Subgroups — #16017423

Order 3328.

650 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#13124491	C₂	1	2	yes	maximal
3	#13116944	C₂	1	4	yes	maximal, complemented
4	#13116927	C₂	1	4	yes	maximal, complemented
5	#13116620	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116557	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
9	#11719530	—	2	2	no	—
10	C₂²×Dic₅₂	C₂²	1	2	yes	—
11	C₁₃×#504	—	2	4	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	#11718798	—	2	4	no	—
17	#11718781	—	2	2	no	—
18	#11718777	C₂²	1	8	yes	—
19	#11718767	C₂²	1	2	yes	—
20	C₂×Dic₁₀₄	—	2	4	no	—
21	C₂×Dic₁₀₄	C₂²	1	2	yes	—
22	C₂×Dic₁₀₄	C₂²	1	2	yes	—
23	#11718097	C₂²	1	2	yes	—
24	#11718096	C₂²	1	2	yes	—
25	#11718095	C₂²	1	8	yes	—
26	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
27	C₂²×Dic₂₆	—	2	2	no	—
28	C₁₃×#131	—	2	2	no	—
29	C₁₃×#131	—	2	2	no	—
30	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	#88198	—	4	2	no	—
35	#88190	—	4	2	no	—
36	#88189	—	4	4	no	—
37	#88180	—	2	2	no	—
38	#88178	—	2	8	no	—
39	C₂×Dic₅₂	—	2	4	no	—
40	C₂×Dic₅₂	—	2	4	no	—
41	C₂×Dic₅₂	—	2	2	no	—
42	C₂×Dic₅₂	C₂³	1	4	yes	—
43	C₁₃×M₃₂	—	2	2	no	—
44	C₁₃×M₃₂	—	2	4	no	—
45	C₂×C₁₃×C₁₆	—	2	4	no	—
46	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
47	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
48	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
50	#88058	C₂³	1	4	yes	—
51	#88057	—	2	2	no	—
52	#88056	—	2	8	no	—
53	Dic₁₀₄	—	2	4	no	—
54	Dic₁₀₄	—	2	4	no	—
55	#33608	—	13	1	no	maximal, Sylow
56	C₁₃×#41	—	2	2	no	—
57	C₁₃×#41	—	2	2	no	—
58	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
59	C₂×D₄×C₁₃	D₈	1	2	yes	—
60	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	—	4	2	no	—
64	C₂×Dic₂₆	—	2	4	no	—
65	C₂×Dic₂₆	—	4	4	no	—
66	C₂×Dic₂₆	—	4	4	no	—
67	C₂×Dic₂₆	—	4	2	no	—
68	C₁₃×M₁₆	—	2	2	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₃	#41	1	2	yes	—
76	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂²⋊Dic₁₃	—	4	8	no	—
78	C₂²⋊Dic₁₃	—	4	2	no	—
79	C₄⋊Dic₁₃	—	2	4	no	—
80	#6101	—	4	8	no	—
81	C₄×Dic₁₃	—	4	2	no	—
82	Dic₅₂	—	2	4	no	—
83	Dic₅₂	—	2	4	no	—
84	Dic₅₂	—	4	4	no	—
85	C₁₃×C₁₆	—	2	2	no	—
86	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
87	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
88	C₂²×Q₃₂	—	13	1	no	—
89	C₂×#504	D₁₃	1	1	yes	characteristic
90	#3059	—	13	2	no	—
91	#2219	—	13	4	no	—
92	#2216	—	13	4	no	—
93	#2158	—	13	1	no	—
94	#2141	—	13	2	no	—
95	C₂³×C₁₃	C₂×D₈	1	2	yes	—
96	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
97	Q₈×C₁₃	—	2	2	no	—
98	D₄×C₁₃	—	2	4	no	—
99	D₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	#121	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#135	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×Dic₁₃	—	4	4	no	—
108	C₂×Dic₁₃	—	4	4	no	—
109	C₂×Dic₁₃	—	8	4	no	—
110	Dic₂₆	—	4	4	no	—
111	Dic₂₆	—	4	4	no	—
112	Dic₂₆	—	8	4	no	—
113	C₈×C₁₃	—	2	2	no	—
114	C₈×C₁₃	#121	1	2	yes	—
115	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₂²×Q₁₆	—	13	2	no	—
118	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
119	#536	—	26	2	no	—
120	C₂×Q₃₂	—	13	2	no	—
121	C₂×Q₃₂	—	13	2	no	—
122	C₂×Q₃₂	—	26	4	no	—
123	#504	—	2	4	no	—
124	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
125	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
126	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
127	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
128	#470	—	26	4	no	—
129	#467	—	13	8	no	—
130	#427	—	13	2	no	—
131	#419	—	26	2	no	—
132	#367	—	13	2	no	—
133	#366	—	13	2	no	—
134	#358	—	13	8	no	—
135	C₂²×C₁₃	—	2	2	no	—
136	C₂²×C₁₃	—	2	4	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	#466	1	2	yes	—
