Subgroups — C₄×#11718573 #16016196

Order 3328.

690 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718573	C₂	1	1	yes	maximal, characteristic
2	C₄×#88166	C₂	1	2	yes	maximal, complemented
3	C₄×Dic₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#13116694	C₂	1	1	yes	maximal, characteristic, complemented
5	#13116691	C₂	1	2	yes	maximal, complemented
6	C₄×#88067	C₂	1	2	yes	maximal, complemented
7	D₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#13116617	C₂	1	1	yes	maximal, characteristic, complemented
9	#13116540	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#88166	C₂²	1	2	yes	—
13	C₄×#6129	—	2	2	no	—
14	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×D₈	C₂²	1	2	yes	—
16	D₈×Dic₁₃	C₂²	1	2	yes	—
17	C₂×#88067	C₂²	1	2	yes	—
18	C₄×#6105	—	2	2	no	—
19	#11718573	C₄	1	8	yes	direct product factor
20	C₂×Dic₁₀₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
23	#11718541	C₂²	1	2	yes	—
24	#11718534	C₂²	1	2	yes	—
25	#11718433	—	2	2	no	—
26	#11718419	—	2	2	no	—
27	C₄×Dic₅₂	C₂²	1	2	yes	—
28	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×#366	C₂²	1	1	yes	characteristic
30	C₁₃×#357	C₂²	1	2	yes	—
31	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
32	#11718154	C₂²	1	2	yes	—
33	#11718096	C₂²	1	1	yes	characteristic
34	#11718095	C₂²	1	2	yes	—
35	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
37	#11718084	C₂²	1	2	yes	—
38	#11718082	C₂²	1	2	yes	—
39	C₄×#6090	C₂²	1	1	yes	characteristic
40	C₂×#6129	—	2	2	no	—
41	C₂×C₁₃×D₈	C₂³	1	2	yes	—
42	C₄×D₄×C₁₃	—	2	2	no	—
43	D₄×Dic₁₃	—	2	2	no	—
44	C₂×#6105	—	2	2	no	—
45	C₄×C₂²⋊D₁₃	—	4	2	no	—
46	C₂×C₄×Dic₁₃	—	4	2	no	—
47	#88166	—	2	4	no	—
48	#88166	C₂×C₄	1	8	yes	complemented
49	C₂×Dic₅₂	C₂³	1	2	yes	—
50	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
51	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
52	#88146	—	2	2	no	—
53	#88141	—	2	2	no	—
54	#88135	—	4	2	no	—
55	#88131	—	4	2	no	—
56	C₄²×D₁₃	D₄	1	1	yes	characteristic
57	C₄×Dic₂₆	—	2	2	no	—
58	C₁₃×D₁₆	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#102	—	2	2	no	—
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	#88072	—	2	2	no	—
65	#88067	C₂×C₄	1	8	yes	complemented
66	#88058	C₂³	1	2	yes	—
67	#88056	—	2	2	no	—
68	C₈×Dic₁₃	D₄	1	1	yes	characteristic
69	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
70	C₂×#6090	C₂³	1	1	yes	characteristic
71	C₂×#6090	C₂³	1	1	yes	characteristic
72	#88050	—	2	2	no	—
73	#88048	—	2	2	no	—
74	C₄×#1055	D₄	1	1	yes	characteristic
75	Dic₁₀₄	C₂×C₄	1	4	yes	complemented
76	C₁₆×D₁₃	—	2	2	no	—
77	C₁₆×D₁₃	C₂×C₄	1	4	yes	complemented
78	C₄×#508	—	13	1	no	maximal, Sylow
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	—	4	2	no	—
81	C₂×C₂²⋊D₁₃	—	4	2	no	—
