Subgroups — #16807256

Order 3328.

490 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	#13131251	C₂	1	8	yes	maximal, complemented
2	#13131171	C₂	1	2	yes	maximal
3	#13131168	C₂	1	1	yes	maximal, characteristic
4	#13131045	C₂	1	2	yes	maximal
5	C₂×#11718231	C₂	1	1	yes	maximal, characteristic
6	#13123771	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#533	C₄	1	1	yes	characteristic
8	C₂×#88106	C₂²	1	2	yes	—
9	C₂²×#6092	C₂²	1	1	yes	characteristic
10	C₂²×#6092	C₂²	1	1	yes	characteristic
11	#11718685	C₄	1	4	yes	—
12	Q₈×#1055	C₂²	1	4	yes	—
13	#11718660	C₄	1	2	yes	—
14	#11718658	C₂²	1	2	yes	—
15	#11718585	C₄	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	#11718231	C₂²	1	4	yes	—
18	#11718231	C₂²	1	4	yes	—
19	C₄⋊#6092	C₂²	1	4	yes	—
20	C₄⋊#6092	C₂²	1	8	yes	—
21	C₄×#6092	C₂²	1	4	yes	—
22	C₁₃×C₄⋊Q₈	C₈	1	2	yes	—
23	C₁₃×#125	C₈	1	2	yes	—
24	C₁₃×C₂²⋊Q₈	C₈	1	4	yes	—
25	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
26	C₁₃×#117	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#6099	C₂×C₄	1	2	yes	—
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂³	1	1	yes	characteristic
31	#88106	—	2	4	no	—
32	C₂×#6092	—	2	2	no	—
33	C₂×#6092	Q₈	1	4	yes	complemented
34	C₂×#6092	C₂³	1	4	yes	—
35	C₂×#6092	C₂³	1	2	yes	—
36	C₂×#6092	C₂³	1	2	yes	—
37	C₄⋊#1055	C₂×C₄	1	8	yes	—
38	C₄⋊#1055	C₂×C₄	1	4	yes	—
39	#88043	C₂×C₄	1	4	yes	—
40	C₄×#1055	C₂×C₄	1	2	yes	—
41	C₄×#1055	C₂³	1	4	yes	—
42	C₄×#1055	C₂³	1	2	yes	—
43	#33475	—	13	1	no	maximal, Sylow
44	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
48	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
49	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
50	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
51	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
54	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
55	#6099	—	2	4	no	—
56	C₂×#1055	—	2	2	no	—
57	C₂×#1055	C₂²×C₄	1	2	yes	—
58	C₂×#1055	C₂²×C₄	1	4	yes	—
59	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
60	C₂×#1055	C₂×Q₈	1	2	yes	—
61	C₂×#1055	#41	1	2	yes	—
62	C₂×#1055	C₂⁴	1	1	yes	characteristic
63	#6092	—	2	2	no	—
64	#6092	—	2	4	no	—
65	#6092	C₂×Q₈	1	4	yes	—
66	#2960	—	13	1	no	—
67	#2183	—	13	8	no	—
68	#2151	—	13	2	no	—
69	#2150	—	13	1	no	—
70	#2108	—	13	2	no	—
71	C₂×#346	—	13	1	no	—
72	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
73	Q₈×C₁₃	—	4	4	no	—
74	C₂×C₄×C₁₃	—	4	2	no	—
75	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
79	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#131	1	2	yes	—
81	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
82	#1055	—	2	2	no	—
83	#1055	—	2	4	no	—
84	#1055	C₄×Q₈	1	2	yes	—
85	#1055	C₂²×Q₈	1	1	yes	characteristic
86	#1055	C₂×#41	1	1	yes	characteristic
87	#533	#261	1	1	yes	characteristic
88	C₂×M₃₂	—	13	2	no	—
89	C₂²×C₁₆	—	13	1	no	—
90	C₂²×C₁₆	—	13	1	no	—
91	#446	—	13	4	no	—
92	C₈×Q₈	—	13	4	no	—
93	#424	—	13	2	no	—
94	#423	—	13	1	no	—
95	#405	—	13	2	no	—
96	C₂×C₄×C₈	—	13	1	no	—
