Subgroups — C₄×#11719112 #16797432

Order 3328.

1022 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719112	C₂	1	2	yes	maximal
2	C₄×#88240	C₂	1	2	yes	maximal, complemented
3	C₄×#88103	C₂	1	1	yes	maximal, characteristic, complemented
4	#13130643	C₂	1	2	yes	maximal, complemented
5	#13130617	C₂	1	4	yes	maximal, complemented
6	C₄×#88118	C₂	1	2	yes	maximal, complemented
7	C₄×#88100	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	#11719112	C₄	1	16	yes	direct product factor
14	C₂×#88103	C₂²	1	2	yes	—
15	C₂×#88100	C₂²	1	2	yes	—
16	C₂×#88100	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	2	yes	—
67	C₄×#1057	D₄	1	2	yes	—
68	#88103	C₂×C₄	1	8	yes	complemented
69	#88100	—	2	4	no	—
70	#88100	C₂×C₄	1	8	yes	complemented
71	C₁₃×#107	C₂×C₄	1	1	yes	characteristic
72	C₁₃×#103	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
74	#88075	C₂×C₄	1	2	yes	—
75	#88061	C₂×C₄	1	2	yes	—
76	#88058	C₂×C₄	1	1	yes	characteristic
77	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
78	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
79	#88049	C₂×C₄	1	2	yes	—
80	#88048	C₂×C₄	1	2	yes	—
81	C₄×#1055	C₂×C₄	1	1	yes	characteristic
82	C₄×#444	—	13	1	no	maximal, Sylow
83	C₂²×#261	—	2	2	no	—
84	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
85	#6131	—	2	4	no	—
86	#6131	C₂²×C₄	1	8	yes	—
87	C₂×D₅₂	—	2	2	no	—
88	C₂×D₅₂	C₂²×C₄	1	2	yes	—
89	C₂×C₄×D₁₃	—	2	2	no	—
90	C₂×C₄×D₁₃	—	2	2	no	—
91	C₂×C₄×D₁₃	—	2	2	no	—
92	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
93	C₂×C₄×D₁₃	#41	1	1	yes	characteristic
94	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
95	C₂×#1057	—	2	4	no	—
96	C₂×#1057	C₂×D₄	1	4	yes	—
97	C₄×#261	—	2	2	no	—
98	C₄×#261	—	4	2	no	—
99	C₄×#261	—	4	2	no	—
100	C₄×#261	C₂²×C₄	1	4	yes	—
101	#6118	—	2	4	no	—
102	#6118	—	2	16	no	—
103	#6118	C₂²×C₄	1	8	yes	—
104	D₁₃⋊C₈	—	2	16	no	—
105	D₁₃⋊C₈	C₂²×C₄	1	4	yes	—
106	D₁₃⋊C₈	#41	1	4	yes	—
107	C₁₃×Q₁₆	C₄²	1	4	yes	complemented
108	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
112	C₄²×C₁₃	—	2	2	no	—
113	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
114	#6106	C₄²	1	8	yes	complemented
115	#6103	—	2	2	no	—
116	#6103	—	2	2	no	—
117	#6103	—	2	2	no	—
118	C₄⋊Dic₁₃	—	2	2	no	—
119	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
120	C₄×Dic₁₃	—	2	2	no	—
121	C₄×Dic₁₃	—	2	2	no	—
122	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
