Subgroups — C₄×#88172 #13123656

Order 1664.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88172	C₂	1	1	yes	maximal, characteristic
2	C₄×#6131	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718845	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#6107	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#6106	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718550	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718542	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718478	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718464	C₂	1	1	yes	maximal, characteristic
12	#11718463	C₂	1	1	yes	maximal, characteristic, complemented
13	#11718450	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#6095	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#6094	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#6131	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
22	Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
23	C₂×#6107	C₂²	1	1	yes	characteristic
24	C₂×#6106	C₂²	1	1	yes	characteristic
25	#88172	C₄	1	8	yes	direct product factor
26	C₂×#6095	C₂²	1	1	yes	characteristic
27	C₂×#6094	C₂²	1	1	yes	characteristic
28	C₂×#6094	C₂²	1	1	yes	characteristic
29	#88147	—	2	1	no	—
30	#88147	C₂²	1	1	yes	characteristic
31	#88146	—	2	1	no	—
32	#88146	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₃	—	2	1	no	—
34	C₄⋊C₄×D₁₃	C₂²	1	1	yes	characteristic
35	#88141	—	2	1	no	—
36	#88141	C₂²	1	1	yes	characteristic
37	C₄×D₅₂	—	2	1	no	—
38	C₄×D₅₂	C₂²	1	1	yes	characteristic
39	C₄²×D₁₃	—	2	1	no	—
40	C₄²×D₁₃	—	2	1	no	—
41	C₄²×D₁₃	C₂²	1	1	yes	characteristic
42	C₄×Dic₂₆	—	2	1	no	—
43	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
44	C₁₃×#107	C₂²	1	1	yes	characteristic
45	C₁₃×#103	C₂²	1	1	yes	characteristic
46	C₁₃×#103	C₂²	1	1	yes	characteristic
47	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
48	#88075	C₂²	1	1	yes	characteristic
49	#88075	C₂²	1	1	yes	characteristic
50	#88061	C₂²	1	1	yes	characteristic
51	#88057	C₂²	1	1	yes	characteristic
52	#88056	C₂²	1	1	yes	characteristic
53	#88055	C₂²	1	1	yes	characteristic
54	#88055	C₂²	1	1	yes	characteristic
55	#88050	C₂²	1	1	yes	characteristic
56	#88049	C₂²	1	1	yes	characteristic
57	#88048	C₂²	1	1	yes	characteristic
58	#88047	C₂²	1	1	yes	characteristic
59	C₄×#1055	C₂²	1	1	yes	characteristic
60	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
62	#6131	—	2	2	no	—
63	#6131	C₂×C₄	1	4	yes	—
64	Q₈×D₁₃	—	2	2	no	—
65	Q₈×D₁₃	C₂×C₄	1	4	yes	complemented
66	C₂×D₅₂	—	2	1	no	—
67	C₂×D₅₂	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₃	—	2	1	no	—
69	C₂×C₄×D₁₃	—	2	1	no	—
70	C₂×C₄×D₁₃	—	2	1	no	—
71	C₂×C₄×D₁₃	—	2	1	no	—
72	C₂×C₄×D₁₃	—	2	1	no	—
73	C₂×C₄×D₁₃	D₄	1	1	yes	characteristic
74	C₂×C₄×D₁₃	D₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
