Subgroups — #15774542

Order 3328.

232 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116639	C₂	1	1	yes	maximal, characteristic
2	#13116592	C₂	1	1	yes	maximal, characteristic
3	#13112906	C₂	1	2	yes	maximal, complemented
4	#13112900	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#431	C₄	1	1	yes	characteristic
7	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88052	—	2	1	no	—
10	C₁₃×#322	C₄	1	2	yes	—
11	C₈²×C₁₃	C₂²	1	2	yes	—
12	C₂²⋊#6090	—	2	1	no	—
13	C₄⋊#6090	—	2	1	no	—
14	C₄⋊#6090	C₂²	1	1	yes	characteristic
15	C₄⋊#6090	C₂²	1	2	yes	—
16	C₄⋊#6090	C₂²	1	1	yes	characteristic
17	#11718053	—	2	1	no	—
18	C₂×C₁₃×M₁₆	C₈	1	2	yes	—
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₁₃×C₄⋊C₈	—	2	1	no	—
23	C₁₃×C₂²⋊C₈	—	2	1	no	—
24	C₁₃×#97	C₈	1	1	yes	characteristic
25	C₁₃×#97	C₈	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	#88052	—	4	2	no	—
33	C₂×#6090	—	2	2	no	—
34	C₂×#6090	—	2	1	no	—
35	C₂×#6090	—	2	1	no	—
36	#12017	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
40	C₁₃×M₁₆	—	2	4	no	—
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	—	2	1	no	—
43	C₂×C₈×C₁₃	—	2	2	no	—
44	C₂×C₈×C₁₃	—	4	1	no	—
45	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
54	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	#6090	—	4	1	no	—
56	#6090	—	4	2	no	—
57	#6090	—	4	1	no	—
58	#2163	—	13	1	no	—
59	#2150	—	13	1	no	—
60	C₈×M₁₆	D₁₃	1	1	yes	characteristic
61	#1370	—	13	2	no	—
62	#1367	—	13	2	no	—
63	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₃	—	2	1	no	—
65	C₂×C₄×C₁₃	—	2	1	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	4	1	no	—
68	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
74	C₈×C₁₃	—	2	1	no	—
75	C₈×C₁₃	—	2	1	no	—
76	C₈×C₁₃	—	2	1	no	—
77	C₈×C₁₃	—	2	2	no	—
78	C₈×C₁₃	—	4	2	no	—
79	C₈×C₁₃	M₃₂	1	4	yes	complemented
80	C₈×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₈×C₁₃	#137	1	1	yes	characteristic
82	C₂×M₃₂	—	26	1	no	—
83	#431	Dic₁₃	1	1	yes	characteristic
84	C₄×M₁₆	Dic₁₃	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
86	C₄⋊C₁₆	—	13	1	no	—
87	C₄⋊C₁₆	—	13	2	no	—
88	C₄⋊C₁₆	—	13	1	no	—
89	C₄⋊C₁₆	—	26	1	no	—
90	C₂²⋊C₁₆	—	26	1	no	—
91	#346	—	26	1	no	—
92	#322	Dic₁₃	1	2	yes	—
93	C₈²	C₂×D₁₃	1	2	yes	—
94	C₂²×C₁₃	—	2	1	no	—
95	C₂²×C₁₃	—	2	1	no	—
96	C₂²×C₁₃	—	2	1	no	—
97	C₂²×C₁₃	C₂×C₄⋊C₈	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	—	2	1	no	—
102	C₄×C₁₃	#428	1	1	yes	characteristic
103	C₄×C₁₃	#430	1	1	yes	characteristic
104	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
105	C₂×M₁₆	#1055	1	2	yes	—
106	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
107	C₂²×C₈	D₅₂	1	1	yes	characteristic
108	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
109	M₃₂	—	52	2	no	—
110	C₂×C₁₆	—	26	1	no	—
111	C₂×C₁₆	—	26	1	no	—
112	C₂×C₁₆	—	26	2	no	—
113	C₄⋊C₈	—	2	1	no	—
114	C₂²⋊C₈	—	2	1	no	—
115	#97	#1055	1	1	yes	characteristic
116	#97	#1055	1	1	yes	characteristic
117	C₄×C₈	Dic₂₆	1	1	yes	characteristic
118	C₄×C₈	D₅₂	1	1	yes	characteristic
119	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
120	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
121	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
122	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
123	C₂×C₁₃	—	4	1	no	—
124	C₂×C₁₃	#1565	1	1	yes	characteristic
125	C₂×C₁₃	#2150	1	1	yes	characteristic
126	C₂×C₁₃	#2153	1	1	yes	characteristic
127	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
128	C₂²×C₄	#6099	1	1	yes	characteristic
129	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
130	M₁₆	—	2	4	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	C₂×#1055	1	2	yes	—
136	C₂×C₈	C₂×#1055	1	2	yes	—
137	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
138	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
140	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
141	C₄²	C₂×#1055	1	1	yes	characteristic
142	C₄²	#6099	1	1	yes	characteristic
143	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
144	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
145	C₁₆	—	52	2	no	—
146	C₁₆	—	52	1	no	—
147	C₁₆	—	52	1	no	—
148	C₁₃	#12017	1	1	yes	characteristic, complemented, Sylow
149	C₂³	C₄⋊#1055	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
155	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
156	C₂×C₄	C₄⋊#1055	1	1	yes	characteristic
157	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
158	C₂×C₄	C₂×#6099	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	4	2	no	—
165	C₈	#88052	1	4	yes	complemented
166	C₈	#88162	1	1	yes	characteristic
167	C₈	#88163	1	1	yes	characteristic
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	C₂×#88052	1	2	yes	—
177	C₄	#11718782	1	1	yes	characteristic
178	C₄	#11718786	1	1	yes	characteristic
179	C₂	—	4	1	no	—
180	C₂	#13113588	1	1	yes	characteristic
181	C₂	#13116592	1	1	yes	characteristic
182	C₂	#13116602	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	#12017
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13112906	C₂	semidirect
4. #13112900	C₂	semidirect
79. C₈×C₁₃	M₃₂	semidirect
148. C₁₃	#12017	semidirect
165. C₈	#88052	semidirect

Maximal subgroups

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

subgroup	index
#13116639	2
#13116592	2
#13112906	2
#13112900	2
C₈×C₁₃×M₁₆	2
#12017	13