Subgroups — #16795244

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131729	C₂	1	2	yes	maximal
2	#13130429	C₂	1	4	yes	maximal, complemented
3	Q₁₆×#1057	C₂	1	4	yes	maximal, complemented
4	#13130383	C₂	1	1	yes	maximal, characteristic
5	#13130138	C₂	1	2	yes	maximal
6	#13130039	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
9	C₂×Q₈×Dic₁₃	C₂²	1	2	yes	—
10	Q₈×#1057	—	2	4	no	—
11	Q₈×#1057	C₂²	1	8	yes	—
12	#11719119	—	2	4	no	—
13	#11719119	C₂²	1	2	yes	—
14	#11719119	C₂²	1	1	yes	characteristic
15	#11719119	C₂²	1	1	yes	characteristic
16	C₂×#88075	C₄	1	2	yes	—
17	Q₁₆×Dic₁₃	C₄	1	2	yes	—
18	Q₁₆×Dic₁₃	C₄	1	2	yes	—
19	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
20	Q₁₆×Dic₁₃	C₂²	1	2	yes	—
21	C₂×#88058	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
23	Q₈⋊#1057	C₂²	1	8	yes	—
24	#11718254	C₂²	1	2	yes	—
25	#11718253	C₂²	1	2	yes	—
26	#11718251	C₂²	1	2	yes	—
27	C₈×#1057	C₂²	1	2	yes	—
28	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₁₃×Q₁₆	C₈	1	8	yes	complemented
30	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	Q₈×Dic₁₃	—	2	2	no	—
34	Q₈×Dic₁₃	—	2	2	no	—
35	Q₈×Dic₁₃	C₂×C₄	1	4	yes	—
36	Q₈×Dic₁₃	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₁₃	—	2	2	no	—
38	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₁₃	—	2	2	no	—
40	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
41	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
42	C₂²⋊#1057	—	2	2	no	—
43	C₂²⋊#1057	—	2	1	no	—
44	C₂²⋊#1057	D₄	1	2	yes	—
45	C₄⋊#1057	—	2	8	no	—
46	C₄⋊#1057	—	2	1	no	—
47	C₄⋊#1057	D₄	1	2	yes	—
48	C₄⋊#1057	C₂³	1	4	yes	—
49	C₄×#1057	—	2	8	no	—
50	C₄×#1057	—	2	2	no	—
51	C₄×#1057	C₂³	1	2	yes	—
52	C₄×#1057	C₂³	1	2	yes	—
53	#88075	C₂×C₄	1	4	yes	—
54	#88075	C₂×C₄	1	4	yes	—
55	#88058	C₂×C₄	1	2	yes	—
56	#88058	C₂×C₄	1	2	yes	—
57	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
58	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
60	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
61	#17856	—	13	1	no	maximal, Sylow
62	C₂×Q₈×C₁₃	—	2	2	no	—
63	C₂×Q₈×C₁₃	—	2	4	no	—
64	C₂×Q₈×C₁₃	C₂×C₈	1	8	yes	—
65	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	2	2	no	—
69	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₁₃	#41	1	1	yes	characteristic
71	C₂×#1057	—	2	4	no	—
72	C₂×#1057	—	2	2	no	—
73	C₂×#1057	C₂×D₄	1	4	yes	—
74	C₁₃×Q₁₆	C₂×C₈	1	8	yes	—
75	C₁₃×Q₁₆	C₂×C₈	1	8	yes	—
76	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₄⋊Dic₁₃	—	2	2	no	—
80	C₄⋊Dic₁₃	—	2	4	no	—
81	C₄⋊Dic₁₃	—	2	4	no	—
82	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
83	C₄×Dic₁₃	—	2	4	no	—
84	C₄×Dic₁₃	—	2	4	no	—
85	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
