Subgroups — #15774508

Order 3328.

242 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13113584	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
3	#13112910	C₂	1	2	yes	maximal, complemented
4	#13112908	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic
7	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	#11718586	—	2	1	no	—
10	C₂×C₄⋊#1055	—	2	1	no	—
11	C₁₃×#346	C₄	1	2	yes	—
12	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
13	#11718050	—	2	1	no	—
14	#11718050	C₂²	1	1	yes	characteristic
15	#11718050	C₂²	1	2	yes	—
16	#11718050	C₂²	1	1	yes	characteristic
17	#11718049	—	2	1	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×#1055	—	4	1	no	—
22	C₁₃×M₃₂	—	2	2	no	—
23	C₁₃×M₃₂	C₈	1	4	yes	complemented
24	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²⋊#1055	—	4	1	no	—
31	C₄⋊#1055	—	2	1	no	—
32	C₄⋊#1055	—	2	1	no	—
33	C₄⋊#1055	—	2	2	no	—
34	C₄⋊#1055	—	4	1	no	—
35	C₄×#1055	—	4	1	no	—
36	#12009	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	Q₁₆	1	1	yes	characteristic
40	C₂×C₈×C₁₃	—	2	2	no	—
41	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
45	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₃	D₈	1	1	yes	characteristic
49	C₄²×C₁₃	Q₁₆	1	1	yes	characteristic
50	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×#1055	—	4	1	no	—
52	C₂×#1055	—	4	1	no	—
53	C₂×#1055	—	4	2	no	—
54	C₂×#1055	—	8	2	no	—
55	C₁₃×C₁₆	—	2	2	no	—
56	C₁₃×C₁₆	C₂×C₈	1	4	yes	complemented
57	C₄×M₃₂	D₁₃	1	1	yes	characteristic
58	#1564	—	13	1	no	—
59	C₂×#335	—	13	1	no	—
60	#1372	—	13	2	no	—
61	#1371	—	13	2	no	—
62	C₂³×C₁₃	#107	1	1	yes	characteristic
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	2	1	no	—
65	C₂×C₄×C₁₃	—	4	1	no	—
66	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
67	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
68	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
69	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
70	C₂×C₄×C₁₃	#108	1	2	yes	—
71	C₂×C₄×C₁₃	C₂×C₄⋊C₄	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	C₈×C₁₃	—	2	1	no	—
75	C₈×C₁₃	—	2	1	no	—
76	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
77	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
78	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
79	#1055	—	8	1	no	—
80	#1055	—	8	2	no	—
81	#1055	—	8	1	no	—
82	C₂×M₃₂	Dic₁₃	1	1	yes	characteristic
83	C₂×M₃₂	Dic₁₃	1	1	yes	characteristic
84	#431	—	26	1	no	—
85	C₂×C₄⋊C₈	—	26	1	no	—
86	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
87	#346	Dic₁₃	1	2	yes	—
88	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
89	#335	—	13	2	no	—
90	#335	—	13	1	no	—
91	#335	—	13	1	no	—
92	#335	—	26	1	no	—
93	#334	—	26	1	no	—
94	C₂²×C₁₃	—	2	1	no	—
95	C₂²×C₁₃	—	2	1	no	—
96	C₂²×C₁₃	#335	1	2	yes	—
97	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
98	C₄×C₁₃	—	2	1	no	—
99	C₄×C₁₃	—	2	1	no	—
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	#335	1	2	yes	—
102	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
104	C₄×C₁₃	#509	1	1	yes	characteristic
105	C₄×C₁₃	#510	1	1	yes	characteristic
106	C₂²×C₈	—	52	1	no	—
107	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
108	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
109	C₂×C₄²	D₅₂	1	1	yes	characteristic
110	M₃₂	—	2	2	no	—
111	M₃₂	#1055	1	4	yes	complemented
112	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
113	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
114	C₄⋊C₈	—	26	1	no	—
115	C₄⋊C₈	—	26	1	no	—
116	C₄⋊C₈	—	26	2	no	—
117	C₄⋊C₈	—	52	1	no	—
118	C₂²⋊C₈	—	52	1	no	—
119	C₄×C₈	—	52	1	no	—
120	C₄×C₈	Dic₂₆	1	1	yes	characteristic
121	C₄×C₈	D₅₂	1	1	yes	characteristic
122	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
124	C₂×C₁₃	—	2	2	no	—
125	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
126	C₂×C₁₃	#2159	1	1	yes	characteristic
127	C₂×C₁₃	#2161	1	1	yes	characteristic
128	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
129	C₂²×C₄	Dic₅₂	1	1	yes	characteristic
130	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	52	1	no	—
133	C₂×C₈	—	52	1	no	—
134	C₂×C₈	—	52	2	no	—
135	C₂×C₈	—	104	2	no	—
136	C₂×C₈	C₂×#1055	1	2	yes	—
137	C₂×C₈	#6099	1	2	yes	—
138	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
139	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
140	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
141	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
142	C₄²	D₁₀₄	1	1	yes	characteristic
143	C₄²	Dic₅₂	1	1	yes	characteristic
144	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
145	C₄²	C₂×D₅₂	1	1	yes	characteristic
146	C₁₆	—	2	2	no	—
147	C₁₆	C₂×#1055	1	4	yes	complemented
148	C₁₃	#12009	1	1	yes	characteristic, complemented, Sylow
149	C₂³	#88058	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₄⋊#1055	1	2	yes	—
154	C₂×C₄	#88058	1	1	yes	characteristic
155	C₂×C₄	#88058	1	1	yes	characteristic
156	C₂×C₄	#88058	1	1	yes	characteristic
157	C₂×C₄	#88059	1	2	yes	—
158	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
159	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	104	1	no	—
164	C₈	—	104	2	no	—
165	C₈	—	104	1	no	—
166	C₈	C₄⋊#1055	1	2	yes	—
167	C₈	C₂²×#1055	1	1	yes	characteristic
168	C₈	C₂×#6099	1	1	yes	characteristic
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#11718050	1	2	yes	—
172	C₂²	C₂×#88058	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	#11718050	1	2	yes	—
177	C₄	#11718569	1	1	yes	characteristic
178	C₄	#11718570	1	1	yes	characteristic
179	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
180	C₄	C₂×#88059	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	C₂×#11718050	1	1	yes	characteristic
183	C₂	#13116624	1	1	yes	characteristic
184	C₂	#13116631	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₄
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×#107
prime radical	C₂³×C₁₃	#107
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#12009
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13112910	C₂	semidirect
4. #13112908	C₂	semidirect
23. C₁₃×M₃₂	C₈	semidirect
56. C₁₃×C₁₆	C₂×C₈	semidirect
111. M₃₂	#1055	semidirect
147. C₁₆	C₂×#1055	semidirect
148. C₁₃	#12009	semidirect

Maximal subgroups

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

subgroup	index
#13113584	2
C₂×#11718050	2
#13112910	2
#13112908	2
C₄×C₁₃×M₃₂	2
#12009	13