Subgroups — #14574274

Order 2432.

362 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927165	C₂	1	2	yes	maximal, complemented
2	#11926949	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926928	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926904	C₂	1	2	yes	maximal, complemented
5	#11926790	C₂	1	2	yes	maximal, complemented
6	#11926785	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926780	C₂	1	1	yes	maximal, characteristic
8	#11926646	C₂	1	2	yes	maximal, complemented
9	C₄×Dic₇₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₁₅₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₉×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#117	C₂²	1	1	yes	characteristic
15	#10597376	C₂²	1	2	yes	—
16	#10597370	C₂²	1	2	yes	—
17	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
18	C₂×D₁₅₂	C₂²	1	1	yes	characteristic
19	C₂×#64516	C₂²	1	2	yes	—
20	#10597344	—	2	2	no	—
21	#10597342	—	2	2	no	—
22	#10597331	—	2	2	no	—
23	#10597329	—	2	2	no	—
24	#10597321	C₂²	1	2	yes	—
25	C₄×D₇₆	C₂²	1	2	yes	—
26	#10597315	C₂²	1	2	yes	—
27	C₄×Dic₃₈	C₂²	1	2	yes	—
28	C₁₉×#97	C₂²	1	1	yes	characteristic
29	C₁₉×#97	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
31	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
32	#10597278	C₂²	1	2	yes	—
33	#10597278	C₂²	1	2	yes	—
34	#10597275	C₂²	1	1	yes	characteristic
35	#10597275	C₂²	1	2	yes	—
36	#10597275	C₂²	1	1	yes	characteristic
37	#10597273	C₂²	1	2	yes	—
38	#10597273	C₂²	1	2	yes	—
39	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₁₉	—	2	2	no	—
41	C₂×D₇₆	C₂³	1	2	yes	—
42	C₂×C₄×D₁₉	—	2	2	no	—
43	C₂×Dic₃₈	—	2	2	no	—
44	C₂×Dic₃₈	C₂³	1	2	yes	—
45	M₁₆×C₁₉	—	2	2	no	—
46	C₂×C₈×C₁₉	—	2	2	no	—
47	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
52	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
53	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
54	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
55	C₄²×C₁₉	C₂³	1	1	yes	characteristic
56	C₄²×C₁₉	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₁₉	—	2	2	no	—
58	(C₂×D₁₉)⋊C₄	—	2	2	no	—
59	(C₂×D₁₉)⋊C₄	—	2	2	no	—
60	(C₂×D₁₉)⋊C₄	—	2	2	no	—
61	(C₂×D₁₉)⋊C₄	—	2	2	no	—
62	C₄⋊Dic₁₉	—	2	2	no	—
63	C₄⋊Dic₁₉	C₂³	1	2	yes	—
64	C₄⋊Dic₁₉	C₂³	1	2	yes	—
65	Dic₁₉⋊C₄	—	2	2	no	—
66	Dic₁₉⋊C₄	—	2	2	no	—
67	C₄×Dic₁₉	—	2	2	no	—
68	Dic₇₆	—	2	2	no	—
69	D₁₅₂	—	2	2	no	—
70	#64516	—	2	4	no	—
71	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×D₁₉	—	2	2	no	—
73	C₂×C₄×C₁₉	—	2	1	no	—
74	C₂×C₄×C₁₉	—	2	1	no	—
75	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₁₉	—	8	2	no	—
82	C₂×Dic₁₉	—	2	2	no	—
83	C₂×Dic₁₉	—	2	2	no	—
84	C₂×Dic₁₉	—	2	2	no	—
85	D₇₆	—	2	2	no	—
86	D₇₆	—	2	2	no	—
87	C₄×D₁₉	—	8	2	no	—
