Subgroups — C₄×#11718080 #15778170

Order 3328.

466 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88158	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718080	C₂	1	2	yes	maximal
3	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112943	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88158	C₂²	1	1	yes	characteristic
7	C₄×#6127	—	2	1	no	—
8	C₂×C₁₃×#106	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
11	#11718774	—	2	1	no	—
12	#11718767	C₂²	1	1	yes	characteristic
13	C₂×#88052	C₂²	1	2	yes	—
14	C₄×Dic₅₂	C₄	1	1	yes	characteristic, complemented
15	C₄×Dic₅₂	C₄	1	1	yes	characteristic, complemented
16	C₄×D₁₀₄	C₄	1	1	yes	characteristic, complemented
17	C₄×D₁₀₄	C₄	1	1	yes	characteristic, complemented
18	C₄×#6095	—	2	1	no	—
19	C₄×#6095	—	2	1	no	—
20	C₁₃×#340	—	2	1	no	—
21	#11718080	C₄	1	8	yes	direct product factor
22	#11718053	—	2	1	no	—
23	C₄×#6090	—	2	1	no	—
24	C₂×#6127	—	2	1	no	—
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄²×C₁₃	—	4	1	no	—
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₄×C₂²⋊D₁₃	—	4	1	no	—
31	C₄×C₂²⋊D₁₃	—	4	1	no	—
32	#88180	—	2	1	no	—
33	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
34	#88158	—	2	2	no	—
35	#88158	C₂×C₄	1	4	yes	—
36	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
37	C₂×#6095	—	2	1	no	—
38	C₄×D₅₂	—	2	1	no	—
39	C₄×D₅₂	—	2	1	no	—
40	#88126	—	4	1	no	—
41	C₄²×D₁₃	—	4	1	no	—
42	C₄×Dic₂₆	—	2	1	no	—
43	C₄×Dic₂₆	—	2	1	no	—
44	C₁₃×#106	—	2	2	no	—
45	C₁₃×#106	C₂×C₄	1	4	yes	complemented
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
50	C₁₃×#95	—	4	1	no	—
51	#88061	—	2	1	no	—
52	#88061	—	2	1	no	—
53	#88058	C₂×C₄	1	1	yes	characteristic
54	#88058	C₂×C₄	1	1	yes	characteristic
55	#88057	—	2	1	no	—
56	#88056	—	2	1	no	—
57	#88056	—	2	1	no	—
58	#88052	—	2	2	no	—
59	#88052	C₂×C₄	1	4	yes	complemented
60	C₂×#6090	—	2	2	no	—
61	C₄×#360	—	13	1	no	maximal, Sylow
62	C₂²×C₄×C₁₃	—	4	2	no	—
63	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₃	—	4	1	no	—
67	#6127	—	2	4	no	—
68	#6127	—	4	2	no	—
69	C₂×D₅₂	—	2	1	no	—
70	C₂×C₄×D₁₃	—	4	1	no	—
71	C₂×C₄×D₁₃	—	4	1	no	—
72	C₂×C₄×D₁₃	—	4	1	no	—
73	C₂×Dic₂₆	—	2	1	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	#41	1	2	yes	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₄²×C₁₃	—	8	2	no	—
84	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
87	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
88	C₂²⋊Dic₁₃	—	4	1	no	—
89	#6103	—	4	1	no	—
90	#6103	—	4	1	no	—
91	C₄⋊Dic₁₃	—	2	1	no	—
92	C₄⋊Dic₁₃	—	2	1	no	—
93	#6101	—	4	1	no	—
94	#6101	—	4	1	no	—
95	C₄×Dic₁₃	—	4	1	no	—
96	C₄×Dic₁₃	—	4	1	no	—
97	C₄×Dic₁₃	—	4	1	no	—
98	Dic₅₂	—	2	1	no	—
99	Dic₅₂	C₄²	1	2	yes	complemented
100	D₁₀₄	—	2	1	no	—
101	D₁₀₄	C₄²	1	2	yes	complemented
102	#6095	—	2	4	no	—
103	#6090	—	2	4	no	—
104	C₄×#135	—	13	1	no	—
105	C₂×#360	—	13	2	no	—
106	C₄×M₃₂	—	13	1	no	—
107	C₄×#106	D₁₃	1	1	yes	characteristic, complemented
108	#1386	—	13	2	no	—
109	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
110	C₂²×D₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	—	8	2	no	—
115	C₂×C₄×C₁₃	—	8	2	no	—
