Subgroups — C₄×#11719110 #16797410

Order 3328.

1086 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719110	C₂	1	2	yes	maximal
2	C₄×Q₈×#261	C₂	1	2	yes	maximal
3	#13130651	C₂	1	2	yes	maximal
4	C₄×#88101	C₂	1	1	yes	maximal, characteristic
5	#13130609	C₂	1	4	yes	maximal
6	C₄×#88117	C₂	1	2	yes	maximal
7	C₄×#88100	C₂	1	1	yes	maximal, characteristic
8	C₄×Q₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×#261	C₂²	1	4	yes	—
10	C₂×Q₁₆×D₁₃	C₂²	1	1	yes	characteristic
11	C₄×Q₈×D₁₃	C₂²	1	2	yes	—
12	C₂×#88117	C₂²	1	4	yes	—
13	#11719110	C₄	1	16	yes	direct product factor
14	C₂×#88101	C₂²	1	2	yes	—
15	C₂×#88100	C₂²	1	2	yes	—
16	C₂×#88100	C₂²	1	2	yes	—
17	C₄⋊C₄×#261	—	2	4	no	—
18	C₄⋊C₄×#261	C₂²	1	4	yes	—
19	C₄×C₄⋊#261	—	2	2	no	—
20	C₄×C₄⋊#261	C₂²	1	2	yes	—
21	C₄²×#261	—	2	2	no	—
22	C₄²×#261	C₂²	1	1	yes	characteristic
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×#6107	C₄	1	2	yes	complemented
27	D₁₃×#107	C₂²	1	1	yes	characteristic
28	#11718534	C₄	1	1	yes	characteristic, complemented
29	D₁₃×#103	C₂²	1	2	yes	—
30	#11718451	C₄	1	2	yes	complemented
31	C₄×Dic₅₂	C₄	1	1	yes	characteristic, complemented
32	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
33	#11718265	C₂²	1	2	yes	—
34	#11718246	C₂²	1	4	yes	—
35	C₂×Q₈×D₁₃	C₂³	1	2	yes	—
36	Q₈×#261	—	2	8	no	—
37	Q₈×#261	C₂×C₄	1	16	yes	—
38	C₂×C₄⋊#261	—	2	4	no	—
39	C₂×C₄⋊#261	C₂³	1	4	yes	—
40	C₂×C₄×#261	—	2	4	no	—
41	C₂×C₄×#261	—	2	4	no	—
42	C₂×C₄×#261	—	2	2	no	—
43	C₂×C₄×#261	D₄	1	2	yes	—
44	C₂×C₄×#261	D₄	1	2	yes	—
45	C₂×C₄×#261	C₂³	1	2	yes	—
46	C₂×D₁₃⋊C₈	C₂³	1	2	yes	—
47	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
48	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
49	Q₈×Dic₁₃	C₂×C₄	1	2	yes	—
50	C₂×#6107	C₂×C₄	1	2	yes	—
51	Q₁₆×D₁₃	C₂×C₄	1	8	yes	—
52	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×D₁₃	—	2	2	no	—
56	C₄⋊C₄×D₁₃	C₂³	1	2	yes	—
57	#88141	—	2	2	no	—
58	#88141	C₂×C₄	1	2	yes	—
59	C₄²×D₁₃	—	2	2	no	—
60	C₄²×D₁₃	C₂³	1	1	yes	characteristic
61	C₄×Dic₂₆	—	2	2	no	—
62	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
63	#88117	C₂×C₄	1	16	yes	—
64	(C₂×#261)⋊C₄	—	2	4	no	—
65	(C₂×#261)⋊C₄	—	2	2	no	—
66	C₄×#1057	—	2	1	no	—
67	C₄×#1057	—	2	1	no	—
68	#88101	C₂×C₄	1	8	yes	—
69	#88100	—	2	4	no	—
70	#88100	C₂×C₄	1	8	yes	—
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	#88058	C₂×C₄	1	1	yes	characteristic
76	#88056	C₂×C₄	1	2	yes	—
77	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
78	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
79	#88050	C₂×C₄	1	2	yes	—
80	#88047	C₂×C₄	1	2	yes	—
81	C₄×#1055	C₂×C₄	1	1	yes	characteristic
82	C₄×#441	—	13	1	no	maximal, Sylow
83	C₂²×#261	—	2	4	no	—
84	C₂²×#261	C₂×D₄	1	4	yes	—
85	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
