Subgroups — C₂×D₄×#11719910 #28328291

Order 13568.

806 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719910	C₂	1	2	yes	maximal, complemented
2	D₄×#11719910	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#11719910	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊#11719910	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×#88322	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719910	—	2	8	no	—
8	C₂²×#11719910	C₂²	1	2	yes	—
9	C₂²×#11719910	C₂²	1	1	yes	characteristic
10	C₂²×#11719910	C₂²	1	1	yes	characteristic
11	C₂²×#11719910	C₂²	1	8	yes	complemented
12	C₄⋊#11719910	C₂²	1	4	yes	complemented
13	C₂²⋊#11719910	C₂²	1	8	yes	complemented
14	C₄×#11719910	C₂²	1	4	yes	complemented
15	C₂³×#88322	C₂²	1	2	yes	—
16	D₄×#88322	C₄	1	4	yes	—
17	D₄×#88322	C₂²	1	4	yes	—
18	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
19	C₂×C₂²⋊#88322	C₄	1	2	yes	—
20	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
22	C₂×#11719910	—	2	4	no	—
23	C₂×#11719910	—	2	2	no	—
24	C₂×#11719910	—	2	16	no	—
25	C₂×#11719910	D₄	1	4	yes	direct product factor
26	C₂×#11719910	C₂³	1	4	yes	—
27	C₂×#11719910	C₂³	1	2	yes	—
28	C₂×#11719910	C₂³	1	2	yes	—
29	C₂²×#88322	—	2	4	no	—
30	C₂²×#88322	—	2	4	no	—
31	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
32	C₂²×#88322	C₂×C₄	1	2	yes	—
33	C₂²×#88322	C₂×C₄	1	4	yes	—
34	C₂²×#88322	C₂³	1	4	yes	—
35	C₂²×#88322	C₂³	1	1	yes	characteristic
36	C₄⋊#88322	C₂×C₄	1	4	yes	—
37	C₂²⋊#88322	C₂×C₄	1	8	yes	—
38	C₄×#88322	C₂×C₄	1	2	yes	—
39	C₄×#88322	C₂³	1	2	yes	—
40	C₂²×D₄×C₅₃	C₈	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
42	C₄×D₄×C₅₃	C₈	1	4	yes	—
43	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂⁴×C₅₃	C₂×C₈	1	2	yes	—
48	C₂×D₄×C₅₃	C₁₆	1	8	yes	complemented
49	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	4	no	—
52	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
59	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
60	C₄²×C₅₃	C₂×C₈	1	2	yes	—
61	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	—	2	8	no	—
65	C₂×#88322	—	2	4	no	—
66	C₂×#88322	—	2	4	no	—
67	C₂×#88322	C₂²×C₄	1	2	yes	—
68	C₂×#88322	C₂²×C₄	1	4	yes	—
69	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
70	C₂×#88322	C₂×D₄	1	2	yes	—
71	C₂×#88322	#41	1	2	yes	—
72	C₂×#88322	C₂⁴	1	1	yes	characteristic
73	#11719910	—	2	4	no	—
74	#11719910	—	2	2	no	—
75	#11719910	C₂×D₄	1	4	yes	direct product factor
76	C₂³×C₅₃	—	2	4	no	—
77	C₂³×C₅₃	—	2	4	no	—
78	C₂³×C₅₃	C₂×C₁₆	1	8	yes	complemented
79	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
80	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
81	D₄×C₅₃	C₂×C₁₆	1	16	yes	complemented
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	2	8	no	—
87	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	complemented
88	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₅₃	#131	1	2	yes	—
