Subgroups — #16020534

Order 3328.

488 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127239	C₂	1	2	yes	maximal, complemented
2	#13117115	C₂	1	1	yes	maximal, characteristic, complemented
3	#13117028	C₂	1	2	yes	maximal, complemented
4	#13117017	C₂	1	2	yes	maximal, complemented
5	#13116994	C₂	1	2	yes	maximal, complemented
6	#13116939	C₂	1	2	yes	maximal, complemented
7	C₄×D₂₀₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116624	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719270	—	2	2	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
13	#11718776	C₂²	1	2	yes	—
14	#11718776	C₂²	1	2	yes	—
15	#11718767	C₂²	1	2	yes	—
16	C₂×#88058	C₂²	1	2	yes	—
17	C₂×D₂₀₈	C₂²	1	1	yes	characteristic
18	#11718508	—	2	2	no	—
19	#11718398	—	2	2	no	—
20	#11718396	—	2	2	no	—
21	C₄×D₁₀₄	C₂²	1	2	yes	—
22	#11718344	C₂²	1	2	yes	—
23	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
25	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
26	#11718101	C₂²	1	2	yes	—
27	#11718101	C₂²	1	2	yes	—
28	#11718101	C₂²	1	2	yes	—
29	#11718097	C₂²	1	2	yes	—
30	#11718097	C₂²	1	2	yes	—
31	#11718096	C₂²	1	1	yes	characteristic
32	#11718096	C₂²	1	2	yes	—
33	#11718096	C₂²	1	1	yes	characteristic
34	#11718096	C₂²	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
37	#88193	—	4	2	no	—
38	#88184	—	2	2	no	—
39	#88184	—	2	2	no	—
40	#88180	—	2	2	no	—
41	C₂×C₄⋊Dic₁₃	—	2	2	no	—
42	C₂×D₁₀₄	C₂³	1	2	yes	—
43	#88152	—	4	2	no	—
44	#88137	—	4	2	no	—
45	#88135	—	4	2	no	—
46	C₄×D₅₂	—	2	2	no	—
47	#88124	—	2	2	no	—
48	C₁₃×M₃₂	—	2	4	no	—
49	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
50	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
51	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
52	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
53	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
54	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	#88061	—	2	2	no	—
56	#88061	—	2	2	no	—
57	#88061	—	2	2	no	—
58	#88058	—	2	2	no	—
59	#88058	C₂³	1	2	yes	—
60	#88058	C₂³	1	2	yes	—
61	#88058	C₂³	1	2	yes	—
62	#88057	—	2	2	no	—
63	#88057	—	2	2	no	—
64	D₂₀₈	—	2	2	no	—
65	#33922	—	13	1	no	maximal, Sylow
66	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
68	C₂×C₂²⋊D₁₃	—	4	2	no	—
69	C₂×C₂²⋊D₁₃	—	4	2	no	—
70	C₂²×Dic₁₃	—	4	2	no	—
71	C₂×D₅₂	—	2	2	no	—
72	C₂×C₄×D₁₃	—	4	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
79	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
80	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²⋊Dic₁₃	—	4	2	no	—
82	#6103	—	4	2	no	—
83	#6103	—	4	2	no	—
84	#6103	—	4	2	no	—
85	C₄⋊Dic₁₃	—	2	2	no	—
86	C₄⋊Dic₁₃	—	2	2	no	—
87	C₄⋊Dic₁₃	—	2	2	no	—
88	C₄⋊Dic₁₃	—	4	2	no	—
89	#6101	—	4	2	no	—
90	#6101	—	4	2	no	—
