Subgroups — #16788701

Order 3328.

262 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131193	C₂	1	2	yes	maximal
2	#13129865	C₂	1	2	yes	maximal, complemented
3	#13129858	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₄×#1055	C₄	1	1	yes	characteristic
7	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
8	C₂²⋊#6092	—	2	2	no	—
9	C₄⋊#6092	C₂²	1	4	yes	—
10	C₄×#6092	—	2	2	no	—
11	C₈×#1055	C₄	1	1	yes	characteristic
12	C₈×#1055	C₄	1	1	yes	characteristic
13	C₈×#1055	C₂²	1	1	yes	characteristic
14	C₈×#1055	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₈	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×#1055	—	2	1	no	—
19	C₂²×#1055	D₄	1	1	yes	characteristic
20	C₂²×#1055	Q₈	1	1	yes	characteristic
21	C₂×#6092	—	2	4	no	—
22	C₄×C₈×C₁₃	C₈	1	1	yes	characteristic
23	C₄×C₈×C₁₃	C₈	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×#1055	—	2	1	no	—
27	C₄×#1055	C₂×C₄	1	1	yes	characteristic
28	C₄×#1055	C₂×C₄	1	1	yes	characteristic
29	C₄×#1055	C₂×C₄	1	1	yes	characteristic
30	C₄×#1055	D₄	1	1	yes	characteristic
31	C₄×#1055	Q₈	1	1	yes	characteristic
32	C₄×#1055	C₂³	1	1	yes	characteristic
33	#12012	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	C₁₆	1	4	yes	complemented
39	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
46	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×#1055	—	2	4	no	—
48	C₂×#1055	—	2	1	no	—
49	C₂×#1055	—	2	1	no	—
50	C₂×#1055	—	2	2	no	—
51	C₂×#1055	—	2	2	no	—
52	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
53	C₂×#1055	C₄⋊C₄	1	1	yes	characteristic
54	C₂×#1055	C₂×D₄	1	1	yes	characteristic
55	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
56	#6092	—	4	4	no	—
57	#2162	—	13	2	no	—
58	C₂×C₈²	—	13	1	no	—
59	#1370	—	13	2	no	—
60	#1367	—	13	2	no	—
61	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	2	1	no	—
65	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
69	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
70	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
73	C₈×C₁₃	—	2	1	no	—
74	C₈×C₁₃	—	2	1	no	—
75	C₈×C₁₃	C₂×C₁₆	1	2	yes	—
76	C₈×C₁₃	C₂×C₁₆	1	2	yes	—
77	#1055	—	2	2	no	—
78	#1055	—	2	2	no	—
79	#1055	—	2	1	no	—
80	#1055	—	2	1	no	—
81	#1055	#108	1	2	yes	—
82	#1055	#135	1	2	yes	—
83	C₂×C₄×C₈	—	13	1	no	—
84	C₂×C₄×C₈	—	13	1	no	—
85	C₂×C₄×C₈	#261	1	1	yes	characteristic
86	C₄⋊C₁₆	—	13	4	no	—
87	C₂²⋊C₁₆	—	26	2	no	—
88	C₄×C₁₆	—	26	2	no	—
89	C₈²	—	13	1	no	—
90	C₈²	—	13	1	no	—
91	C₈²	—	13	1	no	—
92	C₈²	—	13	1	no	—
93	C₂²×C₁₃	—	2	1	no	—
94	C₂²×C₁₃	—	2	1	no	—
95	C₂²×C₁₃	C₄⋊C₁₆	1	2	yes	—
96	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₄×C₁₃	—	2	1	no	—
98	C₄×C₁₃	—	2	1	no	—
99	C₄×C₁₃	C₄⋊C₁₆	1	2	yes	—
100	C₄×C₁₃	#364	1	2	yes	—
101	C₄×C₁₃	#427	1	1	yes	characteristic
102	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
103	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
104	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
105	C₂²×C₈	—	13	1	no	—
106	C₂²×C₈	—	13	1	no	—
107	C₂²×C₈	—	26	1	no	—
