Subgroups — C₄×#11719111 #16797424

Order 3328.

1046 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719111	C₂	1	2	yes	maximal
2	C₄×#88240	C₂	1	2	yes	maximal, complemented
3	C₄×#88104	C₂	1	1	yes	maximal, characteristic, complemented
4	#13130640	C₂	1	2	yes	maximal, complemented
5	#13130613	C₂	1	4	yes	maximal, complemented
6	C₄×#88118	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×#261	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#88173	C₂	1	1	yes	maximal, characteristic
9	C₂×#88240	C₂²	1	4	yes	—
10	C₂×#88173	C₂²	1	1	yes	characteristic
11	C₄×#6131	C₂²	1	2	yes	—
12	C₂×#88118	C₂²	1	4	yes	—
13	#11719111	C₄	1	16	yes	direct product factor
14	C₂×#88104	C₂²	1	2	yes	—
15	C₂×C₈×#261	C₂²	1	2	yes	—
16	C₂×C₈×#261	C₂²	1	2	yes	—
17	#11719072	—	2	4	no	—
18	#11719072	C₂²	1	4	yes	—
19	C₄²×#261	C₂²	1	1	yes	characteristic
20	C₄×#6118	—	2	2	no	—
21	C₄×#6118	C₂²	1	2	yes	—
22	C₄×D₁₃⋊C₈	—	2	2	no	—
23	C₄×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
24	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
25	Q₁₆×Dic₁₃	C₄	1	1	yes	characteristic, complemented
26	C₄×#6106	C₄	1	2	yes	complemented
27	#11718541	C₂²	1	1	yes	characteristic
28	#11718533	C₄	1	1	yes	characteristic, complemented
29	#11718465	C₂²	1	2	yes	—
30	#11718449	C₄	1	2	yes	complemented
31	C₄×D₁₀₄	C₄	1	1	yes	characteristic, complemented
32	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
33	#11718268	C₂²	1	2	yes	—
34	#11718247	C₂²	1	4	yes	—
35	C₂×#6131	C₂³	1	2	yes	—
36	#88240	—	2	8	no	—
37	#88240	C₂×C₄	1	16	yes	complemented
38	C₂×C₄×#261	—	2	1	no	—
39	C₂×C₄×#261	—	2	1	no	—
40	C₂×C₄×#261	C₂³	1	2	yes	—
41	C₂×#6118	—	2	4	no	—
42	C₂×#6118	C₂³	1	4	yes	—
43	C₂×D₁₃⋊C₈	—	2	4	no	—
44	C₂×D₁₃⋊C₈	C₂³	1	2	yes	—
45	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
46	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
47	Q₈×Dic₁₃	C₂×C₄	1	2	yes	—
48	C₂×#6106	C₂×C₄	1	2	yes	—
49	#88173	C₂×C₄	1	8	yes	—
50	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
53	#88147	—	2	2	no	—
54	#88147	C₂×C₄	1	2	yes	—
55	#88146	—	2	2	no	—
56	#88146	C₂³	1	2	yes	—
57	C₄×D₅₂	—	2	2	no	—
58	C₄×D₅₂	C₂×C₄	1	2	yes	—
59	C₄²×D₁₃	—	2	2	no	—
60	C₄²×D₁₃	C₂³	1	1	yes	characteristic
61	#88118	C₂×C₄	1	16	yes	complemented
62	#88110	—	2	4	no	—
63	#88110	—	2	2	no	—
64	C₄×#1057	—	2	4	no	—
65	C₄×#1057	—	2	2	no	—
66	C₄×#1057	D₄	1	1	yes	characteristic
67	C₄×#1057	D₄	1	1	yes	characteristic
68	C₄×#1057	D₄	1	2	yes	—
