Subgroups — C₄×#11718263 #16789301

Order 3328.

522 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718263	C₂	1	2	yes	maximal
2	C₄×#88106	C₂	1	2	yes	maximal, complemented
3	#13129884	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#6094	C₂	1	1	yes	maximal, characteristic
5	C₂²×#6094	C₄	1	1	yes	characteristic
6	C₂²×#6094	C₂²	1	1	yes	characteristic
7	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
8	C₂×#88106	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
10	C₂×#88055	C₄	1	1	yes	characteristic
11	C₂×#88055	C₂²	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₄×#6094	—	2	2	no	—
14	C₄×#6094	—	2	2	no	—
15	#11718263	C₄	1	8	yes	direct product factor
16	#11718231	—	2	2	no	—
17	C₄×#6092	—	2	2	no	—
18	C₂²×C₄×D₁₃	C₈	1	2	yes	—
19	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂³	1	1	yes	characteristic
27	C₂×#6094	—	2	2	no	—
28	C₂×#6094	—	2	2	no	—
29	C₂×#6094	—	2	2	no	—
30	C₂×#6094	—	2	2	no	—
31	C₂×#6094	C₈	1	4	yes	—
32	C₂×#6094	C₂×C₄	1	4	yes	—
33	C₄²×D₁₃	—	2	1	no	—
34	C₄²×D₁₃	—	2	1	no	—
35	C₄²×D₁₃	—	2	2	no	—
36	#88106	—	2	4	no	—
37	#88106	C₂×C₄	1	8	yes	complemented
38	C₂×#6092	—	2	4	no	—
39	C₄×C₈×C₁₃	—	2	1	no	—
40	C₄×C₈×C₁₃	—	2	1	no	—
41	#88055	—	2	2	no	—
42	#88055	—	2	2	no	—
43	C₄×#1055	D₄	1	2	yes	—
44	C₄×#1055	D₄	1	2	yes	—
45	C₄×#349	—	13	1	no	maximal, Sylow
46	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
51	C₂×C₄×D₁₃	—	2	1	no	—
52	C₂×C₄×D₁₃	—	2	1	no	—
53	C₂×C₄×D₁₃	—	2	1	no	—
54	C₂×C₄×D₁₃	—	2	1	no	—
55	C₂×C₄×D₁₃	—	2	2	no	—
56	C₂×C₄×D₁₃	—	2	2	no	—
57	C₂×C₄×D₁₃	—	4	4	no	—
58	C₂×C₄×D₁₃	C₄²	1	2	yes	—
59	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
60	C₂×C₈×C₁₃	—	2	1	no	—
61	C₂×C₈×C₁₃	—	2	1	no	—
62	C₂×C₈×C₁₃	—	2	1	no	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	C₄²	1	2	yes	—
65	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
66	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₁₃	M₁₆	1	2	yes	—
69	C₄×Dic₁₃	—	2	2	no	—
70	C₄×Dic₁₃	—	2	2	no	—
71	C₄×Dic₁₃	—	2	1	no	—
72	C₄×Dic₁₃	—	2	1	no	—
73	C₂×#1055	C₂²⋊C₄	1	2	yes	—
74	C₂×#1055	C₂×C₈	1	2	yes	—
75	C₂×#1055	M₁₆	1	2	yes	—
76	C₂×#1055	C₂²×C₄	1	2	yes	—
77	C₂×#1055	C₂×D₄	1	2	yes	—
78	C₂×#1055	#41	1	2	yes	—
79	#6094	—	2	8	no	—
80	#6094	—	2	8	no	—
81	#6092	—	2	8	no	—
82	C₂×C₄×M₁₆	—	13	1	no	—
83	C₂×#349	—	13	2	no	—
84	C₄×M₃₂	—	13	2	no	—
85	#1377	—	13	2	no	—
86	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
87	C₂²×D₁₃	—	2	1	no	—
88	C₂²×D₁₃	—	2	1	no	—
89	C₂²×D₁₃	—	4	2	no	—
90	C₂²×D₁₃	C₄×C₈	1	2	yes	—
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×Dic₁₃	—	2	1	no	—
101	C₂×Dic₁₃	—	2	1	no	—
102	C₂×Dic₁₃	—	2	2	no	—
103	C₂×Dic₁₃	—	4	2	no	—
104	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
105	C₄×D₁₃	—	2	2	no	—
106	C₄×D₁₃	—	2	2	no	—
107	C₄×D₁₃	—	2	2	no	—
108	C₄×D₁₃	—	2	2	no	—
109	C₄×D₁₃	—	4	4	no	—
110	C₄×D₁₃	—	4	4	no	—
111	C₈×C₁₃	—	2	2	no	—
112	C₈×C₁₃	—	2	2	no	—
113	#1055	C₂²⋊C₈	1	4	yes	—
114	#1055	C₄×D₄	1	4	yes	—
115	C₂²×M₁₆	—	13	1	no	—
116	C₂²×M₁₆	—	13	1	no	—
117	C₂²×C₄²	—	13	1	no	—
118	C₂×M₃₂	—	13	4	no	—
119	C₄×M₁₆	—	26	2	no	—
120	C₄×M₁₆	—	26	2	no	—
121	C₂×#97	—	13	1	no	—
122	C₂×#97	—	13	1	no	—
123	C₂×C₄×C₈	—	13	1	no	—
124	C₂×C₄×C₈	#261	1	1	yes	characteristic
125	#349	—	13	8	no	—
126	#346	—	26	2	no	—
127	C₄×C₁₆	—	26	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
130	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₂²×C₁₃	#431	1	1	yes	characteristic
132	C₂×D₁₃	—	2	2	no	—
133	C₂×D₁₃	—	4	2	no	—
134	C₂×D₁₃	—	4	2	no	—
135	C₂×D₁₃	—	4	2	no	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	#349	1	4	yes	complemented
138	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
140	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₄×C₁₃	#431	1	1	yes	characteristic
142	Dic₁₃	—	2	2	no	—
143	Dic₁₃	—	4	2	no	—