82	C₂²×Dic₁₃	—	4	2	no	—
83	C₂²×Dic₁₃	—	4	2	no	—
84	#6129	—	2	8	no	—
85	#6129	—	4	4	no	—
86	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
87	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
88	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×Dic₂₆	—	2	2	no	—
90	C₁₃×D₈	—	2	2	no	—
91	C₁₃×D₈	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₂²⋊C₄	—	4	2	no	—
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂²⋊Dic₁₃	—	4	2	no	—
98	#6105	—	2	8	no	—
99	#6103	—	4	2	no	—
100	C₄⋊Dic₁₃	—	2	2	no	—
101	#6101	—	4	2	no	—
102	#6101	—	4	2	no	—
103	C₄×Dic₁₃	—	4	2	no	—
104	C₄×Dic₁₃	—	4	2	no	—
105	C₄×Dic₁₃	—	4	2	no	—
106	C₄×Dic₁₃	D₈	1	1	yes	characteristic
107	C₄×Dic₁₃	D₈	1	1	yes	characteristic
108	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×#1055	C₂×D₄	1	1	yes	characteristic
110	C₂×#1055	C₂×D₄	1	1	yes	characteristic
111	Dic₅₂	—	2	2	no	—
112	Dic₅₂	C₂²×C₄	1	4	yes	—
113	C₈×D₁₃	—	2	2	no	—
114	C₈×D₁₃	C₂²×C₄	1	2	yes	—
115	C₈×D₁₃	#41	1	2	yes	—
116	C₁₃×C₁₆	—	2	1	no	—
117	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
118	#6090	—	2	1	no	—
119	#6090	C₂²×C₄	1	2	yes	—
120	C₂×#508	—	13	1	no	—
121	C₄×#135	—	13	2	no	—
122	C₄×Q₃₂	—	13	1	no	—
123	C₄×Q₃₂	—	13	1	no	—
124	C₄×SD₃₂	—	13	2	no	—
125	C₄×SD₃₂	—	13	2	no	—
126	C₄×D₁₆	—	13	1	no	—
127	C₄×D₁₆	D₁₃	1	1	yes	characteristic, complemented
128	#2158	—	13	1	no	—
129	#2138	—	13	2	no	—
130	C₂×C₄×C₁₆	—	13	1	no	—
131	C₂³×C₁₃	—	4	2	no	—
132	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
133	D₄×C₁₃	—	2	4	no	—
134	D₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	8	4	no	—
137	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂²⋊D₁₃	—	4	8	no	—
141	C₂×Dic₁₃	—	4	2	no	—
142	C₂×Dic₁₃	—	4	8	no	—
143	C₂×Dic₁₃	—	4	2	no	—
144	C₂×Dic₁₃	—	8	4	no	—
145	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
148	C₄×D₁₃	—	2	2	no	—
149	C₄×D₁₃	—	2	2	no	—
150	C₄×D₁₃	C₄×D₄	1	2	yes	—
151	C₄×D₁₃	#135	1	2	yes	—
152	Dic₂₆	—	2	4	no	—
153	Dic₂₆	—	4	2	no	—
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
156	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
157	#1055	—	2	1	no	—
158	#1055	C₄×D₄	1	2	yes	—
159	C₂×#135	—	13	2	no	—
160	C₄×#41	—	26	2	no	—
161	#508	—	13	8	no	—
162	C₂×Q₃₂	—	13	1	no	—
163	C₂×SD₃₂	—	13	2	no	—
164	C₂×D₁₆	C₂×D₁₃	1	1	yes	characteristic
165	C₂²×C₁₆	—	13	1	no	—
166	C₂²×C₁₆	—	13	1	no	—
167	C₄×Q₁₆	—	13	2	no	—
168	C₄×Q₁₆	—	13	2	no	—
169	C₄×SD₁₆	—	26	2	no	—
170	C₄×SD₁₆	—	26	2	no	—
171	C₄×D₈	—	13	2	no	—
172	C₄×D₈	C₂×D₁₃	1	2	yes	—
173	#427	—	13	2	no	—
174	#416	—	26	2	no	—
175	C₂×C₄×C₈	—	13	1	no	—
176	#367	—	13	2	no	—
177	#366	—	13	1	no	—
178	#366	C₂×D₁₃	1	1	yes	characteristic
179	#358	—	13	2	no	—
180	#358	—	13	2	no	—
181	#357	—	13	2	no	—