86	Q₈×D₁₃	—	2	4	no	—
87	Q₈×D₁₃	C₂²×C₄	1	8	yes	—
88	C₂×C₄×D₁₃	—	2	2	no	—
89	C₂×C₄×D₁₃	—	2	2	no	—
90	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
93	C₂×Dic₂₆	—	2	2	no	—
94	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
95	C₂×#1057	—	2	2	no	—
96	C₄⋊#261	—	2	16	no	—
97	C₄⋊#261	—	2	4	no	—
98	C₄⋊#261	C₂²×C₄	1	8	yes	—
99	C₄×#261	—	2	16	no	—
100	C₄×#261	—	2	4	no	—
101	C₄×#261	—	4	2	no	—
102	C₄×#261	—	4	8	no	—
103	C₄×#261	—	4	4	no	—
104	C₄×#261	C₂²×C₄	1	4	yes	—
105	C₄×#261	#41	1	4	yes	—
106	D₁₃⋊C₈	—	2	2	no	—
107	D₁₃⋊C₈	C₂²×C₄	1	4	yes	—
108	C₁₃×Q₁₆	C₄²	1	4	yes	complemented
109	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
113	C₄²×C₁₃	—	2	2	no	—
114	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
115	#6107	C₄²	1	8	yes	complemented
116	C₄⋊Dic₁₃	—	2	2	no	—
117	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
118	#6101	—	2	2	no	—
119	#6101	—	2	2	no	—
120	#6101	—	2	2	no	—
121	C₄×Dic₁₃	—	2	2	no	—
122	C₄×Dic₁₃	—	2	2	no	—
123	C₄×Dic₁₃	—	2	2	no	—
124	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
125	C₄×Dic₁₃	#41	1	1	yes	characteristic
126	C₄×Dic₁₃	#41	1	1	yes	characteristic
127	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
128	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
129	Dic₅₂	C₄²	1	4	yes	complemented
130	C₈×D₁₃	—	2	2	no	—
131	C₈×D₁₃	C₂²×C₄	1	4	yes	—
132	C₂×#441	—	13	2	no	—
133	C₂×C₄×Q₁₆	—	13	1	no	—
134	C₄²×Q₈	—	13	2	no	—
135	#1776	—	13	2	no	—
136	C₄×#106	—	13	1	no	—
137	#1763	—	13	4	no	—
138	C₄×#103	—	13	2	no	—
139	C₄×#97	—	13	1	no	—
140	C₂²×D₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×#261	—	2	4	no	—
142	C₂×#261	—	2	4	no	—
143	C₂×#261	—	2	8	no	—
144	C₂×#261	—	4	4	no	—
145	C₂×#261	—	4	4	no	—
146	C₂×#261	C₄×D₄	1	8	yes	—
147	Q₈×C₁₃	—	2	2	no	—
148	Q₈×C₁₃	C₂×C₄²	1	4	yes	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×Dic₁₃	—	2	2	no	—
155	C₂×Dic₁₃	—	2	2	no	—
156	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
157	C₂×Dic₁₃	C₄×D₄	1	1	yes	characteristic
158	C₂×Dic₁₃	C₂×#41	1	1	yes	characteristic
159	C₄×D₁₃	—	2	8	no	—
160	C₄×D₁₃	—	2	2	no	—
161	C₄×D₁₃	—	4	4	no	—
162	C₄×D₁₃	#137	1	4	yes	—
163	C₄×D₁₃	C₂³×C₄	1	2	yes	—
164	C₄×D₁₃	C₂×#41	1	2	yes	—
165	Dic₂₆	—	2	2	no	—
166	Dic₂₆	—	2	8	no	—
167	Dic₂₆	C₂×C₄²	1	4	yes	—
168	#1057	—	2	4	no	—
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	C₂²×Q₁₆	—	13	1	no	—
174	C₂×C₄×Q₈	—	13	4	no	—
175	C₂×C₄×Q₈	—	13	2	no	—
176	#441	—	13	16	no	—
177	C₄×Q₁₆	—	13	1	no	—
178	C₄×Q₁₆	—	13	2	no	—
179	C₄×Q₁₆	—	13	1	no	—
180	C₄×Q₁₆	—	13	2	no	—
181	C₄×Q₁₆	—	13	1	no	—
182	C₄×Q₁₆	#261	1	1	yes	characteristic, complemented
183	C₂×#107	—	13	1	no	—
184	C₂×#106	—	13	2	no	—
185	C₂×#103	—	13	4	no	—
186	C₂×#103	—	13	2	no	—
187	C₂×#97	—	13	2	no	—