91	C₄×Dic₁₃	—	4	2	no	—
92	D₁₀₄	—	2	2	no	—
93	D₁₀₄	—	2	2	no	—
94	C₁₃×C₁₆	—	2	2	no	—
95	C₁₃×C₁₆	—	2	1	no	—
96	C₁₃×C₁₆	#41	1	2	yes	—
97	#3339	—	13	2	no	—
98	#2253	—	13	1	no	—
99	#2234	—	13	2	no	—
100	#2232	—	13	2	no	—
101	#2228	—	13	2	no	—
102	#2218	—	13	2	no	—
103	C₄×D₁₆	—	13	1	no	—
104	#2169	D₁₃	1	1	yes	characteristic, complemented
105	#2159	—	13	2	no	—
106	C₂³×C₁₃	C₂×D₈	1	2	yes	—
107	C₂²×D₁₃	—	4	2	no	—
108	D₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₂²⋊D₁₃	—	8	4	no	—
116	C₂²⋊D₁₃	—	16	2	no	—
117	C₂×Dic₁₃	—	4	2	no	—
118	C₂×Dic₁₃	—	4	2	no	—
119	C₂×Dic₁₃	—	4	2	no	—
120	C₂×Dic₁₃	—	16	2	no	—
121	D₅₂	—	4	2	no	—
122	D₅₂	—	4	2	no	—
123	C₄×D₁₃	—	16	2	no	—
124	C₈×C₁₃	—	2	2	no	—
125	C₈×C₁₃	—	2	1	no	—
126	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
127	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
128	#553	—	26	2	no	—
129	C₂×D₁₆	—	13	1	no	—
130	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
131	#499	—	13	2	no	—
132	#482	—	26	2	no	—
133	#481	—	26	2	no	—
134	#478	—	26	2	no	—
135	#466	—	13	2	no	—
136	#466	—	13	2	no	—
137	C₄×D₈	—	13	2	no	—
138	C₈×D₄	C₂×D₁₃	1	1	yes	characteristic
139	#427	—	13	2	no	—
140	C₂×#107	—	13	2	no	—
141	#367	—	13	2	no	—
142	#367	—	13	2	no	—
143	#366	—	13	1	no	—
144	#366	—	13	2	no	—
145	#366	—	13	1	no	—
146	#366	—	13	1	no	—
147	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
148	#357	—	13	2	no	—
149	#357	—	13	2	no	—
150	#357	—	13	2	no	—
151	C₂²⋊C₁₆	C₂×D₁₃	1	2	yes	—
152	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₁₃	—	8	2	no	—
158	C₂×D₁₃	—	8	2	no	—
159	C₂×D₁₃	—	8	2	no	—
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	#547	1	1	yes	characteristic
164	C₄×C₁₃	#578	1	1	yes	characteristic
165	Dic₁₃	—	8	2	no	—
166	Dic₁₃	—	8	2	no	—
167	Dic₁₃	—	8	2	no	—
168	C₂×D₈	—	13	2	no	—
169	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
170	#126	—	52	2	no	—
171	#125	—	26	2	no	—
172	#123	—	52	2	no	—
173	#121	—	26	2	no	—
174	#121	—	26	2	no	—
175	#121	—	52	2	no	—
176	C₄×D₄	—	26	2	no	—
177	C₄×D₄	—	52	2	no	—
178	C₄×D₄	D₅₂	1	1	yes	characteristic
179	#117	—	26	2	no	—
180	C₂×C₄⋊C₄	—	26	2	no	—
181	D₁₆	—	26	2	no	—
182	M₃₂	—	2	4	no	—
183	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
184	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
185	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
186	#107	—	13	2	no	—
187	#107	—	13	2	no	—
188	#107	—	13	2	no	—
189	#107	—	26	2	no	—
190	#106	—	26	2	no	—
191	#106	—	26	2	no	—
192	C₄⋊C₈	D₅₂	1	1	yes	characteristic
193	#102	—	26	2	no	—
194	#102	—	26	2	no	—
195	#102	—	26	2	no	—
196	C₂²⋊C₈	D₅₂	1	2	yes	—
197	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
198	C₂×C₁₃	—	4	2	no	—
199	C₂×C₁₃	#3334	1	1	yes	characteristic
