Subgroups — #16789986

Order 3328.

440 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718261	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718261	C₂	1	1	yes	maximal, characteristic, complemented
3	#13131080	C₂	1	2	yes	maximal
4	#13129886	C₂	1	2	yes	maximal, complemented
5	#13123258	C₂	1	1	yes	maximal, characteristic
6	C₂×#88161	C₂²	1	1	yes	characteristic
7	C₂×#88155	C₄	1	1	yes	characteristic
8	#11719227	C₄	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	2	yes	—
10	C₂×#88106	C₂²	1	2	yes	—
11	C₁₃×#405	C₄	1	1	yes	characteristic
12	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
14	#11718658	C₂²	1	1	yes	characteristic
15	#11718369	—	2	1	no	—
16	#11718368	—	2	1	no	—
17	#11718350	—	2	1	no	—
18	C₄×C₈×D₁₃	—	2	1	no	—
19	#11718270	—	2	4	no	—
20	#11718261	—	2	1	no	—
21	#11718261	—	2	1	no	—
22	#11718261	—	2	1	no	—
23	#11718261	—	2	1	no	—
24	C₂×#6127	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×#6099	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂³	1	1	yes	characteristic
31	#88161	—	2	1	no	—
32	#88161	—	2	1	no	—
33	#88161	D₄	1	2	yes	—
34	#88161	Q₈	1	2	yes	—
35	#88155	—	2	2	no	—
36	#88155	C₈	1	4	yes	—
37	C₂×#6094	—	2	1	no	—
38	C₂×#6094	—	2	1	no	—
39	C₂×C₈×D₁₃	—	2	1	no	—
40	C₂×C₈×D₁₃	—	2	1	no	—
41	#88147	—	2	1	no	—
42	#88141	—	2	1	no	—
43	#88135	—	2	2	no	—
44	#88126	—	2	1	no	—
45	C₄²×D₁₃	—	2	1	no	—
46	#88106	—	2	2	no	—
47	#88106	—	2	2	no	—
48	#88106	—	2	2	no	—
49	#88106	—	2	2	no	—
50	C₂×#6092	—	2	2	no	—
51	C₂×#6092	—	2	2	no	—
52	C₁₃×#97	—	2	1	no	—
53	C₄×C₈×C₁₃	—	2	1	no	—
54	#88055	—	2	1	no	—
55	C₈×Dic₁₃	—	2	1	no	—
56	C₈×Dic₁₃	D₄	1	1	yes	characteristic
57	C₈×Dic₁₃	D₄	1	1	yes	characteristic
58	C₈×Dic₁₃	D₄	1	1	yes	characteristic
59	C₈×Dic₁₃	D₄	1	1	yes	characteristic
60	#88043	—	2	1	no	—
61	C₄×#1055	—	2	1	no	—
62	#12792	—	13	1	no	maximal, Sylow
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃	—	2	1	no	—
65	C₂²×Dic₁₃	—	2	1	no	—
66	#6127	—	2	1	no	—
67	#6127	—	2	1	no	—
68	#6127	C₄⋊C₄	1	2	yes	—
69	#6127	C₂×C₈	1	2	yes	—
70	C₂×D₅₂	M₁₆	1	1	yes	characteristic
71	C₂×C₄×D₁₃	—	2	1	no	—
72	C₂×C₄×D₁₃	—	2	2	no	—
73	C₂×Dic₂₆	M₁₆	1	1	yes	characteristic
74	C₁₃×M₁₆	—	2	1	no	—
75	C₁₃×M₁₆	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
81	C₁₃×C₄⋊C₄	—	2	1	no	—
82	C₁₃×C₂²⋊C₄	M₁₆	1	2	yes	—
83	C₄²×C₁₃	—	2	1	no	—
84	#6103	—	2	2	no	—
85	#6101	—	2	2	no	—
86	C₄×Dic₁₃	—	2	2	no	—
87	C₄×Dic₁₃	—	2	1	no	—
88	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
90	#6099	—	2	1	no	—
91	#6099	C₂×C₈	1	2	yes	—
92	C₂×#1055	—	2	1	no	—
93	C₂×#1055	C₂×D₄	1	1	yes	characteristic
94	C₂×#1055	C₂×D₄	1	1	yes	characteristic
95	C₂×#1055	C₂×D₄	1	1	yes	characteristic
96	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
97	C₂×#1055	#41	1	1	yes	characteristic
98	C₂×#1055	#41	1	1	yes	characteristic
99	#6094	—	2	4	no	—
100	#6094	—	2	2	no	—
101	#6094	—	2	2	no	—
102	C₈×D₁₃	—	2	4	no	—
