Subgroups — #15754114

Order 3328.

438 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	#13114691	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
4	#13112670	C₂	1	2	yes	maximal
5	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88055	—	2	2	no	—
9	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
10	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
11	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₁₃×#335	C₄	1	4	yes	complemented
15	#11718142	—	2	2	no	—
16	#11718050	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₃	—	2	2	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
25	C₂²×#1055	—	2	2	no	—
26	C₂²×#1055	C₂×C₄	1	2	yes	—
27	C₁₃×C₄⋊C₈	—	2	2	no	—
28	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	#88055	—	4	4	no	—
32	C₈×Dic₁₃	—	2	4	no	—
33	C₈×Dic₁₃	—	2	4	no	—
34	C₄⋊#1055	—	2	2	no	—
35	C₄⋊#1055	C₂×C₄	1	4	yes	—
36	C₄×#1055	—	2	1	no	—
37	C₄×#1055	—	2	1	no	—
38	#8274	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
43	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	4	no	—
46	C₂×C₈×C₁₃	—	4	2	no	—
47	C₂×C₈×C₁₃	—	4	2	no	—
48	C₂×C₈×C₁₃	C₄²	1	2	yes	—
49	C₂×C₈×C₁₃	C₄²	1	2	yes	—
50	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
51	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₁₃	#41	1	1	yes	characteristic
54	C₄²×C₁₃	#41	1	1	yes	characteristic
55	C₄×Dic₁₃	—	2	2	no	—
56	C₄×Dic₁₃	—	2	2	no	—
57	C₄×Dic₁₃	—	2	1	no	—
58	C₄×Dic₁₃	—	2	1	no	—
59	C₄×Dic₁₃	—	2	4	no	—
60	C₄×Dic₁₃	D₈	1	2	yes	—
61	C₄×Dic₁₃	Q₁₆	1	2	yes	—
62	C₂×#1055	—	2	2	no	—
63	C₂×#1055	—	2	2	no	—
64	C₂×#1055	—	2	4	no	—
65	C₂×#1055	—	4	2	no	—
66	C₂×#1055	—	4	2	no	—
67	C₂×#1055	M₁₆	1	4	yes	—
68	#1767	—	13	2	no	—
69	C₄²×C₈	—	13	1	no	—
70	C₂×#335	—	13	1	no	—
71	C₂×#335	D₁₃	1	1	yes	characteristic
72	#1277	—	13	2	no	—
73	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
87	C₂×Dic₁₃	—	2	2	no	—
88	C₂×Dic₁₃	—	2	2	no	—
89	C₂×Dic₁₃	—	2	2	no	—
90	C₂×Dic₁₃	—	2	2	no	—
91	C₂×Dic₁₃	#107	1	4	yes	—
92	C₂×Dic₁₃	#108	1	4	yes	—
93	C₈×C₁₃	—	2	2	no	—
94	C₈×C₁₃	—	4	4	no	—
95	C₈×C₁₃	#97	1	4	yes	complemented
96	#1055	—	2	4	no	—
97	#1055	—	4	4	no	—
98	C₂×C₄⋊C₈	—	13	2	no	—
99	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
100	C₂×#97	—	26	2	no	—
101	C₂×C₄×C₈	—	13	2	no	—
102	C₂×C₄×C₈	—	13	2	no	—
103	C₂×C₄×C₈	—	13	1	no	—
104	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
105	C₄³	—	13	1	no	—
106	#336	—	26	2	no	—
107	#335	—	13	4	no	—
108	#335	Dic₁₃	1	4	yes	complemented
109	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₁₃	#423	1	2	yes	—
111	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
112	C₂²×C₁₃	#427	1	1	yes	characteristic
113	C₂²×C₁₃	C₂×#108	1	2	yes	—
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	—	2	1	no	—
116	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
117	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
118	C₄×C₁₃	#432	1	2	yes	—
119	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
120	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
121	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
122	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
123	Dic₁₃	—	2	4	no	—
124	Dic₁₃	—	2	4	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	13	2	no	—
127	C₂²×C₈	—	26	2	no	—
128	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
129	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
130	C₂×C₄²	—	13	1	no	—
131	C₂×C₄²	—	13	2	no	—
132	C₂×C₄²	—	13	1	no	—
133	C₂×C₄²	—	13	2	no	—
