Subgroups — #16806739

Order 3328.

818 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719162	C₂	1	2	yes	maximal
2	#13131045	C₂	1	6	yes	maximal, complemented
3	C₄×#88106	C₂	1	3	yes	maximal, complemented
4	C₂×#11718231	C₂	1	3	yes	maximal, complemented
5	C₄×#88202	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
7	C₂×#88202	C₄	1	1	yes	characteristic
8	C₂×#88202	C₂²	1	1	yes	characteristic
9	#11719162	C₄	1	16	yes	complemented
10	C₂×#88106	C₂²	1	6	yes	—
11	C₂²×#6092	C₂²	1	6	yes	—
12	#11718658	C₄	1	3	yes	—
13	#11718658	C₂²	1	3	yes	—
14	C₄×#6099	C₄	1	3	yes	—
15	C₂×C₄×#1055	C₂²	1	3	yes	—
16	#11718231	C₂²	1	1	yes	characteristic
17	#11718231	C₂²	1	3	yes	—
18	#11718231	C₂²	1	6	yes	—
19	C₄×#6092	C₂²	1	6	yes	—
20	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₃	C₈	1	2	yes	—
22	C₄×D₄×C₁₃	C₈	1	6	yes	—
23	C₁₃×#117	C₂×C₄	1	3	yes	—
24	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
25	#88202	C₈	1	8	yes	—
26	#88202	C₂×C₄	1	8	yes	—
27	C₂×#6099	C₂×C₄	1	3	yes	—
28	C₂×#6099	C₂×C₄	1	3	yes	—
29	C₂²×#1055	C₂×C₄	1	3	yes	—
30	C₂²×#1055	C₂³	1	3	yes	—
31	#88106	C₂×C₄	1	24	yes	complemented
32	C₂×#6092	C₂×C₄	1	24	yes	complemented
33	C₂×#6092	C₂³	1	6	yes	—
34	C₂×#6092	C₂³	1	2	yes	—
35	#88043	C₂×C₄	1	6	yes	—
36	C₄×#1055	C₂×C₄	1	3	yes	—
37	C₄×#1055	C₂×C₄	1	3	yes	—
38	C₄×#1055	C₂³	1	3	yes	—
39	C₄×#1055	C₂³	1	1	yes	characteristic
40	#33326	—	13	1	no	maximal, Sylow
41	C₁₃×#41	C₄²	1	4	yes	—
42	C₁₃×#41	C₂×C₈	1	4	yes	—
43	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂×D₄×C₁₃	C₂×C₈	1	3	yes	—
45	C₂²×C₄×C₁₃	C₂×C₈	1	6	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
47	C₁₃×C₄⋊C₄	C₂×C₈	1	6	yes	—
48	C₁₃×C₂²⋊C₄	C₂×C₈	1	6	yes	—
49	C₄²×C₁₃	C₂×C₈	1	6	yes	—
50	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
52	#6099	C₄²	1	12	yes	—
53	#6099	C₂×C₈	1	12	yes	—
54	C₂×#1055	C₂×C₈	1	12	yes	—
55	C₂×#1055	C₂²×C₄	1	3	yes	—
56	C₂×#1055	C₂²×C₄	1	12	yes	—
57	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
58	C₂×#1055	C₂²×C₄	1	3	yes	—
59	C₂×#1055	C₂⁴	1	1	yes	characteristic
60	#6092	C₂²×C₄	1	4	yes	—
61	#6092	C₂²×C₄	1	12	yes	—
62	C₂×#504	—	13	2	no	—
63	C₄×#131	—	13	1	no	—
64	#2108	—	13	6	no	—
65	C₄×M₃₂	—	13	3	no	—
66	C₂×#346	—	13	3	no	—
67	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
68	Q₈×C₁₃	C₄×C₈	1	4	yes	—
69	D₄×C₁₃	C₄×C₈	1	12	yes	—
70	C₂×C₄×C₁₃	—	4	6	no	—
71	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
72	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
74	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
75	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
76	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
77	#1055	C₂×C₄²	1	6	yes	—
78	#1055	C₂²×C₈	1	2	yes	—
79	#1055	C₂²×C₈	1	6	yes	—
80	#1055	C₂³×C₄	1	2	yes	—
81	C₂×#131	—	13	1	no	—
82	C₂×#131	—	13	1	no	—
83	C₄×#41	#261	1	1	yes	characteristic
84	#504	—	13	16	no	—
85	C₂×M₃₂	—	13	6	no	—
86	C₂²×C₁₆	—	13	6	no	—
87	#405	—	13	3	no	—