69	#88104	C₂×C₄	1	8	yes	complemented
70	C₈×#261	—	2	4	no	—
71	C₈×#261	C₂×C₄	1	8	yes	complemented
72	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
73	C₁₃×#103	C₂×C₄	1	2	yes	—
74	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
75	#88075	C₂×C₄	1	2	yes	—
76	#88061	C₂×C₄	1	2	yes	—
77	#88058	C₂×C₄	1	1	yes	characteristic
78	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
80	#88049	C₂×C₄	1	2	yes	—
81	#88048	C₂×C₄	1	2	yes	—
82	C₄×#1055	C₂×C₄	1	1	yes	characteristic
83	C₄×#443	—	13	1	no	maximal, Sylow
84	C₂²×#261	—	2	2	no	—
85	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
86	#6131	—	2	4	no	—
87	#6131	C₂²×C₄	1	8	yes	—
88	C₂×D₅₂	—	2	2	no	—
89	C₂×D₅₂	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₁₃	—	2	2	no	—
91	C₂×C₄×D₁₃	—	2	2	no	—
92	C₂×C₄×D₁₃	—	2	2	no	—
93	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
94	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
95	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
96	C₂×#1057	—	2	4	no	—
97	C₂×#1057	C₂×D₄	1	2	yes	—
98	C₂×#1057	C₂×D₄	1	2	yes	—
99	C₄×#261	—	2	4	no	—
100	C₄×#261	—	2	4	no	—
101	C₄×#261	—	2	2	no	—
102	C₄×#261	C₂²×C₄	1	4	yes	—
103	#6118	—	2	4	no	—
104	#6118	—	2	16	no	—
105	#6118	C₂²×C₄	1	8	yes	—
106	D₁₃⋊C₈	—	2	16	no	—
107	D₁₃⋊C₈	C₂²×C₄	1	4	yes	—
108	D₁₃⋊C₈	#41	1	4	yes	—
109	C₁₃×Q₁₆	C₄²	1	4	yes	complemented
110	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₁₃×C₄⋊C₄	—	2	2	no	—
113	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
114	C₄²×C₁₃	—	2	2	no	—
115	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
116	#6106	C₄²	1	8	yes	complemented
117	#6103	—	2	2	no	—
118	#6103	—	2	2	no	—
119	#6103	—	2	2	no	—
120	C₄⋊Dic₁₃	—	2	2	no	—
121	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
122	C₄×Dic₁₃	—	2	2	no	—
123	C₄×Dic₁₃	—	2	2	no	—
124	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
125	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
126	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
127	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
128	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
129	D₁₀₄	C₄²	1	4	yes	complemented
130	C₈×D₁₃	—	2	2	no	—
131	C₈×D₁₃	C₂²×C₄	1	4	yes	—
132	C₂×#443	—	13	2	no	—
133	C₄×#135	—	13	1	no	—
134	C₄×#131	—	13	2	no	—
135	C₄×#108	—	13	1	no	—
136	#1772	—	13	2	no	—
137	#1764	—	13	4	no	—
138	C₄×#104	—	13	2	no	—
139	C₄²×C₈	—	13	1	no	—
140	C₂²×D₁₃	—	2	2	no	—
141	C₂²×D₁₃	C₂×#41	1	1	yes	characteristic