139	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	#466	1	2	yes	—
142	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
143	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
144	Dic₁₃	—	8	4	no	—
145	Dic₁₃	—	8	4	no	—
146	C₂×#41	D₅₂	1	1	yes	characteristic
147	C₂²×Q₈	—	26	2	no	—
148	C₂×Q₁₆	—	13	4	no	—
149	C₂×Q₁₆	—	26	2	no	—
150	C₂×Q₁₆	—	26	4	no	—
151	C₂×Q₁₆	—	26	4	no	—
152	#131	—	2	2	no	—
153	#131	—	2	2	no	—
154	C₂×M₁₆	D₅₂	1	1	yes	characteristic
155	C₂×M₁₆	D₅₂	1	2	yes	—
156	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
157	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
158	C₄⋊Q₈	—	52	2	no	—
159	#124	—	52	2	no	—
160	#123	—	52	4	no	—
161	C₂²⋊Q₈	—	26	8	no	—
162	#117	—	26	2	no	—
163	Q₃₂	—	26	4	no	—
164	Q₃₂	—	26	4	no	—
165	M₃₂	—	2	2	no	—
166	M₃₂	—	2	4	no	—
167	C₂×C₁₆	—	2	4	no	—
168	C₂×C₁₆	C₂²⋊D₁₃	1	4	yes	—
169	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
170	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
171	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
172	#107	—	13	4	no	—
173	#106	—	26	2	no	—
174	#103	—	26	8	no	—
175	C₂×C₁₃	—	2	2	no	—
176	C₂×C₁₃	—	4	2	no	—
177	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
178	C₂×C₁₃	#3417	1	2	yes	—
179	#41	—	2	2	no	—
180	#41	—	2	2	no	—
181	C₂×Q₈	—	26	4	no	—
182	C₂×Q₈	—	52	2	no	—
183	C₂×Q₈	—	52	4	no	—
184	C₂×Q₈	—	52	4	no	—
185	C₂×Q₈	D₁₀₄	1	1	yes	characteristic
186	C₂×D₄	D₁₀₄	1	2	yes	—
187	C₂×D₄	D₁₀₄	1	1	yes	characteristic
188	C₂²×C₄	—	52	2	no	—
189	C₂²×C₄	C₂×D₅₂	1	2	yes	—
190	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
191	Q₁₆	—	26	4	no	—
192	Q₁₆	—	26	4	no	—
193	Q₁₆	—	52	4	no	—
194	M₁₆	—	2	2	no	—
195	M₁₆	—	2	4	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	C₂×D₅₂	1	2	yes	—
199	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
200	C₂×C₈	#6127	1	2	yes	—
201	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
202	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
203	C₄⋊C₄	—	26	4	no	—
204	C₄⋊C₄	—	52	8	no	—
205	C₂²⋊C₄	—	52	2	no	—
206	C₂²⋊C₄	—	52	8	no	—
207	C₄²	—	52	2	no	—
208	C₁₆	—	2	2	no	—
209	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
210	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
211	C₁₃	#33608	1	1	yes	characteristic, complemented, Sylow
212	C₂³	C₂×D₁₀₄	1	2	yes	—
213	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	Q₈	—	52	4	no	—
216	Q₈	—	52	4	no	—
217	Q₈	—	104	4	no	—
218	D₄	—	2	4	no	—
219	D₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	52	4	no	—
223	C₂×C₄	—	52	4	no	—
224	C₂×C₄	—	104	4	no	—
225	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
226	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
227	C₂×C₄	#88158	1	2	yes	—
228	C₂×C₄	#88184	1	2	yes	—
229	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	#88184	1	2	yes	—
232	C₈	C₂×#6127	1	1	yes	characteristic
233	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	#11718776	1	2	yes	—
238	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	104	4	no	—
241	C₄	—	104	4	no	—
242	C₄	#11718776	1	2	yes	—
243	C₄	C₂×#88158	1	1	yes	characteristic
244	C₄	C₂×#88184	1	1	yes	characteristic
245	C₂	—	2	2	no	—
246	C₂	—	4	2	no	—
247	C₂	C₂×#11718776	1	1	yes	characteristic
248	C₂	#13127989	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#504	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×C₁₃×#41	D₄
hypercenter	C₂×#504	D₁₃
hypocenter	C₁₃	#33608
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₁₀₄	C₂	semidirect
3. #13116944	C₂	semidirect
4. #13116927	C₂	semidirect
5. #13116620	C₂	semidirect
6. #13116557	C₂	semidirect
211. C₁₃	#33608	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₁₀₄	2
#13124491	2
#13116944	2
#13116927	2
#13116620	2
#13116557	2
C₂×C₁₃×#504	2
#33608	13