Subgroups — C₄×#11718101 #15778124

Order 3328.

534 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718101	C₂	1	2	yes	maximal
3	C₄×#88058	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112928	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₀₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₅₂	—	2	1	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88061	—	2	1	no	—
11	C₂×#88058	C₂²	1	1	yes	characteristic
12	C₂×#88058	C₂²	1	2	yes	—
13	C₄×C₄⋊Dic₁₃	—	2	1	no	—
14	C₄×D₁₀₄	—	2	2	no	—
15	C₄×D₁₀₄	C₄	1	4	yes	complemented
16	C₄×C₁₃×C₁₆	—	2	2	no	—
17	#11718143	—	2	1	no	—
18	#11718101	C₄	1	8	yes	direct product factor
19	C₂²×D₅₂	—	2	1	no	—
20	C₂²×C₄×D₁₃	—	4	1	no	—
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×#6103	—	4	1	no	—
25	C₂×C₄⋊Dic₁₃	—	2	1	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂×C₄×Dic₁₃	—	4	1	no	—
28	C₂×D₁₀₄	—	2	1	no	—
29	C₂×D₁₀₄	—	2	2	no	—
30	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
31	C₂×D₁₀₄	C₂×C₄	1	4	yes	—
32	C₄×D₅₂	—	2	4	no	—
33	C₄×D₅₂	—	4	2	no	—
34	C₂×C₁₃×C₁₆	—	2	2	no	—
35	C₂×C₁₃×C₁₆	—	2	2	no	—
36	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
37	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₃	D₄	1	2	yes	—
39	#88071	—	4	1	no	—
40	#88061	—	2	4	no	—
41	#88058	—	2	1	no	—
42	#88058	—	2	2	no	—
43	#88058	C₂×C₄	1	2	yes	—
44	#88058	C₂×C₄	1	4	yes	complemented
45	C₄×#357	—	13	1	no	maximal, Sylow
46	C₂³×D₁₃	—	4	1	no	—
47	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	—	4	2	no	—
51	C₂²×Dic₁₃	—	4	1	no	—
52	C₂×D₅₂	—	2	2	no	—
53	C₂×D₅₂	—	2	4	no	—
54	C₂×D₅₂	—	4	1	no	—
55	C₂×D₅₂	—	4	2	no	—
56	C₂×C₄×D₁₃	—	4	4	no	—
57	C₂×C₄×D₁₃	—	8	2	no	—
58	C₂×C₄×D₁₃	—	8	2	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	#41	1	2	yes	—
65	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₁₃	SD₁₆	1	2	yes	—
67	#6103	—	4	4	no	—
68	C₄⋊Dic₁₃	—	2	2	no	—
69	C₄⋊Dic₁₃	—	2	4	no	—
70	C₄⋊Dic₁₃	—	4	1	no	—
71	C₄⋊Dic₁₃	—	4	2	no	—
72	C₄×Dic₁₃	—	8	2	no	—
73	D₁₀₄	—	2	2	no	—
74	D₁₀₄	—	2	4	no	—
75	D₁₀₄	C₄²	1	4	yes	complemented
76	C₁₃×C₁₆	—	2	4	no	—
77	C₂×C₄×D₈	—	13	1	no	—
78	C₂×#357	—	13	2	no	—
79	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic, complemented
80	C₄×#107	—	13	1	no	—
81	#1380	—	13	2	no	—
82	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
83	C₂²×D₁₃	—	4	2	no	—
84	C₂²×D₁₃	—	4	4	no	—
85	C₂²×D₁₃	—	8	1	no	—
86	C₂²×D₁₃	—	8	1	no	—
87	C₂²×D₁₃	—	8	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	#102	1	2	yes	—
90	C₂×C₄×C₁₃	#102	1	2	yes	—
91	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
92	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₁₃	#135	1	2	yes	—
97	C₂×Dic₁₃	—	4	2	no	—
98	C₂×Dic₁₃	—	4	4	no	—
99	C₂×Dic₁₃	—	8	2	no	—
100	C₂×Dic₁₃	—	8	1	no	—
101	C₂×Dic₁₃	—	8	1	no	—
