Subgroups — #16016384

Order 3328.

632 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718840	C₂	1	1	yes	maximal, characteristic
2	#13123754	C₂	1	1	yes	maximal, characteristic
3	#13116725	C₂	1	8	yes	maximal, complemented
4	#13116608	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718157	C₂	1	1	yes	maximal, characteristic, complemented
6	#13116571	C₂	1	1	yes	maximal, characteristic
7	#13116441	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2950	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₁₃×#519	—	2	1	no	—
11	#11719450	—	2	1	no	—
12	C₂×#88158	C₂²	1	1	yes	characteristic
13	C₁₃×#441	C₂²	1	4	yes	complemented
14	C₄×C₁₃×Q₁₆	—	2	2	no	—
15	C₁₃×#427	C₂²	1	1	yes	characteristic
16	C₁₃×#419	—	2	1	no	—
17	C₁₃×#405	C₂²	1	1	yes	characteristic
18	C₂×#88069	C₄	1	2	yes	complemented
19	#11718840	C₄	1	4	yes	complemented
20	#11718840	C₂²	1	4	yes	—
21	C₂×#88068	C₄	1	2	yes	complemented
22	C₂×#88059	C₂²	1	1	yes	characteristic
23	C₂×#88052	C₂²	1	1	yes	characteristic
24	C₂×#88052	C₂²	1	1	yes	characteristic
25	#11718666	—	2	1	no	—
26	#11718378	C₂²	1	4	yes	—
27	#11718361	—	2	2	no	—
28	#11718157	C₂²	1	4	yes	complemented
29	#11718080	C₂²	1	4	yes	—
30	#11718079	C₂²	1	4	yes	complemented
31	#11718058	C₂²	1	4	yes	complemented
32	C₂²×Q₈×C₁₃	—	2	1	no	—
33	C₂×#6127	—	2	1	no	—
34	C₂×C₁₃×Q₁₆	—	2	1	no	—
35	C₂×C₁₃×Q₁₆	—	2	1	no	—
36	C₂×C₁₃×Q₁₆	—	2	1	no	—
37	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
40	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₄×Q₈×C₁₃	—	2	4	no	—
43	C₄×Q₈×C₁₃	—	4	2	no	—
44	C₁₃×#117	—	2	1	no	—
45	C₁₃×#117	—	4	2	no	—
46	C₁₃×#117	D₄	1	1	yes	characteristic
47	C₂×#6099	—	2	1	no	—
48	#88161	—	4	2	no	—
49	M₁₆×D₁₃	—	4	2	no	—
50	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
51	#88158	—	2	1	no	—
52	#88158	—	2	1	no	—
53	#88158	C₂×C₄	1	2	yes	—
54	#88158	C₂³	1	2	yes	—
55	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
56	C₂×#6095	—	2	1	no	—
57	#88155	—	2	4	no	—
58	C₁₃×#107	—	2	1	no	—
59	C₁₃×#106	C₂³	1	2	yes	—
60	C₁₃×#103	—	2	4	no	—
61	C₁₃×#97	C₂³	1	2	yes	—
62	C₄×C₈×C₁₃	D₄	1	2	yes	—
63	#88069	C₂×C₄	1	8	yes	—
64	#88068	C₂×C₄	1	8	yes	—
65	#88059	—	2	1	no	—
66	#88059	C₂³	1	2	yes	—
67	#88052	—	2	1	no	—
68	#88052	C₂×C₄	1	2	yes	—
69	#88052	C₂³	1	2	yes	—
70	#88052	C₂³	1	2	yes	—
71	C₂×#6090	—	2	1	no	—
72	C₂×#6090	C₂×C₄	1	2	yes	—
73	#88045	—	2	4	no	—
74	#33470	—	13	1	no	maximal, Sylow
75	C₂×Q₈×C₁₃	—	2	2	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	4	1	no	—
79	C₂×Q₈×C₁₃	—	4	1	no	—
80	C₂×Q₈×C₁₃	—	4	1	no	—
81	C₂²×C₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₂²⋊D₁₃	—	4	1	no	—
84	#6127	—	2	2	no	—
85	#6127	—	2	2	no	—
86	#6127	—	4	1	no	—
87	#6127	—	4	1	no	—
88	C₂×D₅₂	—	2	1	no	—
89	C₂×C₄×D₁₃	—	4	1	no	—
90	C₂×Dic₂₆	—	2	1	no	—
91	C₁₃×Q₁₆	—	2	2	no	—
92	C₁₃×Q₁₆	—	2	2	no	—
93	C₁₃×Q₁₆	—	2	2	no	—
94	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
95	C₁₃×M₁₆	—	2	1	no	—
96	C₁₃×M₁₆	—	2	1	no	—
97	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	—	2	1	no	—
105	C₁₃×C₄⋊C₄	—	2	2	no	—
106	C₁₃×C₄⋊C₄	—	4	4	no	—
107	C₁₃×C₂²⋊C₄	—	2	1	no	—
108	C₁₃×C₂²⋊C₄	—	4	1	no	—
109	C₄²×C₁₃	—	4	4	no	—
110	C₄²×C₁₃	—	4	1	no	—
111	C₄²×C₁₃	C₂×D₄	1	2	yes	—
112	#6099	—	2	2	no	—
113	#6099	—	4	1	no	—
114	C₂×#1055	—	4	1	no	—
115	Dic₅₂	—	2	1	no	—
116	Dic₅₂	C₂²×C₄	1	2	yes	—
117	D₁₀₄	—	2	1	no	—
118	D₁₀₄	C₂²×C₄	1	2	yes	—
119	#6095	—	2	2	no	—
120	#6095	—	2	2	no	—
121	#6094	—	4	4	no	—
122	C₈×D₁₃	—	4	4	no	—
123	#6090	—	2	2	no	—
