Subgroups — C₂×C₄×#88057 #15860958

Order 3328.

1358 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88057	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊Dic₁₃	C₂	1	2	yes	maximal
3	C₄×#88057	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718143	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
10	C₂²×C₄×Dic₁₃	—	2	2	no	—
11	C₂×C₄×C₈×C₁₃	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
13	C₂×#88071	—	2	2	no	—
14	C₂×#88057	C₄	1	16	yes	direct product factor
15	C₂×#88057	C₂²	1	8	yes	—
16	C₄×C₄⋊Dic₁₃	—	2	4	no	—
17	C₄×C₄⋊Dic₁₃	C₂²	1	8	yes	—
18	#11718143	C₂²	1	8	yes	—
19	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂³×Dic₁₃	—	2	4	no	—
23	C₂²×C₈×C₁₃	—	2	4	no	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₃	D₄	1	2	yes	—
31	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
33	C₂×C₄⋊Dic₁₃	—	2	4	no	—
34	C₂×C₄⋊Dic₁₃	—	2	8	no	—
35	C₂×C₄⋊Dic₁₃	C₂×C₄	1	16	yes	—
36	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
37	C₂×C₄×Dic₁₃	—	2	8	no	—
38	C₂×C₄×Dic₁₃	—	4	8	no	—
39	C₄×C₈×C₁₃	C₂×C₄	1	16	yes	complemented
40	#88071	—	2	8	no	—
41	#88057	C₂×C₄	1	32	yes	direct product factor
42	C₂×C₄×#106	—	13	1	no	maximal, Sylow
43	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₁₃	SD₁₆	1	8	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₁₃	#41	1	4	yes	—
55	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	2	16	no	—
57	C₂²×Dic₁₃	—	2	8	no	—
58	C₂²×Dic₁₃	—	4	8	no	—
59	C₂²×Dic₁₃	—	4	8	no	—
60	C₂×C₈×C₁₃	—	2	8	no	—
61	C₂×C₈×C₁₃	—	2	4	no	—
62	C₂×C₈×C₁₃	C₄²	1	16	yes	complemented
63	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
65	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
66	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
67	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
68	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
69	C₄⋊Dic₁₃	—	2	16	no	—
70	C₄⋊Dic₁₃	—	2	8	no	—
71	C₄⋊Dic₁₃	C₂²×C₄	1	16	yes	—
72	C₄×Dic₁₃	—	4	16	no	—
73	C₂²×#106	—	13	2	no	—
74	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
75	C₂×C₄×C₄⋊C₄	—	13	2	no	—
76	C₄×#106	—	13	8	no	—
77	C₂×#340	—	13	2	no	—
78	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂³×C₁₃	C₄×D₄	1	4	yes	—
80	C₂³×C₁₃	C₂×SD₁₆	1	4	yes	—
81	C₂³×C₁₃	#135	1	4	yes	—
82	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	—	2	4	no	—
84	C₂×C₄×C₁₃	—	2	8	no	—
85	C₂×C₄×C₁₃	#106	1	16	yes	complemented
86	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
87	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
90	C₂×C₄×C₁₃	#117	1	4	yes	—
91	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₁₃	C₄×Q₈	1	8	yes	—
93	C₂×C₄×C₁₃	C₂×SD₁₆	1	8	yes	—
94	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
96	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
98	C₂×Dic₁₃	—	2	16	no	—
99	C₂×Dic₁₃	—	4	8	no	—
100	C₂×Dic₁₃	—	4	8	no	—
101	C₂×Dic₁₃	—	4	32	no	—
102	C₈×C₁₃	—	2	4	no	—
103	C₈×C₁₃	C₂×C₄²	1	8	yes	complemented
104	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
105	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
106	C₂²×C₄⋊C₄	—	13	4	no	—
107	C₂²×C₄²	—	26	2	no	—
108	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
109	C₂×#106	—	13	16	no	—
110	C₂×#106	—	13	8	no	—
111	C₂×C₄×C₈	Dic₁₃	1	8	yes	complemented
112	C₂×C₄×C₈	C₂×D₁₃	1	4	yes	—
113	C₄×C₄⋊C₄	—	13	8	no	—
114	C₄×C₄⋊C₄	—	26	4	no	—
115	C₂×#95	—	26	2	no	—
116	#340	—	13	8	no	—
117	C₂²×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
118	C₂²×C₁₃	C₂×#106	1	4	yes	—
119	C₂²×C₁₃	#427	1	4	yes	—
120	C₂²×C₁₃	C₄×SD₁₆	1	16	yes	—
121	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₁₃	C₂×C₄×D₄	1	2	yes	—
123	C₂²×C₁₃	C₂²×SD₁₆	1	2	yes	—
124	C₂²×C₁₃	C₂×#135	1	2	yes	—
125	C₄×C₁₃	—	2	4	no	—
126	C₄×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
127	C₄×C₁₃	C₂×#106	1	8	yes	complemented
128	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
129	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
130	C₄×C₁₃	C₂×C₄×Q₈	1	2	yes	—
131	Dic₁₃	—	4	16	no	—
132	C₂³×C₄	—	26	4	no	—
133	C₂³×C₄	D₅₂	1	1	yes	characteristic
134	C₂³×C₄	D₅₂	1	1	yes	characteristic
135	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
136	C₂²×C₈	—	2	4	no	—
137	C₂²×C₈	C₄×D₁₃	1	8	yes	—
138	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
139	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
