Subgroups — C₄×#11718074 #15753870

Order 3328.

540 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88123	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718074	C₂	1	2	yes	maximal
4	#13112720	C₂	1	2	yes	maximal
5	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#88123	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₂₆	—	2	1	no	—
8	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	—	2	1	no	—
10	C₄³×C₁₃	C₂²	1	1	yes	characteristic
11	#11718596	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊Dic₁₃	—	2	1	no	—
13	C₄×C₄⋊Dic₁₃	C₄	1	2	yes	—
14	#11718590	—	2	1	no	—
15	#11718589	—	2	1	no	—
16	#11718589	C₄	1	2	yes	—
17	C₂×C₄⋊#1055	C₂²	1	2	yes	—
18	C₂×C₄×#1055	—	2	1	no	—
19	C₁₃×#336	—	2	1	no	—
20	#11718116	—	2	1	no	—
21	#11718074	C₄	1	8	yes	direct product factor
22	C₂²×Dic₂₆	—	2	1	no	—
23	C₂²×C₈×C₁₃	—	2	2	no	—
24	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₃	—	2	1	no	—
32	C₂×C₄⋊Dic₁₃	—	2	1	no	—
33	C₂×C₄⋊Dic₁₃	—	2	1	no	—
34	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
36	C₂×#6101	—	2	2	no	—
37	C₂×C₄×Dic₁₃	—	2	2	no	—
38	C₂²×#1055	—	2	2	no	—
39	#88123	—	2	2	no	—
40	#88123	C₂×C₄	1	4	yes	—
41	C₄×Dic₂₆	—	4	2	no	—
42	C₄×Dic₂₆	—	4	2	no	—
43	C₁₃×C₄⋊C₈	—	2	2	no	—
44	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
45	C₄×C₈×C₁₃	—	4	2	no	—
46	#88071	—	2	2	no	—
47	#88071	—	2	2	no	—
48	C₄⋊#1055	—	2	2	no	—
49	C₄⋊#1055	C₂×C₄	1	4	yes	—
50	C₄×#1055	—	4	2	no	—
51	C₄×#332	—	13	1	no	maximal, Sylow
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	—	2	2	no	—
60	C₂²×Dic₁₃	—	2	2	no	—
61	C₂×Dic₂₆	—	2	4	no	—
62	C₂×Dic₂₆	—	4	1	no	—
63	C₂×Dic₂₆	—	4	1	no	—
64	C₂×Dic₂₆	—	4	2	no	—
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₂×C₈×C₁₃	—	4	2	no	—
67	C₂×C₈×C₁₃	—	4	2	no	—
68	C₄²×C₁₃	—	2	2	no	—
69	C₄²×C₁₃	—	2	2	no	—
70	C₄²×C₁₃	—	2	2	no	—
71	C₄²×C₁₃	—	4	1	no	—
72	C₄²×C₁₃	D₈	1	2	yes	—
73	C₄²×C₁₃	SD₁₆	1	2	yes	—
74	C₄²×C₁₃	SD₁₆	1	2	yes	—
75	C₄²×C₁₃	Q₁₆	1	2	yes	—
76	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₁₃	#41	1	2	yes	—
78	C₄⋊Dic₁₃	—	2	2	no	—
79	C₄⋊Dic₁₃	—	2	4	no	—
80	C₄⋊Dic₁₃	—	4	1	no	—
81	C₄⋊Dic₁₃	—	4	1	no	—
82	C₄⋊Dic₁₃	—	4	2	no	—
83	C₄⋊Dic₁₃	—	4	2	no	—
84	C₄⋊Dic₁₃	C₄²	1	4	yes	—
85	#6101	—	4	4	no	—
86	C₄×Dic₁₃	—	4	4	no	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	—	4	2	no	—
89	C₂×#1055	—	4	2	no	—
90	C₄×C₄⋊Q₈	—	13	1	no	—
91	C₄×C₄⋊C₈	—	13	1	no	—
92	C₄×C₄⋊C₈	D₁₃	1	1	yes	characteristic
93	C₂×#332	—	13	2	no	—
94	#1296	—	13	2	no	—
95	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	#101	1	2	yes	—
106	C₂×C₄×C₁₃	#102	1	2	yes	—
107	C₂×C₄×C₁₃	#102	1	2	yes	—
108	C₂×C₄×C₁₃	#103	1	2	yes	—
109	C₂×C₄×C₁₃	#103	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#135	1	2	yes	—
119	C₂×C₄×C₁₃	#135	1	2	yes	—