87	C₄×Dic₁₃	#41	1	1	yes	characteristic
88	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
89	C₂×#1055	—	2	2	no	—
90	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
91	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
92	#2982	—	13	2	no	—
93	C₂×C₄×Q₁₆	—	13	1	no	—
94	#1579	—	13	4	no	—
95	C₈×Q₁₆	—	13	4	no	—
96	#1564	—	13	1	no	—
97	#1479	—	13	2	no	—
98	#1450	—	13	1	no	—
99	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	4	no	—
101	Q₈×C₁₃	—	2	8	no	—
102	Q₈×C₁₃	C₂²×C₈	1	8	yes	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	8	no	—
105	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₁₃	#131	1	2	yes	—
108	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×Dic₁₃	—	2	2	no	—
110	C₂×Dic₁₃	—	2	4	no	—
111	C₂×Dic₁₃	—	4	1	no	—
112	C₂×Dic₁₃	—	8	2	no	—
113	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
117	#1057	—	2	4	no	—
118	#1057	—	4	2	no	—
119	#1057	—	4	4	no	—
120	C₈×C₁₃	C₂²×C₈	1	2	yes	—
121	C₈×C₁₃	C₂²×C₈	1	2	yes	—
122	#1055	—	2	1	no	—
123	#1055	—	2	1	no	—
124	C₂²×Q₁₆	#261	1	1	yes	characteristic
125	C₂×C₄×Q₈	—	13	2	no	—
126	C₈×Q₈	—	13	8	no	—
127	C₈×Q₈	—	26	4	no	—
128	C₄×Q₁₆	—	13	2	no	—
129	C₄×Q₁₆	—	13	2	no	—
130	C₄×Q₁₆	—	13	2	no	—
131	C₄×Q₁₆	—	13	2	no	—
132	#431	—	13	1	no	—
133	#431	—	13	1	no	—
134	#431	—	13	2	no	—
135	#431	—	26	4	no	—
136	C₂×#107	—	13	1	no	—
137	C₂×#103	—	13	2	no	—
138	C₂×C₄×C₈	—	13	1	no	—
139	#335	—	13	2	no	—
140	#334	—	13	2	no	—
141	Q₈⋊C₈	—	13	8	no	—
142	#322	—	13	2	no	—
143	C₈²	—	13	2	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	1	no	—
146	C₂²×C₁₃	C₈×D₄	1	2	yes	—
147	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₁₃	—	2	8	no	—
149	C₄×C₁₃	C₈×D₄	1	2	yes	—
150	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
151	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
152	Dic₁₃	—	2	1	no	—
153	Dic₁₃	—	2	1	no	—
154	Dic₁₃	—	2	1	no	—
155	Dic₁₃	—	4	4	no	—
156	Dic₁₃	#578	1	2	yes	—
157	C₂²×Q₈	C₂×#261	1	2	yes	—
158	C₂×Q₁₆	#1057	1	8	yes	complemented
159	C₂×Q₁₆	C₂×#261	1	2	yes	—
160	C₂×Q₁₆	C₂×#261	1	2	yes	—
161	C₂²×C₈	—	13	1	no	—
162	C₂²×C₈	C₂×#261	1	1	yes	characteristic
163	C₄×Q₈	—	13	4	no	—
164	C₄×Q₈	—	13	4	no	—
165	C₄×Q₈	—	26	2	no	—
166	C₄×Q₈	—	26	2	no	—
167	C₂×C₄⋊C₄	—	13	2	no	—
168	C₂×C₄⋊C₄	—	26	2	no	—
169	C₂×C₄²	—	13	1	no	—
170	C₂×C₄²	—	26	2	no	—
171	#107	—	13	2	no	—
172	#107	—	13	2	no	—
173	C₄⋊C₈	—	13	2	no	—
174	C₄⋊C₈	—	13	4	no	—
175	C₄⋊C₈	—	26	1	no	—
176	C₄⋊C₈	—	26	8	no	—
177	#103	—	13	4	no	—
178	#103	—	13	4	no	—
179	C₂²⋊C₈	—	13	2	no	—
180	C₂²⋊C₈	—	26	1	no	—
181	C₂²⋊C₈	—	26	2	no	—
182	C₄×C₈	—	13	2	no	—
183	C₄×C₈	—	13	1	no	—
184	C₄×C₈	—	13	1	no	—
185	C₄×C₈	—	13	1	no	—
186	C₄×C₈	—	13	1	no	—
187	C₄×C₈	—	13	2	no	—
