Subgroups — C₄×#11718095 #15778129

Order 3328.

470 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718095	C₂	1	2	yes	maximal
3	C₄×#88058	C₂	1	1	yes	maximal, characteristic
4	#13112928	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₂₆	—	2	1	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88058	C₂²	1	1	yes	characteristic
11	C₂×#88058	C₂²	1	2	yes	—
12	C₂×#88056	—	2	1	no	—
13	C₄×C₄⋊Dic₁₃	—	2	1	no	—
14	C₄×Dic₅₂	—	2	2	no	—
15	C₄×Dic₅₂	C₄	1	4	yes	complemented
16	C₄×C₁₃×C₁₆	—	2	2	no	—
17	#11718143	—	2	1	no	—
18	#11718095	C₄	1	8	yes	direct product factor
19	C₂²×Dic₂₆	—	2	1	no	—
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃	—	2	1	no	—
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×#6101	—	4	1	no	—
26	C₂×C₄×Dic₁₃	—	4	1	no	—
27	C₂×C₄×Dic₁₃	—	4	1	no	—
28	C₂×Dic₅₂	—	2	1	no	—
29	C₂×Dic₅₂	—	2	2	no	—
30	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
31	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
32	C₄×Dic₂₆	—	2	4	no	—
33	C₄×Dic₂₆	—	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	—
37	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₃	D₄	1	2	yes	—
39	#88071	—	4	1	no	—
40	#88058	—	2	1	no	—
41	#88058	—	2	2	no	—
42	#88058	C₂×C₄	1	4	yes	—
43	#88058	C₂×C₄	1	2	yes	—
44	#88056	—	2	4	no	—
45	C₄×#358	—	13	1	no	maximal, Sylow
46	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	—	4	1	no	—
50	C₂²×Dic₁₃	—	4	2	no	—
51	C₂²×Dic₁₃	—	4	1	no	—
52	C₂×Dic₂₆	—	2	2	no	—
53	C₂×Dic₂₆	—	2	4	no	—
54	C₂×Dic₂₆	—	4	1	no	—
55	C₂×Dic₂₆	—	4	2	no	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	#41	1	2	yes	—
62	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₁₃	SD₁₆	1	2	yes	—
64	C₄⋊Dic₁₃	—	2	2	no	—
65	C₄⋊Dic₁₃	—	2	4	no	—
66	C₄⋊Dic₁₃	—	4	1	no	—
67	C₄⋊Dic₁₃	—	4	2	no	—
68	#6101	—	4	4	no	—
69	C₄×Dic₁₃	—	4	4	no	—
70	C₄×Dic₁₃	—	8	2	no	—
71	Dic₅₂	—	2	2	no	—
72	Dic₅₂	—	2	4	no	—
73	Dic₅₂	C₄²	1	4	yes	complemented
74	C₁₃×C₁₆	—	2	4	no	—
75	C₂×C₄×Q₁₆	—	13	1	no	—
76	C₂×#358	—	13	2	no	—
77	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
78	C₄×#107	—	13	1	no	—
79	#1380	—	13	2	no	—
80	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	#102	1	2	yes	—
83	C₂×C₄×C₁₃	#102	1	2	yes	—
84	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
85	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
86	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
88	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#135	1	2	yes	—
90	C₂×Dic₁₃	—	4	2	no	—
91	C₂×Dic₁₃	—	4	4	no	—
92	C₂×Dic₁₃	—	4	2	no	—
93	C₂×Dic₁₃	—	4	4	no	—
94	C₂×Dic₁₃	—	8	2	no	—
95	C₂×Dic₁₃	—	8	2	no	—
96	C₂×Dic₁₃	—	8	2	no	—
97	Dic₂₆	—	2	4	no	—
98	Dic₂₆	—	4	2	no	—
99	Dic₂₆	—	4	4	no	—
100	C₈×C₁₃	—	2	2	no	—
101	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
102	C₈×C₁₃	#117	1	1	yes	characteristic
103	C₈×C₁₃	C₄×D₄	1	2	yes	—
104	C₂²×Q₁₆	—	13	1	no	—
105	C₂×C₄×Q₈	—	26	1	no	—
106	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
107	C₄×Q₁₆	—	13	4	no	—
108	C₄×Q₁₆	—	26	2	no	—
109	C₂×#107	—	13	2	no	—
110	C₂×#107	—	13	1	no	—
111	C₂×#103	—	26	1	no	—
112	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
113	C₄×C₄⋊C₄	—	26	1	no	—
114	#358	—	13	8	no	—
115	C₄×C₁₆	—	2	2	no	—
116	#340	—	26	1	no	—
117	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
118	C₂²×C₁₃	C₄×D₈	1	2	yes	—
119	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
120	C₂²×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
121	C₂²×C₁₃	#508	1	2	yes	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	#358	1	4	yes	complemented
124	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
125	C₄×C₁₃	#416	1	1	yes	characteristic
126	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
127	Dic₁₃	—	4	4	no	—
128	Dic₁₃	—	8	2	no	—
129	Dic₁₃	—	8	4	no	—