103	C₈×D₁₃	—	2	2	no	—
104	C₈×D₁₃	—	2	2	no	—
105	#6092	—	4	2	no	—
106	#6092	—	4	2	no	—
107	C₄×#131	—	13	1	no	—
108	C₂×#364	—	13	1	no	—
109	C₂×#364	—	13	1	no	—
110	#2117	—	13	2	no	—
111	#1378	—	13	2	no	—
112	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
113	C₂²×D₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂²⋊D₁₃	—	2	2	no	—
122	C₂²⋊D₁₃	—	2	2	no	—
123	C₂×Dic₁₃	—	2	1	no	—
124	C₂×Dic₁₃	—	2	1	no	—
125	C₂×Dic₁₃	—	2	1	no	—
126	C₂×Dic₁₃	—	4	2	no	—
127	D₅₂	—	2	1	no	—
128	D₅₂	C₄⋊C₈	1	2	yes	—
129	C₄×D₁₃	—	2	2	no	—
130	C₄×D₁₃	—	2	2	no	—
131	C₄×D₁₃	—	4	4	no	—
132	Dic₂₆	—	2	1	no	—
133	Dic₂₆	C₄⋊C₈	1	2	yes	—
134	C₈×C₁₃	—	2	2	no	—
135	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
136	C₈×C₁₃	C₂²⋊C₈	1	2	yes	—
137	#1055	—	2	2	no	—
138	#1055	#121	1	1	yes	characteristic
139	#1055	#121	1	1	yes	characteristic
140	#1055	C₂²⋊Q₈	1	1	yes	characteristic
141	#1055	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×#131	—	13	1	no	—
143	C₂×#131	—	13	1	no	—
144	C₄×#41	—	13	1	no	—
145	C₂×M₃₂	—	13	2	no	—
146	C₂×M₃₂	—	13	2	no	—
147	#405	—	13	1	no	—
148	#405	—	26	1	no	—
149	#405	—	26	1	no	—
150	#405	#261	1	1	yes	characteristic
151	C₄×M₁₆	—	13	1	no	—
152	C₄×M₁₆	—	26	1	no	—
153	C₂×C₄×C₈	—	13	1	no	—
154	C₂×C₄×C₈	—	26	1	no	—
155	#364	—	26	1	no	—
156	#364	—	26	1	no	—
157	#364	—	26	1	no	—
158	#364	—	26	1	no	—
159	#350	—	26	4	no	—
160	C₂²×C₁₃	—	4	1	no	—
161	C₂²×C₁₃	#407	1	1	yes	characteristic
162	C₂²×C₁₃	#423	1	1	yes	characteristic
163	C₂²×C₁₃	#431	1	1	yes	characteristic
164	C₂×D₁₃	—	2	2	no	—
165	C₂×D₁₃	—	4	2	no	—
166	C₄×C₁₃	—	4	2	no	—
167	C₄×C₁₃	#380	1	1	yes	characteristic
168	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
170	C₄×C₁₃	#432	1	1	yes	characteristic
171	Dic₁₃	—	2	2	no	—
172	Dic₁₃	—	2	2	no	—
173	Dic₁₃	—	4	1	no	—
174	C₂×#41	—	13	1	no	—
175	#131	—	13	2	no	—
176	#131	—	13	2	no	—
177	#131	—	13	4	no	—
178	#131	—	26	1	no	—
179	#131	—	26	1	no	—
180	#131	—	26	2	no	—
181	C₂×M₁₆	—	13	1	no	—
182	C₂×M₁₆	—	26	1	no	—
183	C₂×M₁₆	—	26	1	no	—
184	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
185	C₂²×C₈	—	13	1	no	—
186	C₂²×C₈	—	26	1	no	—
187	C₂²×C₈	—	26	1	no	—
188	C₂²×C₈	C₂×#261	1	1	yes	characteristic
189	C₄×Q₈	—	26	1	no	—
190	C₄×D₄	—	26	1	no	—
191	C₄×D₄	—	26	2	no	—
192	#117	—	26	1	no	—
193	#117	C₂×#261	1	1	yes	characteristic
194	C₂×C₄²	—	13	1	no	—
195	C₂×C₄²	—	26	1	no	—
196	M₃₂	—	26	2	no	—
197	M₃₂	—	26	2	no	—
198	M₃₂	—	26	2	no	—
199	M₃₂	—	26	2	no	—
200	C₂×C₁₆	—	26	2	no	—
201	C₂×C₁₆	—	26	2	no	—
202	#97	—	2	1	no	—
203	#97	—	26	1	no	—
204	#97	—	26	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	13	1	no	—
207	C₄×C₈	—	13	1	no	—
208	C₄×C₈	—	13	1	no	—
209	C₄×C₈	—	13	1	no	—
210	C₄×C₈	—	26	1	no	—
211	C₄×C₈	—	26	1	no	—
212	C₂×C₁₃	—	2	1	no	—
213	C₂×C₁₃	—	2	1	no	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	#1797	1	1	yes	characteristic
216	D₁₃	—	4	2	no	—
