Subgroups — D₄×#6118 #13131573

Order 1664.

544 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6118	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₁₃	C₂	1	1	yes	maximal, characteristic
3	D₄×#1057	C₂	1	2	yes	maximal, complemented
4	#11719127	C₂	1	2	yes	maximal, complemented
5	#11719076	C₂	1	2	yes	maximal, complemented
6	#11719064	C₂	1	4	yes	maximal, complemented
7	#11719052	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#6118	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄×D₁₃	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
11	C₂²×#1057	C₂²	1	4	yes	complemented
12	C₂×#6118	—	2	8	no	—
13	C₂×#6118	C₂²	1	1	yes	characteristic
14	C₂×#6118	C₂²	1	2	yes	—
15	C₂×#6118	C₂²	1	1	yes	characteristic
16	C₂×#6118	C₂²	1	4	yes	complemented
17	C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
18	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
19	C₄×C₂²⋊D₁₃	C₄	1	2	yes	—
20	#88147	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×D₁₃	C₄	1	1	yes	characteristic
22	#88135	C₂²	1	2	yes	—
23	C₂²⋊C₄×D₁₃	C₄	1	2	yes	—
24	C₄×D₅₂	C₄	1	1	yes	characteristic
25	C₄²×D₁₃	C₂²	1	1	yes	characteristic
26	C₂²⋊#1057	C₂²	1	4	yes	complemented
27	#88111	C₂²	1	4	yes	complemented
28	#88110	C₂²	1	2	yes	complemented
29	C₄⋊#1057	C₂²	1	2	yes	complemented
30	C₄⋊#1057	C₂²	1	2	yes	complemented
31	C₄×#1057	C₂²	1	2	yes	complemented
32	C₂³×D₁₃	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
36	C₂²×Dic₁₃	C₂³	1	2	yes	—
37	D₄×D₁₃	—	2	4	no	—
38	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₁₃	—	2	4	no	—
40	C₂×C₄×D₁₃	—	2	4	no	—
41	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₃	C₂³	1	2	yes	—
46	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
47	C₂×#1057	—	2	8	no	—
48	C₂×#1057	C₂³	1	2	yes	—
49	C₂×#1057	C₂³	1	4	yes	—
50	C₂×#1057	C₂³	1	2	yes	—
51	#6118	—	2	4	no	—
52	#6118	—	2	2	no	—
53	#6118	—	2	8	no	—
54	#6118	D₄	1	4	yes	direct product factor
55	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
57	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
58	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
59	#6103	C₂×C₄	1	2	yes	—
60	#6103	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
62	#6101	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
65	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
66	D₄×M₁₆	—	13	1	no	maximal, Sylow
67	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×D₁₃	—	2	4	no	—
69	C₂²×D₁₃	—	2	4	no	—
70	C₂²×D₁₃	—	4	2	no	—
71	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
72	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
73	D₄×C₁₃	M₁₆	1	4	yes	complemented
74	C₂×C₄×C₁₃	—	2	4	no	—
75	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²⋊D₁₃	—	2	4	no	—