88	#405	—	13	3	no	—
89	C₄×M₁₆	—	13	3	no	—
90	C₂×C₄×C₈	—	13	3	no	—
91	#346	—	13	1	no	—
92	#346	—	13	3	no	—
93	#346	—	13	6	no	—
94	C₄×C₁₆	—	13	6	no	—
95	C₂²×C₁₃	—	2	6	no	—
96	C₂²×C₁₃	C₂×C₄×C₈	1	6	yes	—
97	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
98	C₄×C₁₃	—	2	2	no	—
99	C₄×C₁₃	—	2	6	no	—
100	C₄×C₁₃	C₂×C₄×C₈	1	6	yes	—
101	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
102	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
103	C₂×#41	C₂×#261	1	1	yes	characteristic
104	#131	—	13	8	no	—
105	#131	—	13	8	no	—
106	C₂×M₁₆	—	13	3	no	—
107	C₂×M₁₆	—	13	3	no	—
108	C₂²×C₈	—	13	3	no	—
109	C₂²×C₈	—	13	3	no	—
110	C₄×Q₈	#1057	1	2	yes	—
111	C₄×D₄	#1057	1	6	yes	—
112	#117	C₂×#261	1	3	yes	—
113	C₂×C₄²	C₂×#261	1	3	yes	—
114	M₃₂	—	13	24	no	—
115	C₂×C₁₆	—	13	24	no	—
116	C₂×C₁₆	—	13	6	no	—
117	C₂×C₁₆	—	13	2	no	—
118	#97	—	13	6	no	—
119	C₄×C₈	—	13	3	no	—
120	C₄×C₈	—	13	1	no	—
121	C₄×C₈	—	13	3	no	—
122	C₄×C₈	—	13	3	no	—
123	C₂×C₁₃	—	2	6	no	—
124	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
125	C₂×C₁₃	#3407	1	2	yes	—
126	#41	D₁₃⋊C₈	1	4	yes	—
127	#41	C₄×#261	1	4	yes	—
128	C₂×Q₈	C₂×#1057	1	1	yes	characteristic
129	C₂×D₄	C₂×#1057	1	3	yes	—
130	C₂²×C₄	C₂×#1057	1	6	yes	—
131	C₂²×C₄	C₂²×#261	1	3	yes	—
132	M₁₆	—	13	12	no	—
133	M₁₆	—	13	12	no	—
134	C₂×C₈	—	13	12	no	—
135	C₂×C₈	—	13	12	no	—
136	C₂×C₈	—	13	1	no	—
137	C₂×C₈	—	13	1	no	—
138	C₂×C₈	—	13	3	no	—
139	C₂×C₈	—	13	3	no	—
140	C₄⋊C₄	C₂×#1057	1	6	yes	—
141	C₂²⋊C₄	C₂×#1057	1	6	yes	—
142	C₄²	C₂×#1057	1	6	yes	—
143	C₄²	C₂²×#261	1	1	yes	characteristic
144	C₄²	C₂²×#261	1	3	yes	—
145	C₁₆	—	13	4	no	—
146	C₁₆	—	13	12	no	—
147	C₁₃	#33326	1	1	yes	characteristic, complemented, Sylow
148	C₂³	C₂²×#1057	1	3	yes	—
149	Q₈	C₄×#1057	1	4	yes	—
150	D₄	C₄×#1057	1	12	yes	—
151	C₂×C₄	—	4	6	no	—
152	C₂×C₄	C₂×D₁₃⋊C₈	1	6	yes	—
153	C₂×C₄	C₂×C₄×#261	1	6	yes	—
154	C₂×C₄	C₂²×#1057	1	2	yes	—
155	C₂×C₄	C₂²×#1057	1	6	yes	—
156	C₂×C₄	C₂²×#1057	1	3	yes	—
157	C₂×C₄	C₂³×#261	1	1	yes	characteristic
158	C₈	—	13	2	no	—
159	C₈	—	13	6	no	—
160	C₈	—	13	2	no	—
161	C₈	—	13	6	no	—
162	C₂²	—	2	6	no	—
163	C₂²	C₂×C₄×#1057	1	6	yes	—
164	C₂²	C₂³×#1057	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	6	no	—
167	C₄	C₂×C₄×#1057	1	6	yes	—
168	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
169	C₄	C₂²×C₄×#261	1	1	yes	characteristic
170	C₂	—	2	6	no	—
171	C₂	C₂²×C₄×#1057	1	1	yes	characteristic
172	C₂	#13131803	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₄×C₁₃×#41	C₄
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	C₄×#41	#261
hypocenter	C₁₃	#33326
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

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
4. C₂×#11718231	C₂	semidirect
9. #11719162	C₄	semidirect
31. #88106	C₂×C₄	semidirect
32. C₂×#6092	C₂×C₄	semidirect
147. C₁₃	#33326	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719162	2
#13131045	2
C₄×#88106	2
C₂×#11718231	2
C₄×#88202	2
#33326	13