Subgroups — #16784575

Order 3328.

502 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131005	C₂	1	2	yes	maximal
2	C₂×#11718254	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#1057	C₂	1	2	yes	maximal
4	C₂×C₈×#1057	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
7	C₂×Q₈×Dic₁₃	C₂²	1	2	yes	—
8	C₂×C₄⋊#1057	C₂²	1	2	yes	—
9	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
10	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
11	C₂×#88075	C₄	1	2	yes	—
12	Q₁₆×Dic₁₃	—	2	2	no	—
13	Q₁₆×Dic₁₃	—	2	2	no	—
14	C₂×#88058	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	Q₈⋊#1057	—	2	4	no	—
17	#11718290	—	2	4	no	—
18	#11718254	—	2	2	no	—
19	C₈×#1057	—	2	2	no	—
20	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₂²×#1057	—	2	1	no	—
22	C₂²×#1057	—	2	2	no	—
23	C₂²×#1057	D₄	1	2	yes	—
24	C₂×C₁₃×Q₁₆	—	2	2	no	—
25	C₂×C₁₃×Q₁₆	—	2	2	no	—
26	C₂×C₁₃×Q₁₆	C₈	1	4	yes	complemented
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	Q₈×Dic₁₃	—	2	2	no	—
29	Q₈×Dic₁₃	—	2	2	no	—
30	Q₈×Dic₁₃	—	2	2	no	—
31	Q₈×Dic₁₃	—	2	2	no	—
32	C₂×C₄⋊Dic₁₃	—	2	2	no	—
33	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₁₃	—	2	2	no	—
35	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	C₄⋊#1057	—	2	2	no	—
38	C₄⋊#1057	—	2	2	no	—
39	C₄×#1057	—	2	1	no	—
40	C₄×#1057	—	2	1	no	—
41	C₄×#1057	—	2	1	no	—
42	C₄×#1057	D₄	1	2	yes	—
43	#88075	—	2	4	no	—
44	#88058	—	2	2	no	—
45	C₈×Dic₁₃	D₄	1	2	yes	—
46	C₈×Dic₁₃	D₄	1	2	yes	—
47	#9656	—	13	1	no	maximal, Sylow
48	C₂×Q₈×C₁₃	—	2	2	no	—
49	C₂×Q₈×C₁₃	—	2	2	no	—
50	C₂×Q₈×C₁₃	—	2	2	no	—
51	C₂×Q₈×C₁₃	—	2	2	no	—
52	C₂×Q₈×C₁₃	—	2	2	no	—
53	C₂×Q₈×C₁₃	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	2	2	no	—
57	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	#41	1	1	yes	characteristic
59	C₂×#1057	—	2	4	no	—
60	C₂×#1057	—	2	2	no	—
61	C₂×#1057	—	2	2	no	—
62	C₂×#1057	—	2	2	no	—
63	C₂×#1057	—	4	2	no	—
64	C₂×#1057	—	4	4	no	—
65	C₂×#1057	Q₁₆	1	4	yes	complemented
66	C₁₃×Q₁₆	—	2	8	no	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
70	C₄⋊Dic₁₃	—	2	2	no	—
71	C₄⋊Dic₁₃	—	2	2	no	—
72	C₄⋊Dic₁₃	—	4	2	no	—
73	C₄⋊Dic₁₃	—	4	2	no	—
74	C₄×Dic₁₃	—	4	2	no	—
75	C₄×Dic₁₃	—	4	2	no	—
76	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
77	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
78	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
79	C₄×Dic₁₃	#41	1	1	yes	characteristic
80	C₂×#1055	—	2	1	no	—
81	C₂×#1055	—	2	1	no	—
82	C₂×#1055	C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×Q₁₆	—	13	1	no	—
84	#1984	—	13	2	no	—
85	C₂×#335	—	13	1	no	—