125	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
126	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
127	D₁₀₄	C₄²	1	4	yes	complemented
128	C₈×D₁₃	—	2	2	no	—
129	C₈×D₁₃	C₂²×C₄	1	4	yes	—
130	C₂×#444	—	13	2	no	—
131	C₄×#135	—	13	1	no	—
132	C₄×#131	—	13	2	no	—
133	C₄×#108	—	13	1	no	—
134	#1773	—	13	2	no	—
135	#1765	—	13	4	no	—
136	C₄×#104	—	13	2	no	—
137	C₄×#97	—	13	1	no	—
138	C₂²×D₁₃	—	2	2	no	—
139	C₂²×D₁₃	C₂×#41	1	1	yes	characteristic
140	C₂×#261	—	2	4	no	—
141	C₂×#261	—	4	2	no	—
142	C₂×#261	—	4	2	no	—
143	Q₈×C₁₃	—	2	2	no	—
144	Q₈×C₁₃	C₂×C₄²	1	4	yes	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×Dic₁₃	—	2	2	no	—
151	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
154	D₅₂	—	2	8	no	—
155	D₅₂	—	2	2	no	—
156	D₅₂	C₂×C₄²	1	4	yes	—
157	C₄×D₁₃	—	2	2	no	—
158	C₄×D₁₃	—	2	8	no	—
159	C₄×D₁₃	—	2	2	no	—
160	C₄×D₁₃	—	4	4	no	—
161	C₄×D₁₃	—	4	4	no	—
162	C₄×D₁₃	C₂³×C₄	1	2	yes	—
163	C₄×D₁₃	C₂×#41	1	2	yes	—
164	#1057	—	2	8	no	—
165	#1057	C₄×D₄	1	8	yes	—
166	C₈×C₁₃	—	2	1	no	—
167	C₈×C₁₃	C₂×C₄²	1	2	yes	—
168	#1055	—	2	1	no	—
169	#1055	C₂×C₄²	1	2	yes	—
170	C₂×#135	—	13	1	no	—
171	C₂×#131	—	13	4	no	—
172	C₄×#41	—	13	2	no	—
173	#444	—	13	16	no	—
174	C₄×Q₁₆	—	13	1	no	—
175	C₄×Q₁₆	#261	1	1	yes	characteristic, complemented
176	C₄×SD₁₆	—	13	2	no	—
177	C₄×SD₁₆	—	13	2	no	—
178	C₄×D₈	—	13	1	no	—
179	C₄×D₈	—	13	1	no	—
180	C₂×#108	—	13	2	no	—
181	#427	—	13	1	no	—
182	C₂×#104	—	13	4	no	—
183	#416	—	13	2	no	—
184	#405	—	13	4	no	—
185	#405	—	26	4	no	—
186	C₄×M₁₆	—	13	2	no	—
187	C₄×M₁₆	—	26	2	no	—
188	C₂×#97	—	13	2	no	—
189	C₂×#97	—	13	2	no	—
190	C₂×C₄×C₈	—	13	1	no	—
191	C₂×C₄×C₈	—	13	1	no	—
192	C₂×C₄×C₈	—	26	2	no	—
193	C₄³	—	13	1	no	—
194	#341	—	13	2	no	—
195	#339	—	13	4	no	—
196	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
197	C₂×D₁₃	—	2	1	no	—
198	C₂×D₁₃	—	2	1	no	—
199	C₂×D₁₃	—	2	4	no	—
200	C₂×D₁₃	—	4	2	no	—
201	C₂×D₁₃	—	4	2	no	—
202	C₂×D₁₃	C₄×#41	1	2	yes	—
203	#261	—	4	4	no	—
204	C₄×C₁₃	—	2	1	no	—
205	C₄×C₁₃	—	2	4	no	—
206	C₄×C₁₃	—	4	2	no	—
207	C₄×C₁₃	#444	1	2	yes	complemented
208	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
209	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
210	Dic₁₃	—	2	1	no	—
211	Dic₁₃	—	4	2	no	—
212	Dic₁₃	#444	1	2	yes	complemented
213	Dic₁₃	C₂×C₄×D₄	1	2	yes	—
214	C₂×#41	—	13	2	no	—
215	#135	—	13	8	no	—
216	C₂×Q₁₆	C₂×#261	1	1	yes	characteristic
