Subgroups — #16015958

Order 3328.

674 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	C₂×#11718767	C₂	1	2	yes	maximal
2	#13116624	C₂	1	8	yes	maximal, complemented
3	#13116620	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×#88180	—	2	2	no	—
7	C₂×C₁₃×M₃₂	C₄	1	8	yes	complemented
8	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
9	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
10	#11718767	—	2	2	no	—
11	#11718767	—	2	2	no	—
12	#11718767	C₂²	1	4	yes	—
13	#11718767	C₂²	1	4	yes	—
14	C₂×#88058	C₂²	1	4	yes	—
15	C₂×#88057	—	2	2	no	—
16	#11718097	C₂²	1	8	yes	complemented
17	#11718096	C₂²	1	8	yes	complemented
18	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₁₃	D₄	1	2	yes	—
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
23	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊Dic₁₃	—	4	2	no	—
26	#88180	—	2	4	no	—
27	#88180	—	2	4	no	—
28	#88180	—	4	2	no	—
29	#88180	—	4	2	no	—
30	C₂×C₄⋊Dic₁₃	—	2	4	no	—
31	C₂×C₄×Dic₁₃	—	4	2	no	—
32	C₁₃×M₃₂	C₂×C₄	1	16	yes	complemented
33	C₂×C₁₃×C₁₆	C₂×C₄	1	8	yes	complemented
34	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
35	#88058	—	2	4	no	—
36	#88058	C₂³	1	8	yes	—
37	#88057	—	2	4	no	—
38	#88057	—	2	4	no	—
39	#33409	—	13	1	no	maximal, Sylow
40	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₃	D₈	1	2	yes	—
43	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	Q₁₆	1	4	yes	—
45	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₁₃	—	4	4	no	—
48	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
49	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
50	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
52	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
55	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
57	C₂²⋊Dic₁₃	—	4	4	no	—
58	C₂²⋊Dic₁₃	—	4	4	no	—
59	C₄⋊Dic₁₃	—	2	8	no	—
60	C₄⋊Dic₁₃	—	4	4	no	—
61	C₄×Dic₁₃	—	4	4	no	—
62	C₄×Dic₁₃	—	4	4	no	—
63	C₁₃×C₁₆	C₂²×C₄	1	8	yes	—
64	C₂²×M₃₂	D₁₃	1	1	yes	characteristic
65	C₂×#427	—	13	2	no	—
66	#2159	—	13	8	no	—
67	#2158	—	13	4	no	—
68	C₂³×C₁₃	—	4	2	no	—
69	C₂³×C₁₃	#107	1	4	yes	—
70	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
71	C₂³×C₁₃	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₁₃	#107	1	8	yes	—
73	C₂×C₄×C₁₃	#107	1	4	yes	—
74	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
77	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
79	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
80	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₁₃	—	4	8	no	—
82	C₂×Dic₁₃	—	8	8	no	—
83	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
85	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
86	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
87	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
88	C₂×#117	—	26	2	no	—
89	C₂×M₃₂	Dic₁₃	1	8	yes	complemented
90	C₂×M₃₂	C₂×D₁₃	1	4	yes	—
91	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
92	#427	—	13	4	no	—
93	#427	—	13	4	no	—
94	#427	—	26	2	no	—
95	#427	—	26	2	no	—
96	C₂×#107	—	13	4	no	—
97	C₂×#106	—	26	2	no	—
98	#367	—	13	8	no	—
99	#366	—	13	8	no	—
100	C₂²×C₁₃	—	2	4	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	—	4	4	no	—
103	C₂²×C₁₃	C₂×#107	1	2	yes	—
104	C₂²×C₁₃	C₂×#107	1	4	yes	—
105	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
106	C₄×C₁₃	C₂×#107	1	4	yes	—
107	C₄×C₁₃	C₂×#107	1	2	yes	—
108	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
110	Dic₁₃	—	8	8	no	—
111	C₂³×C₄	D₅₂	1	1	yes	characteristic
112	C₂²×C₈	Dic₂₆	1	4	yes	—
113	C₂²×C₈	D₅₂	1	2	yes	—
114	C₂²×C₈	D₅₂	1	1	yes	characteristic
115	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