86	C₂×Q₈⋊C₈	—	13	2	no	—
87	C₂×C₈²	—	13	1	no	—
88	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
89	Q₈×C₁₃	—	2	4	no	—
90	Q₈×C₁₃	—	2	4	no	—
91	Q₈×C₁₃	—	4	2	no	—
92	Q₈×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
102	C₂×Dic₁₃	—	2	2	no	—
103	C₂×Dic₁₃	—	4	2	no	—
104	C₂×Dic₁₃	—	4	4	no	—
105	C₂×Dic₁₃	#121	1	1	yes	characteristic
106	C₂×Dic₁₃	#121	1	1	yes	characteristic
107	C₂×Dic₁₃	#124	1	1	yes	characteristic
108	C₂×Dic₁₃	#127	1	1	yes	characteristic
109	C₂×Dic₁₃	C₂×Q₁₆	1	2	yes	—
110	C₂×Dic₁₃	#135	1	2	yes	—
111	#1057	—	2	4	no	—
112	#1057	—	2	4	no	—
113	#1057	—	8	2	no	—
114	C₈×C₁₃	C₂²⋊C₈	1	4	yes	—
115	#1055	—	2	2	no	—
116	C₂²×Q₁₆	#261	1	1	yes	characteristic
117	C₂×C₄×Q₈	—	13	2	no	—
118	C₄×Q₁₆	—	26	2	no	—
119	C₄×Q₁₆	—	26	2	no	—
120	C₂×#107	—	13	1	no	—
121	C₂×C₄⋊C₈	—	13	2	no	—
122	C₂×#103	—	13	2	no	—
123	C₂×C₄×C₈	—	13	1	no	—
124	C₂×C₄×C₈	—	13	1	no	—
125	C₂×C₄×C₈	—	13	1	no	—
126	#336	—	26	4	no	—
127	#335	—	26	2	no	—
128	Q₈⋊C₈	—	26	4	no	—
129	C₈²	—	26	2	no	—
130	C₂²×C₁₃	#390	1	1	yes	characteristic
131	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
132	C₂²×C₁₃	#436	1	1	yes	characteristic
133	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
134	C₂²×C₁₃	#443	1	2	yes	—
135	C₄×C₁₃	—	4	4	no	—
136	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
137	C₄×C₁₃	#408	1	1	yes	characteristic
138	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
139	C₄×C₁₃	#436	1	1	yes	characteristic
140	Dic₁₃	—	2	1	no	—
141	Dic₁₃	—	2	1	no	—
142	Dic₁₃	—	8	2	no	—
143	Dic₁₃	#467	1	2	yes	—
144	Dic₁₃	#494	1	1	yes	characteristic
145	Dic₁₃	#495	1	1	yes	characteristic
146	C₂²×Q₈	C₂×#261	1	2	yes	—
147	C₂×Q₁₆	—	2	2	no	—
148	C₂×Q₁₆	—	2	2	no	—
149	C₂×Q₁₆	#1057	1	4	yes	complemented
150	C₂²×C₈	—	13	2	no	—
151	C₂²×C₈	—	13	1	no	—
152	C₂²×C₈	—	26	1	no	—
153	C₂²×C₈	—	26	2	no	—
154	C₂²×C₈	C₂×#261	1	1	yes	characteristic
155	C₄×Q₈	—	26	2	no	—
156	C₄×Q₈	—	26	2	no	—
157	C₄×Q₈	—	26	2	no	—
158	C₄×Q₈	—	26	2	no	—
159	C₂×C₄⋊C₄	—	13	2	no	—
160	C₂×C₄⋊C₄	—	26	2	no	—
161	C₂×C₄²	—	13	1	no	—
162	C₂×C₄²	—	26	2	no	—
163	#107	—	26	2	no	—
164	C₄⋊C₈	—	26	2	no	—
165	C₄⋊C₈	—	26	2	no	—
166	#103	—	26	4	no	—
167	C₄×C₈	—	13	2	no	—
168	C₄×C₈	—	13	2	no	—
169	C₄×C₈	—	13	2	no	—
170	C₄×C₈	—	26	1	no	—
171	C₄×C₈	—	26	1	no	—
172	C₄×C₈	—	26	1	no	—
173	C₂×C₁₃	C₈×Q₁₆	1	2	yes	—
174	C₂×C₁₃	#1591	1	2	yes	—
175	C₂×C₁₃	#1954	1	1	yes	characteristic
176	C₂×C₁₃	#1969	1	1	yes	characteristic
177	C₂×C₁₃	#1974	1	1	yes	characteristic
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	C₂×#1057	1	4	yes	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	13	1	no	—
