Subgroups — #16017416

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₀₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#13124484	C₂	1	2	yes	maximal, complemented
3	#13116939	C₂	1	4	yes	maximal, complemented
4	#13116922	C₂	1	4	yes	maximal, complemented
5	#13116620	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116553	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
9	#11719535	—	2	2	no	—
10	C₂²×D₁₀₄	C₂²	1	2	yes	—
11	C₁₃×#504	—	2	4	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₃₂	C₂²	1	2	yes	complemented
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	complemented
16	#11718796	—	2	4	no	—
17	#11718792	—	2	2	no	—
18	#11718776	C₂²	1	8	yes	complemented
19	#11718767	C₂²	1	2	yes	—
20	C₂×D₂₀₈	—	2	4	no	—
21	C₂×D₂₀₈	C₂²	1	2	yes	—
22	C₂×D₂₀₈	C₂²	1	2	yes	—
23	#11718101	C₂²	1	8	yes	complemented
24	#11718097	C₂²	1	2	yes	complemented
25	#11718096	C₂²	1	2	yes	complemented
26	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
27	C₂²×D₅₂	—	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	#88201	—	4	2	no	—
35	#88194	—	4	2	no	—
36	#88191	—	4	4	no	—
37	#88184	—	2	8	no	—
38	#88180	—	2	2	no	—
39	C₂×D₁₀₄	—	2	4	no	—
40	C₂×D₁₀₄	—	2	4	no	—
41	C₂×D₁₀₄	—	2	2	no	—
42	C₂×D₁₀₄	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	complemented
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	#88061	—	2	8	no	—
51	#88058	C₂³	1	4	yes	—
52	#88057	—	2	2	no	—
53	D₂₀₈	—	2	4	no	—
54	D₂₀₈	—	2	4	no	—
55	#33607	—	13	1	no	maximal, Sylow
56	C₂³×D₁₃	—	4	2	no	—
57	C₁₃×#41	—	2	2	no	—
58	C₁₃×#41	—	2	2	no	—
59	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
60	C₂×D₄×C₁₃	D₈	1	2	yes	—
61	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃	—	4	8	no	—
65	C₂×D₅₂	—	2	4	no	—
66	C₂×D₅₂	—	4	4	no	—
67	C₂×D₅₂	—	4	4	no	—
68	C₂×D₅₂	—	4	2	no	—
69	C₁₃×M₁₆	—	2	2	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃	#41	1	2	yes	—
77	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₂²⋊Dic₁₃	—	4	2	no	—
79	#6103	—	4	8	no	—
80	C₄⋊Dic₁₃	—	2	4	no	—
81	C₄×Dic₁₃	—	4	2	no	—
82	D₁₀₄	—	2	4	no	—
83	D₁₀₄	—	2	4	no	—
84	D₁₀₄	—	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×D₁₆	—	13	1	no	—
89	C₂×#504	D₁₃	1	1	yes	characteristic, complemented
90	#3058	—	13	2	no	—
91	#2218	—	13	4	no	—
92	#2215	—	13	4	no	—
93	#2158	—	13	1	no	—
94	#2140	—	13	2	no	—
95	C₂³×C₁₃	C₂×D₈	1	2	yes	—
96	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
97	C₂²×D₁₃	—	4	4	no	—
98	C₂²×D₁₃	—	8	4	no	—
99	C₂²×D₁₃	—	16	4	no	—
100	Q₈×C₁₃	—	2	2	no	—
101	D₄×C₁₃	—	2	4	no	—
102	D₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	#121	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#135	1	2	yes	—
109	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
110	C₂²⋊D₁₃	—	16	8	no	—
111	C₂×Dic₁₃	—	4	4	no	—
112	D₅₂	—	4	4	no	—
113	D₅₂	—	4	4	no	—
114	D₅₂	—	8	4	no	—
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	#121	1	2	yes	—
117	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂²×D₈	—	13	2	no	—
120	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
121	#535	—	26	2	no	—
122	C₂×D₁₆	—	13	2	no	—
123	C₂×D₁₆	—	13	2	no	—
124	C₂×D₁₆	—	26	4	no	—
125	#504	—	2	4	no	—
126	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
127	C₂×M₃₂	C₂×D₁₃	1	2	yes	complemented
128	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
129	C₂²×C₁₆	C₂×D₁₃	1	2	yes	complemented
130	#469	—	26	4	no	—
131	#466	—	13	8	no	—
132	#427	—	13	2	no	—
133	#418	—	26	2	no	—
134	#367	—	13	2	no	—
135	#366	—	13	2	no	—
136	#357	—	13	8	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	2	4	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	#466	1	2	yes	—
141	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₁₃	—	8	8	no	—
143	C₂×D₁₃	—	8	4	no	—
144	C₂×D₁₃	—	16	8	no	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	#466	1	2	yes	—
147	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
148	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
