—
17	#11718254	C₂²	1	6	yes	—
18	#11718253	C₂²	1	3	yes	—
19	#11718253	C₂²	1	6	yes	—
20	#11718253	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
22	C₂²×#1057	—	2	6	no	—
23	C₁₃×#131	C₈	1	8	yes	complemented
24	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
26	#88202	—	2	4	no	—
27	Q₈×Dic₁₃	D₄	1	1	yes	characteristic
28	Q₈×Dic₁₃	Q₈	1	1	yes	characteristic
29	D₄×Dic₁₃	D₄	1	3	yes	—
30	D₄×Dic₁₃	Q₈	1	3	yes	—
31	#88180	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
33	C₂×#6099	C₂×C₄	1	3	yes	—
34	C₂²×#1055	C₂×C₄	1	3	yes	—
35	C₂²⋊#1057	—	2	6	no	—
36	C₄⋊#1057	—	2	6	no	—
37	C₄⋊#1057	C₂³	1	6	yes	—
38	C₄⋊#1057	C₂³	1	2	yes	—
39	C₄×#1057	—	2	6	no	—
40	#88055	C₂×C₄	1	6	yes	—
41	C₈×Dic₁₃	C₂×C₄	1	3	yes	—
42	C₈×Dic₁₃	C₂×C₄	1	3	yes	—
43	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
44	C₈×Dic₁₃	C₂³	1	3	yes	—
45	#17788	—	13	1	no	maximal, Sylow
46	C₁₃×#41	C₂×C₈	1	4	yes	—
47	C₁₃×#41	M₁₆	1	4	yes	—
48	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×D₄×C₁₃	C₄⋊C₄	1	3	yes	—
50	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
51	C₂²×Dic₁₃	C₂×D₄	1	3	yes	—
52	C₂²×Dic₁₃	C₂×Q₈	1	3	yes	—
53	C₂×#1057	—	2	2	no	—
54	C₂×#1057	—	2	6	no	—
55	C₂×#1057	—	8	6	no	—
56	C₁₃×M₁₆	C₂×C₈	1	12	yes	complemented
57	C₂×C₈×C₁₃	C₂×C₈	1	12	yes	complemented
58	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
60	C₂²⋊Dic₁₃	C₄⋊C₄	1	6	yes	—
61	C₄⋊Dic₁₃	C₄⋊C₄	1	3	yes	—
62	C₄⋊Dic₁₃	C₄⋊C₄	1	3	yes	—
63	C₄×Dic₁₃	C₂²×C₄	1	3	yes	—
64	C₄×Dic₁₃	C₂×D₄	1	3	yes	—
65	C₄×Dic₁₃	C₂×Q₈	1	3	yes	—
66	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
67	#6099	—	2	6	no	—
68	C₂×#1055	—	2	6	no	—
69	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
70	C₂×#1055	C₂²×C₄	1	3	yes	—
71	#2988	—	13	2	no	—
72	C₄×#131	—	13	1	no	—
73	#1565	—	13	6	no	—
74	#1564	—	13	3	no	—
75	C₂×#334	—	13	3	no	—
76	C₂³×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
77	Q₈×C₁₃	C₄⋊C₈	1	4	yes	—
78	D₄×C₁₃	C₄⋊C₈	1	12	yes	—
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
80	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
81	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
82	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
83	C₂×Dic₁₃	—	2	6	no	—
84	C₂×Dic₁₃	—	4	3	no	—
85	C₂×Dic₁₃	C₂×C₄⋊C₄	1	3	yes	—
86	C₂×Dic₁₃	C₂×C₄⋊C₄	1	3	yes	—
87	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
89	#1057	—	4	2	no	—
90	#1057	—	4	6	no	—
91	C₈×C₁₃	C₂²×C₈	1	2	yes	—
92	C₈×C₁₃	C₂²×C₈	1	6	yes	—
93	#1055	—	2	1	no	—
94	#1055	—	2	3	no	—
95	C₂×#131	—	13	1	no	—
96	C₂×#131	#261	1	1	yes	characteristic
97	C₄×#41	—	13	1	no	—
98	C₈×Q₈	—	26	2	no	—
99	C₈×D₄	—	26	6	no	—
100	#431	—	13	6	no	—
101	C₂×C₄⋊C₈	—	13	6	no	—
102	#405	—	13	3	no	—
103	#405	—	13	3	no	—
104	C₄×M₁₆	—	13	3	no	—
105	C₂×C₄×C₈	—	13	3	no	—
106	#335	—	13	6	no	—
107	#334	—	13	3	no	—
108	#334	—	13	6	no	—
109	#334	—	13	1	no	—
110	C₂²×C₁₃	—	2	3	no	—
111	C₂²×C₁₃	—	2	3	no	—
112	C₂²×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
113	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₄×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
115	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
116	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