144	C₂³×C₄	—	13	1	no	—
145	C₂³×C₄	—	13	2	no	—
146	C₂×M₁₆	—	13	4	no	—
147	C₂×M₁₆	—	13	4	no	—
148	C₂×M₁₆	—	26	2	no	—
149	C₂×M₁₆	—	26	2	no	—
150	C₂×M₁₆	—	26	2	no	—
151	C₂×M₁₆	—	26	2	no	—
152	C₂²×C₈	—	13	1	no	—
153	C₂²×C₈	—	13	1	no	—
154	C₂²×C₈	C₂×#261	1	1	yes	characteristic
155	C₂²×C₈	C₂×#261	1	1	yes	characteristic
156	C₂×C₄²	—	13	2	no	—
157	C₂×C₄²	—	13	1	no	—
158	C₂×C₄²	—	26	2	no	—
159	C₂×C₄²	—	26	1	no	—
160	C₂×C₄²	—	26	1	no	—
161	C₂×C₄²	C₂×#261	1	1	yes	characteristic
162	M₃₂	—	13	8	no	—
163	M₃₂	—	26	4	no	—
164	C₂×C₁₆	—	26	4	no	—
165	#97	—	26	2	no	—
166	#97	—	26	2	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	13	2	no	—
170	C₄×C₈	—	13	2	no	—
171	C₂×C₁₃	—	2	1	no	—
172	C₂×C₁₃	—	2	1	no	—
173	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
174	C₂×C₁₃	C₂×#349	1	2	yes	—
175	D₁₃	—	4	2	no	—
176	C₂⁴	—	13	1	no	—
177	C₂²×C₄	—	13	1	no	—
178	C₂²×C₄	—	13	2	no	—
179	C₂²×C₄	—	13	2	no	—
180	C₂²×C₄	—	13	2	no	—
181	C₂²×C₄	—	26	1	no	—
182	C₂²×C₄	—	26	1	no	—
183	C₂²×C₄	—	26	1	no	—
184	C₂²×C₄	—	26	1	no	—
185	C₂²×C₄	—	26	2	no	—
186	C₂²×C₄	—	26	2	no	—
187	C₂²×C₄	—	52	4	no	—
188	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
189	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
190	M₁₆	—	26	8	no	—
191	M₁₆	—	26	8	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	13	2	no	—
197	C₂×C₈	—	13	2	no	—
198	C₂×C₈	—	13	2	no	—
199	C₂×C₈	—	13	2	no	—
200	C₂×C₈	—	13	2	no	—
201	C₂×C₈	—	13	2	no	—
202	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
203	C₂×C₈	C₄×#261	1	2	yes	—
204	C₄²	—	26	2	no	—
205	C₄²	—	26	2	no	—
206	C₄²	—	26	1	no	—
207	C₄²	—	26	1	no	—
208	C₄²	#6118	1	2	yes	—
209	C₄²	C₂²⋊#261	1	1	yes	characteristic
210	C₄²	C₂²⋊#261	1	1	yes	characteristic
211	C₁₆	—	26	8	no	—
212	C₁₃	C₄×#349	1	1	yes	characteristic, complemented, Sylow
213	C₂³	—	13	2	no	—
214	C₂³	—	26	1	no	—
215	C₂³	—	26	1	no	—
216	C₂³	—	52	2	no	—
217	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	13	2	no	—
220	C₂×C₄	—	26	1	no	—
221	C₂×C₄	—	26	1	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	26	2	no	—
224	C₂×C₄	—	26	2	no	—
225	C₂×C₄	—	26	2	no	—
226	C₂×C₄	—	26	2	no	—
227	C₂×C₄	—	52	4	no	—
228	C₂×C₄	—	52	4	no	—
229	C₂×C₄	—	52	2	no	—
230	C₂×C₄	#88111	1	2	yes	—
231	C₂×C₄	#88111	1	2	yes	—
232	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
233	C₂×C₄	C₂×#6118	1	1	yes	characteristic
234	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
236	C₂×C₄	#88237	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	—	13	4	no	—
241	C₈	—	13	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	26	2	no	—
244	C₂²	—	52	2	no	—
245	C₂²	—	52	2	no	—
246	C₂²	—	52	2	no	—
247	C₂²	C₄×D₁₃⋊C₈	1	1	yes	characteristic
248	C₂²	#11719049	1	1	yes	characteristic
249	C₂²	C₂×#88111	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	26	2	no	—
252	C₄	—	52	2	no	—
253	C₄	#11718263	1	4	yes	direct product factor
254	C₄	C₄×#6118	1	1	yes	characteristic
255	C₄	#11719051	1	1	yes	characteristic
256	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
257	C₄	C₄×C₂²⋊#261	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	52	2	no	—
261	C₂	C₄×#88111	1	1	yes	characteristic
262	C₂	C₂×#11718263	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊#261
commutator	C₂²×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₄×#349
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄×#88106	C₂	semidirect
3. #13129884	C₂	semidirect
15. #11718263	C₄	direct (N × H)
37. #88106	C₂×C₄	semidirect
137. C₄×C₁₃	#349	semidirect
212. C₁₃	C₄×#349	semidirect
253. C₄	#11718263	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×#11718263	2
C₄×#88106	2
#13129884	2
C₂×C₄×#6094	2
C₄×#349	13

Subgroups — C4×#11718263 #16789301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718263 #16789301