120	C₂×Dic₁₃	—	2	4	no	—
121	C₂×Dic₁₃	—	4	2	no	—
122	C₂×Dic₁₃	—	4	2	no	—
123	C₂×Dic₁₃	—	4	2	no	—
124	C₂×Dic₁₃	—	4	2	no	—
125	C₂×Dic₁₃	—	4	2	no	—
126	Dic₂₆	—	4	4	no	—
127	Dic₂₆	—	4	4	no	—
128	C₈×C₁₃	—	4	4	no	—
129	#1055	—	4	4	no	—
130	C₂×C₄⋊Q₈	—	13	1	no	—
131	C₂×C₄×Q₈	—	26	1	no	—
132	C₂×C₄⋊C₈	—	13	2	no	—
133	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
134	C₂×C₄×C₈	—	2	1	no	—
135	C₂×C₄×C₈	—	26	1	no	—
136	#391	—	26	1	no	—
137	#389	—	13	2	no	—
138	#389	—	26	1	no	—
139	#384	—	13	1	no	—
140	C₄×C₄⋊C₄	—	13	2	no	—
141	C₄×C₄⋊C₄	—	26	1	no	—
142	C₄³	C₂×D₁₃	1	1	yes	characteristic
143	#336	—	2	1	no	—
144	#336	—	26	1	no	—
145	#332	—	13	8	no	—
146	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₁₃	C₂×#101	1	1	yes	characteristic
148	C₂²×C₁₃	#413	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
150	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
151	C₂²×C₁₃	#416	1	1	yes	characteristic
152	C₂²×C₁₃	#416	1	1	yes	characteristic
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	—	2	1	no	—
156	C₄×C₁₃	—	2	1	no	—
157	C₄×C₁₃	—	4	1	no	—
158	C₄×C₁₃	#332	1	4	yes	complemented
159	C₄×C₁₃	C₄×D₈	1	2	yes	—
160	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
161	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
162	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
163	Dic₁₃	—	4	4	no	—
164	Dic₁₃	—	8	2	no	—
165	C₂²×Q₈	—	26	1	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	26	2	no	—
168	C₄⋊Q₈	—	13	4	no	—
169	C₄⋊Q₈	—	26	2	no	—
170	C₄×Q₈	—	52	2	no	—
171	C₄×Q₈	—	52	2	no	—
172	C₂×C₄⋊C₄	—	13	2	no	—
173	C₂×C₄⋊C₄	—	13	2	no	—
174	C₂×C₄⋊C₄	—	26	1	no	—
175	C₂×C₄⋊C₄	—	26	2	no	—
176	C₂×C₄⋊C₄	—	26	1	no	—
177	C₂×C₄⋊C₄	—	26	1	no	—
178	C₂×C₄²	—	26	2	no	—
179	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
180	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
181	C₂×C₄²	D₅₂	1	1	yes	characteristic
182	C₂×C₄²	D₅₂	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
186	C₄⋊C₈	—	2	2	no	—
187	C₄⋊C₈	—	13	4	no	—
188	C₄⋊C₈	—	26	2	no	—
189	C₄⋊C₈	C₄×D₁₃	1	4	yes	—
190	C₄×C₈	—	4	2	no	—
191	C₄×C₈	—	52	2	no	—
192	#95	—	26	2	no	—
193	#95	—	26	2	no	—
194	C₂×C₁₃	C₂×#332	1	1	yes	characteristic
195	C₂×C₁₃	#1540	1	1	yes	characteristic
196	C₂×C₁₃	#1542	1	1	yes	characteristic
197	C₂×C₁₃	#1544	1	1	yes	characteristic
198	C₂×C₁₃	C₄×#101	1	1	yes	characteristic
199	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
200	C₂×C₁₃	C₄×#103	1	1	yes	characteristic
201	C₂×Q₈	—	26	4	no	—
202	C₂×Q₈	—	52	1	no	—
203	C₂×Q₈	—	52	1	no	—
204	C₂×Q₈	—	52	2	no	—
205	C₂²×C₄	—	26	2	no	—
206	C₂²×C₄	—	26	2	no	—
207	C₂²×C₄	#6103	1	1	yes	characteristic
208	C₂²×C₄	#6103	1	1	yes	characteristic
209	C₂²×C₄	#6103	1	1	yes	characteristic
210	C₂²×C₄	#6103	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	26	4	no	—