217	C₂×SD₁₆	—	13	2	no	—
218	C₂×D₈	—	13	1	no	—
219	#131	—	13	16	no	—
220	#131	—	26	8	no	—
221	C₂×M₁₆	—	13	4	no	—
222	C₂×M₁₆	—	26	4	no	—
223	C₂²×C₈	—	13	2	no	—
224	C₂²×C₈	—	13	1	no	—
225	C₂²×C₈	—	13	1	no	—
226	C₂²×C₈	—	26	4	no	—
227	C₄×Q₈	—	13	2	no	—
228	C₄×Q₈	C₂×#261	1	2	yes	—
229	C₄×D₄	—	13	2	no	—
230	C₄×D₄	—	13	2	no	—
231	C₄×D₄	—	26	2	no	—
232	C₄×D₄	—	26	2	no	—
233	#117	—	13	2	no	—
234	#117	—	26	2	no	—
235	C₂×C₄²	—	13	1	no	—
236	C₂×C₄²	—	13	2	no	—
237	C₂×C₄²	—	26	2	no	—
238	C₂×C₄²	—	26	1	no	—
239	C₂×C₄²	—	26	1	no	—
240	#108	—	13	8	no	—
241	#107	—	13	1	no	—
242	#107	C₂×#261	1	1	yes	characteristic
243	#106	—	13	2	no	—
244	#104	—	13	16	no	—
245	#103	—	13	2	no	—
246	#103	C₂×#261	1	2	yes	—
247	#102	—	13	2	no	—
248	#102	—	13	2	no	—
249	#97	—	13	8	no	—
250	#97	—	26	2	no	—
251	#97	—	26	4	no	—
252	#97	—	26	4	no	—
253	C₄×C₈	—	13	2	no	—
254	C₄×C₈	—	13	2	no	—
255	C₄×C₈	—	13	1	no	—
256	C₄×C₈	—	13	1	no	—
257	C₄×C₈	—	13	1	no	—
258	C₄×C₈	—	26	2	no	—
259	C₄×C₈	—	26	4	no	—
260	C₄×C₈	C₂×#261	1	1	yes	characteristic
261	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
262	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
263	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
264	D₁₃	—	2	2	no	—
265	D₁₃	—	4	4	no	—
266	#41	—	13	8	no	—
267	#41	—	26	4	no	—
268	C₂×Q₈	C₂²×#261	1	2	yes	—
269	C₂×D₄	—	13	2	no	—
270	C₂×D₄	—	26	2	no	—
271	C₂²×C₄	—	13	1	no	—
272	C₂²×C₄	—	13	1	no	—
273	C₂²×C₄	—	13	1	no	—
274	C₂²×C₄	—	26	2	no	—
275	C₂²×C₄	—	26	2	no	—
276	C₂²×C₄	—	26	2	no	—
277	C₂²×C₄	—	26	2	no	—
278	Q₁₆	C₄×#261	1	4	yes	complemented
279	SD₁₆	—	13	8	no	—
280	D₈	—	13	4	no	—
281	M₁₆	—	13	8	no	—
282	M₁₆	—	26	16	no	—
283	M₁₆	—	26	4	no	—
284	C₂×C₈	—	13	4	no	—
285	C₂×C₈	—	13	4	no	—
286	C₂×C₈	—	13	4	no	—
287	C₂×C₈	—	13	1	no	—
288	C₂×C₈	—	13	4	no	—
289	C₂×C₈	—	13	1	no	—
290	C₂×C₈	—	26	2	no	—
291	C₂×C₈	—	26	16	no	—
292	C₂×C₈	—	26	4	no	—
293	C₂×C₈	C₂²×#261	1	1	yes	characteristic
294	C₂×C₈	C₂²×#261	1	1	yes	characteristic
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	13	2	no	—
297	C₄⋊C₄	—	26	2	no	—
298	C₄⋊C₄	C₂²×#261	1	2	yes	—
299	C₂²⋊C₄	—	26	2	no	—
300	C₂²⋊C₄	—	26	2	no	—
301	C₂²⋊C₄	—	26	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	13	1	no	—
304	C₄²	—	13	1	no	—
305	C₄²	—	13	1	no	—
306	C₄²	—	13	4	no	—
307	C₄²	—	26	2	no	—
308	C₄²	—	26	2	no	—
309	C₄²	—	26	2	no	—
310	C₄²	—	52	2	no	—