186	C₂²×C₄	—	13	1	no	—
187	C₂²×C₄	—	26	2	no	—
188	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
189	Q₁₆	—	2	8	no	—
190	C₂×C₈	—	13	4	no	—
191	C₂×C₈	—	13	2	no	—
192	C₂×C₈	—	26	2	no	—
193	C₂×C₈	—	26	2	no	—
194	C₂×C₈	—	26	2	no	—
195	C₂×C₈	—	26	4	no	—
196	C₂×C₈	—	26	1	no	—
197	C₂×C₈	—	26	1	no	—
198	C₂×C₈	—	52	2	no	—
199	C₂×C₈	—	52	4	no	—
200	C₂×C₈	C₂×#1057	1	2	yes	—
201	C₂×C₈	#6122	1	2	yes	—
202	C₂×C₈	C₂²⋊#261	1	2	yes	—
203	C₄⋊C₄	—	26	2	no	—
204	C₄⋊C₄	—	26	2	no	—
205	C₄⋊C₄	—	52	2	no	—
206	C₄⋊C₄	—	52	2	no	—
207	C₄²	—	13	1	no	—
208	C₄²	—	13	1	no	—
209	C₄²	—	13	1	no	—
210	C₄²	—	13	1	no	—
211	C₄²	—	52	2	no	—
212	C₄²	—	52	2	no	—
213	C₁₃	#9656	1	1	yes	characteristic, complemented, Sylow
214	C₂³	D₄×#261	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	2	4	no	—
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	13	2	no	—
223	C₂×C₄	—	13	2	no	—
224	C₂×C₄	—	13	1	no	—
225	C₂×C₄	—	13	1	no	—
226	C₂×C₄	—	13	1	no	—
227	C₂×C₄	—	13	1	no	—
228	C₂×C₄	—	26	2	no	—
229	C₂×C₄	—	52	4	no	—
230	C₂×C₄	—	52	2	no	—
231	C₂×C₄	#88238	1	1	yes	characteristic
232	C₂×C₄	#88238	1	1	yes	characteristic
233	C₂×C₄	D₄×#261	1	1	yes	characteristic
234	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
235	C₂×C₄	C₂×#6122	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
237	C₈	—	26	4	no	—
238	C₈	—	26	4	no	—
239	C₈	—	26	2	no	—
240	C₈	—	104	2	no	—
241	C₈	C₂²⋊#1057	1	4	yes	—
242	C₂²	Q₁₆×#261	1	2	yes	—
243	C₂²	#11719111	1	2	yes	—
244	C₂²	D₄×#1057	1	1	yes	characteristic
245	C₂²	#11719145	1	1	yes	characteristic
246	C₂²	#11719151	1	1	yes	characteristic
247	C₄	—	4	4	no	—
248	C₄	—	13	2	no	—
249	C₄	—	13	1	no	—
250	C₄	—	13	1	no	—
251	C₄	—	26	1	no	—
252	C₄	—	26	1	no	—
253	C₄	—	104	2	no	—
254	C₄	D₄×#1057	1	1	yes	characteristic
255	C₄	#11719145	1	1	yes	characteristic
256	C₄	#11719147	1	1	yes	characteristic
257	C₄	C₂×C₂²⋊#1057	1	1	yes	characteristic
258	C₂	Q₁₆×#1057	1	2	yes	—
259	C₂	#13130476	1	2	yes	—
260	C₂	#13130957	1	1	yes	characteristic
261	C₂	#13130982	1	1	yes	characteristic
262	C₂	#13130987	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂²×C₁₃×Q₁₆	C₄
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂²×Q₁₆	#261
hypocenter	C₁₃	#9656
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₂×C₁₃×Q₁₆	C₈	semidirect
65. C₂×#1057	Q₁₆	semidirect
149. C₂×Q₁₆	#1057	semidirect
213. C₁₃	#9656	semidirect

Maximal subgroups

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

subgroup	index
#13131005	2
C₂×#11718254	2
C₂×Q₈⋊#1057	2
C₂×C₈×#1057	2
C₂×Q₁₆×Dic₁₃	2
#9656	13