76	C₂×Dic₂₆	—	2	1	no	—
77	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
78	C₁₃×Q₁₆	C₂×C₄	1	4	yes	complemented
79	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
81	C₁₃×C₄⋊C₄	—	2	1	no	—
82	C₁₃×C₄⋊C₄	—	2	1	no	—
83	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
85	C₄²×C₁₃	—	2	1	no	—
86	C₄²×C₁₃	—	2	1	no	—
87	C₄²×C₁₃	C₂³	1	1	yes	characteristic
88	#6107	C₂×C₄	1	4	yes	complemented
89	#6106	C₂×C₄	1	4	yes	complemented
90	#6103	—	2	1	no	—
91	#6103	—	2	1	no	—
92	#6103	—	2	1	no	—
93	C₄⋊Dic₁₃	—	2	1	no	—
94	C₄⋊Dic₁₃	—	2	1	no	—
95	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
97	#6101	—	2	1	no	—
98	#6101	—	2	1	no	—
99	#6101	—	2	1	no	—
100	C₄×Dic₁₃	—	2	1	no	—
101	C₄×Dic₁₃	—	2	1	no	—
102	C₄×Dic₁₃	—	2	1	no	—
103	C₄×Dic₁₃	—	2	1	no	—
104	C₄×Dic₁₃	—	2	1	no	—
105	C₄×Dic₁₃	D₄	1	1	yes	characteristic
106	C₄×Dic₁₃	D₄	1	1	yes	characteristic
107	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
108	C₂×#1055	C₂³	1	1	yes	characteristic
109	C₂×#1055	C₂³	1	1	yes	characteristic
110	#6095	C₂×C₄	1	4	yes	complemented
111	#6094	—	2	2	no	—
112	#6094	C₂×C₄	1	4	yes	complemented
113	C₄×#137	—	13	1	no	maximal, Sylow
114	C₂²×D₁₃	—	2	1	no	—
115	C₂²×D₁₃	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₁₃	—	2	1	no	—
117	Q₈×C₁₃	—	2	1	no	—
118	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
119	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
127	C₂×Dic₁₃	—	2	1	no	—
128	C₂×Dic₁₃	—	2	1	no	—
129	C₂×Dic₁₃	—	2	1	no	—
130	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₃	C₂×D₄	1	1	yes	characteristic
133	D₅₂	—	2	4	no	—
134	D₅₂	—	2	1	no	—
135	D₅₂	C₂²×C₄	1	2	yes	—
136	C₄×D₁₃	—	2	4	no	—
137	C₄×D₁₃	—	2	2	no	—
138	C₄×D₁₃	—	2	4	no	—
139	C₄×D₁₃	—	4	1	no	—
140	C₄×D₁₃	—	4	2	no	—
141	C₄×D₁₃	—	4	2	no	—
142	C₄×D₁₃	—	4	2	no	—
143	C₄×D₁₃	C₂²×C₄	1	2	yes	—
144	C₄×D₁₃	#41	1	2	yes	—
145	Dic₂₆	—	2	1	no	—
146	Dic₂₆	—	2	4	no	—
147	Dic₂₆	C₂²×C₄	1	2	yes	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	C₂²×C₄	1	2	yes	—
150	#1055	—	2	1	no	—
151	#1055	C₂²×C₄	1	2	yes	—
152	C₂×#137	—	13	1	no	—
153	C₄×#41	—	13	1	no	—
154	C₂×C₄×Q₈	—	13	1	no	—
155	#441	—	13	1	no	—
156	#441	—	13	1	no	—
157	#440	—	13	1	no	—
158	#440	—	13	1	no	—
159	C₄×Q₁₆	—	13	1	no	—
160	C₄×Q₁₆	D₁₃	1	1	yes	characteristic, complemented
161	C₄×SD₁₆	—	13	1	no	—
162	C₄×SD₁₆	—	13	1	no	—
163	#428	—	13	1	no	—
164	#419	—	13	1	no	—
165	#417	—	13	1	no	—
166	C₄×M₁₆	—	13	1	no	—
167	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×D₁₃	—	2	1	no	—
169	C₂×D₁₃	—	2	1	no	—
170	C₂×D₁₃	—	2	2	no	—
171	C₂×D₁₃	—	4	1	no	—
172	C₂×D₁₃	—	4	1	no	—
173	C₂×D₁₃	C₄×D₄	1	2	yes	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	2	2	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	4	1	no	—
178	C₄×C₁₃	—	4	1	no	—