80	C₂×Dic₁₃	—	2	2	no	—
81	C₂×Dic₁₃	—	2	2	no	—
82	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
86	D₅₂	—	2	2	no	—
87	C₄×D₁₃	—	2	4	no	—
88	C₄×D₁₃	—	2	4	no	—
89	C₄×D₁₃	—	2	2	no	—
90	C₄×D₁₃	—	2	2	no	—
91	C₄×D₁₃	—	2	2	no	—
92	C₄×D₁₃	—	4	1	no	—
93	C₄×D₁₃	C₂×D₄	1	2	yes	—
94	C₄×D₁₃	#41	1	2	yes	—
95	#1057	—	2	4	no	—
96	#1057	—	2	2	no	—
97	#1057	C₂×D₄	1	4	yes	complemented
98	C₂²×M₁₆	—	13	2	no	—
99	C₂×C₄×D₄	—	13	1	no	—
100	#436	—	13	2	no	—
101	#435	—	13	4	no	—
102	C₈×D₄	—	13	2	no	—
103	#423	—	13	1	no	—
104	#407	—	13	2	no	—
105	C₄×M₁₆	—	13	1	no	—
106	C₂²×C₁₃	—	2	2	no	—
107	C₂²×C₁₃	—	2	2	no	—
108	C₂²×C₁₃	C₂×M₁₆	1	4	yes	complemented
109	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×D₁₃	—	2	1	no	—
111	C₂×D₁₃	—	2	1	no	—
112	C₂×D₁₃	—	2	2	no	—
113	C₂×D₁₃	—	2	2	no	—
114	C₂×D₁₃	—	2	2	no	—
115	C₂×D₁₃	—	4	8	no	—
116	C₂×D₁₃	C₄×D₄	1	2	yes	—
117	C₄×C₁₃	—	2	2	no	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	C₄×D₄	1	2	yes	—
120	C₄×C₁₃	C₂×M₁₆	1	2	yes	complemented
121	Dic₁₃	—	2	2	no	—
122	Dic₁₃	—	2	1	no	—
123	Dic₁₃	—	2	1	no	—
124	Dic₁₃	C₂²×D₄	1	1	yes	characteristic
125	Dic₁₃	C₂×#41	1	1	yes	characteristic
126	C₂²×D₄	—	13	1	no	—
127	C₂³×C₄	—	13	2	no	—
128	C₂×M₁₆	—	13	1	no	—
129	C₂×M₁₆	—	13	2	no	—
130	C₂×M₁₆	—	13	4	no	—
131	C₂×M₁₆	—	13	1	no	—
132	C₂×M₁₆	—	26	8	no	—
133	C₂²×C₈	—	13	4	no	—
134	C₄×D₄	—	13	1	no	—
135	C₄×D₄	—	13	2	no	—
136	C₄×D₄	—	13	1	no	—
137	C₄×D₄	—	13	2	no	—
138	C₄×D₄	—	13	1	no	—
139	C₄×D₄	#261	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	13	1	no	—
141	C₂×C₂²⋊C₄	—	13	2	no	—
142	C₂×C₄²	—	13	1	no	—
143	C₄⋊C₈	—	13	2	no	—
144	C₄⋊C₈	—	13	2	no	—
145	C₂²⋊C₈	—	13	4	no	—
146	C₂²⋊C₈	—	13	4	no	—
147	#97	—	13	2	no	—
148	C₄×C₈	—	13	2	no	—
149	C₂×C₁₃	—	2	4	no	—
150	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
152	C₂×C₁₃	#568	1	1	yes	characteristic
153	D₁₃	—	2	2	no	—
154	D₁₃	—	4	2	no	—
155	C₂⁴	—	13	2	no	—
156	C₂×D₄	—	13	2	no	—
157	C₂×D₄	—	13	1	no	—
158	C₂×D₄	—	26	4	no	—
159	C₂×D₄	C₂×#261	1	1	yes	characteristic
160	C₂²×C₄	—	13	1	no	—
161	C₂²×C₄	—	13	2	no	—
162	C₂²×C₄	—	13	2	no	—
163	C₂²×C₄	—	13	1	no	—
164	C₂²×C₄	—	13	2	no	—
165	C₂²×C₄	—	13	2	no	—
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	—	26	4	no	—
168	C₂²×C₄	—	26	4	no	—
169	C₂²×C₄	C₂×#261	1	2	yes	—
170	M₁₆	—	13	4	no	—
171	M₁₆	—	26	8	no	—
172	M₁₆	—	26	4	no	—
173	M₁₆	—	26	2	no	—
174	C₂×C₈	—	13	2	no	—
175	C₂×C₈	—	13	2	no	—
176	C₂×C₈	—	13	4	no	—
177	C₂×C₈	—	26	8	no	—
178	C₄⋊C₄	—	13	2	no	—
179	C₄⋊C₄	—	13	1	no	—
180	C₄⋊C₄	C₂×#261	1	1	yes	characteristic
181	C₂²⋊C₄	—	13	2	no	—
182	C₂²⋊C₄	—	13	2	no	—
183	C₂²⋊C₄	—	13	2	no	—
184	C₂²⋊C₄	C₂×#261	1	2	yes	—
185	C₄²	—	13	1	no	—