200	C₂×C₁₃	#3414	1	1	yes	characteristic
201	C₂×C₁₃	#3415	1	1	yes	characteristic
202	D₁₃	—	16	2	no	—
203	C₂×D₄	—	26	2	no	—
204	C₂×D₄	—	52	2	no	—
205	C₂×D₄	—	52	2	no	—
206	C₂×D₄	D₁₀₄	1	1	yes	characteristic
207	C₂²×C₄	—	52	2	no	—
208	C₂²×C₄	—	52	2	no	—
209	C₂²×C₄	C₂×D₅₂	1	2	yes	—
210	D₈	—	26	2	no	—
211	D₈	—	26	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
215	C₂×C₈	C₂×D₅₂	1	2	yes	—
216	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
217	C₄⋊C₄	—	26	2	no	—
218	C₄⋊C₄	—	26	2	no	—
219	C₄⋊C₄	—	26	2	no	—
220	C₄⋊C₄	—	52	2	no	—
221	C₄⋊C₄	—	52	2	no	—
222	C₄⋊C₄	—	52	2	no	—
223	C₄⋊C₄	D₁₀₄	1	1	yes	characteristic
224	C₂²⋊C₄	—	52	2	no	—
225	C₂²⋊C₄	—	52	2	no	—
226	C₂²⋊C₄	—	52	2	no	—
227	C₂²⋊C₄	—	52	2	no	—
228	C₂²⋊C₄	D₁₀₄	1	2	yes	—
229	C₄²	—	52	2	no	—
230	C₄²	C₂×D₅₂	1	1	yes	characteristic
231	C₁₆	—	2	2	no	—
232	C₁₆	—	2	1	no	—
233	C₁₆	#6129	1	2	yes	—
234	C₁₃	#33922	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	52	2	no	—
236	C₂³	C₂×D₁₀₄	1	2	yes	—
237	D₄	—	4	1	no	—
238	D₄	—	52	2	no	—
239	D₄	—	52	2	no	—
240	D₄	—	104	4	no	—
241	D₄	—	208	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	52	2	no	—
245	C₂×C₄	—	52	2	no	—
246	C₂×C₄	—	52	2	no	—
247	C₂×C₄	—	208	2	no	—
248	C₂×C₄	—	208	2	no	—
249	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
250	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
251	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	—	2	2	no	—
255	C₈	C₂×#6129	1	1	yes	characteristic
256	C₈	#88257	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	104	2	no	—
261	C₂²	—	104	2	no	—
262	C₂²	—	104	2	no	—
263	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	104	2	no	—
268	C₄	—	104	2	no	—
269	C₄	—	104	2	no	—
270	C₄	#11719260	1	1	yes	characteristic
271	C₄	#11719458	1	1	yes	characteristic
272	C₂	—	4	2	no	—
273	C₂	—	208	2	no	—
274	C₂	#13127184	1	1	yes	characteristic
275	C₂	#13127972	1	1	yes	characteristic
276	C₂	#13127977	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2169	C₂
socle	C₂²×C₁₃	C₂²×D₈
hypercenter	#2169	D₁₃
hypocenter	C₁₃	#33922
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #2169

Complements

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

N	H ≅ G/N	type
1. #13127239	C₂	semidirect
2. #13117115	C₂	semidirect
3. #13117028	C₂	semidirect
4. #13117017	C₂	semidirect
5. #13116994	C₂	semidirect
6. #13116939	C₂	semidirect
7. C₄×D₂₀₈	C₂	semidirect
8. #13116624	C₂	semidirect
9. C₁₃×#2169	C₂	semidirect
104. #2169	D₁₃	semidirect
234. C₁₃	#33922	semidirect

Maximal subgroups

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

subgroup	index
#13127239	2
#13117115	2
#13117028	2
#13117017	2
#13116994	2
#13116939	2
C₄×D₂₀₈	2
#13116624	2
C₁₃×#2169	2
#33922	13