Subgroups — #16018449

Order 3328.

482 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125350	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718344	C₂	1	1	yes	maximal, characteristic
3	#13117122	C₂	1	4	yes	maximal, complemented
4	#13117115	C₂	1	4	yes	maximal, complemented
5	#13116624	C₂	1	4	yes	maximal
6	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
7	#11719175	—	2	1	no	—
8	C₂×#88124	—	2	1	no	—
9	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	#11718767	—	2	1	no	—
12	#11718767	C₂²	1	2	yes	—
13	C₂×#88058	C₂²	1	2	yes	—
14	C₂×#88057	—	2	1	no	—
15	#11718344	—	2	1	no	—
16	#11718344	—	2	1	no	—
17	#11718344	—	2	1	no	—
18	#11718344	C₂²	1	2	yes	—
19	#11718344	C₂²	1	2	yes	—
20	#11718344	C₂²	1	2	yes	—
21	#11718343	C₂²	1	2	yes	—
22	#11718341	—	2	1	no	—
23	C₁₃×#346	C₂²	1	2	yes	—
24	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
25	#11718097	C₂²	1	8	yes	—
26	#11718096	C₂²	1	8	yes	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	#88180	—	2	2	no	—
31	C₂×C₄⋊Dic₁₃	—	2	2	no	—
32	#88178	—	4	2	no	—
33	C₂×#6101	—	4	2	no	—
34	#88124	—	2	2	no	—
35	#88124	—	2	2	no	—
36	#88124	—	2	2	no	—
37	#88124	—	4	2	no	—
38	#88123	—	2	2	no	—
39	C₄×Dic₂₆	—	4	2	no	—
40	C₁₃×M₃₂	Q₈	1	8	yes	complemented
41	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
42	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	#88058	—	2	2	no	—
47	#88058	—	2	2	no	—
48	#88058	C₂³	1	4	yes	—
49	#88058	C₂³	1	4	yes	—
50	#88057	—	2	2	no	—
51	#88057	—	2	2	no	—
52	#88057	—	2	2	no	—
53	#88057	—	2	2	no	—
54	#33732	—	13	1	no	maximal, Sylow
55	C₂²×C₄×C₁₃	D₈	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	—	4	2	no	—
58	C₂×Dic₂₆	—	4	2	no	—
59	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
64	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₄²×C₁₃	D₈	1	2	yes	—
66	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²⋊Dic₁₃	—	4	2	no	—
69	C₄⋊Dic₁₃	—	2	4	no	—
70	C₄⋊Dic₁₃	—	2	4	no	—
71	C₄⋊Dic₁₃	—	4	2	no	—
72	C₄⋊Dic₁₃	—	4	2	no	—
73	#6101	—	4	4	no	—
74	#6101	—	4	4	no	—
75	#6101	—	4	4	no	—
76	C₄×Dic₁₃	—	4	2	no	—
77	C₁₃×C₁₆	C₂×Q₈	1	8	yes	—
78	#3160	—	13	1	no	—
79	C₂×#499	—	13	1	no	—
80	#2255	—	13	4	no	—
81	#2253	—	13	4	no	—
82	#2159	—	13	4	no	—
83	C₄×M₃₂	D₁₃	1	1	yes	characteristic
84	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	—	4	2	no	—
86	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
87	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
91	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₁₃	—	4	4	no	—
93	C₂×Dic₁₃	—	4	4	no	—
94	C₂×Dic₁₃	—	8	4	no	—
95	Dic₂₆	—	8	4	no	—
96	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
97	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
98	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
99	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
100	#533	—	26	1	no	—
101	C₂×#125	—	26	1	no	—
102	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
103	#500	—	13	2	no	—
104	#499	—	13	2	no	—
105	#499	—	13	2	no	—
106	#499	—	13	2	no	—
107	#499	—	26	1	no	—
108	#499	—	26	1	no	—
109	#499	—	26	1	no	—
110	#498	—	26	1	no	—
111	#427	—	13	2	no	—
112	#427	—	26	1	no	—
113	C₂×#107	—	13	2	no	—
114	C₂×#106	—	26	1	no	—
115	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
116	#367	—	13	8	no	—
117	#366	—	13	8	no	—
118	#346	C₂×D₁₃	1	2	yes	—
119	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	#500	1	2	yes	—
