Subgroups — C₄×#11718103 #15778238

Order 3328.

398 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718103	C₂	1	2	yes	maximal
2	C₄×#88052	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	#13112917	C₂	1	2	yes	maximal
5	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88052	C₂²	1	2	yes	—
13	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic
15	C₁₃×#346	—	2	1	no	—
16	C₄×C₁₃×C₁₆	—	2	1	no	—
17	#11718103	C₄	1	8	yes	direct product factor
18	#11718053	—	2	1	no	—
19	C₄×#6090	—	2	1	no	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	—	2	2	no	—
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₁₃×M₃₂	—	2	2	no	—
29	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
30	C₂×C₁₃×C₁₆	—	2	2	no	—
31	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
35	C₈×Dic₁₃	—	2	2	no	—
36	C₈×Dic₁₃	—	2	2	no	—
37	C₈×Dic₁₃	C₈	1	4	yes	—
38	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
39	#88052	—	2	2	no	—
40	#88052	C₂×C₄	1	4	yes	—
41	C₂×#6090	—	2	2	no	—
42	C₄×#1055	C₈	1	4	yes	—
43	C₄×#364	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	—	2	1	no	—
47	C₂²×Dic₁₃	—	2	1	no	—
48	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₁₃	#41	1	2	yes	—
56	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₁₃	C₂×C₈	1	2	yes	—
59	C₄×Dic₁₃	—	2	8	no	—
60	C₄×Dic₁₃	—	2	1	no	—
61	C₄×Dic₁₃	—	2	1	no	—
62	C₄×Dic₁₃	C₄²	1	2	yes	—
63	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
64	C₂×#1055	—	2	1	no	—
65	C₂×#1055	—	2	1	no	—
66	C₂×#1055	C₄²	1	2	yes	—
67	C₂×#1055	C₂×C₈	1	2	yes	—
68	C₂×#1055	C₂×C₈	1	2	yes	—
69	C₂×#1055	C₂×C₈	1	2	yes	—
70	C₁₃×C₁₆	—	2	4	no	—
71	#6090	—	2	4	no	—
72	C₂×#364	—	13	2	no	—
73	C₄×M₃₂	—	13	1	no	—
74	C₄×M₃₂	D₁₃	1	1	yes	characteristic
75	C₄²×C₈	—	13	1	no	—
76	#1376	—	13	2	no	—
77	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃	—	2	4	no	—
79	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×Dic₁₃	—	2	4	no	—
88	C₂×Dic₁₃	—	2	4	no	—
89	C₂×Dic₁₃	—	2	2	no	—
90	C₂×Dic₁₃	—	2	2	no	—
91	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
92	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
93	C₈×C₁₃	C₄×D₄	1	2	yes	—
94	C₈×C₁₃	C₄×Q₈	1	2	yes	—
95	#1055	—	2	2	no	—
96	#1055	C₄×C₈	1	4	yes	—
97	C₂×M₃₂	—	13	2	no	—
98	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
99	C₂×C₄×C₈	—	13	1	no	—
100	C₂×C₄×C₈	—	13	1	no	—
101	C₂×C₄×C₈	—	13	1	no	—
102	C₂×C₄×C₈	—	13	1	no	—
103	C₂×C₄×C₈	—	13	1	no	—
104	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
105	C₄³	—	13	1	no	—
106	#364	—	13	8	no	—
107	#346	—	2	1	no	—
108	#346	—	26	1	no	—
109	C₄×C₁₆	—	2	1	no	—
110	C₄×C₁₆	—	26	1	no	—
111	C₂²×C₁₃	—	2	2	no	—
112	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
113	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₂²×C₁₃	#431	1	1	yes	characteristic
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	#364	1	4	yes	complemented
117	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
119	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₄×C₁₃	#431	1	1	yes	characteristic
121	Dic₁₃	—	2	4	no	—
122	Dic₁₃	—	2	4	no	—
123	C₂²×C₈	—	13	1	no	—
124	C₂²×C₈	—	13	1	no	—
125	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
127	C₂×C₄²	—	13	1	no	—
128	C₂×C₄²	—	13	1	no	—
129	C₂×C₄²	—	26	2	no	—