124	#6090	C₂²×C₄	1	4	yes	—
125	C₂×#510	—	13	1	no	—
126	#2957	—	13	1	no	—
127	#2950	D₁₃	1	1	yes	characteristic, complemented
128	#2180	—	13	8	no	—
129	#2155	—	13	1	no	—
130	C₂×#362	—	13	1	no	—
131	#2145	—	13	1	no	—
132	#2110	—	13	1	no	—
133	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂²×D₁₃	—	4	1	no	—
135	Q₈×C₁₃	—	2	4	no	—
136	Q₈×C₁₃	—	4	2	no	—
137	Q₈×C₁₃	—	4	2	no	—
138	Q₈×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₂²⋊D₁₃	—	4	2	no	—
151	C₂²⋊D₁₃	—	4	2	no	—
152	C₂×Dic₁₃	—	4	1	no	—
153	D₅₂	—	2	2	no	—
154	D₅₂	—	4	1	no	—
155	C₄×D₁₃	—	4	2	no	—
156	C₄×D₁₃	—	4	2	no	—
157	Dic₂₆	—	2	2	no	—
158	Dic₂₆	—	4	1	no	—
159	C₈×C₁₃	—	2	2	no	—
160	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
161	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
162	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
163	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
164	#1055	—	4	2	no	—
165	C₂×#135	—	13	1	no	—
166	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
167	#568	—	26	1	no	—
168	#519	—	2	1	no	—
169	#510	—	13	4	no	—
170	#510	—	13	4	no	—
171	C₂×Q₃₂	—	13	2	no	—
172	C₂×SD₃₂	—	13	2	no	—
173	C₂×M₃₂	—	13	1	no	—
174	C₂×M₃₂	—	13	1	no	—
175	#444	—	13	4	no	—
176	#443	—	26	2	no	—
177	#441	C₂×D₁₃	1	4	yes	complemented
178	C₄×Q₁₆	—	2	2	no	—
179	C₂×#108	—	13	1	no	—
180	#427	C₂×D₁₃	1	1	yes	characteristic
181	#421	—	26	1	no	—
182	#419	—	2	1	no	—
183	#405	C₂×D₁₃	1	1	yes	characteristic
184	#365	—	13	4	no	—
185	#362	—	13	4	no	—
186	#360	—	13	4	no	—
187	#347	—	13	4	no	—
188	C₂²×C₁₃	—	4	1	no	—
189	C₂²×C₁₃	#442	1	1	yes	characteristic
190	C₂²×C₁₃	#442	1	1	yes	characteristic
191	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₂×D₁₃	—	4	2	no	—
193	C₂×D₁₃	—	16	1	no	—
194	C₄×C₁₃	—	4	2	no	—
195	C₄×C₁₃	—	4	4	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	#442	1	1	yes	characteristic
198	C₄×C₁₃	#442	1	1	yes	characteristic
199	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
200	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
201	Dic₁₃	—	4	2	no	—
202	C₂×#41	—	26	1	no	—
203	C₂²×Q₈	—	2	1	no	—
204	#135	—	13	2	no	—
205	#135	—	13	2	no	—
206	#135	—	26	1	no	—
207	#135	—	26	1	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	—	2	1	no	—
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	—	13	1	no	—
212	C₂×Q₁₆	C₄×D₁₃	1	2	yes	—
213	C₂×Q₁₆	C₄×D₁₃	1	2	yes	—
214	C₂×Q₁₆	C₂²×D₁₃	1	2	yes	—
215	C₂×SD₁₆	—	26	1	no	—
216	C₂×D₈	—	13	1	no	—
217	#131	—	26	4	no	—
218	#131	—	52	2	no	—
219	C₂×M₁₆	—	26	1	no	—
220	C₂×M₁₆	—	52	2	no	—
221	C₂×M₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
223	C₄×Q₈	—	2	4	no	—
224	C₄×Q₈	—	4	2	no	—
225	#117	—	2	1	no	—
226	#117	—	4	2	no	—
227	#117	C₂²⋊D₁₃	1	1	yes	characteristic
228	Q₃₂	—	13	8	no	—
229	SD₃₂	—	13	8	no	—
230	M₃₂	—	13	2	no	—
231	M₃₂	—	13	2	no	—
232	M₃₂	—	13	2	no	—
233	M₃₂	—	26	1	no	—
234	C₂×C₁₆	—	13	2	no	—
235	C₂×C₁₆	—	26	1	no	—
236	#108	—	13	2	no	—
237	#108	—	26	1	no	—
238	#107	—	2	1	no	—
239	#106	C₂²×D₁₃	1	2	yes	—
240	#104	—	26	4	no	—
241	#103	—	2	4	no	—
242	#97	C₂²×D₁₃	1	2	yes	—
243	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
244	C₂×C₁₃	—	2	1	no	—
245	C₂×C₁₃	—	2	1	no	—
246	C₂×C₁₃	—	2	1	no	—
247	C₂×C₁₃	C₂×#442	1	1	yes	characteristic
248	D₁₃	—	8	2	no	—
249	#41	—	26	2	no	—
250	#41	—	26	2	no	—
251	#41	—	52	1	no	—
252	#41	—	52	1	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	26	1	no	—
260	C₂×D₄	—	26	1	no	—
261	C₂×D₄	—	52	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	52	1	no	—