108	C₂²×C₈	#1057	1	1	yes	characteristic
109	C₂²×C₈	#1057	1	1	yes	characteristic
110	C₂×C₄²	C₂×#261	1	1	yes	characteristic
111	C₂×C₁₆	—	26	4	no	—
112	C₄×C₈	—	13	1	no	—
113	C₄×C₈	—	13	1	no	—
114	C₄×C₈	—	13	1	no	—
115	C₄×C₈	—	13	1	no	—
116	C₄×C₈	—	13	1	no	—
117	C₄×C₈	—	13	1	no	—
118	C₄×C₈	—	26	1	no	—
119	C₄×C₈	#1057	1	1	yes	characteristic
120	C₄×C₈	#1057	1	1	yes	characteristic
121	C₄×C₈	C₂×#261	1	1	yes	characteristic
122	C₄×C₈	C₂×#261	1	1	yes	characteristic
123	C₂×C₁₃	—	2	2	no	—
124	C₂×C₁₃	#1564	1	1	yes	characteristic
125	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
126	C₂×C₁₃	C₂×#364	1	1	yes	characteristic
127	C₂²×C₄	#6118	1	1	yes	characteristic
128	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
129	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	13	1	no	—
132	C₂×C₈	—	13	1	no	—
133	C₂×C₈	—	13	1	no	—
134	C₂×C₈	—	13	1	no	—
135	C₂×C₈	—	26	2	no	—
136	C₂×C₈	—	26	2	no	—
137	C₂×C₈	—	26	4	no	—
138	C₂×C₈	—	26	1	no	—
139	C₂×C₈	—	26	1	no	—
140	C₂×C₈	#6092	1	4	yes	complemented
141	C₂×C₈	C₂×#1057	1	1	yes	characteristic
142	C₂×C₈	C₂×#1057	1	1	yes	characteristic
143	C₂×C₈	C₂×#1057	1	1	yes	characteristic
144	C₂×C₈	C₂×#1057	1	1	yes	characteristic
145	C₄²	#6118	1	1	yes	characteristic
146	C₄²	C₄⋊#261	1	1	yes	characteristic
147	C₄²	C₂×#1057	1	1	yes	characteristic
148	C₄²	C₂²×#261	1	1	yes	characteristic
149	C₁₆	—	52	4	no	—
150	C₁₃	#12012	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₄⋊#1057	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	C₂×#6092	1	2	yes	—
156	C₂×C₄	#88106	1	2	yes	—
157	C₂×C₄	C₄⋊#1057	1	1	yes	characteristic
158	C₂×C₄	C₄⋊#1057	1	1	yes	characteristic
159	C₂×C₄	C₄⋊#1057	1	1	yes	characteristic
160	C₂×C₄	C₂×#6118	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
162	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	13	2	no	—
166	C₈	—	13	2	no	—
167	C₈	—	26	2	no	—
168	C₈	—	26	2	no	—
169	C₈	—	26	1	no	—
170	C₈	—	26	1	no	—
171	C₈	C₂×#6092	1	2	yes	—
172	C₈	C₂×#6092	1	2	yes	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊#6092	1	2	yes	—
176	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	C₄⋊#6092	1	2	yes	—
180	C₄	#11718261	1	2	yes	—
181	C₄	#11719092	1	1	yes	characteristic
182	C₄	D₁₃⋊#108	1	1	yes	characteristic
183	C₄	C₂²×#6092	1	1	yes	characteristic
184	C₄	C₂×#88106	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#13130383	1	1	yes	characteristic
187	C₂	C₂×C₄⋊#6092	1	1	yes	characteristic
188	C₂	C₂×#11718261	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₄⋊#261
commutator	C₄×C₁₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₄×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#12012
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13129865	C₂	semidirect
3. #13129858	C₂	semidirect
38. C₂×C₈×C₁₃	C₁₆	semidirect
140. C₂×C₈	#6092	semidirect
150. C₁₃	#12012	semidirect

Maximal subgroups

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

subgroup	index
#13131193	2
#13129865	2
#13129858	2
C₂×C₈×#1055	2
#12012	13