94	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
95	#88322	—	2	4	no	—
96	#88322	—	2	2	no	—
97	#88322	C₄×D₄	1	2	yes	—
98	#88322	C₂²×D₄	1	1	yes	characteristic
99	#88322	C₂×#41	1	1	yes	characteristic
100	C₂×D₄×C₁₆	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	4	no	—
102	C₂²×C₅₃	—	2	4	no	—
103	C₂²×C₅₃	—	2	16	no	—
104	C₂²×C₅₃	C₈×D₄	1	2	yes	—
105	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
106	C₂²×C₅₃	C₂²×C₁₆	1	8	yes	complemented
107	C₂²×C₅₃	#504	1	2	yes	—
108	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	—	2	4	no	—
111	C₄×C₅₃	C₈×D₄	1	2	yes	—
112	C₄×C₅₃	C₂²×C₁₆	1	4	yes	complemented
113	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
114	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
115	C₂³×C₁₆	—	53	2	no	—
116	C₂×C₈×D₄	—	53	1	no	—
117	D₄×C₁₆	—	53	8	no	—
118	C₂×C₄⋊C₁₆	—	53	1	no	—
119	C₂×C₂²⋊C₁₆	—	53	2	no	—
120	C₂×C₄×C₁₆	—	53	1	no	—
121	C₂×C₅₃	—	2	8	no	—
122	C₂×C₅₃	D₄×C₁₆	1	4	yes	complemented
123	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
124	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
125	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
126	C₂³×C₈	—	53	2	no	—
127	C₂×C₄×D₄	#6157	1	1	yes	characteristic
128	C₂²×C₁₆	—	53	1	no	—
129	C₂²×C₁₆	—	53	1	no	—
130	C₂²×C₁₆	—	53	2	no	—
131	C₂²×C₁₆	—	53	8	no	—
132	C₂²×C₁₆	—	106	8	no	—
133	C₈×D₄	—	53	4	no	—
134	C₈×D₄	—	53	4	no	—
135	C₂×C₄⋊C₈	—	53	1	no	—
136	C₂×C₂²⋊C₈	—	53	2	no	—
137	C₂×C₄×C₈	—	53	1	no	—
138	C₄⋊C₁₆	—	53	4	no	—
139	C₂²⋊C₁₆	—	53	8	no	—
140	C₄×C₁₆	—	53	4	no	—
141	C₅₃	C₂×D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
142	C₂²×D₄	#88324	1	1	yes	characteristic
143	C₂³×C₄	C₂×#6157	1	2	yes	—
144	C₂²×C₈	—	53	1	no	—
145	C₂²×C₈	—	53	4	no	—
146	C₂²×C₈	—	53	4	no	—
147	C₂²×C₈	—	53	2	no	—
148	C₂²×C₈	—	53	1	no	—
149	C₂²×C₈	—	106	4	no	—
150	C₂²×C₈	—	106	4	no	—
151	C₄×D₄	#88324	1	4	yes	—
152	C₄×D₄	C₂×#6157	1	4	yes	—
153	C₂×C₄⋊C₄	#88324	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	#88324	1	2	yes	—
155	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
156	C₂×C₁₆	—	53	2	no	—
157	C₂×C₁₆	—	53	4	no	—
158	C₂×C₁₆	—	53	2	no	—
159	C₂×C₁₆	—	53	4	no	—
160	C₂×C₁₆	—	106	16	no	—
161	C₂×C₁₆	—	106	2	no	—
162	C₂×C₁₆	—	106	4	no	—
163	C₄⋊C₈	—	53	4	no	—
164	C₂²⋊C₈	—	53	8	no	—
165	C₄×C₈	—	53	2	no	—
166	C₄×C₈	—	53	2	no	—
167	C₂⁴	C₂×#88324	1	2	yes	—
168	C₂×D₄	#11719910	1	8	yes	direct product factor
169	C₂×D₄	C₂×#88324	1	4	yes	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
173	C₂²×C₄	C₂×#88324	1	2	yes	—
174	C₂²×C₄	C₂×#88324	1	4	yes	—
175	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
176	C₂²×C₄	C₂²×#6157	1	4	yes	—
177	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
178	C₂×C₈	—	53	2	no	—
179	C₂×C₈	—	53	2	no	—
180	C₂×C₈	—	53	1	no	—
181	C₂×C₈	—	53	1	no	—
182	C₂×C₈	—	53	2	no	—
183	C₂×C₈	—	53	4	no	—
184	C₂×C₈	—	106	4	no	—
185	C₂×C₈	—	106	8	no	—