188	C₄×C₈	—	26	2	no	—
189	C₄×C₈	—	26	8	no	—
190	C₂×C₁₃	—	2	2	no	—
191	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
192	C₂×C₁₃	#2950	1	1	yes	characteristic
193	C₂×C₁₃	#2957	1	1	yes	characteristic
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	C₂×#1057	1	8	yes	—
197	C₂×Q₈	C₂²×#261	1	4	yes	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	13	1	no	—
200	C₂²×C₄	—	13	1	no	—
201	C₂²×C₄	—	26	2	no	—
202	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
203	Q₁₆	C₂×#1057	1	8	yes	—
204	Q₁₆	C₂×#1057	1	8	yes	—
205	C₂×C₈	—	13	4	no	—
206	C₂×C₈	—	13	1	no	—
207	C₂×C₈	—	13	1	no	—
208	C₂×C₈	—	26	2	no	—
209	C₂×C₈	—	26	2	no	—
210	C₂×C₈	—	26	4	no	—
211	C₂×C₈	C₂×#1057	1	4	yes	—
212	C₂×C₈	C₂²×#261	1	1	yes	characteristic
213	C₂×C₈	C₂²×#261	1	1	yes	characteristic
214	C₄⋊C₄	—	13	4	no	—
215	C₄⋊C₄	—	26	4	no	—
216	C₄⋊C₄	—	26	4	no	—
217	C₄⋊C₄	—	26	2	no	—
218	C₄²	—	13	1	no	—
219	C₄²	—	13	1	no	—
220	C₄²	—	13	1	no	—
221	C₄²	—	13	1	no	—
222	C₄²	—	26	4	no	—
223	C₄²	—	26	4	no	—
224	C₁₃	#17856	1	1	yes	characteristic, complemented, Sylow
225	C₂³	D₄×#261	1	1	yes	characteristic
226	Q₈	—	2	4	no	—
227	Q₈	—	2	8	no	—
228	Q₈	C₂²×#1057	1	8	yes	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	13	1	no	—
232	C₂×C₄	—	13	1	no	—
233	C₂×C₄	—	13	1	no	—
234	C₂×C₄	—	13	1	no	—
235	C₂×C₄	—	26	2	no	—
236	C₂×C₄	—	26	4	no	—
237	C₂×C₄	—	52	1	no	—
238	C₂×C₄	—	104	2	no	—
239	C₂×C₄	#88238	1	2	yes	—
240	C₂×C₄	D₄×#261	1	1	yes	characteristic
241	C₂×C₄	C₂²×#1057	1	2	yes	—
242	C₂×C₄	C₂³×#261	1	1	yes	characteristic
243	C₈	—	26	4	no	—
244	C₈	—	26	1	no	—
245	C₈	—	26	1	no	—
246	C₈	—	52	4	no	—
247	C₈	—	52	2	no	—
248	C₈	C₂²×#1057	1	2	yes	—
249	C₈	C₂²×#1057	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	D₄×#1057	1	2	yes	—
253	C₂²	C₂×D₄×#261	1	1	yes	characteristic
254	C₄	—	2	8	no	—
255	C₄	—	13	2	no	—
256	C₄	—	26	1	no	—
257	C₄	—	26	1	no	—
258	C₄	—	26	1	no	—
259	C₄	—	52	4	no	—
260	C₄	D₄×#1057	1	2	yes	—
261	C₄	C₂×#88238	1	1	yes	characteristic
262	C₄	C₂³×#1057	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	C₂×D₄×#1057	1	1	yes	characteristic
265	C₂	#13131597	1	1	yes	characteristic
266	C₂	#13131616	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₁₃×Q₁₆	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂²×Q₁₆	#261
hypocenter	C₁₃	#17856
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #13130429	C₂	semidirect
3. Q₁₆×#1057	C₂	semidirect
29. C₂×C₁₃×Q₁₆	C₈	semidirect
158. C₂×Q₁₆	#1057	semidirect
224. C₁₃	#17856	semidirect

Maximal subgroups

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

subgroup	index
#13131729	2
#13130429	2
Q₁₆×#1057	2
#13130383	2
#13130138	2
#13130039	2
C₂×Q₁₆×Dic₁₃	2
#17856	13