116	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#135	1	2	yes	—
124	C₂²⋊D₁₃	—	4	4	no	—
125	C₂×Dic₁₃	—	4	1	no	—
126	C₂×Dic₁₃	—	4	1	no	—
127	C₂×Dic₁₃	—	4	1	no	—
128	D₅₂	—	2	2	no	—
129	D₅₂	—	4	1	no	—
130	C₄×D₁₃	—	4	4	no	—
131	C₄×D₁₃	—	8	2	no	—
132	C₄×D₁₃	—	8	2	no	—
133	Dic₂₆	—	2	2	no	—
134	Dic₂₆	—	4	1	no	—
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	—	2	1	no	—
137	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
138	C₈×C₁₃	#117	1	1	yes	characteristic
139	C₈×C₁₃	C₄×D₄	1	2	yes	—
140	C₂×#135	—	13	1	no	—
141	C₄×#41	—	26	1	no	—
142	C₂×M₃₂	—	13	2	no	—
143	C₄×Q₁₆	—	13	1	no	—
144	C₄×Q₁₆	—	13	1	no	—
145	C₄×SD₁₆	—	26	1	no	—
146	C₄×SD₁₆	—	26	1	no	—
147	C₄×D₈	—	13	1	no	—
148	C₄×D₈	—	13	1	no	—
149	#427	—	13	1	no	—
150	C₂×#106	C₂×D₁₃	1	2	yes	—
151	#416	—	26	1	no	—
152	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
153	C₄×C₄⋊C₄	—	2	1	no	—
154	#360	—	13	8	no	—
155	#346	—	26	1	no	—
156	C₄×C₁₆	—	26	1	no	—
157	#340	—	2	1	no	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
161	C₂²×C₁₃	C₄×D₈	1	2	yes	—
162	C₂×D₁₃	—	4	2	no	—
163	C₂×D₁₃	—	8	1	no	—
164	C₂×D₁₃	—	8	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	8	4	no	—
169	C₄×C₁₃	#360	1	2	yes	complemented
170	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
171	C₄×C₁₃	#416	1	1	yes	characteristic
172	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
173	Dic₁₃	—	4	2	no	—
174	Dic₁₃	—	8	1	no	—
175	Dic₁₃	—	8	1	no	—
176	C₂×#41	—	26	1	no	—
177	#135	—	13	4	no	—
178	#135	—	26	2	no	—
179	C₂×Q₁₆	—	13	1	no	—
180	C₂×SD₁₆	—	26	1	no	—
181	C₂×D₈	—	13	1	no	—
182	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
184	C₄×Q₈	—	26	1	no	—
185	C₄×Q₈	—	26	1	no	—
186	C₄×D₄	—	26	1	no	—
187	C₄×D₄	—	26	1	no	—
188	C₄×D₄	—	52	1	no	—
189	C₄×D₄	—	52	1	no	—
190	#117	—	26	1	no	—
191	#117	—	52	1	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄²	—	4	1	no	—
194	C₂×C₄²	—	52	1	no	—
195	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
196	M₃₂	—	13	4	no	—
197	M₃₂	—	26	2	no	—
198	C₂×C₁₆	—	26	2	no	—
199	#107	—	13	1	no	—
200	#107	—	13	1	no	—
201	#106	—	2	2	no	—
202	#106	—	26	1	no	—
203	#106	C₄×D₁₃	1	4	yes	complemented
204	#103	—	26	1	no	—
205	#103	—	26	1	no	—
206	#102	—	26	1	no	—
207	#102	—	26	1	no	—
208	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
209	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
210	C₄×C₈	D₅₂	1	1	yes	characteristic
211	C₄×C₈	D₅₂	1	1	yes	characteristic
212	#95	—	4	1	no	—
213	C₂×C₁₃	—	2	2	no	—
214	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
215	C₂×C₁₃	C₂×#360	1	1	yes	characteristic
216	C₂×C₁₃	#2145	1	1	yes	characteristic
217	D₁₃	—	8	2	no	—
218	#41	—	26	4	no	—
219	#41	—	52	2	no	—
220	C₂×Q₈	—	26	1	no	—
221	C₂×D₄	—	26	1	no	—
222	C₂×D₄	—	52	1	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	52	1	no	—
225	C₂²×C₄	—	52	1	no	—
226	C₂²×C₄	—	52	1	no	—
227	C₂²×C₄	#6104	1	1	yes	characteristic
228	C₂²×C₄	#6104	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
230	Q₁₆	—	13	2	no	—
231	Q₁₆	—	26	1	no	—
232	SD₁₆	—	26	4	no	—
233	D₈	—	13	2	no	—
234	D₈	—	26	1	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	#6103	1	1	yes	characteristic
