Subgroups — #16794914

Order 3328.

778 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719119	C₂	1	2	yes	maximal
2	#13130387	C₂	1	8	yes	maximal, complemented
3	#13130383	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
7	C₂²×#6099	C₄	1	1	yes	characteristic
8	C₂×C₂²⋊#1057	—	2	2	no	—
9	#11719119	—	2	2	no	—
10	#11719119	—	2	2	no	—
11	#11719119	C₂²	1	4	yes	—
12	#11719119	C₂²	1	4	yes	—
13	C₂×C₄⋊#1057	C₂²	1	4	yes	—
14	C₂×C₄×#1057	—	2	2	no	—
15	M₁₆×Dic₁₃	C₄	1	4	yes	—
16	M₁₆×Dic₁₃	C₂²	1	4	yes	—
17	C₂×#88055	C₄	1	2	yes	—
18	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
19	#11718254	C₂²	1	8	yes	complemented
20	#11718253	C₂²	1	8	yes	complemented
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×Dic₁₃	D₄	1	1	yes	characteristic
23	C₂³×Dic₁₃	Q₈	1	1	yes	characteristic
24	C₂²×#1057	—	2	4	no	—
25	C₂×C₁₃×M₁₆	C₈	1	8	yes	complemented
26	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
31	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
34	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
36	C₂×#6099	—	2	4	no	—
37	C₂×#6099	C₂×C₄	1	4	yes	—
38	C₂²×#1055	C₂×C₄	1	2	yes	—
39	C₂²⋊#1057	—	2	4	no	—
40	C₂²⋊#1057	—	2	4	no	—
41	C₄⋊#1057	—	2	4	no	—
42	C₄⋊#1057	C₂³	1	8	yes	—
43	C₄×#1057	—	2	4	no	—
44	C₄×#1057	—	2	4	no	—
45	#88055	C₂×C₄	1	8	yes	—
46	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
47	C₈×Dic₁₃	C₂³	1	4	yes	—
48	#17787	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂²×Dic₁₃	—	4	2	no	—
57	C₂²×Dic₁₃	—	4	2	no	—
58	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
59	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
62	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
64	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
66	C₂×#1057	—	2	8	no	—
67	C₂×#1057	—	4	8	no	—
68	C₁₃×M₁₆	C₂×C₈	1	16	yes	complemented
69	C₂×C₈×C₁₃	C₂×C₈	1	8	yes	complemented
70	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
71	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
72	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
73	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
74	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
76	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
77	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
78	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
79	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
80	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
81	#6099	—	2	8	no	—
82	C₂×#1055	—	2	4	no	—
83	C₂×#1055	C₂²×C₄	1	4	yes	—
84	C₂×#431	—	13	2	no	—
85	C₂×C₄×M₁₆	—	13	1	no	—
86	#1565	—	13	8	no	—
87	#1564	—	13	4	no	—
88	C₂³×C₁₃	—	4	1	no	—
89	C₂³×C₁₃	—	4	1	no	—
90	C₂³×C₁₃	C₄⋊C₈	1	4	yes	—
91	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