264	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
265	Q₁₆	—	2	2	no	—
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	13	2	no	—
269	Q₁₆	—	26	1	no	—
270	Q₁₆	C₂×C₄×D₁₃	1	4	yes	—
271	SD₁₆	—	26	2	no	—
272	SD₁₆	—	26	2	no	—
273	D₈	—	13	2	no	—
274	D₈	—	26	1	no	—
275	M₁₆	—	2	1	no	—
276	M₁₆	—	2	1	no	—
277	M₁₆	—	26	2	no	—
278	M₁₆	—	52	4	no	—
279	M₁₆	—	52	1	no	—
280	C₂×C₈	—	52	4	no	—
281	C₂×C₈	—	52	1	no	—
282	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
283	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
284	C₂×C₈	#6127	1	1	yes	characteristic
285	C₂×C₈	#6127	1	1	yes	characteristic
286	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
287	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
288	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	4	no	—
292	C₂²⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	4	1	no	—
294	C₄²	—	4	4	no	—
295	C₄²	—	4	1	no	—
296	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
297	C₁₆	—	13	4	no	—
298	C₁₆	—	26	2	no	—
299	C₁₃	#33470	1	1	yes	characteristic, complemented, Sylow
300	C₂³	—	52	1	no	—
301	C₂³	#88186	1	1	yes	characteristic
302	Q₈	—	2	4	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	26	2	no	—
307	Q₈	—	52	1	no	—
308	D₄	—	26	2	no	—
309	D₄	—	52	2	no	—
310	D₄	—	52	2	no	—
311	D₄	—	52	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	52	1	no	—
318	C₂×C₄	—	52	2	no	—
319	C₂×C₄	—	52	2	no	—
320	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
321	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
322	C₂×C₄	#88186	1	1	yes	characteristic
323	C₂×C₄	#88203	1	1	yes	characteristic
324	C₂×C₄	#88203	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	52	2	no	—
328	C₈	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
329	C₈	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
330	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
331	C₈	C₂×#6127	1	1	yes	characteristic
332	C₂²	—	4	1	no	—
333	C₂²	—	52	2	no	—
334	C₂²	—	208	1	no	—
335	C₂²	#11718675	1	1	yes	characteristic
336	C₂²	#11718675	1	1	yes	characteristic
337	C₂²	C₂×#88186	1	1	yes	characteristic
338	C₄	—	4	2	no	—
339	C₄	—	4	4	no	—
340	C₄	—	8	2	no	—
341	C₄	—	52	2	no	—
342	C₄	#11718675	1	1	yes	characteristic
343	C₄	#11718675	1	1	yes	characteristic
344	C₄	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic
345	C₄	C₂×#88203	1	1	yes	characteristic
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	104	2	no	—
350	C₂	C₂×#11718675	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11718675
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#2950	C₂
socle	C₂×C₁₃	C₂×#442
prime radical	C₂×#88158	C₂²
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	#2950	D₁₃
hypocenter	C₁₃	#33470
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2950

Complements

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

N	H ≅ G/N	type
3. #13116725	C₂	semidirect
4. #13116608	C₂	semidirect
5. C₂×#11718157	C₂	semidirect
7. #13116441	C₂	semidirect
8. C₁₃×#2950	C₂	semidirect
13. C₁₃×#441	C₂²	semidirect
18. C₂×#88069	C₄	semidirect
19. #11718840	C₄	semidirect
21. C₂×#88068	C₄	semidirect
28. #11718157	C₂²	semidirect
30. #11718079	C₂²	semidirect
31. #11718058	C₂²	semidirect
127. #2950	D₁₃	semidirect
177. #441	C₂×D₁₃	semidirect
299. C₁₃	#33470	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718840	2
#13123754	2
#13116725	2
#13116608	2
C₂×#11718157	2
#13116571	2
#13116441	2
C₁₃×#2950	2
#33470	13