218	C₂×C₈	—	52	2	no	—
219	C₂×C₈	—	52	2	no	—
220	C₄⋊C₄	—	13	4	no	—
221	C₄⋊C₄	—	26	2	no	—
222	C₄⋊C₄	—	26	4	no	—
223	C₄⋊C₄	—	52	2	no	—
224	C₄⋊C₄	—	52	2	no	—
225	C₄⋊C₄	—	52	1	no	—
226	C₄⋊C₄	—	52	1	no	—
227	C₄⋊C₄	—	52	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	4	1	no	—
232	C₄²	—	52	4	no	—
233	C₄²	#6095	1	2	yes	—
234	C₄²	Dic₅₂	1	2	yes	—
235	C₄²	#6104	1	2	yes	—
236	C₄²	#6106	1	2	yes	—
237	C₄²	C₂×C₄×D₁₃	1	2	yes	—
238	C₄²	#6127	1	2	yes	—
239	C₁₃	C₄×#332	1	1	yes	characteristic, complemented, Sylow
240	C₂³	C₂×#6103	1	1	yes	characteristic
241	Q₈	—	52	4	no	—
242	Q₈	—	52	4	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	26	4	no	—
253	C₂×C₄	—	52	2	no	—
254	C₂×C₄	—	52	2	no	—
255	C₂×C₄	—	52	2	no	—
256	C₂×C₄	—	52	2	no	—
257	C₂×C₄	—	52	2	no	—
258	C₂×C₄	#88049	1	2	yes	—
259	C₂×C₄	#88049	1	2	yes	—
260	C₂×C₄	#88056	1	2	yes	—
261	C₂×C₄	#88056	1	2	yes	—
262	C₂×C₄	#88064	1	2	yes	—
263	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
264	C₂×C₄	#88126	1	1	yes	characteristic
265	C₂×C₄	C₄×D₅₂	1	2	yes	—
266	C₂×C₄	C₂×#6095	1	1	yes	characteristic
267	C₂×C₄	#88158	1	2	yes	—
268	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
269	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
270	C₂×C₄	C₂×#6104	1	1	yes	characteristic
271	C₂×C₄	C₂×#6106	1	1	yes	characteristic
272	C₂×C₄	#88204	1	2	yes	—
273	C₈	—	4	4	no	—
274	C₈	—	52	4	no	—
275	C₂²	C₄×#6103	1	1	yes	characteristic
276	C₂²	C₂×#88049	1	1	yes	characteristic
277	C₂²	#11718663	1	1	yes	characteristic
278	C₂²	C₂×#88056	1	1	yes	characteristic
279	C₂²	#11718774	1	1	yes	characteristic
280	C₂²	#11718793	1	1	yes	characteristic
281	C₂²	C₂×#88064	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	4	1	no	—
287	C₄	—	52	4	no	—
288	C₄	—	104	2	no	—
289	C₄	#11718074	1	4	yes	direct product factor
290	C₄	C₄×#6095	1	2	yes	—
291	C₄	C₄×Dic₅₂	1	2	yes	—
292	C₄	C₄×#6104	1	2	yes	—
293	C₄	C₄×#6106	1	2	yes	—
294	C₂	C₂×#11718074	1	1	yes	characteristic
295	C₂	#13113486	1	1	yes	characteristic
296	C₂	#13113491	1	1	yes	characteristic
297	C₂	#13113500	1	1	yes	characteristic
298	C₂	C₄×#88064	1	1	yes	characteristic
299	C₂	C₄×#88049	1	1	yes	characteristic
300	C₂	C₄×#88056	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#6103
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₃
hypocenter	C₁₃	C₄×#332
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
21. #11718074	C₄	direct (N × H)
158. C₄×C₁₃	#332	semidirect
239. C₁₃	C₄×#332	semidirect
289. C₄	#11718074	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×#88123	2
C₄×C₄⋊#1055	2
C₂×#11718074	2
#13112720	2
C₄×C₁₃×C₄⋊C₈	2
C₄×#332	13

Subgroups — C4×#11718074 #15753870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718074 #15753870