Subgroups — D₄×#6099 #13123573

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	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6099	C₂	1	2	yes	maximal, complemented
3	#11718868	C₂	1	2	yes	maximal, complemented
4	#11718673	C₂	1	2	yes	maximal, complemented
5	#11718672	C₂	1	4	yes	maximal, complemented
6	D₄×#1055	C₂	1	2	yes	maximal, complemented
7	#11718585	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#6099	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₁₃	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
11	C₄×D₄×C₁₃	C₄	1	2	yes	—
12	C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
13	C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
14	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₁₃	C₂²	1	2	yes	—
16	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
18	C₂×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×#6099	—	2	8	no	—
21	C₂×#6099	C₂²	1	2	yes	—
22	C₂×#6099	C₂²	1	1	yes	characteristic
23	C₂×#6099	C₂²	1	1	yes	characteristic
24	C₂×#6099	C₂²	1	4	yes	complemented
25	C₂²×#1055	C₂²	1	4	yes	complemented
26	C₂²⋊#1055	C₂²	1	4	yes	complemented
27	C₂²⋊#1055	C₂²	1	4	yes	complemented
28	C₄⋊#1055	C₂²	1	2	yes	complemented
29	C₄⋊#1055	C₂²	1	2	yes	complemented
30	#88043	C₂²	1	2	yes	complemented
31	C₄×#1055	C₂²	1	2	yes	complemented
32	C₂⁴×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₁₃	—	2	4	no	—
34	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	—	2	4	no	—
38	C₂²×C₄×C₁₃	—	2	4	no	—
39	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
51	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
52	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
53	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
54	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	#6099	—	2	4	no	—
59	#6099	—	2	2	no	—
60	#6099	—	2	8	no	—
61	#6099	D₄	1	4	yes	direct product factor
62	C₂×#1055	—	2	8	no	—
63	C₂×#1055	C₂³	1	2	yes	—
64	C₂×#1055	C₂³	1	4	yes	—
65	C₂×#1055	C₂³	1	2	yes	—
66	D₄×M₁₆	—	13	1	no	maximal, Sylow
67	C₂³×C₁₃	—	2	4	no	—
68	C₂³×C₁₃	—	2	4	no	—
69	C₂³×C₁₃	—	4	2	no	—
70	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
73	D₄×C₁₃	—	2	4	no	—
74	D₄×C₁₃	—	2	2	no	—
75	D₄×C₁₃	M₁₆	1	4	yes	complemented
76	C₂×C₄×C₁₃	—	2	4	no	—
77	C₂×C₄×C₁₃	—	2	4	no	—
78	C₂×C₄×C₁₃	—	2	4	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	4	1	no	—
85	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₃	#41	1	2	yes	—
94	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	#1055	—	2	4	no	—
96	#1055	—	2	2	no	—
97	#1055	C₂×D₄	1	4	yes	complemented
98	C₂²×M₁₆	—	13	2	no	—
99	C₂×C₄×D₄	D₁₃	1	1	yes	characteristic
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	1	no	—
108	C₂²×C₁₃	—	2	1	no	—
109	C₂²×C₁₃	—	2	2	no	—
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	—	2	2	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	4	8	no	—
114	C₂²×C₁₃	C₄×D₄	1	2	yes	—
115	C₂²×C₁₃	C₂×M₁₆	1	4	yes	complemented
116	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₄×C₁₃	—	2	2	no	—
118	C₄×C₁₃	—	2	2	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₄×C₁₃	C₂×M₁₆	1	2	yes	complemented
124	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂²×D₄	Dic₁₃	1	1	yes	characteristic
127	C₂³×C₄	C₂×D₁₃	1	2	yes	—
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	Dic₁₃	1	2	yes	—
135	C₄×D₄	Dic₁₃	1	1	yes	characteristic
136	C₄×D₄	Dic₁₃	1	1	yes	characteristic
137	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
138	C₄×D₄	C₂×D₁₃	1	2	yes	—
139	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
140	C₂×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	Dic₁₃	1	2	yes	—
142	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
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	2	no	—
150	C₂×C₁₃	—	2	4	no	—
151	C₂×C₁₃	—	4	2	no	—
152	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
154	C₂×C₁₃	#568	1	1	yes	characteristic
155	C₂⁴	C₂×Dic₁₃	1	2	yes	—
156	C₂×D₄	—	2	4	no	—
157	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
158	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
159	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
163	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
164	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
165	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
166	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
168	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
169	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
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	C₂×Dic₁₃	1	2	yes	—
179	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
180	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
181	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
182	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
183	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
184	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
185	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
186	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
187	C₄²	C₂²×D₁₃	1	1	yes	characteristic
188	C₄²	C₂²×D₁₃	1	1	yes	characteristic
189	C₁₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	4	2	no	—
193	C₂³	C₂²×Dic₁₃	1	2	yes	—
194	C₂³	C₂²×Dic₁₃	1	2	yes	—
195	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
196	D₄	—	2	4	no	—
197	D₄	—	2	2	no	—
198	D₄	#6099	1	4	yes	direct product factor
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	D₄×D₁₃	1	2	yes	—
209	C₂×C₄	#6129	1	2	yes	—
210	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
212	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
213	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
214	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
215	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
216	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
217	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
218	C₈	—	13	4	no	—
219	C₈	—	26	4	no	—
220	C₈	—	26	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	8	no	—
229	C₂²	C₂×#6099	1	4	yes	complemented
230	C₂²	D₄×Dic₁₃	1	2	yes	—
231	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	C₂×#6099	1	2	yes	complemented
238	C₄	D₄×Dic₁₃	1	2	yes	—
239	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
240	C₄	C₂×#6129	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	4	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂²×#6099	1	1	yes	characteristic
245	C₂	C₂×D₄×Dic₁₃	1	1	yes	characteristic
246	C₂	#11719524	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂²×#6099	C₂	semidirect
3. #11718868	C₂	semidirect
4. #11718673	C₂	semidirect
5. #11718672	C₂	semidirect
6. D₄×#1055	C₂	semidirect
7. #11718585	C₂	semidirect
8. C₄×#6099	C₂	semidirect
24. C₂×#6099	C₂²	semidirect
25. C₂²×#1055	C₂²	semidirect
26. C₂²⋊#1055	C₂²	semidirect
27. C₂²⋊#1055	C₂²	semidirect
28. C₄⋊#1055	C₂²	semidirect
29. C₄⋊#1055	C₂²	semidirect
30. #88043	C₂²	semidirect
31. C₄×#1055	C₂²	semidirect
61. #6099	D₄	direct (N × H)
75. D₄×C₁₃	M₁₆	semidirect
97. #1055	C₂×D₄	semidirect
115. C₂²×C₁₃	C₂×M₁₆	semidirect
123. C₄×C₁₃	C₂×M₁₆	semidirect
189. C₁₃	D₄×M₁₆	semidirect
198. D₄	#6099	direct (N × H)
229. C₂²	C₂×#6099	semidirect
237. C₄	C₂×#6099	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₁₃	2
C₂²×#6099	2
#11718868	2
#11718673	2
#11718672	2
D₄×#1055	2
#11718585	2
C₄×#6099	2
D₄×M₁₆	13

Subgroups — D4×#6099 #13123573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#6099 #13123573