102	C₂×Dic₁₃	—	8	2	no	—
103	D₅₂	—	2	4	no	—
104	D₅₂	—	4	2	no	—
105	D₅₂	—	4	4	no	—
106	C₄×D₁₃	—	8	8	no	—
107	C₈×C₁₃	—	2	2	no	—
108	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
109	C₈×C₁₃	#117	1	1	yes	characteristic
110	C₈×C₁₃	C₄×D₄	1	2	yes	—
111	C₂²×D₈	—	13	1	no	—
112	C₂×C₄×D₄	—	26	1	no	—
113	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
114	C₄×D₈	—	13	4	no	—
115	C₄×D₈	—	26	2	no	—
116	C₂×#107	—	13	2	no	—
117	C₂×#107	—	13	1	no	—
118	C₂×#102	—	26	1	no	—
119	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
120	C₄×C₄⋊C₄	—	26	1	no	—
121	#357	—	13	8	no	—
122	C₄×C₁₆	—	2	2	no	—
123	#340	—	26	1	no	—
124	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
125	C₂²×C₁₃	C₄×D₈	1	2	yes	—
126	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
128	C₂²×C₁₃	#508	1	2	yes	—
129	C₂×D₁₃	—	4	4	no	—
130	C₂×D₁₃	—	8	2	no	—
131	C₂×D₁₃	—	8	2	no	—
132	C₂×D₁₃	—	8	4	no	—
133	C₂×D₁₃	—	8	4	no	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	#357	1	4	yes	complemented
136	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₄×C₁₃	#416	1	1	yes	characteristic
138	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
139	Dic₁₃	—	8	2	no	—
140	Dic₁₃	—	8	4	no	—
141	C₂²×D₄	—	26	1	no	—
142	C₂³×C₄	—	52	1	no	—
143	C₂×D₈	—	13	4	no	—
144	C₂×D₈	—	13	2	no	—
145	C₂×D₈	—	26	2	no	—
146	C₂×D₈	—	26	1	no	—
147	C₂²×C₈	D₅₂	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
149	C₄×D₄	—	26	4	no	—
150	C₄×D₄	—	52	2	no	—
151	C₂×C₄⋊C₄	—	26	1	no	—
152	C₂×C₄⋊C₄	—	26	2	no	—
153	C₂×C₂²⋊C₄	—	52	1	no	—
154	C₂×C₄²	—	52	1	no	—
155	C₂×C₄²	D₅₂	1	1	yes	characteristic
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
159	#107	—	13	4	no	—
160	#107	—	13	2	no	—
161	#107	—	26	2	no	—
162	#107	—	26	1	no	—
163	#102	—	26	4	no	—
164	C₄×C₈	C₄×D₁₃	1	2	yes	—
165	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
166	#95	—	52	1	no	—
167	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
168	C₂×C₁₃	C₂×#357	1	1	yes	characteristic
169	C₂×C₁₃	#2138	1	1	yes	characteristic
170	C₂×C₁₃	C₄×D₁₆	1	2	yes	—
171	C₂×C₁₃	C₄×SD₃₂	1	2	yes	—
172	D₁₃	—	8	4	no	—
173	C₂⁴	—	52	1	no	—
174	C₂×D₄	—	26	4	no	—
175	C₂×D₄	—	26	2	no	—
176	C₂×D₄	—	52	1	no	—
177	C₂×D₄	—	52	2	no	—
178	C₂²×C₄	—	52	2	no	—
179	C₂²×C₄	—	52	1	no	—
180	C₂²×C₄	—	52	4	no	—
181	C₂²×C₄	—	104	2	no	—
182	C₂²×C₄	—	104	2	no	—
183	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
184	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
186	D₈	—	13	4	no	—
187	D₈	—	26	2	no	—
188	D₈	—	26	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	#6103	1	2	yes	—
191	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
192	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
193	C₂×C₈	#6127	1	2	yes	—
194	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
