Subgroups — #16807252

Order 3328.

506 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	#13131251	C₂	1	4	yes	maximal, complemented
2	Q₈×#6092	C₂	1	4	yes	maximal, complemented
3	#13131171	C₂	1	2	yes	maximal
4	#13131168	C₂	1	1	yes	maximal, characteristic
5	#13131045	C₂	1	2	yes	maximal
6	C₂×C₄×#6092	C₂	1	1	yes	maximal, characteristic
7	#13123771	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#533	C₄	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	2	yes	—
10	C₂²×#6092	C₂²	1	1	yes	characteristic
11	C₂²×#6092	C₂²	1	1	yes	characteristic
12	#11718685	C₄	1	4	yes	—
13	Q₈×#1055	C₂²	1	4	yes	—
14	#11718660	C₄	1	2	yes	—
15	#11718658	C₂²	1	2	yes	—
16	#11718585	C₄	1	1	yes	characteristic
17	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
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	2	yes	—
22	C₄×#6092	C₂²	1	2	yes	—
23	C₄×#6092	C₂²	1	4	yes	—
24	C₁₃×C₄⋊Q₈	C₈	1	2	yes	—
25	C₁₃×#125	C₈	1	2	yes	—
26	C₁₃×C₂²⋊Q₈	C₈	1	4	yes	—
27	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
28	C₁₃×#117	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#6099	C₂×C₄	1	2	yes	—
31	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂³	1	1	yes	characteristic
33	#88106	—	2	4	no	—
34	C₂×#6092	—	2	2	no	—
35	C₂×#6092	Q₈	1	4	yes	complemented
36	C₂×#6092	C₂³	1	2	yes	—
37	C₂×#6092	C₂³	1	4	yes	—
38	C₂×#6092	C₂³	1	1	yes	characteristic
39	C₂×#6092	C₂³	1	1	yes	characteristic
40	C₄⋊#1055	C₂×C₄	1	8	yes	—
41	C₄⋊#1055	C₂×C₄	1	4	yes	—
42	#88043	C₂×C₄	1	4	yes	—
43	C₄×#1055	C₂×C₄	1	2	yes	—
44	C₄×#1055	C₂³	1	4	yes	—
45	C₄×#1055	C₂³	1	2	yes	—
46	#33474	—	13	1	no	maximal, Sylow
47	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
48	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
51	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
52	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
53	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
54	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
56	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
57	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
58	#6099	—	2	4	no	—
59	C₂×#1055	—	2	2	no	—
60	C₂×#1055	C₂²×C₄	1	2	yes	—
61	C₂×#1055	C₂²×C₄	1	4	yes	—
62	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
63	C₂×#1055	C₂×Q₈	1	2	yes	—
64	C₂×#1055	#41	1	2	yes	—
65	C₂×#1055	C₂⁴	1	1	yes	characteristic
66	#6092	—	2	2	no	—
67	#6092	—	2	4	no	—
68	#6092	C₂×Q₈	1	2	yes	—
69	#6092	C₂×Q₈	1	2	yes	—
70	#2960	—	13	1	no	—
71	#2183	—	13	4	no	—
72	Q₈×C₁₆	—	13	4	no	—
73	#2151	—	13	2	no	—
74	#2150	—	13	1	no	—
75	#2108	—	13	2	no	—
76	C₂×C₄×C₁₆	—	13	1	no	—
77	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
78	Q₈×C₁₃	—	2	8	no	—
79	C₂×C₄×C₁₃	—	2	4	no	—
80	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
83	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₁₃	#131	1	2	yes	—
86	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
87	#1055	—	2	2	no	—
88	#1055	—	2	4	no	—
89	#1055	C₄×Q₈	1	2	yes	—
90	#1055	C₂²×Q₈	1	1	yes	characteristic
91	#1055	C₂×#41	1	1	yes	characteristic
92	#533	#261	1	1	yes	characteristic
93	C₂×M₃₂	—	13	2	no	—
94	C₂²×C₁₆	—	13	1	no	—
95	C₂²×C₁₆	—	13	1	no	—
96	#446	—	13	4	no	—