94	C₂×C₄×C₁₃	C₄⋊C₈	1	8	yes	—
95	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
99	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
101	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×Dic₁₃	—	2	4	no	—
103	C₂×Dic₁₃	—	2	2	no	—
104	C₂×Dic₁₃	—	2	4	no	—
105	C₂×Dic₁₃	—	2	2	no	—
106	C₂×Dic₁₃	—	4	1	no	—
107	C₂×Dic₁₃	—	4	2	no	—
108	C₂×Dic₁₃	—	4	1	no	—
109	C₂×Dic₁₃	—	4	2	no	—
110	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
116	#1057	—	4	8	no	—
117	C₈×C₁₃	C₂²×C₈	1	8	yes	—
118	#1055	—	2	4	no	—
119	C₂²×M₁₆	—	13	1	no	—
120	C₂²×M₁₆	#261	1	1	yes	characteristic
121	C₂²×C₄²	—	13	1	no	—
122	#431	—	13	4	no	—
123	#431	—	13	4	no	—
124	#431	—	26	2	no	—
125	#431	—	26	2	no	—
126	C₂×C₄⋊C₈	—	13	4	no	—
127	C₂×C₂²⋊C₈	—	26	2	no	—
128	C₄×M₁₆	—	13	4	no	—
129	C₄×M₁₆	—	13	4	no	—
130	C₂×#97	—	13	2	no	—
131	C₂×C₄×C₈	—	13	2	no	—
132	C₂×C₄×C₈	—	26	2	no	—
133	#335	—	13	8	no	—
134	#334	—	13	8	no	—
135	C₂²×C₁₃	—	2	2	no	—
136	C₂²×C₁₃	—	2	1	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	—	4	4	no	—
140	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
141	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
142	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
144	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
145	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
146	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
147	Dic₁₃	—	2	2	no	—
148	Dic₁₃	—	2	1	no	—
149	Dic₁₃	—	2	2	no	—
150	Dic₁₃	—	2	1	no	—
151	Dic₁₃	#430	1	2	yes	—
152	Dic₁₃	#575	1	2	yes	—
153	C₂³×C₄	—	13	1	no	—
154	C₂³×C₄	—	13	1	no	—
155	C₂³×C₄	C₂×#261	1	1	yes	characteristic
156	C₂×M₁₆	—	13	4	no	—
157	C₂×M₁₆	—	26	4	no	—
158	C₂×M₁₆	#1057	1	8	yes	complemented
159	C₂×M₁₆	C₂×#261	1	4	yes	—
160	C₂²×C₈	—	13	2	no	—
161	C₂²×C₈	—	26	4	no	—
162	C₂²×C₈	C₂×#261	1	2	yes	—
163	C₂×C₄²	—	13	2	no	—
164	C₂×C₄²	—	13	1	no	—
165	C₂×C₄²	—	13	1	no	—
166	C₂×C₄²	—	13	2	no	—
167	C₂×C₄²	—	13	1	no	—
168	C₂×C₄²	—	13	2	no	—
169	C₂×C₄²	—	13	1	no	—
170	C₂×C₄²	—	13	2	no	—
171	C₄⋊C₈	—	13	8	no	—
172	C₄⋊C₈	—	26	4	no	—
173	C₂²⋊C₈	—	26	4	no	—
174	C₂²⋊C₈	—	26	4	no	—
175	#97	—	13	8	no	—
176	C₄×C₈	—	13	4	no	—
177	C₄×C₈	—	13	4	no	—
178	C₄×C₈	—	26	4	no	—
179	C₄×C₈	—	26	4	no	—
180	C₂×C₁₃	—	2	4	no	—
181	C₂×C₁₃	—	2	2	no	—
182	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
183	C₂×C₁₃	#2911	1	1	yes	characteristic
184	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
185	C₂⁴	C₄⋊#261	1	1	yes	characteristic
186	C₂²×C₄	—	13	1	no	—
187	C₂²×C₄	—	13	2	no	—
188	C₂²×C₄	—	13	2	no	—
189	C₂²×C₄	—	13	1	no	—
190	C₂²×C₄	—	13	1	no	—
191	C₂²×C₄	—	13	1	no	—
192	C₂²×C₄	—	13	2	no	—
193	C₂²×C₄	—	13	2	no	—
194	C₂²×C₄	—	52	2	no	—
195	C₂²×C₄	—	52	2	no	—
196	C₂²×C₄	#6118	1	4	yes	—