142	C₂×#261	—	2	4	no	—
143	C₂×#261	—	2	4	no	—
144	C₂×#261	—	2	4	no	—
145	Q₈×C₁₃	—	2	2	no	—
146	Q₈×C₁₃	C₂×C₄²	1	4	yes	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
152	C₂×Dic₁₃	—	2	2	no	—
153	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
154	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
155	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
156	D₅₂	—	2	8	no	—
157	D₅₂	—	2	2	no	—
158	D₅₂	C₂×C₄²	1	4	yes	—
159	C₄×D₁₃	—	2	2	no	—
160	C₄×D₁₃	—	2	8	no	—
161	C₄×D₁₃	—	2	2	no	—
162	C₄×D₁₃	—	4	4	no	—
163	C₄×D₁₃	—	4	4	no	—
164	C₄×D₁₃	C₂³×C₄	1	2	yes	—
165	C₄×D₁₃	C₂×#41	1	2	yes	—
166	#1057	—	2	8	no	—
167	#1057	C₄×D₄	1	4	yes	—
168	#1057	C₄×D₄	1	4	yes	—
169	C₈×C₁₃	—	2	1	no	—
170	C₈×C₁₃	C₂×C₄²	1	2	yes	—
171	#1055	—	2	1	no	—
172	#1055	C₂×C₄²	1	2	yes	—
173	C₂×#135	—	13	1	no	—
174	C₂×#131	—	13	4	no	—
175	C₄×#41	—	13	2	no	—
176	#443	—	13	16	no	—
177	C₄×Q₁₆	—	13	1	no	—
178	C₄×Q₁₆	#261	1	1	yes	characteristic, complemented
179	C₄×SD₁₆	—	13	2	no	—
180	C₄×SD₁₆	—	13	2	no	—
181	C₄×D₈	—	13	1	no	—
182	C₄×D₈	—	13	1	no	—
183	C₂×#108	—	13	2	no	—
184	#427	—	13	1	no	—
185	C₂×#104	—	13	4	no	—
186	#416	—	13	2	no	—
187	#405	—	13	4	no	—
188	#405	—	26	4	no	—
189	C₄×M₁₆	—	13	2	no	—
190	C₄×M₁₆	—	26	2	no	—
191	C₂×C₄×C₈	—	13	1	no	—
192	C₂×C₄×C₈	—	13	2	no	—
193	C₂×C₄×C₈	—	13	1	no	—
194	C₂×C₄×C₈	—	13	2	no	—
195	C₂×C₄×C₈	—	26	2	no	—
196	C₄³	—	13	1	no	—
197	#341	—	13	2	no	—
198	#339	—	13	4	no	—
199	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂×D₁₃	—	2	1	no	—
201	C₂×D₁₃	—	2	1	no	—
202	C₂×D₁₃	—	2	4	no	—
203	C₂×D₁₃	—	4	2	no	—
204	C₂×D₁₃	—	4	2	no	—
205	C₂×D₁₃	C₄×#41	1	2	yes	—
206	#261	—	2	8	no	—
207	C₄×C₁₃	—	2	1	no	—
208	C₄×C₁₃	—	2	4	no	—
209	C₄×C₁₃	—	4	2	no	—
210	C₄×C₁₃	#443	1	2	yes	complemented
211	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
212	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
213	Dic₁₃	—	2	1	no	—
214	Dic₁₃	—	4	2	no	—
215	Dic₁₃	#443	1	2	yes	complemented
216	Dic₁₃	C₂×C₄×D₄	1	2	yes	—
217	C₂×#41	—	13	2	no	—
218	#135	—	13	8	no	—
219	C₂×Q₁₆	C₂×#261	1	1	yes	characteristic
220	C₂×SD₁₆	—	13	2	no	—
221	C₂×D₈	—	13	1	no	—
222	#131	—	13	16	no	—
223	#131	—	26	8	no	—
224	C₂×M₁₆	—	13	4	no	—
225	C₂×M₁₆	—	26	4	no	—
226	C₂²×C₈	—	13	2	no	—
227	C₂²×C₈	—	13	1	no	—
228	C₂²×C₈	—	13	1	no	—
229	C₂²×C₈	—	26	4	no	—
230	C₄×Q₈	—	13	2	no	—
231	C₄×Q₈	C₂×#261	1	2	yes	—
232	C₄×D₄	—	13	2	no	—
233	C₄×D₄	—	13	2	no	—
234	C₄×D₄	—	26	2	no	—