149	Dic₁₃	—	8	4	no	—
150	C₂×#41	D₅₂	1	1	yes	characteristic
151	C₂²×D₄	—	26	2	no	—
152	C₂×D₈	—	13	4	no	—
153	C₂×D₈	—	26	2	no	—
154	C₂×D₈	—	26	4	no	—
155	C₂×D₈	—	26	4	no	—
156	#131	—	2	2	no	—
157	#131	—	2	2	no	—
158	C₂×M₁₆	D₅₂	1	1	yes	characteristic
159	C₂×M₁₆	D₅₂	1	2	yes	—
160	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
161	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
162	#127	—	52	2	no	—
163	#124	—	52	2	no	—
164	#121	—	26	8	no	—
165	#120	—	52	4	no	—
166	#117	—	26	2	no	—
167	D₁₆	—	26	4	no	—
168	D₁₆	—	26	4	no	—
169	M₃₂	—	2	2	no	—
170	M₃₂	—	2	4	no	—
171	C₂×C₁₆	—	2	4	no	—
172	C₂×C₁₆	C₂²⋊D₁₃	1	4	yes	complemented
173	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
174	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
175	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
176	#107	—	13	4	no	—
177	#106	—	26	2	no	—
178	#102	—	26	8	no	—
179	C₂×C₁₃	—	2	2	no	—
180	C₂×C₁₃	—	4	2	no	—
181	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
182	C₂×C₁₃	#3415	1	2	yes	—
183	D₁₃	—	16	4	no	—
184	C₂⁴	—	52	2	no	—
185	#41	—	2	2	no	—
186	#41	—	2	2	no	—
187	C₂×Q₈	D₁₀₄	1	1	yes	characteristic
188	C₂×D₄	—	26	4	no	—
189	C₂×D₄	—	52	8	no	—
190	C₂×D₄	—	52	2	no	—
191	C₂×D₄	—	52	4	no	—
192	C₂×D₄	—	52	4	no	—
193	C₂×D₄	D₁₀₄	1	2	yes	—
194	C₂×D₄	D₁₀₄	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₅₂	1	2	yes	—
196	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
197	D₈	—	26	4	no	—
198	D₈	—	26	4	no	—
199	D₈	—	52	4	no	—
200	M₁₆	—	2	2	no	—
201	M₁₆	—	2	4	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	C₂×D₅₂	1	2	yes	—
205	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
206	C₂×C₈	#6127	1	2	yes	—
207	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
208	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
209	C₄⋊C₄	—	26	4	no	—
210	C₂²⋊C₄	—	52	2	no	—
211	C₂²⋊C₄	—	52	8	no	—
212	C₄²	—	52	2	no	—
213	C₁₆	—	2	2	no	—
214	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
215	C₁₆	C₂×C₂²⋊D₁₃	1	2	yes	—
216	C₁₃	#33607	1	1	yes	characteristic, complemented, Sylow
217	C₂³	—	52	4	no	—
218	C₂³	—	104	4	no	—
219	C₂³	—	208	4	no	—
220	C₂³	C₂×D₁₀₄	1	2	yes	—
221	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	2	no	—
225	D₄	—	52	4	no	—
226	D₄	—	52	4	no	—
227	D₄	—	104	4	no	—
228	D₄	—	208	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	52	4	no	—
232	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
233	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
234	C₂×C₄	#88158	1	2	yes	—
235	C₂×C₄	#88184	1	2	yes	—
236	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	#88184	1	2	yes	—
239	C₈	C₂×#6127	1	1	yes	characteristic
240	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	104	8	no	—
245	C₂²	—	104	4	no	—
246	C₂²	—	208	8	no	—
247	C₂²	#11718776	1	2	yes	—
248	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	104	4	no	—
251	C₄	#11718776	1	2	yes	—
252	C₄	C₂×#88158	1	1	yes	characteristic
253	C₄	C₂×#88184	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	4	2	no	—
256	C₂	—	208	4	no	—
257	C₂	C₂×#11718776	1	1	yes	characteristic
258	C₂	#13127977	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
hypercenter	C₂×#504	D₁₃
hypocenter	C₁₃	#33607
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×D₂₀₈	C₂	semidirect
2. #13124484	C₂	semidirect
3. #13116939	C₂	semidirect
4. #13116922	C₂	semidirect
5. #13116620	C₂	semidirect
6. #13116553	C₂	semidirect
7. C₂×C₁₃×#504	C₂	semidirect
13. C₂×C₁₃×M₃₂	C₂²	semidirect
15. C₂²×C₁₃×C₁₆	C₂²	semidirect
18. #11718776	C₂²	semidirect
23. #11718101	C₂²	semidirect
24. #11718097	C₂²	semidirect
25. #11718096	C₂²	semidirect
46. C₂×C₁₃×C₁₆	D₄	semidirect
89. C₂×#504	D₁₃	semidirect
127. C₂×M₃₂	C₂×D₁₃	semidirect
129. C₂²×C₁₆	C₂×D₁₃	semidirect
172. C₂×C₁₆	C₂²⋊D₁₃	semidirect
216. C₁₃	#33607	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₂₀₈	2
#13124484	2
#13116939	2
#13116922	2
#13116620	2
#13116553	2
C₂×C₁₃×#504	2
#33607	13