186	C₄²	—	13	1	no	—
187	C₄²	—	13	1	no	—
188	C₄²	C₂×#261	1	1	yes	characteristic
189	C₁₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
190	C₂³	—	13	1	no	—
191	C₂³	—	13	2	no	—
192	C₂³	—	26	4	no	—
193	C₂³	—	26	4	no	—
194	C₂³	—	52	2	no	—
195	C₂³	C₂²×#261	1	2	yes	—
196	D₄	—	26	4	no	—
197	D₄	—	26	2	no	—
198	D₄	#6118	1	4	yes	direct product factor
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	13	1	no	—
201	C₂×C₄	—	13	2	no	—
202	C₂×C₄	—	13	2	no	—
203	C₂×C₄	—	13	1	no	—
204	C₂×C₄	—	13	1	no	—
205	C₂×C₄	—	13	2	no	—
206	C₂×C₄	—	26	2	no	—
207	C₂×C₄	—	26	2	no	—
208	C₂×C₄	—	26	4	no	—
209	C₂×C₄	—	26	2	no	—
210	C₂×C₄	—	26	2	no	—
211	C₂×C₄	—	26	2	no	—
212	C₂×C₄	—	26	4	no	—
213	C₂×C₄	—	52	1	no	—
214	C₂×C₄	C₂²×#261	1	1	yes	characteristic
215	C₂×C₄	C₂²×#261	1	2	yes	—
216	C₂×C₄	C₂²×#261	1	1	yes	characteristic
217	C₂×C₄	C₂²×#261	1	1	yes	characteristic
218	C₈	—	13	4	no	—
219	C₈	—	26	4	no	—
220	C₈	—	26	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	13	2	no	—
224	C₂²	—	26	1	no	—
225	C₂²	—	26	1	no	—
226	C₂²	—	26	2	no	—
227	C₂²	—	26	2	no	—
228	C₂²	—	26	2	no	—
229	C₂²	—	52	8	no	—
230	C₂²	C₂×#6118	1	4	yes	complemented
231	C₂²	C₂³×#261	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	—	13	1	no	—
235	C₄	—	13	1	no	—
236	C₄	—	26	1	no	—
237	C₄	—	26	2	no	—
238	C₄	—	26	1	no	—
239	C₄	C₂×#6118	1	2	yes	complemented
240	C₄	D₄×#261	1	2	yes	—
241	C₂	—	2	4	no	—
242	C₂	—	26	2	no	—
243	C₂	—	52	2	no	—
244	C₂	C₂²×#6118	1	1	yes	characteristic
245	C₂	C₂×D₄×#261	1	1	yes	characteristic
246	C₂	#11719629	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₄×D₄×C₁₃	C₄
prime radical	C₂×D₄×D₁₃	C₄
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	C₄×D₄	#261
hypocenter	C₁₃	D₄×M₁₆
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#6118	C₂	semidirect
3. D₄×#1057	C₂	semidirect
4. #11719127	C₂	semidirect
5. #11719076	C₂	semidirect
6. #11719064	C₂	semidirect
7. #11719052	C₂	semidirect
8. C₄×#6118	C₂	semidirect
11. C₂²×#1057	C₂²	semidirect
16. C₂×#6118	C₂²	semidirect
26. C₂²⋊#1057	C₂²	semidirect
27. #88111	C₂²	semidirect
28. #88110	C₂²	semidirect
29. C₄⋊#1057	C₂²	semidirect
30. C₄⋊#1057	C₂²	semidirect
31. C₄×#1057	C₂²	semidirect
54. #6118	D₄	direct (N × H)
73. D₄×C₁₃	M₁₆	semidirect
97. #1057	C₂×D₄	semidirect
108. C₂²×C₁₃	C₂×M₁₆	semidirect
120. C₄×C₁₃	C₂×M₁₆	semidirect
189. C₁₃	D₄×M₁₆	semidirect
198. D₄	#6118	direct (N × H)
230. C₂²	C₂×#6118	semidirect
239. C₄	C₂×#6118	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#6118	2
C₄×D₄×D₁₃	2
D₄×#1057	2
#11719127	2
#11719076	2
#11719064	2
#11719052	2
C₄×#6118	2
D₄×M₁₆	13

Subgroups — D4×#6118 #13131573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#6118 #13131573