Subgroups — #16806729

Order 3328.

826 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88106	C₂	1	2	yes	maximal
2	#13131045	C₂	1	8	yes	maximal, complemented
3	C₄×#88106	C₂	1	4	yes	maximal, complemented
4	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
6	C₂²×#6099	C₄	1	1	yes	characteristic
7	C₂³×#1055	C₂²	1	1	yes	characteristic
8	C₂×#88106	C₄	1	16	yes	complemented
9	C₂×#88106	C₂²	1	4	yes	—
10	C₂×#88106	C₂²	1	4	yes	—
11	C₂²×#6092	C₂²	1	4	yes	—
12	#11718658	C₄	1	4	yes	—
13	#11718658	C₂²	1	4	yes	—
14	C₂×#88043	C₄	1	2	yes	—
15	C₂×C₄×#1055	C₂²	1	2	yes	—
16	#11718231	C₂²	1	8	yes	complemented
17	C₄×#6092	C₂²	1	8	yes	complemented
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₁₃×#117	C₈	1	4	yes	—
20	C₁₃×#117	C₂×C₄	1	4	yes	—
21	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
22	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
24	C₂×#6099	C₈	1	8	yes	—
25	C₂×#6099	C₂×C₄	1	4	yes	—
26	C₂²×#1055	C₂×C₄	1	2	yes	—
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	C₂×C₄	1	8	yes	—
29	C₂²×#1055	C₂×C₄	1	2	yes	—
30	C₂²×#1055	C₂³	1	2	yes	—
31	C₂²×#1055	C₂³	1	1	yes	characteristic
32	#88106	C₂×C₄	1	16	yes	—
33	#88106	C₂×C₄	1	16	yes	complemented
34	C₂×#6092	C₂×C₄	1	16	yes	complemented
35	C₂×#6092	C₂³	1	8	yes	—
36	#88043	C₂×C₄	1	8	yes	—
37	C₄×#1055	C₂×C₄	1	4	yes	—
38	C₄×#1055	C₂³	1	4	yes	—
39	#33324	—	13	1	no	maximal, Sylow
40	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
48	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
49	C₁₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
50	C₄²×C₁₃	C₂×C₈	1	4	yes	—
51	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
52	#6099	C₂×C₈	1	16	yes	—
53	C₂×#1055	C₄²	1	8	yes	—
54	C₂×#1055	C₄²	1	4	yes	—
55	C₂×#1055	C₂×C₈	1	8	yes	—
56	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
57	C₂×#1055	C₂²×C₄	1	2	yes	—
58	C₂×#1055	C₂²×C₄	1	4	yes	—
59	C₂×#1055	C₂²×C₄	1	4	yes	—
60	C₂×#1055	C₂²×C₄	1	4	yes	—
61	C₂×#1055	C₂²×C₄	1	4	yes	—
62	C₂×#1055	C₂⁴	1	1	yes	characteristic
63	#6092	C₂²×C₄	1	16	yes	—
64	C₂²×M₃₂	—	13	2	no	—
65	C₂×#405	—	13	1	no	—
66	#2108	—	13	8	no	—
67	C₄×M₃₂	—	13	4	no	—
68	C₂³×C₁₃	—	4	2	no	—
69	C₂³×C₁₃	C₄×C₈	1	4	yes	—
70	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
71	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	4	4	no	—
73	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
74	C₂×C₄×C₁₃	C₄×C₈	1	8	yes	—
75	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
76	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
77	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
78	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
80	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
83	#1055	C₂×C₄²	1	2	yes	—
84	#1055	C₂×C₄²	1	4	yes	—
85	#1055	C₂²×C₈	1	8	yes	—
86	#1055	C₂³×C₄	1	2	yes	—
87	C₂²×M₁₆	—	13	1	no	—
88	C₂³×C₈	—	13	1	no	—
89	C₂×#117	#261	1	1	yes	characteristic
90	C₂×M₃₂	—	13	4	no	—
91	C₂×M₃₂	—	13	4	no	—
92	C₂×M₃₂	—	13	16	no	—
93	C₂²×C₁₆	—	13	4	no	—
94	#405	—	13	4	no	—
95	#405	—	13	4	no	—
96	C₂×#97	—	13	2	no	—
97	C₂×C₄×C₈	—	13	2	no	—
98	#346	—	13	8	no	—
99	C₄×C₁₆	—	13	8	no	—
100	C₂²×C₁₃	—	2	2	no	—
101	C₂²×C₁₃	—	2	4	no	—
102	C₂²×C₁₃	—	4	4	no	—
103	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