188	C₂×#97	—	13	2	no	—
189	C₂×C₄×C₈	—	13	1	no	—
190	C₂×C₄×C₈	—	13	1	no	—
191	C₄×C₄⋊C₄	—	13	2	no	—
192	C₄×C₄⋊C₄	—	13	4	no	—
193	C₄×C₄⋊C₄	—	26	2	no	—
194	C₄×C₄⋊C₄	—	26	4	no	—
195	C₄³	—	13	1	no	—
196	C₄³	—	26	2	no	—
197	#340	—	13	4	no	—
198	#340	—	13	2	no	—
199	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂×D₁₃	C₂×C₄×D₄	1	2	yes	—
201	C₂×D₁₃	C₂×#137	1	2	yes	—
202	C₂×D₁₃	#575	1	2	yes	—
203	#261	—	2	8	no	—
204	#261	—	4	8	no	—
205	C₄×C₁₃	—	2	1	no	—
206	C₄×C₁₃	—	2	4	no	—
207	C₄×C₁₃	—	4	2	no	—
208	C₄×C₁₃	#441	1	2	yes	complemented
209	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
210	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
211	Dic₁₃	—	2	1	no	—
212	Dic₁₃	—	2	4	no	—
213	Dic₁₃	—	4	2	no	—
214	Dic₁₃	#441	1	2	yes	complemented
215	Dic₁₃	C₄×#41	1	2	yes	—
216	C₂²×Q₈	—	13	2	no	—
217	C₂×Q₁₆	—	13	2	no	—
218	C₂×Q₁₆	—	13	8	no	—
219	C₂×Q₁₆	—	13	1	no	—
220	C₂×Q₁₆	C₂×#261	1	1	yes	characteristic
221	C₂²×C₈	—	13	1	no	—
222	C₂²×C₈	—	13	2	no	—
223	C₂²×C₈	—	13	1	no	—
224	C₄×Q₈	—	13	16	no	—
225	C₄×Q₈	—	13	2	no	—
226	C₄×Q₈	—	13	2	no	—
227	C₄×Q₈	—	13	2	no	—
228	C₄×Q₈	—	26	2	no	—
229	C₄×Q₈	—	26	2	no	—
230	C₄×Q₈	—	26	8	no	—
231	C₄×Q₈	C₂×#261	1	2	yes	—
232	C₂×C₄⋊C₄	—	13	4	no	—
233	C₂×C₄⋊C₄	—	13	2	no	—
234	C₂×C₄⋊C₄	—	26	2	no	—
235	C₂×C₄⋊C₄	—	26	4	no	—
236	C₂×C₄²	—	13	2	no	—
237	C₂×C₄²	—	13	1	no	—
238	C₂×C₄²	—	13	2	no	—
239	C₂×C₄²	—	13	2	no	—
240	C₂×C₄²	—	26	2	no	—
241	C₂×C₄²	—	26	4	no	—
242	C₂×C₄²	—	26	2	no	—
243	C₂×C₄²	—	26	4	no	—
244	#107	—	13	1	no	—
245	#107	—	13	2	no	—
246	#107	C₂×#261	1	1	yes	characteristic
247	#106	—	13	8	no	—
248	#103	—	13	16	no	—
249	#103	—	13	2	no	—
250	#103	—	13	2	no	—
251	#103	—	13	2	no	—
252	#103	C₂×#261	1	2	yes	—
253	#97	—	13	8	no	—
254	#97	—	26	4	no	—
255	C₄×C₈	—	13	1	no	—
256	C₄×C₈	—	13	1	no	—
257	C₄×C₈	—	13	1	no	—
258	C₄×C₈	—	26	1	no	—
259	C₄×C₈	—	26	1	no	—
260	C₄×C₈	C₂×#261	1	1	yes	characteristic
261	#95	—	26	4	no	—
262	#95	—	26	2	no	—
263	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
264	C₂×C₁₃	C₂×#441	1	1	yes	characteristic
265	C₂×C₁₃	#2943	1	1	yes	characteristic
266	D₁₃	C₄×#137	1	4	yes	—
267	C₂×Q₈	—	13	8	no	—
268	C₂×Q₈	—	13	2	no	—
269	C₂×Q₈	—	26	2	no	—
270	C₂×Q₈	—	26	4	no	—
271	C₂×Q₈	C₂²×#261	1	2	yes	—
272	C₂²×C₄	—	13	1	no	—
273	C₂²×C₄	—	13	1	no	—
274	C₂²×C₄	—	13	4	no	—
275	C₂²×C₄	—	13	1	no	—
276	C₂²×C₄	—	26	2	no	—
277	C₂²×C₄	—	26	2	no	—
278	C₂²×C₄	—	26	4	no	—
279	Q₁₆	—	13	8	no	—
280	Q₁₆	—	13	4	no	—
281	Q₁₆	C₄×#261	1	4	yes	complemented
282	C₂×C₈	—	13	1	no	—
283	C₂×C₈	—	13	4	no	—
284	C₂×C₈	—	13	4	no	—
285	C₂×C₈	—	13	1	no	—
286	C₂×C₈	—	26	2	no	—
287	C₂×C₈	—	26	2	no	—
288	C₂×C₈	—	26	2	no	—
289	C₂×C₈	C₂²×#261	1	1	yes	characteristic
290	C₂×C₈	C₂²×#261	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	13	8	no	—