182	#357	C₂×D₁₃	1	2	yes	—
183	C₄×C₁₆	—	13	1	no	—
184	C₄×C₁₆	—	13	1	no	—
185	C₄×C₁₆	—	13	1	no	—
186	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
187	C₂²×C₁₃	—	4	4	no	—
188	C₂²×C₁₃	—	8	2	no	—
189	C₂²×C₁₃	—	8	2	no	—
190	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
191	C₂×D₁₃	—	2	1	no	—
192	C₂×D₁₃	—	2	1	no	—
193	C₂×D₁₃	C₄×D₈	1	2	yes	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	8	2	no	—
196	C₄×C₁₃	#508	1	2	yes	complemented
197	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
198	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
199	Dic₁₃	—	2	1	no	—
200	Dic₁₃	—	4	4	no	—
201	Dic₁₃	—	8	2	no	—
202	Dic₁₃	C₄×D₈	1	2	yes	—
203	Dic₁₃	#508	1	2	yes	complemented
204	C₂×#41	—	26	2	no	—
205	#135	—	13	8	no	—
206	#135	—	26	4	no	—
207	C₂×Q₁₆	—	13	2	no	—
208	C₂×SD₁₆	—	26	2	no	—
209	C₂×D₈	C₂²×D₁₃	1	2	yes	—
210	C₂²×C₈	—	13	1	no	—
211	C₂²×C₈	—	13	1	no	—
212	C₄×Q₈	—	26	2	no	—
213	C₄×Q₈	—	26	2	no	—
214	C₄×D₄	—	2	2	no	—
215	C₄×D₄	—	26	2	no	—
216	C₄×D₄	—	52	2	no	—
217	C₄×D₄	—	52	2	no	—
218	#117	—	26	2	no	—
219	#117	—	52	2	no	—
220	C₂×C₄²	—	13	1	no	—
221	C₂×C₄²	—	52	2	no	—
222	Q₃₂	—	13	4	no	—
223	SD₃₂	—	13	8	no	—
224	D₁₆	C₄×D₁₃	1	4	yes	complemented
225	C₂×C₁₆	—	13	1	no	—
226	C₂×C₁₆	—	13	4	no	—
227	C₂×C₁₆	—	13	1	no	—
228	C₂×C₁₆	—	26	2	no	—
229	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
230	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
231	#107	—	13	2	no	—
232	#107	C₂²×D₁₃	1	2	yes	—
233	#106	—	26	2	no	—
234	#103	—	26	2	no	—
235	#103	—	26	2	no	—
236	#102	—	2	2	no	—
237	#102	—	26	2	no	—
238	C₄×C₈	—	13	1	no	—
239	C₄×C₈	—	13	1	no	—
240	C₄×C₈	—	13	1	no	—
241	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
242	C₂×C₁₃	—	8	4	no	—
243	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
244	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
245	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
246	D₁₃	—	2	2	no	—
247	#41	—	26	8	no	—
248	#41	—	52	4	no	—
249	C₂×Q₈	—	26	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	52	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	13	1	no	—
254	C₂²×C₄	—	13	1	no	—
255	C₂²×C₄	—	13	1	no	—
256	C₂²×C₄	—	52	2	no	—
257	C₂²×C₄	—	52	2	no	—
258	Q₁₆	—	13	4	no	—
259	Q₁₆	—	26	2	no	—
260	SD₁₆	—	26	8	no	—
261	D₈	—	2	2	no	—
262	D₈	C₂×C₄×D₁₃	1	4	yes	—
263	C₂×C₈	—	13	1	no	—
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	13	2	no	—
266	C₂×C₈	—	13	1	no	—
267	C₂×C₈	—	26	2	no	—
268	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	26	2	no	—
272	C₄⋊C₄	—	52	2	no	—
273	C₄⋊C₄	—	52	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	52	2	no	—
276	C₂²⋊C₄	—	52	2	no	—
277	C₄²	—	13	1	no	—
278	C₄²	—	13	1	no	—
279	C₄²	—	13	1	no	—
280	C₄²	—	52	2	no	—
281	C₄²	—	52	2	no	—