97	C₈×Q₈	—	13	4	no	—
98	#424	—	13	2	no	—
99	#423	—	13	1	no	—
100	#405	—	13	2	no	—
101	C₂×C₄×C₈	—	13	1	no	—
102	C₄⋊C₁₆	—	13	4	no	—
103	C₄⋊C₁₆	—	13	8	no	—
104	#346	—	13	4	no	—
105	C₄×C₁₆	—	13	2	no	—
106	C₄×C₁₆	—	13	2	no	—
107	C₄×C₁₆	—	13	4	no	—
108	C₂²×C₁₃	—	2	2	no	—
109	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
110	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
111	C₄×C₁₃	—	2	2	no	—
112	C₄×C₁₃	—	2	8	no	—
113	C₄×C₁₃	C₈×Q₈	1	2	yes	—
114	C₄×C₁₃	#504	1	4	yes	—
115	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
116	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
117	C₂×M₁₆	—	13	2	no	—
118	C₂²×C₈	—	13	1	no	—
119	C₂²×C₈	—	13	1	no	—
120	C₄⋊Q₈	#1057	1	2	yes	—
121	#125	#1057	1	2	yes	—
122	C₂²⋊Q₈	#1057	1	4	yes	—
123	C₄×Q₈	C₂×#261	1	4	yes	—
124	#117	C₂×#261	1	2	yes	—
125	C₂×C₄²	C₂×#261	1	1	yes	characteristic
126	M₃₂	—	26	4	no	—
127	C₂×C₁₆	—	13	4	no	—
128	C₂×C₁₆	—	13	1	no	—
129	C₂×C₁₆	—	13	1	no	—
130	C₂×C₁₆	—	13	2	no	—
131	C₂×C₁₆	—	13	4	no	—
132	C₂×C₁₆	—	26	2	no	—
133	C₄⋊C₈	—	13	4	no	—
134	C₄⋊C₈	—	13	8	no	—
135	#97	—	13	4	no	—
136	C₄×C₈	—	13	2	no	—
137	C₄×C₈	—	13	4	no	—
138	C₄×C₈	—	13	2	no	—
139	C₂×C₁₃	—	2	2	no	—
140	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
141	C₂×C₁₃	C₂×#504	1	2	yes	—
142	C₂×Q₈	C₂×#1057	1	4	yes	—
143	C₂²×C₄	C₂×#1057	1	2	yes	—
144	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
145	M₁₆	—	26	4	no	—
146	C₂×C₈	—	13	2	no	—
147	C₂×C₈	—	13	2	no	—
148	C₂×C₈	—	13	1	no	—
149	C₂×C₈	—	13	1	no	—
150	C₂×C₈	—	13	2	no	—
151	C₂×C₈	—	13	4	no	—
152	C₂×C₈	—	26	2	no	—
153	C₄⋊C₄	C₂×#1057	1	4	yes	—
154	C₄⋊C₄	C₂×#1057	1	8	yes	—
155	C₄⋊C₄	C₂×#1057	1	4	yes	—
156	C₂²⋊C₄	C₂×#1057	1	4	yes	—
157	C₄²	C₂×#1057	1	1	yes	characteristic
158	C₄²	C₂×#1057	1	1	yes	characteristic
159	C₄²	C₂²×#261	1	2	yes	—
160	C₄²	C₂²×#261	1	4	yes	—
161	C₁₆	—	13	2	no	—
162	C₁₆	—	13	2	no	—
163	C₁₆	—	26	2	no	—
164	C₁₆	—	26	4	no	—
165	C₁₃	#33474	1	1	yes	characteristic, complemented, Sylow
166	C₂³	C₂²×#1057	1	1	yes	characteristic
167	Q₈	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	#88240	1	2	yes	—
170	C₂×C₄	Q₈×#261	1	2	yes	—
171	C₂×C₄	C₂²×#1057	1	2	yes	—
172	C₂×C₄	C₂²×#1057	1	8	yes	—
173	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
174	C₂×C₄	C₂²×#1057	1	2	yes	—
175	C₂×C₄	C₂³×#261	1	1	yes	characteristic
176	C₈	—	13	1	no	—
177	C₈	—	13	1	no	—
178	C₈	—	13	2	no	—
179	C₈	—	26	2	no	—
180	C₈	—	26	4	no	—
181	C₂²	—	2	2	no	—
182	C₂²	Q₈×#1057	1	2	yes	—
183	C₂²	C₂³×#1057	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	8	no	—
186	C₄	Q₈×#1057	1	2	yes	—
187	C₄	#11719162	1	4	yes	—
188	C₄	C₂×#88240	1	1	yes	characteristic
189	C₄	C₂×Q₈×#261	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×Q₈×#1057	1	1	yes	characteristic
192	C₂	C₂×#11719162	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	#33474
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
2. Q₈×#6092	C₂	semidirect
35. C₂×#6092	Q₈	semidirect
165. C₁₃	#33474	semidirect

Maximal subgroups

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

subgroup	index
#13131251	2
Q₈×#6092	2
#13131171	2
#13131168	2
#13131045	2
C₂×C₄×#6092	2
#13123771	2
#33474	13