217	#41	—	13	2	no	—
218	#41	—	13	2	no	—
219	#41	—	26	1	no	—
220	#41	—	26	1	no	—
221	C₂×Q₈	—	13	1	no	—
222	C₂×D₄	—	13	1	no	—
223	C₂×D₄	—	26	1	no	—
224	C₂²×C₄	—	26	2	no	—
225	C₂²×C₄	—	26	1	no	—
226	C₂²×C₄	—	26	1	no	—
227	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
228	M₁₆	—	2	1	no	—
229	M₁₆	—	13	2	no	—
230	M₁₆	—	26	2	no	—
231	M₁₆	—	26	2	no	—
232	M₁₆	—	26	1	no	—
233	M₁₆	—	26	4	no	—
234	M₁₆	C₄⋊#261	1	2	yes	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	13	1	no	—
237	C₂×C₈	—	13	1	no	—
238	C₂×C₈	—	13	1	no	—
239	C₂×C₈	—	13	1	no	—
240	C₂×C₈	—	13	1	no	—
241	C₂×C₈	—	13	1	no	—
242	C₂×C₈	—	26	2	no	—
243	C₂×C₈	—	26	2	no	—
244	C₂×C₈	—	26	1	no	—
245	C₂×C₈	—	26	4	no	—
246	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
247	C₂×C₈	#6118	1	2	yes	—
248	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	26	2	no	—
252	C₂²⋊C₄	—	26	2	no	—
253	C₂²⋊C₄	#6118	1	2	yes	—
254	C₄²	—	2	1	no	—
255	C₄²	—	13	1	no	—
256	C₄²	—	13	1	no	—
257	C₄²	—	26	2	no	—
258	C₄²	—	26	1	no	—
259	C₁₆	—	52	2	no	—
260	C₁₆	—	52	2	no	—
261	C₁₃	#12792	1	1	yes	characteristic, complemented, Sylow
262	C₂³	—	26	1	no	—
263	C₂³	C₂×#6118	1	1	yes	characteristic
264	Q₈	—	13	2	no	—
265	Q₈	—	26	1	no	—
266	D₄	—	13	2	no	—
267	D₄	—	26	2	no	—
268	D₄	—	26	2	no	—
269	D₄	—	26	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	26	1	no	—
272	C₂×C₄	—	26	1	no	—
273	C₂×C₄	—	26	2	no	—
274	C₂×C₄	—	26	2	no	—
275	C₂×C₄	—	26	1	no	—
276	C₂×C₄	—	52	4	no	—
277	C₂×C₄	—	52	2	no	—
278	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
279	C₂×C₄	C₂×#6118	1	1	yes	characteristic
280	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
282	C₂×C₄	#88237	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	13	1	no	—
286	C₈	—	13	1	no	—
287	C₈	—	13	1	no	—
288	C₈	—	13	1	no	—
289	C₈	—	26	2	no	—
290	C₈	#88111	1	2	yes	—
291	C₈	#88111	1	2	yes	—
292	C₂²	—	4	1	no	—
293	C₂²	—	26	2	no	—
294	C₂²	—	52	2	no	—
295	C₂²	#11719051	1	1	yes	characteristic
296	C₂²	#11719052	1	1	yes	characteristic
297	C₂²	#11719127	1	1	yes	characteristic
298	C₄	—	4	2	no	—
299	C₄	—	26	2	no	—
300	C₄	—	26	2	no	—
301	C₄	—	52	1	no	—
302	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
303	C₄	#11719054	1	1	yes	characteristic
304	C₄	C₂×#88111	1	1	yes	characteristic
305	C₄	#11719136	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	52	2	no	—
310	C₂	#13130691	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11719136
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#405	C₄
socle	C₂×C₁₃	#1797
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂²×#1055	C₂³
hypercenter	#405	#261
hypocenter	C₁₃	#12792
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. C₂×#11718261	C₂	semidirect
2. C₂×#11718261	C₂	semidirect
4. #13129886	C₂	semidirect
261. C₁₃	#12792	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718261	2
#13131080	2
#13129886	2
#13123258	2
#12792	13