Subgroups — #16789046

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88106	C₂	1	2	yes	maximal
2	#13129875	C₂	1	4	yes	maximal, complemented
3	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
6	C₂³×#1055	C₂²	1	1	yes	characteristic
7	C₂×#88106	—	2	4	no	—
8	C₂×#88106	—	2	4	no	—
9	C₂×#88106	C₄	1	8	yes	complemented
10	C₂²×#6092	C₂²	1	4	yes	—
11	C₂×C₈×Dic₁₃	—	2	4	no	—
12	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
13	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂³×Dic₁₃	C₈	1	2	yes	—
16	C₂²×C₈×C₁₃	—	2	2	no	—
17	C₂²×C₈×C₁₃	—	2	2	no	—
18	C₂²×C₈×C₁₃	C₈	1	4	yes	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₁₃	—	2	2	no	—
21	C₂×C₄×Dic₁₃	—	2	2	no	—
22	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
23	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	4	yes	—
26	C₂²×#1055	D₄	1	2	yes	—
27	C₂²×#1055	D₄	1	4	yes	—
28	C₂²×#1055	Q₈	1	2	yes	—
29	C₂²×#1055	C₂³	1	1	yes	characteristic
30	#88106	—	2	16	no	—
31	C₂×#6092	—	2	4	no	—
32	C₂×#6092	—	2	4	no	—
33	C₂×#6092	C₂×C₄	1	8	yes	complemented
34	C₈×Dic₁₃	—	2	4	no	—
35	C₈×Dic₁₃	—	2	4	no	—
36	#12544	—	13	1	no	maximal, Sylow
37	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
44	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	—	2	4	no	—
47	C₂²×Dic₁₃	—	2	8	no	—
48	C₂²×Dic₁₃	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₁₃	—	2	8	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
53	C₄×Dic₁₃	—	2	4	no	—
54	C₄×Dic₁₃	—	2	2	no	—
55	C₄×Dic₁₃	—	2	2	no	—
56	C₂×#1055	C₄²	1	2	yes	—
57	C₂×#1055	C₂²⋊C₄	1	2	yes	—
58	C₂×#1055	C₂²⋊C₄	1	8	yes	—
59	C₂×#1055	C₄⋊C₄	1	2	yes	—
60	C₂×#1055	C₄⋊C₄	1	8	yes	—
61	C₂×#1055	C₂²×C₄	1	2	yes	—
62	C₂×#1055	C₂×D₄	1	1	yes	characteristic
63	C₂×#1055	C₂×D₄	1	2	yes	—
64	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
65	#6092	—	2	8	no	—
66	C₂²×M₃₂	—	13	2	no	—
67	C₂²×C₄×C₈	—	13	1	no	—
68	#1374	—	13	4	no	—
69	C₂³×C₁₃	—	2	4	no	—
70	C₂³×C₁₃	C₄×C₈	1	2	yes	—
71	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
72	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
73	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#95	1	4	yes	—
75	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
76	C₂×C₄×C₁₃	#97	1	4	yes	—
77	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
79	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
81	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×Dic₁₃	—	2	16	no	—
88	C₂×Dic₁₃	—	2	4	no	—
89	C₂×Dic₁₃	—	2	4	no	—
90	C₂×Dic₁₃	M₃₂	1	8	yes	complemented
91	C₈×C₁₃	—	2	4	no	—
92	#1055	#95	1	4	yes	—
93	#1055	C₂×C₂²⋊C₄	1	2	yes	—
94	#1055	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₈	—	13	1	no	—
96	C₂³×C₈	#261	1	1	yes	characteristic
97	C₂²×C₄²	—	13	1	no	—
98	C₂×M₃₂	—	13	8	no	—
99	C₂×M₃₂	—	26	4	no	—
100	C₂×M₃₂	—	26	4	no	—
101	C₂²×C₁₆	—	13	4	no	—
102	C₂×C₄×C₈	—	13	2	no	—
103	C₂×C₄×C₈	—	13	2	no	—
104	C₂×C₄×C₈	—	26	4	no	—
105	C₂²×C₁₃	—	2	4	no	—
106	C₂²×C₁₃	—	2	4	no	—
107	C₂²×C₁₃	—	2	4	no	—
108	C₂²×C₁₃	#336	1	4	yes	—
109	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
110	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
111	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₁₃	C₂×M₃₂	1	4	yes	—
113	C₄×C₁₃	#336	1	4	yes	—
114	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
115	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
