Subgroups — #16807367

Order 3328.

486 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131253	C₂	1	8	yes	maximal, complemented
2	#13131198	C₂	1	2	yes	maximal
3	#13131197	C₂	1	2	yes	maximal
4	#13131194	C₂	1	2	yes	maximal
5	C₂×#11718491	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
8	C₂×#88054	C₄	1	1	yes	characteristic
9	C₂×#88054	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	#11718491	C₄	1	4	yes	—
14	#11718491	C₂²	1	4	yes	—
15	C₂²⋊#6092	C₂²	1	4	yes	—
16	#11718260	C₂²	1	4	yes	—
17	#11718260	C₂²	1	4	yes	—
18	#11718258	C₂²	1	4	yes	—
19	#11718257	C₂²	1	4	yes	—
20	#11718231	C₂²	1	4	yes	—
21	C₄⋊#6092	C₂²	1	4	yes	—
22	C₂²×Dic₂₆	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
27	C₂×#6101	C₈	1	1	yes	characteristic
28	C₂×#6101	C₈	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂³	1	1	yes	characteristic
33	C₄×Dic₂₆	C₈	1	4	yes	—
34	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
35	C₂×#6092	C₂³	1	8	yes	—
36	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
37	#88054	C₂×C₄	1	4	yes	—
38	#88054	C₂×C₄	1	4	yes	—
39	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
40	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
41	C₈×Dic₁₃	C₂³	1	2	yes	—
42	C₈×Dic₁₃	C₂³	1	2	yes	—
43	C₄×#1055	C₂×C₄	1	2	yes	—
44	C₄×#1055	C₂³	1	2	yes	—
45	#33497	—	13	1	no	maximal, Sylow
46	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
53	C₂×Dic₂₆	—	2	4	no	—
54	C₂×Dic₂₆	C₂×C₈	1	4	yes	—
55	C₂×C₈×C₁₃	—	4	1	no	—
56	C₂×C₈×C₁₃	—	4	1	no	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
59	C₄²×C₁₃	C₂×C₈	1	2	yes	—
60	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
61	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
62	#6101	C₂×C₈	1	4	yes	—
63	#6101	C₂×C₈	1	4	yes	—
64	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
65	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
66	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
67	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
68	C₂×#1055	—	2	2	no	—
69	C₂×#1055	—	4	1	no	—
70	C₂×#1055	C₂²×C₄	1	2	yes	—
71	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
72	C₂×#1055	C₂⁴	1	1	yes	characteristic
73	#6092	—	2	8	no	—
74	C₂×C₈×Q₈	—	13	1	no	—
75	#2183	—	13	8	no	—
76	#2163	—	13	2	no	—
77	#2163	—	13	2	no	—
78	#2162	—	13	2	no	—
79	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	4	1	no	—
84	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
88	C₂×Dic₁₃	—	2	2	no	—
89	C₂×Dic₁₃	—	4	1	no	—
90	C₂×Dic₁₃	—	4	1	no	—
91	C₂×Dic₁₃	M₃₂	1	4	yes	—
92	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
93	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
94	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
95	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
96	Dic₂₆	—	2	8	no	—
97	C₈×C₁₃	—	2	2	no	—
98	C₈×C₁₃	—	2	2	no	—
99	#1055	—	2	1	no	—
100	#1055	—	2	2	no	—
101	#1055	2₋¹⁺⁴	1	2	yes	—
102	C₂×C₄×Q₈	—	13	1	no	—
103	C₈×Q₈	—	13	4	no	—
104	C₈×Q₈	—	13	4	no	—
105	C₂×C₄⋊C₈	—	13	1	no	—
106	C₂×C₄⋊C₈	—	13	1	no	—
107	C₂×C₄⋊C₈	#261	1	1	yes	characteristic
108	C₂×C₄×C₈	—	13	1	no	—
109	C₂×C₄×C₈	—	13	1	no	—
110	C₂×C₄×C₈	—	13	1	no	—
111	C₄⋊C₁₆	—	13	4	no	—
112	C₄⋊C₁₆	—	13	4	no	—
113	C₄⋊C₁₆	—	13	4	no	—
114	C₂²⋊C₁₆	—	13	4	no	—
115	#346	—	13	4	no	—
116	#346	—	13	4	no	—
117	C₄×C₁₆	—	13	4	no	—
