Subgroups — #16807364

Order 3328.

486 conjugacy classes of subgroups.

The subgroup lattice has 178 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	#13131197	C₂	1	4	yes	maximal
3	#13131193	C₂	1	2	yes	maximal
4	C₂×Q₈×#1055	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
6	Q₈×#1055	C₄	1	4	yes	—
7	Q₈×#1055	C₂²	1	4	yes	—
8	C₂×C₄⋊#1055	C₄	1	2	yes	—
9	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
10	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
11	C₂×C₄×#1055	C₂²	1	2	yes	—
12	C₂²⋊#6092	C₂²	1	4	yes	—
13	#11718231	C₂²	1	8	yes	—
14	C₄⋊#6092	C₂²	1	4	yes	—
15	C₄⋊#6092	C₂²	1	8	yes	—
16	C₄×#6092	C₂²	1	4	yes	—
17	C₂²×Q₈×C₁₃	C₈	1	1	yes	characteristic
18	C₄×Q₈×C₁₃	C₈	1	4	yes	—
19	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
20	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
21	C₂×C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	2	yes	—
26	C₂²×#1055	C₂³	1	1	yes	characteristic
27	C₂×#6092	C₂³	1	8	yes	—
28	C₄⋊#1055	C₂×C₄	1	8	yes	—
29	C₄⋊#1055	C₂×C₄	1	4	yes	—
30	C₄×#1055	C₂×C₄	1	2	yes	—
31	C₄×#1055	C₂×C₄	1	4	yes	—
32	C₄×#1055	C₂³	1	2	yes	—
33	C₄×#1055	C₂³	1	4	yes	—
34	#33497	—	13	1	no	maximal, Sylow
35	C₂×Q₈×C₁₃	—	2	4	no	—
36	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
37	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
43	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
44	C₄²×C₁₃	C₂×C₈	1	4	yes	—
45	C₄²×C₁₃	C₂×C₈	1	2	yes	—
46	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
47	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
48	C₂×#1055	—	2	2	no	—
49	C₂×#1055	—	4	1	no	—
50	C₂×#1055	—	4	2	no	—
51	C₂×#1055	C₂²×C₄	1	2	yes	—
52	C₂×#1055	C₂²×C₄	1	4	yes	—
53	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
54	C₂×#1055	C₂⁴	1	1	yes	characteristic
55	#6092	—	2	8	no	—
56	C₂×C₈×Q₈	—	13	1	no	—
57	#2183	—	13	8	no	—
58	#2163	—	13	4	no	—
59	#2162	—	13	2	no	—
60	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
61	Q₈×C₁₃	—	2	8	no	—
62	C₂×C₄×C₁₃	—	2	4	no	—
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	2	1	no	—
65	C₂×C₄×C₁₃	—	4	2	no	—
66	C₂×C₄×C₁₃	—	4	1	no	—
67	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
68	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
70	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
72	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
74	#1055	—	2	1	no	—
75	#1055	—	2	2	no	—
76	#1055	—	2	4	no	—
77	#1055	2₋¹⁺⁴	1	2	yes	—
78	C₂×C₄×Q₈	#261	1	1	yes	characteristic
79	C₈×Q₈	—	13	4	no	—
80	C₈×Q₈	—	13	4	no	—
81	C₂×C₄⋊C₈	—	13	2	no	—
82	C₂×C₄⋊C₈	—	13	1	no	—
83	C₂×C₄×C₈	—	13	1	no	—
84	C₂×C₄×C₈	—	13	2	no	—
85	C₄⋊C₁₆	—	13	4	no	—
86	C₄⋊C₁₆	—	13	8	no	—
87	C₂²⋊C₁₆	—	13	4	no	—
88	#346	—	13	8	no	—
89	C₄×C₁₆	—	13	4	no	—
90	C₂²×C₁₃	—	2	1	no	—
91	C₂²×C₁₃	—	2	1	no	—
92	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