235	C₄×D₄	—	26	2	no	—
236	#117	—	13	2	no	—
237	#117	—	26	2	no	—
238	C₂×C₄²	—	13	1	no	—
239	C₂×C₄²	—	13	2	no	—
240	C₂×C₄²	—	26	2	no	—
241	C₂×C₄²	—	26	1	no	—
242	C₂×C₄²	—	26	1	no	—
243	#108	—	13	8	no	—
244	#107	—	13	1	no	—
245	#107	C₂×#261	1	1	yes	characteristic
246	#106	—	13	2	no	—
247	#104	—	13	16	no	—
248	#103	—	13	2	no	—
249	#103	C₂×#261	1	2	yes	—
250	#102	—	13	2	no	—
251	#102	—	13	2	no	—
252	#97	—	26	2	no	—
253	#97	—	26	4	no	—
254	C₄×C₈	—	13	8	no	—
255	C₄×C₈	—	13	1	no	—
256	C₄×C₈	—	13	1	no	—
257	C₄×C₈	—	13	2	no	—
258	C₄×C₈	—	13	1	no	—
259	C₄×C₈	—	13	1	no	—
260	C₄×C₈	—	13	1	no	—
261	C₄×C₈	—	26	2	no	—
262	C₄×C₈	—	26	4	no	—
263	C₄×C₈	—	26	4	no	—
264	C₄×C₈	C₂×#261	1	1	yes	characteristic
265	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
266	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
267	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
268	D₁₃	—	2	2	no	—
269	D₁₃	—	4	4	no	—
270	#41	—	13	8	no	—
271	#41	—	26	4	no	—
272	C₂×Q₈	C₂²×#261	1	2	yes	—
273	C₂×D₄	—	13	2	no	—
274	C₂×D₄	—	26	2	no	—
275	C₂²×C₄	—	13	1	no	—
276	C₂²×C₄	—	13	1	no	—
277	C₂²×C₄	—	13	1	no	—
278	C₂²×C₄	—	26	2	no	—
279	C₂²×C₄	—	26	2	no	—
280	C₂²×C₄	—	26	2	no	—
281	C₂²×C₄	—	26	2	no	—
282	Q₁₆	C₄×#261	1	4	yes	complemented
283	SD₁₆	—	13	8	no	—
284	D₈	—	13	4	no	—
285	M₁₆	—	13	8	no	—
286	M₁₆	—	26	16	no	—
287	M₁₆	—	26	4	no	—
288	C₂×C₈	—	13	2	no	—
289	C₂×C₈	—	13	2	no	—
290	C₂×C₈	—	13	4	no	—
291	C₂×C₈	—	13	4	no	—
292	C₂×C₈	—	13	1	no	—
293	C₂×C₈	—	13	4	no	—
294	C₂×C₈	—	13	1	no	—
295	C₂×C₈	—	26	2	no	—
296	C₂×C₈	—	26	16	no	—
297	C₂×C₈	—	26	4	no	—
298	C₂×C₈	C₂²×#261	1	1	yes	characteristic
299	C₂×C₈	C₂²×#261	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	13	2	no	—
302	C₄⋊C₄	—	26	2	no	—
303	C₄⋊C₄	C₂²×#261	1	2	yes	—
304	C₂²⋊C₄	—	26	2	no	—
305	C₂²⋊C₄	—	26	2	no	—
306	C₂²⋊C₄	—	26	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	13	1	no	—
309	C₄²	—	13	1	no	—
310	C₄²	—	13	1	no	—
311	C₄²	—	13	4	no	—
312	C₄²	—	26	2	no	—
313	C₄²	—	26	2	no	—
314	C₄²	—	26	2	no	—
315	C₄²	—	26	4	no	—
316	C₄²	—	26	4	no	—
317	C₄²	C₂²×#261	1	1	yes	characteristic
318	C₁₃	C₄×#443	1	1	yes	characteristic, complemented, Sylow
319	C₂³	—	13	1	no	—
320	C₂³	—	26	2	no	—
321	Q₈	—	2	2	no	—
322	Q₈	C₂×C₄×#261	1	4	yes	—
323	D₄	—	13	4	no	—
324	D₄	—	26	8	no	—
325	D₄	—	26	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	13	1	no	—
329	C₂×C₄	—	13	1	no	—
330	C₂×C₄	—	13	2	no	—
331	C₂×C₄	—	13	2	no	—