134	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	13	4	no	—
137	C₄⋊C₈	—	26	2	no	—
138	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
139	#97	—	52	4	no	—
140	C₄×C₈	—	26	4	no	—
141	C₄×C₈	—	26	4	no	—
142	C₄×C₈	—	26	1	no	—
143	C₄×C₈	—	26	1	no	—
144	C₄×C₈	C₄×D₁₃	1	2	yes	—
145	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
146	C₂×C₁₃	#1567	1	2	yes	—
147	C₂×C₁₃	#1568	1	2	yes	—
148	C₂×C₁₃	#1767	1	1	yes	characteristic
149	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
150	C₂×C₁₃	C₄×#108	1	1	yes	characteristic
151	C₂²×C₄	—	13	2	no	—
152	C₂²×C₄	—	13	2	no	—
153	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
154	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
155	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	13	4	no	—
161	C₂×C₈	—	26	2	no	—
162	C₂×C₈	—	26	2	no	—
163	C₂×C₈	—	26	4	no	—
164	C₂×C₈	—	52	2	no	—
165	C₂×C₈	—	52	2	no	—
166	C₂×C₈	#6094	1	4	yes	—
167	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
168	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
169	C₄²	—	13	2	no	—
170	C₄²	—	13	2	no	—
171	C₄²	—	26	4	no	—
172	C₄²	—	26	2	no	—
173	C₄²	—	26	2	no	—
174	C₄²	—	26	1	no	—
175	C₄²	—	26	1	no	—
176	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
177	C₄²	#6129	1	1	yes	characteristic
178	C₄²	#6131	1	1	yes	characteristic
179	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
180	C₁₃	#8274	1	1	yes	characteristic, complemented, Sylow
181	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	13	4	no	—
184	C₂×C₄	—	13	4	no	—
185	C₂×C₄	—	26	2	no	—
186	C₂×C₄	—	26	2	no	—
187	C₂×C₄	—	26	2	no	—
188	C₂×C₄	—	26	2	no	—
189	C₂×C₄	#88141	1	1	yes	characteristic
190	C₂×C₄	#88146	1	1	yes	characteristic
191	C₂×C₄	#88147	1	1	yes	characteristic
192	C₂×C₄	C₂×#6094	1	2	yes	—
193	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
194	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
195	C₂×C₄	#88166	1	1	yes	characteristic
196	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
197	C₂×C₄	#88173	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
199	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
200	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	4	4	no	—
203	C₈	—	26	4	no	—
204	C₈	—	52	4	no	—
205	C₈	#88055	1	4	yes	complemented
206	C₂²	#11718498	1	2	yes	—
207	C₂²	D₁₃×#107	1	1	yes	characteristic
208	C₂²	#11718541	1	1	yes	characteristic
209	C₂²	D₁₃×#108	1	2	yes	—
210	C₂²	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	26	4	no	—
214	C₄	—	26	4	no	—
215	C₄	#11718496	1	2	yes	—
216	C₄	#11718533	1	1	yes	characteristic
217	C₄	#11718534	1	1	yes	characteristic
218	C₄	C₂×#88055	1	1	yes	characteristic
219	C₄	M₁₆×Dic₁₃	1	1	yes	characteristic
220	C₄	D₈×Dic₁₃	1	1	yes	characteristic
221	C₄	Q₁₆×Dic₁₃	1	1	yes	characteristic
222	C₂	#13113593	1	2	yes	—
223	C₂	#13113598	1	2	yes	—
224	C₂	#13114691	1	1	yes	characteristic
225	C₂	#107×Dic₁₃	1	1	yes	characteristic
226	C₂	#108×Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#335	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#335	D₁₃
hypocenter	C₁₃	#8274
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
14. C₁₃×#335	C₄	semidirect
16. #11718050	C₄	semidirect
95. C₈×C₁₃	#97	semidirect
108. #335	Dic₁₃	semidirect
180. C₁₃	#8274	semidirect
205. C₈	#88055	semidirect

Maximal subgroups

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

subgroup	index
#13114691	2
C₄×C₈×Dic₁₃	2
C₂×#11718050	2
#13112670	2
C₂×C₁₃×#335	2
#8274	13