293	C₄⋊C₄	—	13	2	no	—
294	C₄⋊C₄	—	26	2	no	—
295	C₄⋊C₄	—	26	2	no	—
296	C₄⋊C₄	—	26	2	no	—
297	C₄⋊C₄	—	26	4	no	—
298	C₄⋊C₄	—	26	16	no	—
299	C₄⋊C₄	—	26	2	no	—
300	C₄⋊C₄	C₂²×#261	1	2	yes	—
301	C₄²	—	2	2	no	—
302	C₄²	—	13	4	no	—
303	C₄²	—	13	4	no	—
304	C₄²	—	13	1	no	—
305	C₄²	—	13	1	no	—
306	C₄²	—	13	1	no	—
307	C₄²	—	26	2	no	—
308	C₄²	—	26	2	no	—
309	C₄²	—	26	4	no	—
310	C₄²	—	26	16	no	—
311	C₄²	—	26	2	no	—
312	C₄²	—	52	4	no	—
313	C₄²	—	52	2	no	—
314	C₄²	—	52	8	no	—
315	C₄²	C₂²×#261	1	1	yes	characteristic
316	C₁₃	C₄×#441	1	1	yes	characteristic, complemented, Sylow
317	C₂³	—	13	1	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	13	4	no	—
320	Q₈	—	26	8	no	—
321	Q₈	—	26	2	no	—
322	Q₈	C₂×C₄×#261	1	4	yes	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	13	1	no	—
326	C₂×C₄	—	13	1	no	—
327	C₂×C₄	—	13	2	no	—
328	C₂×C₄	—	13	4	no	—
329	C₂×C₄	—	13	2	no	—
330	C₂×C₄	—	13	8	no	—
331	C₂×C₄	—	13	1	no	—
332	C₂×C₄	—	26	8	no	—
333	C₂×C₄	—	26	2	no	—
334	C₂×C₄	—	26	4	no	—
335	C₂×C₄	—	26	4	no	—
336	C₂×C₄	—	26	2	no	—
337	C₂×C₄	—	26	8	no	—
338	C₂×C₄	—	26	2	no	—
339	C₂×C₄	—	52	4	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	2	no	—
347	C₈	—	26	4	no	—
348	C₈	—	26	1	no	—
349	C₈	C₂×C₄×#261	1	2	yes	—
350	C₂²	—	13	2	no	—
351	C₂²	—	13	2	no	—
352	C₂²	—	13	2	no	—
353	C₂²	C₂×D₄×#261	1	1	yes	characteristic
354	C₄	—	2	4	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	2	no	—
357	C₄	—	13	2	no	—
358	C₄	—	13	2	no	—
359	C₄	—	26	8	no	—
360	C₄	—	26	4	no	—
361	C₄	—	26	1	no	—
362	C₄	—	52	8	no	—
363	C₄	—	52	2	no	—
364	C₄	#11719110	1	2	yes	direct product factor
365	C₄	C₂²×C₄×#261	1	1	yes	characteristic
366	C₄	#41×#261	1	1	yes	characteristic
367	C₂	—	13	4	no	—
368	C₂	C₄×D₄×#261	1	1	yes	characteristic
369	C₂	C₂×#11719110	1	1	yes	characteristic
370	C₂	#13131590	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	C₂²×D₁₃	C₂²×D₄
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₄×Q₁₆	#261
hypocenter	C₁₃	C₄×#441
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
13. #11719110	C₄	direct (N × H)
24. C₄×C₁₃×Q₁₆	C₄	semidirect
25. Q₁₆×Dic₁₃	C₄	semidirect
26. C₄×#6107	C₄	semidirect
28. #11718534	C₄	semidirect
30. #11718451	C₄	semidirect
31. C₄×Dic₅₂	C₄	semidirect
108. C₁₃×Q₁₆	C₄²	semidirect
115. #6107	C₄²	semidirect
129. Dic₅₂	C₄²	semidirect
182. C₄×Q₁₆	#261	semidirect
208. C₄×C₁₃	#441	semidirect
214. Dic₁₃	#441	semidirect
281. Q₁₆	C₄×#261	semidirect
316. C₁₃	C₄×#441	semidirect
364. C₄	#11719110	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×#11719110	2
C₄×Q₈×#261	2
#13130651	2
C₄×#88101	2
#13130609	2
C₄×#88117	2
C₄×#88100	2
C₄×Q₁₆×D₁₃	2
C₄×#441	13

Subgroups — C4×#11719110 #16797410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11719110 #16797410