Subgroups — #15779364

Order 3328.

350 conjugacy classes of subgroups.

The subgroup lattice has 194 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	2	yes	maximal
2	C₂×#11718096	C₂	1	2	yes	maximal
3	#13115000	C₂	1	1	yes	maximal, characteristic
4	#13114996	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88058	—	2	1	no	—
9	C₂×#88058	C₂²	1	2	yes	—
10	C₂×#88058	C₂²	1	1	yes	characteristic
11	C₂×#88058	C₂²	1	1	yes	characteristic
12	C₂×#88057	—	2	1	no	—
13	#11718597	—	2	1	no	—
14	#11718596	—	2	1	no	—
15	C₁₃×#346	C₄	1	4	yes	complemented
16	#11718097	—	2	4	no	—
17	#11718096	—	2	4	no	—
18	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₃	—	2	1	no	—
22	C₂×C₄⋊Dic₁₃	—	2	1	no	—
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	—	4	2	no	—
25	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
26	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
27	C₂×C₁₃×C₁₆	Q₈	1	4	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	—	2	2	no	—
34	#88058	—	2	2	no	—
35	#88058	—	4	2	no	—
36	#88057	—	4	2	no	—
37	#12908	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	D₈	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₁₃	—	4	1	no	—
41	C₂²×Dic₁₃	—	4	1	no	—
42	C₂²×Dic₁₃	—	4	2	no	—
43	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₃	D₈	1	2	yes	—
45	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
51	C₄²×C₁₃	Q₁₆	1	2	yes	—
52	C₄⋊Dic₁₃	—	4	1	no	—
53	C₄⋊Dic₁₃	—	4	1	no	—
54	C₄⋊Dic₁₃	—	4	2	no	—
55	C₄⋊Dic₁₃	—	4	1	no	—
56	C₄⋊Dic₁₃	—	4	1	no	—
57	C₄⋊Dic₁₃	—	4	2	no	—
58	C₄⋊Dic₁₃	—	8	4	no	—
59	C₁₃×C₁₆	C₄⋊C₄	1	8	yes	complemented
60	C₂×#367	—	13	2	no	—
61	C₂×#366	—	13	2	no	—
62	C₂×#346	D₁₃	1	1	yes	characteristic
63	#1846	—	13	1	no	—
64	#1845	—	13	1	no	—
65	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	#107	1	4	yes	—
69	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×Dic₁₃	—	8	1	no	—
76	C₂×Dic₁₃	—	8	2	no	—
77	C₂×Dic₁₃	—	8	1	no	—
78	C₂×Dic₁₃	—	8	4	no	—
79	C₂×Dic₁₃	—	8	2	no	—
80	C₂×Dic₁₃	—	8	2	no	—
81	C₈×C₁₃	#107	1	4	yes	—
82	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
83	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
84	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
85	C₂×#107	—	13	2	no	—
86	C₂×#107	—	13	1	no	—
87	C₂×#107	—	13	1	no	—
88	C₂×#107	—	26	1	no	—
89	C₂×#106	—	26	1	no	—
90	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
91	#384	—	26	1	no	—
92	#382	—	26	1	no	—
93	#367	—	26	4	no	—
94	#366	—	26	4	no	—
95	#346	Dic₁₃	1	4	yes	complemented
96	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
97	C₂²×C₁₃	#493	1	1	yes	characteristic
98	C₂²×C₁₃	#500	1	1	yes	characteristic
99	C₂²×C₁₃	#509	1	2	yes	—
100	C₂²×C₁₃	#510	1	2	yes	—
101	C₄×C₁₃	—	2	4	no	—
102	C₄×C₁₃	#384	1	1	yes	characteristic
103	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
104	C₄×C₁₃	#494	1	1	yes	characteristic
105	C₄×C₁₃	#500	1	1	yes	characteristic
106	Dic₁₃	—	16	2	no	—
107	Dic₁₃	—	16	1	no	—
108	Dic₁₃	—	16	1	no	—