237	C₂×C₈	#6103	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
240	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
241	C₂×C₈	#6127	1	2	yes	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	26	1	no	—
245	C₄⋊C₄	—	26	1	no	—
246	C₄⋊C₄	—	52	1	no	—
247	C₄⋊C₄	—	52	1	no	—
248	C₂²⋊C₄	—	52	1	no	—
249	C₂²⋊C₄	—	52	1	no	—
250	C₂²⋊C₄	—	52	1	no	—
251	C₄²	—	8	2	no	—
252	C₄²	—	52	1	no	—
253	C₄²	—	52	1	no	—
254	C₄²	—	52	1	no	—
255	C₄²	#6103	1	1	yes	characteristic
256	C₄²	#6103	1	1	yes	characteristic
257	C₄²	#6106	1	1	yes	characteristic
258	C₄²	#6106	1	1	yes	characteristic
259	C₁₆	—	26	4	no	—
260	C₁₃	C₄×#360	1	1	yes	characteristic, complemented, Sylow
261	C₂³	—	52	1	no	—
262	C₂³	C₂×#6104	1	1	yes	characteristic
263	Q₈	—	26	2	no	—
264	Q₈	—	52	1	no	—
265	D₄	—	26	2	no	—
266	D₄	—	52	4	no	—
267	D₄	—	52	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	52	1	no	—
274	C₂×C₄	—	52	4	no	—
275	C₂×C₄	—	52	1	no	—
276	C₂×C₄	—	52	1	no	—
277	C₂×C₄	—	104	2	no	—
278	C₂×C₄	—	104	2	no	—
279	C₂×C₄	#88049	1	1	yes	characteristic
280	C₂×C₄	#88049	1	1	yes	characteristic
281	C₂×C₄	#88049	1	1	yes	characteristic
282	C₂×C₄	#88049	1	1	yes	characteristic
283	C₂×C₄	C₂×#6103	1	1	yes	characteristic
284	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
285	C₂×C₄	C₂×#6106	1	1	yes	characteristic
286	C₂×C₄	#88204	1	2	yes	—
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	C₄²×D₁₃	1	1	yes	characteristic
290	C₈	#88126	1	1	yes	characteristic
291	C₈	C₄×D₅₂	1	2	yes	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	52	2	no	—
295	C₂²	—	104	1	no	—
296	C₂²	—	104	1	no	—
297	C₂²	C₂×#88049	1	1	yes	characteristic
298	C₂²	C₄×#6104	1	2	yes	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	8	4	no	—
303	C₄	—	52	2	no	—
304	C₄	—	104	1	no	—
305	C₄	—	104	1	no	—
306	C₄	#11718080	1	2	yes	direct product factor
307	C₄	C₄×#6103	1	1	yes	characteristic
308	C₄	#11718663	1	1	yes	characteristic
309	C₄	C₄×#6106	1	2	yes	—
310	C₂	—	2	2	no	—
311	C₂	—	104	2	no	—
312	C₂	C₄×#88049	1	1	yes	characteristic
313	C₂	C₂×#11718080	1	1	yes	characteristic
314	C₂	#13116571	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88049
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×#106	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂×#88158	C₂²
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×#106	D₁₃
hypocenter	C₁₃	C₄×#360
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
3. C₄×#88052	C₂	semidirect
4. #13112943	C₂	semidirect
5. C₄×C₁₃×#106	C₂	semidirect
14. C₄×Dic₅₂	C₄	semidirect
15. C₄×Dic₅₂	C₄	semidirect
16. C₄×D₁₀₄	C₄	semidirect
17. C₄×D₁₀₄	C₄	semidirect
21. #11718080	C₄	direct (N × H)
45. C₁₃×#106	C₂×C₄	semidirect
59. #88052	C₂×C₄	semidirect
99. Dic₅₂	C₄²	semidirect
101. D₁₀₄	C₄²	semidirect
107. C₄×#106	D₁₃	semidirect
169. C₄×C₁₃	#360	semidirect
203. #106	C₄×D₁₃	semidirect
260. C₁₃	C₄×#360	semidirect
306. C₄	#11718080	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#88158	2
C₂×#11718080	2
C₄×#88052	2
#13112943	2
C₄×C₁₃×#106	2
C₄×#360	13

Subgroups — C4×#11718080 #15778170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718080 #15778170