130	C₂²×Q₈	—	26	1	no	—
131	C₂×Q₁₆	—	13	4	no	—
132	C₂×Q₁₆	—	13	2	no	—
133	C₂×Q₁₆	—	26	2	no	—
134	C₂×Q₁₆	—	26	1	no	—
135	C₂²×C₈	D₅₂	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
137	C₄×Q₈	—	26	4	no	—
138	C₄×Q₈	—	52	2	no	—
139	C₂×C₄⋊C₄	—	26	1	no	—
140	C₂×C₄⋊C₄	—	26	2	no	—
141	C₂×C₄⋊C₄	—	52	1	no	—
142	C₂×C₄²	—	52	1	no	—
143	C₂×C₄²	—	52	1	no	—
144	C₂×C₄²	D₅₂	1	1	yes	characteristic
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	C₄×D₁₃	1	4	yes	—
148	#107	—	13	4	no	—
149	#107	—	13	2	no	—
150	#107	—	26	2	no	—
151	#107	—	26	1	no	—
152	#103	—	26	4	no	—
153	C₄×C₈	C₄×D₁₃	1	2	yes	—
154	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
155	#95	—	52	1	no	—
156	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
157	C₂×C₁₃	C₂×#358	1	1	yes	characteristic
158	C₂×C₁₃	#2138	1	1	yes	characteristic
159	C₂×C₁₃	C₄×SD₃₂	1	2	yes	—
160	C₂×C₁₃	C₄×Q₃₂	1	2	yes	—
161	C₂×Q₈	—	26	4	no	—
162	C₂×Q₈	—	26	2	no	—
163	C₂×Q₈	—	52	1	no	—
164	C₂×Q₈	—	52	2	no	—
165	C₂²×C₄	—	52	1	no	—
166	C₂²×C₄	—	52	2	no	—
167	C₂²×C₄	—	52	1	no	—
168	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
169	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
170	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
171	Q₁₆	—	13	4	no	—
172	Q₁₆	—	26	2	no	—
173	Q₁₆	—	26	4	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	#6103	1	2	yes	—
176	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
177	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
178	C₂×C₈	#6127	1	2	yes	—
179	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
180	C₄⋊C₄	—	26	4	no	—
181	C₄⋊C₄	—	26	2	no	—
182	C₄⋊C₄	—	52	1	no	—
183	C₄⋊C₄	—	52	2	no	—
184	C₄⋊C₄	—	52	4	no	—
185	C₄²	—	52	4	no	—
186	C₄²	—	104	2	no	—
187	C₄²	#6095	1	2	yes	—
188	C₄²	#6103	1	2	yes	—
189	C₁₆	—	2	4	no	—
190	C₁₃	C₄×#358	1	1	yes	characteristic, complemented, Sylow
191	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
192	Q₈	—	26	4	no	—
193	Q₈	—	52	2	no	—
194	Q₈	—	52	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	52	2	no	—
197	C₂×C₄	—	52	4	no	—
198	C₂×C₄	—	52	2	no	—
199	C₂×C₄	—	52	4	no	—
200	C₂×C₄	—	104	2	no	—
201	C₂×C₄	—	104	2	no	—
202	C₂×C₄	—	104	2	no	—
203	C₂×C₄	#88040	1	2	yes	—
204	C₂×C₄	Dic₁₀₄	1	2	yes	—
205	C₂×C₄	#88061	1	2	yes	—
206	C₂×C₄	#88061	1	2	yes	—
207	C₂×C₄	C₄×D₅₂	1	2	yes	—
208	C₂×C₄	C₂×#6095	1	1	yes	characteristic
209	C₂×C₄	#88158	1	2	yes	—
210	C₂×C₄	C₂×#6103	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	C₄²×D₁₃	1	1	yes	characteristic
213	C₈	#88126	1	1	yes	characteristic
214	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
215	C₂²	C₄×D₁₀₄	1	2	yes	—
216	C₂²	C₂×#88040	1	1	yes	characteristic
217	C₂²	#11718565	1	2	yes	—
218	C₂²	C₂×Dic₁₀₄	1	1	yes	characteristic
219	C₂²	C₂×#88061	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	52	4	no	—
222	C₄	—	104	4	no	—
223	C₄	—	104	2	no	—
224	C₄	#11718095	1	4	yes	direct product factor
225	C₄	C₄×#6095	1	2	yes	—
226	C₄	C₄×#6103	1	1	yes	characteristic
227	C₄	#11718774	1	1	yes	characteristic
228	C₂	C₄×#88061	1	1	yes	characteristic
229	C₂	C₂×#11718095	1	1	yes	characteristic
230	C₂	#13116544	1	1	yes	characteristic
231	C₂	C₄×#88040	1	2	yes	—
232	C₂	C₄×Dic₁₀₄	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, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	C₄×#358
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
15. C₄×Dic₅₂	C₄	semidirect
18. #11718095	C₄	direct (N × H)
73. Dic₅₂	C₄²	semidirect
123. C₄×C₁₃	#358	semidirect
190. C₁₃	C₄×#358	semidirect
224. C₄	#11718095	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×Dic₅₂	2
C₂×#11718095	2
C₄×#88058	2
#13112928	2
C₂×C₄×C₁₃×C₁₆	2
C₄×#358	13

Subgroups — C4×#11718095 #15778129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718095 #15778129