179	C₄×C₁₃	#137	1	2	yes	complemented
180	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
181	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
182	Dic₁₃	—	2	2	no	—
183	Dic₁₃	—	2	1	no	—
184	Dic₁₃	—	2	1	no	—
185	Dic₁₃	—	4	1	no	—
186	Dic₁₃	—	4	1	no	—
187	Dic₁₃	C₄×D₄	1	2	yes	—
188	C₂×#41	—	13	1	no	—
189	C₂²×Q₈	—	13	1	no	—
190	#137	—	13	8	no	—
191	C₂×Q₁₆	—	13	1	no	—
192	C₂×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
193	C₂×SD₁₆	—	13	1	no	—
194	C₂×SD₁₆	—	13	1	no	—
195	C₂×M₁₆	—	13	1	no	—
196	C₂×M₁₆	—	13	1	no	—
197	C₄×Q₈	—	13	1	no	—
198	C₄×Q₈	—	13	1	no	—
199	C₄×Q₈	—	13	1	no	—
200	C₄×Q₈	—	13	1	no	—
201	C₄×Q₈	—	26	1	no	—
202	C₄×Q₈	—	26	1	no	—
203	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
204	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
205	C₄×D₄	—	13	1	no	—
206	C₄×D₄	—	13	1	no	—
207	C₄×D₄	—	26	1	no	—
208	C₄×D₄	—	26	1	no	—
209	#117	—	13	1	no	—
210	#117	—	26	1	no	—
211	C₂×C₄⋊C₄	—	13	1	no	—
212	C₂×C₄⋊C₄	—	26	1	no	—
213	C₂×C₄²	—	13	1	no	—
214	C₂×C₄²	—	26	1	no	—
215	C₂×C₄²	—	26	1	no	—
216	#107	—	13	1	no	—
217	#107	C₂×D₁₃	1	1	yes	characteristic
218	#106	—	13	1	no	—
219	#106	—	13	1	no	—
220	#103	—	13	1	no	—
221	#103	—	13	1	no	—
222	#103	—	13	1	no	—
223	#103	—	13	1	no	—
224	#103	C₂×D₁₃	1	1	yes	characteristic
225	#103	C₂×D₁₃	1	1	yes	characteristic
226	#102	—	13	1	no	—
227	#102	—	13	1	no	—
228	#97	—	13	1	no	—
229	#97	—	13	1	no	—
230	C₄×C₈	—	13	1	no	—
231	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
232	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
233	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
234	C₂×C₁₃	#575	1	1	yes	characteristic
235	D₁₃	—	2	2	no	—
236	D₁₃	—	4	2	no	—
237	#41	—	13	4	no	—
238	#41	—	26	2	no	—
239	C₂×Q₈	—	13	4	no	—
240	C₂×Q₈	—	13	1	no	—
241	C₂×Q₈	—	26	1	no	—
242	C₂×Q₈	—	26	2	no	—
243	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
244	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
245	C₂×D₄	—	13	1	no	—
246	C₂×D₄	—	26	1	no	—
247	C₂²×C₄	—	13	1	no	—
248	C₂²×C₄	—	13	1	no	—
249	C₂²×C₄	—	13	1	no	—
250	C₂²×C₄	—	26	1	no	—
251	C₂²×C₄	—	26	1	no	—
252	C₂²×C₄	—	26	1	no	—
253	C₂²×C₄	—	26	1	no	—
254	C₂²×C₄	—	26	1	no	—
255	Q₁₆	—	13	4	no	—
256	Q₁₆	C₄×D₁₃	1	4	yes	complemented
257	SD₁₆	—	13	4	no	—
258	SD₁₆	—	13	4	no	—
259	M₁₆	—	13	4	no	—
260	M₁₆	—	26	2	no	—
261	C₂×C₈	—	13	1	no	—
262	C₂×C₈	—	13	1	no	—
263	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	13	1	no	—
268	C₄⋊C₄	—	13	1	no	—
269	C₄⋊C₄	—	26	1	no	—
270	C₄⋊C₄	—	26	1	no	—
271	C₄⋊C₄	—	26	1	no	—
272	C₄⋊C₄	—	26	1	no	—
273	C₄⋊C₄	—	26	1	no	—
274	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
276	C₂²⋊C₄	—	26	1	no	—
277	C₂²⋊C₄	—	26	1	no	—
278	C₂²⋊C₄	—	26	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	13	1	no	—
282	C₄²	—	13	1	no	—
283	C₄²	—	13	1	no	—
284	C₄²	—	26	1	no	—