282	C₄²	—	52	2	no	—
283	C₄²	D₄×D₁₃	1	1	yes	characteristic
284	C₁₆	—	2	1	no	—
285	C₁₆	—	13	2	no	—
286	C₁₆	—	26	1	no	—
287	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
288	C₁₃	C₄×#508	1	1	yes	characteristic, complemented, Sylow
289	C₂³	—	4	2	no	—
290	C₂³	—	13	1	no	—
291	Q₈	—	26	4	no	—
292	Q₈	—	52	2	no	—
293	D₄	—	2	4	no	—
294	D₄	—	4	2	no	—
295	D₄	—	52	8	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	—	13	1	no	—
299	C₂×C₄	—	13	1	no	—
300	C₂×C₄	—	13	2	no	—
301	C₂×C₄	—	13	2	no	—
302	C₂×C₄	—	13	1	no	—
303	C₂×C₄	—	26	2	no	—
304	C₂×C₄	—	26	2	no	—
305	C₂×C₄	—	52	8	no	—
306	C₂×C₄	—	52	2	no	—
307	C₂×C₄	—	52	2	no	—
308	C₂×C₄	—	104	4	no	—
309	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
310	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
311	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	13	2	no	—
314	C₈	—	26	1	no	—
315	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
316	C₈	#41×D₁₃	1	1	yes	characteristic
317	C₂²	—	4	4	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	13	2	no	—
321	C₂²	—	26	1	no	—
322	C₂²	—	26	1	no	—
323	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	8	2	no	—
326	C₄	—	13	2	no	—
327	C₄	—	13	2	no	—
328	C₄	—	26	1	no	—
329	C₄	—	52	4	no	—
330	C₄	—	104	2	no	—
331	C₄	#11718573	1	2	yes	direct product factor
332	C₄	C₄×D₄×D₁₃	1	1	yes	characteristic
333	C₄	D₁₃×#135	1	1	yes	characteristic
334	C₂	—	8	4	no	—
335	C₂	—	26	2	no	—
336	C₂	C₄×D₈×D₁₃	1	1	yes	characteristic
337	C₂	D₁₃×#508	1	1	yes	characteristic
338	C₂	C₂×#11718573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×D₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×#11718573	C₂
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×D₁₆	D₁₃
hypocenter	C₁₃	C₄×#508
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#88166	C₂	semidirect
3. C₄×Dic₁₀₄	C₂	semidirect
4. #13116694	C₂	semidirect
5. #13116691	C₂	semidirect
6. C₄×#88067	C₂	semidirect
7. D₁₆×Dic₁₃	C₂	semidirect
8. #13116617	C₂	semidirect
9. #13116540	C₂	semidirect
10. C₄×C₁₆×D₁₃	C₂	semidirect
11. C₄×C₁₃×D₁₆	C₂	semidirect
19. #11718573	C₄	direct (N × H)
48. #88166	C₂×C₄	semidirect
58. C₁₃×D₁₆	C₂×C₄	semidirect
65. #88067	C₂×C₄	semidirect
75. Dic₁₀₄	C₂×C₄	semidirect
77. C₁₆×D₁₃	C₂×C₄	semidirect
127. C₄×D₁₆	D₁₃	semidirect
196. C₄×C₁₃	#508	semidirect
203. Dic₁₃	#508	semidirect
224. D₁₆	C₄×D₁₃	semidirect
288. C₁₃	C₄×#508	semidirect
331. C₄	#11718573	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11718573	2
C₄×#88166	2
C₄×Dic₁₀₄	2
#13116694	2
#13116691	2
C₄×#88067	2
D₁₆×Dic₁₃	2
#13116617	2
#13116540	2
C₄×C₁₆×D₁₃	2
C₄×C₁₃×D₁₆	2
C₄×#508	13

Subgroups — C4×#11718573 #16016196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718573 #16016196