116	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
117	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
118	#117	—	26	4	no	—
119	#117	—	26	4	no	—
120	#117	—	52	2	no	—
121	#117	—	52	2	no	—
122	C₂×C₄⋊C₄	—	26	4	no	—
123	C₂×C₂²⋊C₄	—	52	2	no	—
124	C₂×C₄²	—	52	2	no	—
125	M₃₂	C₂×Dic₁₃	1	16	yes	complemented
126	C₂×C₁₆	C₂×Dic₁₃	1	8	yes	complemented
127	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
128	#107	—	13	8	no	—
129	#107	—	26	4	no	—
130	#106	—	26	4	no	—
131	#106	—	26	4	no	—
132	C₂×C₁₃	—	2	4	no	—
133	C₂×C₁₃	—	2	2	no	—
134	C₂×C₁₃	C₂²×#107	1	1	yes	characteristic
135	C₂×C₁₃	#3414	1	2	yes	—
136	C₂⁴	D₁₀₄	1	1	yes	characteristic
137	C₂²×C₄	—	52	4	no	—
138	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
139	C₂²×C₄	D₁₀₄	1	2	yes	—
140	C₂²×C₄	Dic₅₂	1	4	yes	—
141	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
142	C₂²×C₄	C₂×D₅₂	1	2	yes	—
143	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
144	C₂×C₈	C₄⋊Dic₁₃	1	8	yes	—
145	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
146	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
147	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
148	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
149	C₂×C₈	C₂×D₅₂	1	2	yes	—
150	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
151	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
152	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
153	C₄⋊C₄	—	26	8	no	—
154	C₄⋊C₄	—	52	4	no	—
155	C₂²⋊C₄	—	52	4	no	—
156	C₂²⋊C₄	—	52	4	no	—
157	C₄²	—	52	4	no	—
158	C₄²	—	52	4	no	—
159	C₁₆	C₂²×Dic₁₃	1	8	yes	—
160	C₁₃	#33409	1	1	yes	characteristic, complemented, Sylow
161	C₂³	—	4	2	no	—
162	C₂³	#88058	1	4	yes	—
163	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
164	C₂³	C₂×D₁₀₄	1	2	yes	—
165	C₂×C₄	—	52	8	no	—
166	C₂×C₄	—	104	8	no	—
167	C₂×C₄	#88058	1	4	yes	—
168	C₂×C₄	#88058	1	8	yes	—
169	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
170	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
171	C₂×C₄	C₂×Dic₅₂	1	4	yes	—
172	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
173	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
174	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
175	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
176	C₈	C₂×C₄⋊Dic₁₃	1	2	yes	—
177	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
178	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
179	C₈	C₂³×Dic₁₃	1	1	yes	characteristic
180	C₂²	—	2	4	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	4	4	no	—
183	C₂²	C₂×#88058	1	2	yes	—
184	C₂²	C₂×#88058	1	4	yes	—
185	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
186	C₄	—	104	8	no	—
187	C₄	C₂×#88058	1	4	yes	—
188	C₄	C₂×#88058	1	2	yes	—
189	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
190	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
191	C₂	—	2	4	no	—
192	C₂	—	2	2	no	—
193	C₂	C₂²×#88058	1	1	yes	characteristic
194	C₂	#13127972	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₄
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂⁴×C₁₃	D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×M₃₂	D₁₃
hypocenter	C₁₃	#33409
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #13116624	C₂	semidirect
3. #13116620	C₂	semidirect
7. C₂×C₁₃×M₃₂	C₄	semidirect
16. #11718097	C₂²	semidirect
17. #11718096	C₂²	semidirect
32. C₁₃×M₃₂	C₂×C₄	semidirect
33. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
89. C₂×M₃₂	Dic₁₃	semidirect
125. M₃₂	C₂×Dic₁₃	semidirect
126. C₂×C₁₆	C₂×Dic₁₃	semidirect
160. C₁₃	#33409	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718767	2
#13116624	2
#13116620	2
C₂²×C₁₃×M₃₂	2
#33409	13