195	C₄⋊C₄	—	26	4	no	—
196	C₄⋊C₄	—	26	2	no	—
197	C₄⋊C₄	—	52	1	no	—
198	C₄⋊C₄	—	52	2	no	—
199	C₂²⋊C₄	—	52	4	no	—
200	C₄²	—	104	2	no	—
201	C₄²	#6095	1	2	yes	—
202	C₄²	#6103	1	2	yes	—
203	C₁₆	—	2	4	no	—
204	C₁₃	C₄×#357	1	1	yes	characteristic, complemented, Sylow
205	C₂³	—	52	2	no	—
206	C₂³	—	52	4	no	—
207	C₂³	—	104	1	no	—
208	C₂³	—	104	1	no	—
209	C₂³	—	104	2	no	—
210	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
211	D₄	—	26	4	no	—
212	D₄	—	52	2	no	—
213	D₄	—	52	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	52	2	no	—
216	C₂×C₄	—	52	4	no	—
217	C₂×C₄	—	104	2	no	—
218	C₂×C₄	—	104	1	no	—
219	C₂×C₄	—	104	1	no	—
220	C₂×C₄	—	104	2	no	—
221	C₂×C₄	—	104	8	no	—
222	C₂×C₄	D₂₀₈	1	2	yes	—
223	C₂×C₄	#88040	1	2	yes	—
224	C₂×C₄	#88061	1	2	yes	—
225	C₂×C₄	#88061	1	2	yes	—
226	C₂×C₄	C₄×D₅₂	1	2	yes	—
227	C₂×C₄	C₂×#6095	1	1	yes	characteristic
228	C₂×C₄	#88158	1	2	yes	—
229	C₂×C₄	C₂×#6103	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	C₄²×D₁₃	1	1	yes	characteristic
232	C₈	#88126	1	1	yes	characteristic
233	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
234	C₂²	—	52	4	no	—
235	C₂²	—	104	4	no	—
236	C₂²	—	104	4	no	—
237	C₂²	—	104	2	no	—
238	C₂²	—	104	2	no	—
239	C₂²	C₄×D₁₀₄	1	2	yes	—
240	C₂²	C₂×D₂₀₈	1	1	yes	characteristic
241	C₂²	C₂×#88040	1	1	yes	characteristic
242	C₂²	#11718565	1	2	yes	—
243	C₂²	C₂×#88061	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	104	4	no	—
246	C₄	—	104	2	no	—
247	C₄	#11718101	1	4	yes	direct product factor
248	C₄	C₄×#6095	1	2	yes	—
249	C₄	C₄×#6103	1	1	yes	characteristic
250	C₄	#11718774	1	1	yes	characteristic
251	C₂	—	104	4	no	—
252	C₂	C₄×#88061	1	1	yes	characteristic
253	C₂	C₂×#11718101	1	1	yes	characteristic
254	C₂	#13116544	1	1	yes	characteristic
255	C₂	C₄×D₂₀₈	1	2	yes	—
256	C₂	C₄×#88040	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₁₀₄
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle	C₂³×C₁₃	C₂×D₈
prime radical	C₂²×D₁₀₄	C₂²
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	C₄×#357
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×#88058	C₂	semidirect
4. #13112928	C₂	semidirect
5. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
15. C₄×D₁₀₄	C₄	semidirect
18. #11718101	C₄	direct (N × H)
36. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
44. #88058	C₂×C₄	semidirect
75. D₁₀₄	C₄²	semidirect
79. C₂×C₄×C₁₆	D₁₃	semidirect
135. C₄×C₁₃	#357	semidirect
158. C₂×C₁₆	C₄×D₁₃	semidirect
204. C₁₃	C₄×#357	semidirect
247. C₄	#11718101	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×C₄×D₁₀₄	2
C₂×#11718101	2
C₄×#88058	2
#13112928	2
C₂×C₄×C₁₃×C₁₆	2
C₄×#357	13

Subgroups — C4×#11718101 #15778124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718101 #15778124