311	C₄²	—	52	2	no	—
312	C₄²	C₂²×#261	1	1	yes	characteristic
313	C₁₃	C₄×#444	1	1	yes	characteristic, complemented, Sylow
314	C₂³	—	13	1	no	—
315	C₂³	—	26	2	no	—
316	Q₈	—	2	2	no	—
317	Q₈	C₂×C₄×#261	1	4	yes	—
318	D₄	—	13	4	no	—
319	D₄	—	26	8	no	—
320	D₄	—	26	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	13	1	no	—
324	C₂×C₄	—	13	1	no	—
325	C₂×C₄	—	13	2	no	—
326	C₂×C₄	—	13	2	no	—
327	C₂×C₄	—	13	1	no	—
328	C₂×C₄	—	26	2	no	—
329	C₂×C₄	—	26	8	no	—
330	C₂×C₄	—	26	2	no	—
331	C₂×C₄	—	26	4	no	—
332	C₂×C₄	—	26	2	no	—
333	C₂×C₄	—	52	4	no	—
334	C₂×C₄	—	52	2	no	—
335	C₂×C₄	—	52	2	no	—
336	C₂×C₄	—	52	4	no	—
337	C₂×C₄	D₄×#261	1	1	yes	characteristic
338	C₂×C₄	D₄×#261	1	1	yes	characteristic
339	C₂×C₄	C₂³×#261	1	1	yes	characteristic
340	C₈	—	2	1	no	—
341	C₈	—	13	8	no	—
342	C₈	—	13	2	no	—
343	C₈	—	26	8	no	—
344	C₈	—	26	1	no	—
345	C₈	C₂×C₄×#261	1	2	yes	—
346	C₂²	—	13	2	no	—
347	C₂²	—	26	1	no	—
348	C₂²	—	26	1	no	—
349	C₂²	—	26	4	no	—
350	C₂²	—	52	2	no	—
351	C₂²	—	52	2	no	—
352	C₂²	C₂×D₄×#261	1	1	yes	characteristic
353	C₄	—	2	4	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	2	no	—
356	C₄	—	13	2	no	—
357	C₄	—	13	2	no	—
358	C₄	—	26	1	no	—
359	C₄	—	52	4	no	—
360	C₄	—	52	2	no	—
361	C₄	#11719112	1	2	yes	direct product factor
362	C₄	C₂²×C₄×#261	1	1	yes	characteristic
363	C₄	#41×#261	1	1	yes	characteristic
364	C₂	—	26	2	no	—
365	C₂	—	52	4	no	—
366	C₂	C₄×D₄×#261	1	1	yes	characteristic
367	C₂	C₂×#11719112	1	1	yes	characteristic
368	C₂	D₁₃⋊#444	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×#444
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×#88103	C₂	semidirect
4. #13130643	C₂	semidirect
5. #13130617	C₂	semidirect
6. C₄×#88118	C₂	semidirect
7. C₄×#88100	C₂	semidirect
13. #11719112	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
68. #88103	C₂×C₄	semidirect
70. #88100	C₂×C₄	semidirect
107. C₁₃×Q₁₆	C₄²	semidirect
114. #6106	C₄²	semidirect
127. D₁₀₄	C₄²	semidirect
175. C₄×Q₁₆	#261	semidirect
207. C₄×C₁₃	#444	semidirect
212. Dic₁₃	#444	semidirect
278. Q₁₆	C₄×#261	semidirect
313. C₁₃	C₄×#444	semidirect
361. C₄	#11719112	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×#11719112	2
C₄×#88240	2
C₄×#88103	2
#13130643	2
#13130617	2
C₄×#88118	2
C₄×#88100	2
C₄×#88173	2
C₄×#444	13

Subgroups — C4×#11719112 #16797432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11719112 #16797432