93	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
94	C₄×C₁₃	—	2	4	no	—
95	C₄×C₁₃	—	2	4	no	—
96	C₄×C₁₃	—	2	2	no	—
97	C₄×C₁₃	—	2	1	no	—
98	C₄×C₁₃	C₂×M₃₂	1	4	yes	—
99	C₄×C₁₃	#519	1	1	yes	characteristic
100	C₄×C₁₃	#568	1	1	yes	characteristic
101	C₂²×Q₈	#1057	1	1	yes	characteristic
102	C₂²×C₈	—	13	1	no	—
103	C₂²×C₈	—	13	1	no	—
104	C₂²×C₈	—	13	2	no	—
105	C₄×Q₈	#1057	1	4	yes	—
106	C₄×Q₈	C₂×#261	1	4	yes	—
107	C₂×C₄⋊C₄	#1057	1	2	yes	—
108	C₂×C₄⋊C₄	#1057	1	1	yes	characteristic
109	C₂×C₄²	C₂×#261	1	2	yes	—
110	C₂×C₄²	C₂×#261	1	1	yes	characteristic
111	C₂×C₁₆	—	13	8	no	—
112	C₄⋊C₈	—	13	8	no	—
113	C₄⋊C₈	—	13	4	no	—
114	C₄×C₈	—	13	2	no	—
115	C₄×C₈	—	13	4	no	—
116	C₄×C₈	—	13	2	no	—
117	C₄×C₈	—	13	4	no	—
118	C₂×C₁₃	—	2	2	no	—
119	C₂×C₁₃	#2982	1	1	yes	characteristic
120	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
121	C₂×C₁₃	#3407	1	1	yes	characteristic
122	C₂×Q₈	—	2	4	no	—
123	C₂×Q₈	C₂×#1057	1	4	yes	—
124	C₂²×C₄	C₂×#1057	1	2	yes	—
125	C₂²×C₄	C₂×#1057	1	2	yes	—
126	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
127	C₂²×C₄	C₂×#1057	1	1	yes	characteristic
128	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
129	C₂×C₈	—	13	1	no	—
130	C₂×C₈	—	13	1	no	—
131	C₂×C₈	—	13	2	no	—
132	C₂×C₈	—	13	4	no	—
133	C₂×C₈	—	26	2	no	—
134	C₂×C₈	—	52	1	no	—
135	C₂×C₈	—	52	2	no	—
136	C₄⋊C₄	C₂×#1057	1	8	yes	—
137	C₄⋊C₄	C₂×#1057	1	4	yes	—
138	C₄²	C₂×#1057	1	4	yes	—
139	C₄²	C₂×#1057	1	2	yes	—
140	C₄²	C₂²×#261	1	4	yes	—
141	C₄²	C₂²×#261	1	2	yes	—
142	C₁₆	—	26	8	no	—
143	C₁₃	#33497	1	1	yes	characteristic, complemented, Sylow
144	C₂³	C₂²×#1057	1	1	yes	characteristic
145	Q₈	—	2	8	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	#88106	1	4	yes	—
152	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
153	C₂×C₄	C₂²×#1057	1	4	yes	—
154	C₂×C₄	C₂²×#1057	1	2	yes	—
155	C₂×C₄	C₂²×#1057	1	4	yes	—
156	C₂×C₄	C₂²×#1057	1	2	yes	—
157	C₂×C₄	C₂³×#261	1	1	yes	characteristic
158	C₈	—	13	2	no	—
159	C₈	—	26	2	no	—
160	C₈	—	26	4	no	—
161	C₈	—	26	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	C₂×#88106	1	2	yes	—
165	C₂²	C₂³×#1057	1	1	yes	characteristic
166	C₄	—	2	4	no	—
167	C₄	—	2	4	no	—
168	C₄	—	2	2	no	—
169	C₄	—	2	1	no	—
170	C₄	C₂×#88106	1	4	yes	—
171	C₄	#11719625	1	1	yes	characteristic
172	C₄	#11719627	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	#13131729	1	1	yes	characteristic
175	C₂	C₂²×#88106	1	1	yes	characteristic
176	C₂	#13131803	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₄×Q₈×C₁₃	C₄
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄×Q₈	#261
hypocenter	C₁₃	#33497
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #13131251	C₂	semidirect
143. C₁₃	#33497	semidirect

Maximal subgroups

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

subgroup	index
#13131251	2
#13131197	2
#13131193	2
C₂×Q₈×#1055	2
#33497	13