197	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
198	C₂²×C₄	C₄⋊#261	1	2	yes	—
199	C₂²×C₄	C₂×#1057	1	4	yes	—
200	C₂²×C₄	C₂²×#261	1	2	yes	—
201	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
202	M₁₆	—	26	8	no	—
203	M₁₆	C₂×#1057	1	16	yes	complemented
204	C₂×C₈	—	13	4	no	—
205	C₂×C₈	—	26	4	no	—
206	C₂×C₈	—	26	8	no	—
207	C₂×C₈	—	52	8	no	—
208	C₂×C₈	C₂×#1057	1	8	yes	complemented
209	C₂×C₈	C₂²×#261	1	4	yes	—
210	C₄²	—	13	2	no	—
211	C₄²	—	13	2	no	—
212	C₄²	—	13	2	no	—
213	C₄²	—	13	2	no	—
214	C₄²	—	13	1	no	—
215	C₄²	—	13	2	no	—
216	C₄²	—	13	1	no	—
217	C₄²	—	13	1	no	—
218	C₄²	—	13	1	no	—
219	C₄²	—	13	2	no	—
220	C₁₃	#17787	1	1	yes	characteristic, complemented, Sylow
221	C₂³	—	4	1	no	—
222	C₂³	—	4	1	no	—
223	C₂³	C₄⋊#1057	1	4	yes	—
224	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
225	C₂³	C₂×C₄⋊#261	1	2	yes	—
226	C₂×C₄	—	13	1	no	—
227	C₂×C₄	—	13	1	no	—
228	C₂×C₄	—	13	1	no	—
229	C₂×C₄	—	13	2	no	—
230	C₂×C₄	—	13	2	no	—
231	C₂×C₄	—	13	1	no	—
232	C₂×C₄	—	26	4	no	—
233	C₂×C₄	—	26	2	no	—
234	C₂×C₄	—	26	4	no	—
235	C₂×C₄	—	26	2	no	—
236	C₂×C₄	—	52	1	no	—
237	C₂×C₄	—	52	2	no	—
238	C₂×C₄	—	52	1	no	—
239	C₂×C₄	—	52	2	no	—
240	C₂×C₄	C₄⋊#1057	1	4	yes	—
241	C₂×C₄	C₄⋊#1057	1	8	yes	—
242	C₂×C₄	C₂×#6118	1	2	yes	—
243	C₂×C₄	C₂×#6118	1	4	yes	—
244	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
245	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
246	C₂×C₄	C₂²×#1057	1	2	yes	—
247	C₂×C₄	C₂²×#1057	1	4	yes	—
248	C₂×C₄	C₂³×#261	1	1	yes	characteristic
249	C₈	—	26	4	no	—
250	C₈	—	52	8	no	—
251	C₈	C₂²×#1057	1	8	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₂×C₄⋊#1057	1	2	yes	—
258	C₂²	C₂×C₄⋊#1057	1	4	yes	—
259	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
260	C₄	—	13	2	no	—
261	C₄	—	13	2	no	—
262	C₄	—	26	2	no	—
263	C₄	—	26	1	no	—
264	C₄	—	26	2	no	—
265	C₄	—	26	1	no	—
266	C₄	C₂×C₄⋊#1057	1	4	yes	—
267	C₄	C₂×C₄⋊#1057	1	2	yes	—
268	C₄	C₂²×#6118	1	1	yes	characteristic
269	C₄	C₂³×#1057	1	1	yes	characteristic
270	C₂	—	2	4	no	—
271	C₂	—	2	2	no	—
272	C₂	C₂²×C₄⋊#1057	1	1	yes	characteristic
273	C₂	#13131489	1	1	yes	characteristic
274	C₂	D₁₃⋊#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₁₃×M₁₆	C₄
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂²×M₁₆	#261
hypocenter	C₁₃	#17787
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13130387	C₂	semidirect
3. #13130383	C₂	semidirect
19. #11718254	C₂²	semidirect
20. #11718253	C₂²	semidirect
25. C₂×C₁₃×M₁₆	C₈	semidirect
68. C₁₃×M₁₆	C₂×C₈	semidirect
69. C₂×C₈×C₁₃	C₂×C₈	semidirect
158. C₂×M₁₆	#1057	semidirect
203. M₁₆	C₂×#1057	semidirect
208. C₂×C₈	C₂×#1057	semidirect
220. C₁₃	#17787	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719119	2
#13130387	2
#13130383	2
C₂×M₁₆×Dic₁₃	2
#17787	13