332	C₂×C₄	—	13	1	no	—
333	C₂×C₄	—	26	2	no	—
334	C₂×C₄	—	26	8	no	—
335	C₂×C₄	—	26	4	no	—
336	C₂×C₄	—	26	4	no	—
337	C₂×C₄	—	26	4	no	—
338	C₂×C₄	—	26	2	no	—
339	C₂×C₄	—	26	2	no	—
340	C₂×C₄	—	52	4	no	—
341	C₂×C₄	—	52	4	no	—
342	C₂×C₄	D₄×#261	1	1	yes	characteristic
343	C₂×C₄	D₄×#261	1	1	yes	characteristic
344	C₂×C₄	C₂³×#261	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	13	4	no	—
347	C₈	—	13	4	no	—
348	C₈	—	13	2	no	—
349	C₈	—	26	8	no	—
350	C₈	—	26	1	no	—
351	C₈	C₂×C₄×#261	1	2	yes	—
352	C₂²	—	13	2	no	—
353	C₂²	—	26	1	no	—
354	C₂²	—	26	1	no	—
355	C₂²	—	26	4	no	—
356	C₂²	—	52	2	no	—
357	C₂²	—	52	2	no	—
358	C₂²	C₂×D₄×#261	1	1	yes	characteristic
359	C₄	—	2	4	no	—
360	C₄	—	2	1	no	—
361	C₄	—	4	2	no	—
362	C₄	—	13	2	no	—
363	C₄	—	13	2	no	—
364	C₄	—	26	8	no	—
365	C₄	—	26	1	no	—
366	C₄	—	52	2	no	—
367	C₄	#11719111	1	2	yes	direct product factor
368	C₄	C₂²×C₄×#261	1	1	yes	characteristic
369	C₄	#41×#261	1	1	yes	characteristic
370	C₂	—	26	2	no	—
371	C₂	—	52	4	no	—
372	C₂	C₄×D₄×#261	1	1	yes	characteristic
373	C₂	D₁₃⋊#443	1	1	yes	characteristic
374	C₂	C₂×#11719111	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#261
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#261
Fitting	C₄×C₁₃×Q₁₆	C₄
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88173	C₂²
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₄×Q₁₆	#261
hypocenter	C₁₃	C₄×#443
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×#88240	C₂	semidirect
3. C₄×#88104	C₂	semidirect
4. #13130640	C₂	semidirect
5. #13130613	C₂	semidirect
6. C₄×#88118	C₂	semidirect
7. C₄×C₈×#261	C₂	semidirect
13. #11719111	C₄	direct (N × H)
24. C₄×C₁₃×Q₁₆	C₄	semidirect
25. Q₁₆×Dic₁₃	C₄	semidirect
26. C₄×#6106	C₄	semidirect
28. #11718533	C₄	semidirect
30. #11718449	C₄	semidirect
31. C₄×D₁₀₄	C₄	semidirect
37. #88240	C₂×C₄	semidirect
61. #88118	C₂×C₄	semidirect
69. #88104	C₂×C₄	semidirect
71. C₈×#261	C₂×C₄	semidirect
109. C₁₃×Q₁₆	C₄²	semidirect
116. #6106	C₄²	semidirect
129. D₁₀₄	C₄²	semidirect
178. C₄×Q₁₆	#261	semidirect
210. C₄×C₁₃	#443	semidirect
215. Dic₁₃	#443	semidirect
282. Q₁₆	C₄×#261	semidirect
318. C₁₃	C₄×#443	semidirect
367. C₄	#11719111	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×#11719111	2
C₄×#88240	2
C₄×#88104	2
#13130640	2
#13130613	2
C₄×#88118	2
C₄×C₈×#261	2
C₄×#88173	2
C₄×#443	13

Subgroups — C4×#11719111 #16797424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11719111 #16797424