104	C₂²×C₁₃	C₂×C₄×C₈	1	4	yes	—
105	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
106	C₄×C₁₃	—	2	8	no	—
107	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
108	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
109	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
110	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
111	C₂³×C₄	C₂×#261	1	1	yes	characteristic
112	C₂×M₁₆	—	13	8	no	—
113	C₂×M₁₆	—	13	4	no	—
114	C₂²×C₈	—	13	8	no	—
115	C₂²×C₈	—	13	1	no	—
116	C₂²×C₈	—	13	2	no	—
117	C₂²×C₈	—	13	1	no	—
118	C₂²×C₈	—	13	2	no	—
119	C₂²×C₈	—	13	2	no	—
120	#117	#1057	1	4	yes	—
121	#117	C₂×#261	1	4	yes	—
122	C₂×C₄⋊C₄	#1057	1	2	yes	—
123	C₂×C₂²⋊C₄	#1057	1	2	yes	—
124	C₂×C₄²	C₂×#261	1	2	yes	—
125	M₃₂	—	13	16	no	—
126	M₃₂	—	13	16	no	—
127	C₂×C₁₆	—	13	16	no	—
128	C₂×C₁₆	—	13	8	no	—
129	#97	—	13	8	no	—
130	C₄×C₈	—	13	4	no	—
131	C₄×C₈	—	13	4	no	—
132	C₂×C₁₃	—	2	2	no	—
133	C₂×C₁₃	—	2	4	no	—
134	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
135	C₂×C₁₃	#3407	1	2	yes	—
136	C₂⁴	C₂×#1057	1	1	yes	characteristic
137	C₂²×C₄	D₁₃⋊C₈	1	4	yes	—
138	C₂²×C₄	C₄×#261	1	4	yes	—
139	C₂²×C₄	C₂×#1057	1	4	yes	—
140	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
141	C₂²×C₄	C₂×#1057	1	2	yes	—
142	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
143	C₂²×C₄	C₂²×#261	1	2	yes	—
144	M₁₆	—	13	16	no	—
145	C₂×C₈	—	13	8	no	—
146	C₂×C₈	—	13	4	no	—
147	C₂×C₈	—	13	4	no	—
148	C₂×C₈	—	13	4	no	—
149	C₂×C₈	—	13	4	no	—
150	C₂×C₈	—	13	8	no	—
151	C₂×C₈	—	13	1	no	—
152	C₂×C₈	—	13	1	no	—
153	C₂×C₈	—	13	4	no	—
154	C₂×C₈	—	13	2	no	—
155	C₄⋊C₄	C₂×#1057	1	8	yes	—
156	C₂²⋊C₄	C₂×#1057	1	8	yes	—
157	C₄²	C₂×#1057	1	4	yes	—
158	C₄²	C₂²×#261	1	4	yes	—
159	C₁₆	—	13	16	no	—
160	C₁₃	#33324	1	1	yes	characteristic, complemented, Sylow
161	C₂³	—	4	2	no	—
162	C₂³	C₄×#1057	1	4	yes	—
163	C₂³	C₂²×#1057	1	2	yes	—
164	C₂³	C₂²×#1057	1	1	yes	characteristic
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	C₄×#1057	1	4	yes	—
167	C₂×C₄	C₄×#1057	1	8	yes	—
168	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
169	C₂×C₄	C₂×D₁₃⋊C₈	1	4	yes	—
170	C₂×C₄	C₂×C₄×#261	1	4	yes	—
171	C₂×C₄	C₂×C₄×#261	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	8	no	—
177	C₈	—	13	2	no	—
178	C₈	—	13	4	no	—
179	C₈	—	13	2	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	4	no	—
183	C₂²	C₂×C₄×#1057	1	2	yes	—
184	C₂²	C₂×C₄×#1057	1	4	yes	—
185	C₂²	C₂³×#1057	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	C₂×C₄×#1057	1	2	yes	—
188	C₄	C₂×C₄×#1057	1	4	yes	—
189	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
190	C₄	C₂²×C₄×#261	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	4	no	—
193	C₂	C₂²×C₄×#1057	1	1	yes	characteristic
194	C₂	#13131803	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₂×C₁₃×#117	C₄
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×#117	#261
hypocenter	C₁₃	#33324
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #13131045	C₂	semidirect
3. C₄×#88106	C₂	semidirect
8. C₂×#88106	C₄	semidirect
16. #11718231	C₂²	semidirect
17. C₄×#6092	C₂²	semidirect
33. #88106	C₂×C₄	semidirect
34. C₂×#6092	C₂×C₄	semidirect
160. C₁₃	#33324	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88106	2
#13131045	2
C₄×#88106	2
C₂×#11718658	2
#33324	13