97	C₄⋊C₁₆	—	13	4	no	—
98	C₄⋊C₁₆	—	13	8	no	—
99	#346	—	13	4	no	—
100	#346	—	13	4	no	—
101	C₄×C₁₆	—	13	4	no	—
102	C₂²×C₁₃	—	2	2	no	—
103	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
104	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
105	C₄×C₁₃	—	2	2	no	—
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	—	2	8	no	—
108	C₄×C₁₃	C₈×Q₈	1	2	yes	—
109	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
110	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
111	C₂×M₁₆	—	13	2	no	—
112	C₂²×C₈	—	13	1	no	—
113	C₂²×C₈	—	13	1	no	—
114	C₄⋊Q₈	#1057	1	2	yes	—
115	#125	#1057	1	2	yes	—
116	C₂²⋊Q₈	#1057	1	4	yes	—
117	C₄×Q₈	C₂×#261	1	4	yes	—
118	#117	C₂×#261	1	2	yes	—
119	C₂×C₄²	C₂×#261	1	1	yes	characteristic
120	M₃₂	—	26	4	no	—
121	C₂×C₁₆	—	13	4	no	—
122	C₂×C₁₆	—	13	2	no	—
123	C₂×C₁₆	—	13	4	no	—
124	C₂×C₁₆	—	13	2	no	—
125	C₂×C₁₆	—	26	2	no	—
126	C₄⋊C₈	—	13	4	no	—
127	C₄⋊C₈	—	13	8	no	—
128	#97	—	13	4	no	—
129	C₄×C₈	—	13	4	no	—
130	C₄×C₈	—	13	2	no	—
131	C₄×C₈	—	13	2	no	—
132	C₂×C₁₃	—	2	2	no	—
133	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
134	C₂×C₁₃	#3407	1	2	yes	—
135	C₂×Q₈	C₂×#1057	1	4	yes	—
136	C₂²×C₄	C₂×#1057	1	2	yes	—
137	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
138	M₁₆	—	26	4	no	—
139	C₂×C₈	—	13	2	no	—
140	C₂×C₈	—	13	2	no	—
141	C₂×C₈	—	13	1	no	—
142	C₂×C₈	—	13	1	no	—
143	C₂×C₈	—	13	2	no	—
144	C₂×C₈	—	13	4	no	—
145	C₂×C₈	—	26	2	no	—
146	C₄⋊C₄	C₂×#1057	1	4	yes	—
147	C₄⋊C₄	C₂×#1057	1	8	yes	—
148	C₄⋊C₄	C₂×#1057	1	4	yes	—
149	C₂²⋊C₄	C₂×#1057	1	4	yes	—
150	C₄²	C₂×#1057	1	1	yes	characteristic
151	C₄²	C₂×#1057	1	1	yes	characteristic
152	C₄²	C₂²×#261	1	2	yes	—
153	C₄²	C₂²×#261	1	4	yes	—
154	C₁₆	—	13	4	no	—
155	C₁₆	—	26	2	no	—
156	C₁₆	—	26	4	no	—
157	C₁₃	#33475	1	1	yes	characteristic, complemented, Sylow
158	C₂³	C₂²×#1057	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	#88240	1	2	yes	—
162	C₂×C₄	Q₈×#261	1	2	yes	—
163	C₂×C₄	C₂²×#1057	1	2	yes	—
164	C₂×C₄	C₂²×#1057	1	2	yes	—
165	C₂×C₄	C₂²×#1057	1	8	yes	—
166	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
167	C₂×C₄	C₂³×#261	1	1	yes	characteristic
168	C₈	—	13	1	no	—
169	C₈	—	13	1	no	—
170	C₈	—	13	2	no	—
171	C₈	—	26	2	no	—
172	C₈	—	26	4	no	—
173	C₂²	—	2	2	no	—
174	C₂²	Q₈×#1057	1	2	yes	—
175	C₂²	C₂³×#1057	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	Q₈×#1057	1	2	yes	—
180	C₄	C₂×#88240	1	1	yes	characteristic
181	C₄	C₂×Q₈×#261	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×Q₈×#1057	1	1	yes	characteristic
184	C₂	#13131803	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#533	C₄
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#533	#261
hypocenter	C₁₃	#33475
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #13131251	C₂	semidirect
33. C₂×#6092	Q₈	semidirect
157. C₁₃	#33475	semidirect

Maximal subgroups

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

subgroup	index
#13131251	2
#13131171	2
#13131168	2
#13131045	2
C₂×#11718231	2
#13123771	2
#33475	13