140	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
141	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
142	C₂×C₄⋊C₄	—	13	16	no	—
143	C₂×C₄⋊C₄	—	13	8	no	—
144	C₂×C₄⋊C₄	—	26	4	no	—
145	C₂×C₄⋊C₄	—	26	8	no	—
146	C₂×C₄²	—	26	8	no	—
147	C₂×C₄²	—	52	8	no	—
148	C₂×C₄²	Dic₂₆	1	4	yes	—
149	C₂×C₄²	D₅₂	1	2	yes	—
150	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
151	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
152	#106	—	13	32	no	—
153	C₄×C₈	C₂×Dic₁₃	1	16	yes	complemented
154	#95	—	26	8	no	—
155	C₂×C₁₃	C₄×#106	1	8	yes	complemented
156	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₁₃	C₂²×#106	1	1	yes	characteristic
158	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
159	C₂×C₁₃	C₂×C₄×SD₁₆	1	4	yes	—
160	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	26	16	no	—
163	C₂²×C₄	—	26	8	no	—
164	C₂²×C₄	—	52	8	no	—
165	C₂²×C₄	—	52	8	no	—
166	C₂²×C₄	#6095	1	8	yes	—
167	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
168	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
169	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
170	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
171	C₂²×C₄	C₂×D₅₂	1	2	yes	—
172	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₅₂	1	2	yes	—
174	C₂²×C₄	#6127	1	4	yes	—
175	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
176	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
177	C₂×C₈	—	2	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	C₄×Dic₁₃	1	16	yes	complemented
180	C₂×C₈	C₂×C₄×D₁₃	1	8	yes	—
181	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
182	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
183	C₄⋊C₄	—	13	16	no	—
184	C₄⋊C₄	—	26	16	no	—
185	C₄⋊C₄	—	26	8	no	—
186	C₄²	—	52	16	no	—
187	C₄²	C₄⋊Dic₁₃	1	8	yes	—
188	C₄²	C₂×Dic₂₆	1	4	yes	—
189	C₄²	C₂²×Dic₁₃	1	4	yes	—
190	C₁₃	C₂×C₄×#106	1	1	yes	characteristic, complemented, Sylow
191	C₂³	C₄×D₅₂	1	4	yes	—
192	C₂³	C₂×#6095	1	4	yes	—
193	C₂³	#88158	1	4	yes	—
194	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
195	C₂³	C₂²×D₅₂	1	1	yes	characteristic
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	26	16	no	—
199	C₂×C₄	—	52	8	no	—
200	C₂×C₄	—	52	8	no	—
201	C₂×C₄	—	52	32	no	—
202	C₂×C₄	#88057	1	16	yes	direct product factor
203	C₂×C₄	C₄×Dic₂₆	1	8	yes	—
204	C₂×C₄	C₄×D₅₂	1	4	yes	—
205	C₂×C₄	C₂×#6095	1	8	yes	—
206	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
210	C₂×C₄	#88180	1	4	yes	—
211	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
213	C₂×C₄	C₂×#6127	1	2	yes	—
214	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	C₂×C₄×Dic₁₃	1	8	yes	complemented
217	C₂²	C₄×#6095	1	16	yes	—
218	C₂²	C₄×C₄⋊Dic₁₃	1	4	yes	—
219	C₂²	C₂×#88057	1	4	yes	—
220	C₂²	#11718767	1	4	yes	—
221	C₂²	C₂×C₄×D₅₂	1	2	yes	—
222	C₂²	C₂²×#6095	1	2	yes	—
223	C₂²	C₂×#88158	1	2	yes	—
224	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	52	16	no	—
227	C₄	C₄×C₄⋊Dic₁₃	1	4	yes	—
228	C₄	C₂×#88057	1	8	yes	direct product factor
229	C₄	C₂×C₄×Dic₂₆	1	2	yes	—
230	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
231	C₄	C₂×#88180	1	1	yes	characteristic
232	C₂	C₄×#88057	1	8	yes	direct product factor
233	C₂	C₂×C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
234	C₂	C₂²×#88057	1	1	yes	characteristic
235	C₂	C₂×#11718767	1	1	yes	characteristic
236	C₂	C₂×C₄×#6095	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	D₅₂
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×C₄×#106
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#88057	C₂	direct (N × H)
11. C₂×C₄×C₈×C₁₃	C₄	semidirect
14. C₂×#88057	C₄	direct (N × H)
39. C₄×C₈×C₁₃	C₂×C₄	semidirect
41. #88057	C₂×C₄	direct (N × H)
62. C₂×C₈×C₁₃	C₄²	semidirect
85. C₂×C₄×C₁₃	#106	semidirect
103. C₈×C₁₃	C₂×C₄²	semidirect
111. C₂×C₄×C₈	Dic₁₃	semidirect
127. C₄×C₁₃	C₂×#106	semidirect
153. C₄×C₈	C₂×Dic₁₃	semidirect
155. C₂×C₁₃	C₄×#106	semidirect
179. C₂×C₈	C₄×Dic₁₃	semidirect
190. C₁₃	C₂×C₄×#106	semidirect
202. C₂×C₄	#88057	direct (N × H)
216. C₈	C₂×C₄×Dic₁₃	semidirect
228. C₄	C₂×#88057	direct (N × H)
232. C₂	C₄×#88057	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#88057	2
C₂×C₄×C₄⋊Dic₁₃	2
C₄×#88057	2
C₂×#11718143	2
C₂²×C₄×C₈×C₁₃	2
C₂×C₄×#106	13

Subgroups — C2×C4×#88057 #15860958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#88057 #15860958