130	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
131	M₃₂	—	2	2	no	—
132	M₃₂	—	13	4	no	—
133	M₃₂	—	26	2	no	—
134	M₃₂	C₄×D₁₃	1	4	yes	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	26	2	no	—
137	C₄×C₈	—	13	4	no	—
138	C₄×C₈	—	13	4	no	—
139	C₄×C₈	—	13	4	no	—
140	C₄×C₈	—	26	2	no	—
141	C₄×C₈	—	26	2	no	—
142	C₄×C₈	Dic₂₆	1	1	yes	characteristic
143	C₄×C₈	Dic₂₆	1	1	yes	characteristic
144	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
146	C₂×C₁₃	—	2	1	no	—
147	C₂×C₁₃	—	2	1	no	—
148	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
149	C₂×C₁₃	C₂×#364	1	2	yes	—
150	C₂²×C₄	—	26	1	no	—
151	C₂²×C₄	—	26	1	no	—
152	C₂²×C₄	#6094	1	2	yes	—
153	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
154	C₂×C₈	—	13	2	no	—
155	C₂×C₈	—	13	2	no	—
156	C₂×C₈	—	13	2	no	—
157	C₂×C₈	—	13	2	no	—
158	C₂×C₈	—	26	1	no	—
159	C₂×C₈	—	26	1	no	—
160	C₂×C₈	C₈×D₁₃	1	2	yes	—
161	C₂×C₈	#6094	1	2	yes	—
162	C₂×C₈	#6101	1	1	yes	characteristic
163	C₂×C₈	#6101	1	1	yes	characteristic
164	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
165	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
166	C₂×C₈	#6127	1	2	yes	—
167	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
168	C₄²	—	13	2	no	—
169	C₄²	—	13	2	no	—
170	C₄²	—	26	8	no	—
171	C₄²	—	26	1	no	—
172	C₄²	—	26	1	no	—
173	C₄²	C₈×D₁₃	1	2	yes	—
174	C₄²	#6101	1	1	yes	characteristic
175	C₄²	#6101	1	1	yes	characteristic
176	C₁₆	—	2	4	no	—
177	C₁₆	—	26	4	no	—
178	C₁₃	C₄×#364	1	1	yes	characteristic, complemented, Sylow
179	C₂³	C₂×#6094	1	1	yes	characteristic
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	26	4	no	—
182	C₂×C₄	—	26	4	no	—
183	C₂×C₄	—	26	2	no	—
184	C₂×C₄	—	26	2	no	—
185	C₂×C₄	#88054	1	2	yes	—
186	C₂×C₄	#88054	1	2	yes	—
187	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
188	C₂×C₄	#88126	1	1	yes	characteristic
189	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
190	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
191	C₂×C₄	C₂×#6094	1	1	yes	characteristic
192	C₂×C₄	C₂×#6101	1	1	yes	characteristic
193	C₈	—	13	4	no	—
194	C₈	—	26	2	no	—
195	C₈	#88054	1	2	yes	—
196	C₈	#88054	1	2	yes	—
197	C₈	C₄×Dic₂₆	1	2	yes	—
198	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	#11718346	1	1	yes	characteristic
201	C₂²	C₄×#6094	1	1	yes	characteristic
202	C₂²	C₂×#88054	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	26	4	no	—
205	C₄	—	26	4	no	—
206	C₄	#11718103	1	4	yes	direct product factor
207	C₄	C₄×C₈×D₁₃	1	1	yes	characteristic
208	C₄	#11718346	1	1	yes	characteristic
209	C₄	C₄×#6101	1	1	yes	characteristic
210	C₄	C₂×#88054	1	1	yes	characteristic
211	C₂	—	2	1	no	—
212	C₂	—	2	1	no	—
213	C₂	C₄×#88054	1	1	yes	characteristic
214	C₂	C₂×#11718103	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₂×M₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	C₄×#364
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
17. #11718103	C₄	direct (N × H)
116. C₄×C₁₃	#364	semidirect
178. C₁₃	C₄×#364	semidirect
206. C₄	#11718103	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×#11718103	2
C₄×#88052	2
C₄×C₈×Dic₁₃	2
#13112917	2
C₄×C₁₃×M₃₂	2
C₄×#364	13

Subgroups — C4×#11718103 #15778238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718103 #15778238