285	C₄²	—	26	1	no	—
286	C₄²	—	26	1	no	—
287	C₄²	—	26	1	no	—
288	C₄²	—	26	1	no	—
289	C₄²	C₂²×D₁₃	1	1	yes	characteristic
290	C₁₃	C₄×#137	1	1	yes	characteristic, complemented, Sylow
291	C₂³	—	13	1	no	—
292	C₂³	—	26	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	13	2	no	—
296	Q₈	—	26	4	no	—
297	Q₈	—	26	1	no	—
298	Q₈	C₂×C₄×D₁₃	1	2	yes	—
299	Q₈	C₂×C₄×D₁₃	1	2	yes	—
300	D₄	—	13	2	no	—
301	D₄	—	26	4	no	—
302	D₄	—	26	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	13	1	no	—
308	C₂×C₄	—	13	2	no	—
309	C₂×C₄	—	13	2	no	—
310	C₂×C₄	—	13	1	no	—
311	C₂×C₄	—	13	1	no	—
312	C₂×C₄	—	26	4	no	—
313	C₂×C₄	—	26	4	no	—
314	C₂×C₄	—	26	1	no	—
315	C₂×C₄	—	26	2	no	—
316	C₂×C₄	—	26	1	no	—
317	C₂×C₄	—	26	1	no	—
318	C₂×C₄	—	52	1	no	—
319	C₂×C₄	—	52	2	no	—
320	C₂×C₄	—	52	2	no	—
321	C₂×C₄	—	52	2	no	—
322	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
323	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	13	2	no	—
327	C₈	—	26	1	no	—
328	C₈	C₂×C₄×D₁₃	1	2	yes	—
329	C₂²	—	13	2	no	—
330	C₂²	—	26	1	no	—
331	C₂²	—	26	1	no	—
332	C₂²	—	26	2	no	—
333	C₂²	—	52	1	no	—
334	C₂²	—	52	1	no	—
335	C₂²	C₂×D₄×D₁₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	13	2	no	—
342	C₄	—	26	1	no	—
343	C₄	—	26	1	no	—
344	C₄	—	26	2	no	—
345	C₄	—	52	1	no	—
346	C₄	—	52	1	no	—
347	C₄	#88172	1	2	yes	direct product factor
348	C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
349	C₄	#41×D₁₃	1	1	yes	characteristic
350	C₂	—	26	2	no	—
351	C₂	—	52	2	no	—
352	C₂	C₄×D₄×D₁₃	1	1	yes	characteristic
353	C₂	C₂×#88172	1	1	yes	characteristic
354	C₂	#11719473	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×#6131	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₄×Q₁₆	D₁₃
hypocenter	C₁₃	C₄×#137
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
3. C₄×Q₈×D₁₃	C₂	semidirect
4. C₄×C₁₃×Q₁₆	C₂	semidirect
5. #11718845	C₂	semidirect
6. C₄×#6107	C₂	semidirect
7. C₄×#6106	C₂	semidirect
8. #11718550	C₂	semidirect
9. #11718542	C₂	semidirect
10. #11718478	C₂	semidirect
12. #11718463	C₂	semidirect
13. #11718450	C₂	semidirect
14. C₄×#6095	C₂	semidirect
15. C₄×#6094	C₂	semidirect
25. #88172	C₄	direct (N × H)
65. Q₈×D₁₃	C₂×C₄	semidirect
78. C₁₃×Q₁₆	C₂×C₄	semidirect
88. #6107	C₂×C₄	semidirect
89. #6106	C₂×C₄	semidirect
110. #6095	C₂×C₄	semidirect
112. #6094	C₂×C₄	semidirect
160. C₄×Q₁₆	D₁₃	semidirect
179. C₄×C₁₃	#137	semidirect
256. Q₁₆	C₄×D₁₃	semidirect
290. C₁₃	C₄×#137	semidirect
347. C₄	#88172	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×#88172	2
C₄×#6131	2
C₄×Q₈×D₁₃	2
C₄×C₁₃×Q₁₆	2
#11718845	2
C₄×#6107	2
C₄×#6106	2
#11718550	2
#11718542	2
#11718478	2
#11718464	2
#11718463	2
#11718450	2
C₄×#6095	2
C₄×#6094	2
C₄×#137	13

Subgroups — C4×#88172 #13123656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88172 #13123656