122	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	#500	1	2	yes	—
126	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
128	Dic₁₃	—	8	4	no	—
129	Dic₁₃	—	8	4	no	—
130	C₂²×C₈	D₅₂	1	1	yes	characteristic
131	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
132	C₄⋊Q₈	—	26	2	no	—
133	#125	—	26	2	no	—
134	#125	—	26	2	no	—
135	#125	—	26	2	no	—
136	#125	—	52	2	no	—
137	C₂²⋊Q₈	—	52	2	no	—
138	C₄×Q₈	—	52	2	no	—
139	#117	—	26	2	no	—
140	C₂×C₄⋊C₄	—	26	2	no	—
141	C₂×C₄⋊C₄	—	52	2	no	—
142	C₂×C₄²	D₅₂	1	1	yes	characteristic
143	M₃₂	Dic₂₆	1	8	yes	complemented
144	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
145	#107	—	13	4	no	—
146	#107	—	13	4	no	—
147	#107	—	26	2	no	—
148	#107	—	26	2	no	—
149	#106	—	26	2	no	—
150	#106	—	26	2	no	—
151	#106	—	26	2	no	—
152	#106	—	26	2	no	—
153	C₄×C₈	D₅₂	1	1	yes	characteristic
154	C₄×C₈	D₅₂	1	1	yes	characteristic
155	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
156	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
157	C₂×C₁₃	—	2	2	no	—
158	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
159	C₂×C₁₃	#3414	1	2	yes	—
160	C₂×Q₈	—	52	2	no	—
161	C₂²×C₄	—	52	2	no	—
162	C₂²×C₄	D₁₀₄	1	2	yes	—
163	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
164	C₂×C₈	Dic₅₂	1	4	yes	—
165	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
166	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
167	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
168	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
169	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
170	C₄⋊C₄	—	26	4	no	—
171	C₄⋊C₄	—	26	4	no	—
172	C₄⋊C₄	—	52	2	no	—
173	C₄⋊C₄	—	52	4	no	—
174	C₄⋊C₄	—	52	4	no	—
175	C₄⋊C₄	—	52	4	no	—
176	C₄⋊C₄	—	52	2	no	—
177	C₂²⋊C₄	—	52	2	no	—
178	C₄²	—	52	2	no	—
179	C₄²	D₁₀₄	1	2	yes	—
180	C₄²	C₂×D₅₂	1	1	yes	characteristic
181	C₄²	C₂×D₅₂	1	1	yes	characteristic
182	C₁₆	C₂×Dic₂₆	1	8	yes	—
183	C₁₃	#33732	1	1	yes	characteristic, complemented, Sylow
184	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
185	Q₈	—	104	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	52	4	no	—
188	C₂×C₄	—	52	4	no	—
189	C₂×C₄	—	104	4	no	—
190	C₂×C₄	#88123	1	2	yes	—
191	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
192	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
193	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
194	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
195	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
196	C₈	#88123	1	2	yes	—
197	C₈	C₂×Dic₅₂	1	2	yes	—
198	C₈	C₂×Dic₅₂	1	2	yes	—
199	C₈	C₂²×Dic₂₆	1	2	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	#11718343	1	2	yes	—
202	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	104	4	no	—
206	C₄	—	104	4	no	—
207	C₄	#11718343	1	2	yes	—
208	C₄	C₂×#88123	1	1	yes	characteristic
209	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	C₂×#11718343	1	1	yes	characteristic
212	C₂	#13127972	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#33732
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #13117122	C₂	semidirect
4. #13117115	C₂	semidirect
40. C₁₃×M₃₂	Q₈	semidirect
143. M₃₂	Dic₂₆	semidirect
183. C₁₃	#33732	semidirect

Maximal subgroups

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

subgroup	index
#13125350	2
C₂×#11718344	2
#13117122	2
#13117115	2
#13116624	2
C₄×C₁₃×M₃₂	2
#33732	13