109	C₂²×C₈	D₅₂	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	26	1	no	—
112	C₂×C₄⋊C₄	—	26	1	no	—
113	C₂×C₄⋊C₄	—	26	2	no	—
114	C₂×C₄⋊C₄	—	52	2	no	—
115	C₂×C₄²	D₅₂	1	1	yes	characteristic
116	C₂×C₁₆	Dic₂₆	1	4	yes	—
117	C₂×C₁₆	D₅₂	1	4	yes	—
118	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
119	#107	—	26	2	no	—
120	#107	—	26	2	no	—
121	#107	—	26	2	no	—
122	#107	—	26	2	no	—
123	#107	—	52	2	no	—
124	#106	—	52	2	no	—
125	C₄×C₈	Dic₂₆	1	2	yes	—
126	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
127	#95	—	52	2	no	—
128	C₂×C₁₃	#1845	1	1	yes	characteristic
129	C₂×C₁₃	#2159	1	2	yes	—
130	C₂×C₁₃	#2250	1	1	yes	characteristic
131	C₂×C₁₃	#2251	1	1	yes	characteristic
132	C₂×C₁₃	#2255	1	2	yes	—
133	C₂²×C₄	—	52	1	no	—
134	C₂²×C₄	—	52	1	no	—
135	C₂²×C₄	—	52	2	no	—
136	C₂²×C₄	D₁₀₄	1	2	yes	—
137	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
138	C₂×C₈	D₁₀₄	1	2	yes	—
139	C₂×C₈	Dic₅₂	1	2	yes	—
140	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
141	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
143	C₂×C₈	C₂×D₅₂	1	2	yes	—
144	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
145	C₄⋊C₄	—	52	1	no	—
146	C₄⋊C₄	—	52	2	no	—
147	C₄⋊C₄	—	52	1	no	—
148	C₄⋊C₄	—	52	1	no	—
149	C₄⋊C₄	—	52	1	no	—
150	C₄⋊C₄	—	52	2	no	—
151	C₄⋊C₄	—	104	4	no	—
152	C₄²	Dic₅₂	1	2	yes	—
153	C₄²	C₄⋊Dic₁₃	1	2	yes	—
154	C₁₆	C₄⋊Dic₁₃	1	8	yes	complemented
155	C₁₃	#12908	1	1	yes	characteristic, complemented, Sylow
156	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	104	4	no	—
160	C₂×C₄	—	104	2	no	—
161	C₂×C₄	—	104	2	no	—
162	C₂×C₄	—	104	1	no	—
163	C₂×C₄	—	104	2	no	—
164	C₂×C₄	—	104	1	no	—
165	C₂×C₄	#88058	1	4	yes	—
166	C₂×C₄	#88123	1	1	yes	characteristic
167	C₂×C₄	#88128	1	1	yes	characteristic
168	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
169	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
172	C₈	#88058	1	4	yes	—
173	C₈	#88123	1	2	yes	—
174	C₈	C₂×C₄⋊Dic₁₃	1	2	yes	—
175	C₂²	#11718343	1	1	yes	characteristic
176	C₂²	#11718356	1	1	yes	characteristic
177	C₂²	#11718569	1	2	yes	—
178	C₂²	#11718570	1	2	yes	—
179	C₂²	C₂×#88058	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	208	2	no	—
182	C₄	—	208	1	no	—
183	C₄	—	208	1	no	—
184	C₄	#11718343	1	1	yes	characteristic
185	C₄	#11718360	1	1	yes	characteristic
186	C₄	#11718596	1	1	yes	characteristic
187	C₄	C₂×#88058	1	1	yes	characteristic
188	C₂	#13114996	1	1	yes	characteristic
189	C₂	#13116624	1	2	yes	—
190	C₂	#13117102	1	1	yes	characteristic
191	C₂	#13117107	1	1	yes	characteristic
192	C₂	#13117122	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
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	#12908
LCS	G ▷ C₂×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
15. C₁₃×#346	C₄	semidirect
59. C₁₃×C₁₆	C₄⋊C₄	semidirect
95. #346	Dic₁₃	semidirect
154. C₁₆	C₄⋊Dic₁₃	semidirect
155. C₁₃	#12908	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
#13115000	2
#13114996	2
C₂×C₁₃×#346	2
#12908	13