116	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
118	Dic₁₃	—	2	8	no	—
119	C₂³×C₄	—	13	2	no	—
120	C₂³×C₄	C₂×#261	1	1	yes	characteristic
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	13	4	no	—
124	C₂²×C₈	—	13	2	no	—
125	C₂²×C₈	—	13	2	no	—
126	C₂²×C₈	—	13	4	no	—
127	C₂²×C₈	—	13	1	no	—
128	C₂²×C₈	—	13	1	no	—
129	C₂²×C₈	#1057	1	4	yes	—
130	C₂²×C₈	C₂×#261	1	2	yes	—
131	C₂×C₄²	—	13	2	no	—
132	C₂×C₄²	—	13	2	no	—
133	C₂×C₄²	—	26	2	no	—
134	C₂×C₄²	—	26	2	no	—
135	M₃₂	—	26	16	no	—
136	C₂×C₁₆	—	13	8	no	—
137	C₂×C₁₆	—	26	4	no	—
138	C₂×C₁₆	—	26	4	no	—
139	C₄×C₈	—	26	4	no	—
140	C₄×C₈	—	26	4	no	—
141	C₂×C₁₃	—	2	4	no	—
142	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
143	C₂×C₁₃	C₄×M₃₂	1	2	yes	—
144	C₂×C₁₃	#2112	1	2	yes	—
145	C₂×C₁₃	#2150	1	2	yes	—
146	C₂⁴	D₁₃⋊C₈	1	1	yes	characteristic
147	C₂²×C₄	—	13	4	no	—
148	C₂²×C₄	—	26	8	no	—
149	C₂²×C₄	—	26	4	no	—
150	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
151	C₂²×C₄	#6118	1	2	yes	—
152	C₂²×C₄	C₄×#261	1	2	yes	—
153	C₂²×C₄	C₄⋊#261	1	2	yes	—
154	C₂²×C₄	C₂×#1057	1	2	yes	—
155	C₂²×C₄	#6122	1	2	yes	—
156	C₂²×C₄	C₂²⋊#261	1	2	yes	—
157	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	13	8	no	—
162	C₂×C₈	—	13	8	no	—
163	C₂×C₈	—	13	2	no	—
164	C₂×C₈	—	13	2	no	—
165	C₂×C₈	—	13	1	no	—
166	C₂×C₈	—	13	2	no	—
167	C₂×C₈	—	13	1	no	—
168	C₂×C₈	—	13	2	no	—
169	C₂×C₈	—	13	2	no	—
170	C₂×C₈	C₂×#1057	1	4	yes	—
171	C₄²	—	26	4	no	—
172	C₄²	—	26	2	no	—
173	C₄²	—	26	2	no	—
174	C₁₆	—	26	8	no	—
175	C₁₃	#12544	1	1	yes	characteristic, complemented, Sylow
176	C₂³	—	2	4	no	—
177	C₂³	C₄×#1057	1	2	yes	—
178	C₂³	#88111	1	2	yes	—
179	C₂³	#88112	1	2	yes	—
180	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
181	C₂×C₄	—	13	8	no	—
182	C₂×C₄	—	26	16	no	—
183	C₂×C₄	—	26	4	no	—
184	C₂×C₄	—	26	4	no	—
185	C₂×C₄	C₄×#1057	1	2	yes	—
186	C₂×C₄	C₄⋊#1057	1	4	yes	—
187	C₂×C₄	#88110	1	4	yes	—
188	C₂×C₄	#88111	1	2	yes	—
189	C₂×C₄	#88112	1	2	yes	—
190	C₂×C₄	(C₂×#261)⋊C₄	1	4	yes	—
191	C₂×C₄	C₂²⋊#1057	1	4	yes	—
192	C₂×C₄	C₂×#6118	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
195	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
196	C₂×C₄	C₂×#6122	1	1	yes	characteristic
197	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	—	13	2	no	—
200	C₈	—	13	4	no	—
201	C₈	—	13	2	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	#11718290	1	4	yes	—
206	C₂²	D₁₃⋊M₃₂	1	4	yes	—
207	C₂²	C₂×C₄×#1057	1	1	yes	characteristic
208	C₂²	C₂×#88111	1	1	yes	characteristic
209	C₂²	C₂×#88112	1	1	yes	characteristic
210	C₄	—	26	8	no	—
211	C₄	#11718290	1	4	yes	—
212	C₄	C₂×C₄⋊#1057	1	1	yes	characteristic
213	C₄	C₂×#88110	1	1	yes	characteristic
214	C₄	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
215	C₄	C₂×C₂²⋊#1057	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	C₂×#11718290	1	1	yes	characteristic
218	C₂	#13131042	1	2	yes	—
219	C₂	#13131065	1	2	yes	—
220	C₂	#13131170	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂³×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂³×C₈	#261
hypocenter	C₁₃	#12544
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #13129875	C₂	semidirect
9. C₂×#88106	C₄	semidirect
33. C₂×#6092	C₂×C₄	semidirect
90. C₂×Dic₁₃	M₃₂	semidirect
175. C₁₃	#12544	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88106	2
#13129875	2
C₂²×C₈×Dic₁₃	2
#12544	13