117	Dic₁₃	—	2	3	no	—
118	Dic₁₃	—	2	1	no	—
119	Dic₁₃	—	2	3	no	—
120	Dic₁₃	#577	1	2	yes	—
121	C₂×#41	C₂×#261	1	1	yes	characteristic
122	#131	—	26	4	no	—
123	#131	#1057	1	8	yes	complemented
124	C₂×M₁₆	—	13	3	no	—
125	C₂×M₁₆	C₂×#261	1	3	yes	—
126	C₂²×C₈	—	13	3	no	—
127	C₂²×C₈	—	26	6	no	—
128	C₂²×C₈	C₂×#261	1	3	yes	—
129	C₄×Q₈	—	13	1	no	—
130	C₄×Q₈	—	13	1	no	—
131	C₄×D₄	—	13	3	no	—
132	C₄×D₄	—	13	3	no	—
133	#117	—	13	3	no	—
134	C₂×C₄²	—	13	3	no	—
135	C₄⋊C₈	—	13	6	no	—
136	C₄⋊C₈	—	13	2	no	—
137	C₄⋊C₈	—	26	6	no	—
138	C₂²⋊C₈	—	26	6	no	—
139	#97	—	13	6	no	—
140	C₄×C₈	—	13	1	no	—
141	C₄×C₈	—	13	3	no	—
142	C₄×C₈	—	13	3	no	—
143	C₄×C₈	—	13	3	no	—
144	C₄×C₈	—	26	6	no	—
145	C₂×C₁₃	—	2	6	no	—
146	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
147	C₂×C₁₃	#2912	1	1	yes	characteristic
148	C₂×C₁₃	#2916	1	1	yes	characteristic
149	#41	#6118	1	4	yes	—
150	#41	C₂×#1057	1	4	yes	—
151	C₂×Q₈	C₄⋊#261	1	1	yes	characteristic
152	C₂×D₄	C₄⋊#261	1	3	yes	—
153	C₂²×C₄	—	13	3	no	—
154	C₂²×C₄	—	13	3	no	—
155	C₂²×C₄	C₂²×#261	1	3	yes	—
156	M₁₆	—	26	6	no	—
157	M₁₆	C₂×#1057	1	12	yes	complemented
158	C₂×C₈	—	13	1	no	—
159	C₂×C₈	—	13	3	no	—
160	C₂×C₈	—	26	6	no	—
161	C₂×C₈	—	26	2	no	—
162	C₂×C₈	—	26	6	no	—
163	C₂×C₈	—	104	6	no	—
164	C₂×C₈	C₂×#1057	1	12	yes	complemented
165	C₂×C₈	C₂²×#261	1	1	yes	characteristic
166	C₂×C₈	C₂²×#261	1	3	yes	—
167	C₄⋊C₄	—	13	3	no	—
168	C₄⋊C₄	—	13	3	no	—
169	C₂²⋊C₄	—	13	6	no	—
170	C₄²	—	13	3	no	—
171	C₄²	—	13	3	no	—
172	C₄²	—	13	1	no	—
173	C₄²	—	13	3	no	—
174	C₁₃	#17788	1	1	yes	characteristic, complemented, Sylow
175	C₂³	C₂×C₄⋊#261	1	3	yes	—
176	Q₈	C₄⋊#1057	1	4	yes	—
177	D₄	C₄⋊#1057	1	12	yes	—
178	C₂×C₄	—	13	1	no	—
179	C₂×C₄	—	13	1	no	—
180	C₂×C₄	—	13	3	no	—
181	C₂×C₄	—	13	3	no	—
182	C₂×C₄	—	26	6	no	—
183	C₂×C₄	—	52	3	no	—
184	C₂×C₄	C₂×#6118	1	6	yes	—
185	C₂×C₄	C₂×C₄⋊#261	1	3	yes	—
186	C₂×C₄	C₂²×#1057	1	6	yes	—
187	C₂×C₄	C₂³×#261	1	1	yes	characteristic
188	C₈	—	26	1	no	—
189	C₈	—	26	3	no	—
190	C₈	—	52	6	no	—
191	C₈	—	52	2	no	—
192	C₈	C₂²×#1057	1	2	yes	—
193	C₈	C₂²×#1057	1	6	yes	—
194	C₂²	—	2	3	no	—
195	C₂²	—	2	3	no	—
196	C₂²	C₂×C₄⋊#1057	1	6	yes	—
197	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
198	C₄	—	13	2	no	—
199	C₄	—	26	3	no	—
200	C₄	—	26	3	no	—
201	C₄	—	26	1	no	—
202	C₄	C₂×C₄⋊#1057	1	6	yes	—
203	C₄	C₂²×#6118	1	1	yes	characteristic
204	C₄	C₂³×#1057	1	1	yes	characteristic
205	C₂	—	2	6	no	—
206	C₂	C₂²×C₄⋊#1057	1	1	yes	characteristic
207	C₂	#13131490	1	1	yes	characteristic
208	C₂	#13131497	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×#131	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂×#131	#261
hypocenter	C₁₃	#17788
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #13130387	C₂	semidirect
3. #13130383	C₂	semidirect
4. C₂×#11718253	C₂	semidirect
23. C₁₃×#131	C₈	semidirect
56. C₁₃×M₁₆	C₂×C₈	semidirect
57. C₂×C₈×C₁₃	C₂×C₈	semidirect
123. #131	#1057	semidirect
157. M₁₆	C₂×#1057	semidirect
164. C₂×C₈	C₂×#1057	semidirect
174. C₁₃	#17788	semidirect

Maximal subgroups

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

subgroup	index
#13131751	2
#13130387	2
#13130383	2
C₂×#11718253	2
#131×Dic₁₃	2
#17788	13