186	C₂×C₈	—	106	4	no	—
187	C₂×C₈	—	106	4	no	—
188	C₂×C₈	—	106	2	no	—
189	C₄⋊C₄	C₂×#88324	1	4	yes	—
190	C₂²⋊C₄	C₂×#88324	1	8	yes	—
191	C₄²	C₂×#88324	1	2	yes	—
192	C₄²	C₂²×#6157	1	2	yes	—
193	C₁₆	—	53	4	no	—
194	C₁₆	—	106	4	no	—
195	C₁₆	—	106	2	no	—
196	C₂³	—	2	4	no	—
197	C₂³	—	2	4	no	—
198	C₂³	C₂×#11719910	1	8	yes	complemented
199	C₂³	C₂²×#88324	1	4	yes	—
200	C₂³	C₂²×#88324	1	1	yes	characteristic
201	D₄	C₂×#11719910	1	16	yes	direct product factor
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	C₂×#11719910	1	4	yes	complemented
208	C₂×C₄	D₄×#6157	1	2	yes	—
209	C₂×C₄	C₂²×#88324	1	2	yes	—
210	C₂×C₄	C₂²×#88324	1	4	yes	—
211	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
212	C₂×C₄	C₂²×#88324	1	2	yes	—
213	C₂×C₄	#13133304	1	2	yes	—
214	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
215	C₈	—	53	2	no	—
216	C₈	—	53	1	no	—
217	C₈	—	53	1	no	—
218	C₈	—	106	4	no	—
219	C₈	—	106	2	no	—
220	C₂²	—	2	16	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	D₄×#88324	1	2	yes	—
224	C₂²	C₂²×#11719910	1	2	yes	—
225	C₂²	C₂²×#11719910	1	8	yes	complemented
226	C₂²	#16811681	1	2	yes	—
227	C₂²	C₂³×#88324	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	D₄×#88324	1	2	yes	—
231	C₄	C₂²×#11719910	1	4	yes	complemented
232	C₄	C₂×D₄×#6157	1	1	yes	characteristic
233	C₄	C₂×#13133304	1	1	yes	characteristic
234	C₂	—	2	8	no	—
235	C₂	D₄×#11719910	1	4	yes	direct product factor
236	C₂	C₂×D₄×#88324	1	1	yes	characteristic
237	C₂	C₂³×#11719910	1	1	yes	characteristic
238	C₂	C₂×#16811681	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×D₄×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×C₄×D₄	#6157
hypocenter	C₅₃	C₂×D₄×C₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#11719910	C₂	semidirect
2. D₄×#11719910	C₂	direct (N × H)
3. C₂×C₄⋊#11719910	C₂	semidirect
4. C₂×C₂²⋊#11719910	C₂	semidirect
5. C₂×C₄×#11719910	C₂	semidirect
11. C₂²×#11719910	C₂²	semidirect
12. C₄⋊#11719910	C₂²	semidirect
13. C₂²⋊#11719910	C₂²	semidirect
14. C₄×#11719910	C₂²	semidirect
25. C₂×#11719910	D₄	direct (N × H)
48. C₂×D₄×C₅₃	C₁₆	semidirect
75. #11719910	C₂×D₄	direct (N × H)
78. C₂³×C₅₃	C₂×C₁₆	semidirect
81. D₄×C₅₃	C₂×C₁₆	semidirect
87. C₂×C₄×C₅₃	C₂×C₁₆	semidirect
106. C₂²×C₅₃	C₂²×C₁₆	semidirect
112. C₄×C₅₃	C₂²×C₁₆	semidirect
122. C₂×C₅₃	D₄×C₁₆	semidirect
141. C₅₃	C₂×D₄×C₁₆	semidirect
168. C₂×D₄	#11719910	direct (N × H)
198. C₂³	C₂×#11719910	semidirect
201. D₄	C₂×#11719910	direct (N × H)
207. C₂×C₄	C₂×#11719910	semidirect
225. C₂²	C₂²×#11719910	semidirect
231. C₄	C₂²×#11719910	semidirect
235. C₂	D₄×#11719910	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719910	2
D₄×#11719910	2
C₂×C₄⋊#11719910	2
C₂×C₂²⋊#11719910	2
C₂×C₄×#11719910	2
C₂×D₄×#88322	2
C₂×D₄×C₁₆	53

Subgroups — C2×D4×#11719910 #28328291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×#11719910 #28328291