118	C₂²×C₁₃	—	2	1	no	—
119	C₂²×C₁₃	—	2	1	no	—
120	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
121	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	1	no	—
125	C₄×C₁₃	#519	1	1	yes	characteristic
126	C₄×C₁₃	#568	1	1	yes	characteristic
127	Dic₁₃	—	2	2	no	—
128	Dic₁₃	—	2	2	no	—
129	Dic₁₃	—	2	2	no	—
130	Dic₁₃	C₂×M₃₂	1	4	yes	—
131	C₂²×Q₈	—	13	1	no	—
132	C₂²×C₈	—	13	1	no	—
133	C₂²×C₈	—	13	1	no	—
134	C₂²×C₈	C₂×#261	1	1	yes	characteristic
135	C₂²×C₈	C₂×#261	1	1	yes	characteristic
136	C₄×Q₈	—	13	4	no	—
137	C₄×Q₈	—	13	4	no	—
138	C₂×C₄⋊C₄	—	13	1	no	—
139	C₂×C₄⋊C₄	—	13	1	no	—
140	C₂×C₄⋊C₄	—	13	1	no	—
141	C₂×C₄²	—	13	1	no	—
142	C₂×C₄²	—	13	1	no	—
143	C₂×C₄²	C₂×#261	1	1	yes	characteristic
144	C₂×C₁₆	—	13	8	no	—
145	C₄⋊C₈	—	13	4	no	—
146	C₄⋊C₈	—	13	4	no	—
147	C₄⋊C₈	C₂×#261	1	4	yes	—
148	C₄×C₈	—	13	2	no	—
149	C₄×C₈	—	13	2	no	—
150	C₄×C₈	—	13	2	no	—
151	C₄×C₈	—	13	2	no	—
152	C₄×C₈	—	13	2	no	—
153	C₄×C₈	—	13	2	no	—
154	C₂×C₁₃	—	2	2	no	—
155	C₂×C₁₃	#2982	1	1	yes	characteristic
156	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
157	C₂×C₁₃	#3407	1	1	yes	characteristic
158	C₂×Q₈	—	13	4	no	—
159	C₂×Q₈	—	26	4	no	—
160	C₂²×C₄	—	13	1	no	—
161	C₂²×C₄	—	13	1	no	—
162	C₂²×C₄	—	13	1	no	—
163	C₂²×C₄	—	13	1	no	—
164	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
165	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
166	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	13	1	no	—
170	C₂×C₈	—	13	1	no	—
171	C₂×C₈	—	13	2	no	—
172	C₂×C₈	—	26	2	no	—
173	C₂×C₈	—	52	1	no	—
174	C₂×C₈	C₂²×#261	1	2	yes	—
175	C₂×C₈	C₂²×#261	1	2	yes	—
176	C₄⋊C₄	—	13	4	no	—
177	C₄⋊C₄	—	13	4	no	—
178	C₄⋊C₄	—	13	4	no	—
179	C₄²	—	13	2	no	—
180	C₄²	—	13	2	no	—
181	C₄²	—	13	2	no	—
182	C₄²	—	13	2	no	—
183	C₄²	D₁₃⋊C₈	1	2	yes	—
184	C₄²	C₂²×#261	1	2	yes	—
185	C₁₆	—	26	8	no	—
186	C₁₃	#33497	1	1	yes	characteristic, complemented, Sylow
187	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
188	Q₈	—	26	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	13	4	no	—
194	C₂×C₄	—	13	2	no	—
195	C₂×C₄	—	13	2	no	—
196	C₂×C₄	—	13	2	no	—
197	C₂×C₄	—	13	2	no	—
198	C₂×C₄	—	26	2	no	—
199	C₂×C₄	—	52	1	no	—
200	C₂×C₄	—	52	1	no	—
201	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
202	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
203	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
204	C₂×C₄	C₂³×#261	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	13	2	no	—
208	C₈	—	26	2	no	—
209	C₈	—	26	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	D₁₃⋊M₃₂	1	2	yes	—
213	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	13	4	no	—
218	C₄	—	26	2	no	—
219	C₄	—	26	2	no	—
220	C₄	—	26	2	no	—
221	C₄	#11719621	1	1	yes	characteristic
222	C₄	#11719627	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	#13131731	1	1	yes	characteristic
225	C₂	C₂×D₁₃⋊M₃₂	1	1	yes	characteristic
226	C₂	#13131805	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₁₃×C₄⋊C₈	C₄
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄⋊C₈	#261
hypocenter	C₁₃	#33497
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #13131253	C₂	semidirect
186. C₁₃	#33497	semidirect

Maximal subgroups

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

subgroup	index
#13131253	2
#13131198	2
#13131197	2
#13131194	2
C₂×#11718491	2
#33497	13