88	Dic₃₈	—	2	2	no	—
89	Dic₃₈	—	2	2	no	—
90	Dic₃₈	—	8	2	no	—
91	C₈×C₁₉	—	2	1	no	—
92	C₈×C₁₉	—	2	1	no	—
93	C₈×C₁₉	#41	1	2	yes	—
94	C₈×C₁₉	#41	1	2	yes	—
95	#3175	—	19	1	no	maximal, Sylow
96	C₂²×C₁₉	—	2	1	no	—
97	C₂²×C₁₉	—	2	1	no	—
98	C₂²×C₁₉	—	2	1	no	—
99	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
100	C₂×D₁₉	—	4	2	no	—
101	C₂×D₁₉	—	4	2	no	—
102	C₂×D₁₉	—	4	2	no	—
103	C₄×C₁₉	—	2	1	no	—
104	C₄×C₁₉	—	2	1	no	—
105	C₄×C₁₉	—	2	1	no	—
106	C₄×C₁₉	—	2	1	no	—
107	C₄×C₁₉	—	2	1	no	—
108	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
109	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
110	Dic₁₉	—	4	2	no	—
111	Dic₁₉	—	4	2	no	—
112	Dic₁₉	—	4	2	no	—
113	#542	—	19	2	no	—
114	#500	—	19	1	no	—
115	#495	—	19	1	no	—
116	#489	—	19	2	no	—
117	#468	—	19	2	no	—
118	#467	—	19	1	no	—
119	#466	—	19	1	no	—
120	#440	—	19	2	no	—
121	C₄×Q₁₆	—	19	1	no	—
122	C₄×D₈	—	19	1	no	—
123	#405	D₁₉	1	1	yes	characteristic, complemented
124	C₂×C₁₉	—	4	1	no	—
125	C₂×C₁₉	#532	1	1	yes	characteristic
126	C₂×C₁₉	#576	1	1	yes	characteristic
127	C₂×C₁₉	#578	1	1	yes	characteristic
128	D₁₉	—	8	2	no	—
129	C₂×Q₁₆	—	19	1	no	—
130	C₂×SD₁₆	—	19	2	no	—
131	C₂×D₈	—	19	1	no	—
132	C₂×M₁₆	C₂×D₁₉	1	1	yes	characteristic
133	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
134	C₄⋊Q₈	—	19	2	no	—
135	#126	—	38	2	no	—
136	#124	—	19	2	no	—
137	#124	—	38	2	no	—
138	#123	—	38	2	no	—
139	C₂²⋊Q₈	—	19	2	no	—
140	C₂²⋊Q₈	—	38	2	no	—
141	#121	—	19	2	no	—
142	C₄×Q₈	—	19	2	no	—
143	C₄×D₄	—	19	2	no	—
144	#117	C₂×D₁₉	1	1	yes	characteristic
145	#107	—	19	1	no	—
146	#107	—	19	1	no	—
147	#107	—	19	2	no	—
148	#103	—	19	2	no	—
149	#103	—	19	2	no	—
150	#102	—	19	2	no	—
151	#102	—	19	2	no	—
152	#97	C₂×D₁₉	1	1	yes	characteristic
153	#97	C₂×D₁₉	1	1	yes	characteristic
154	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
155	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
156	C₁₉	#3175	1	1	yes	characteristic, complemented, Sylow
157	C₂×Q₈	—	19	2	no	—
158	C₂×Q₈	—	38	2	no	—
159	C₂×D₄	—	19	2	no	—
160	C₂×D₄	—	38	2	no	—
161	C₂²×C₄	—	38	2	no	—
162	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
163	Q₁₆	—	38	2	no	—
164	SD₁₆	—	38	4	no	—
165	D₈	—	38	2	no	—
166	M₁₆	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
169	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
170	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
172	C₄⋊C₄	—	19	2	no	—
173	C₄⋊C₄	—	19	2	no	—
174	C₄⋊C₄	—	38	2	no	—
175	C₄⋊C₄	—	38	2	no	—
176	C₄⋊C₄	—	38	2	no	—
177	C₄⋊C₄	D₇₆	1	1	yes	characteristic
178	C₄⋊C₄	D₇₆	1	1	yes	characteristic
179	C₂²⋊C₄	—	38	2	no	—
180	C₂²⋊C₄	—	38	2	no	—
181	C₂²⋊C₄	—	38	2	no	—
182	C₂²⋊C₄	—	38	2	no	—
183	C₂²⋊C₄	—	38	2	no	—
184	C₂²⋊C₄	D₇₆	1	1	yes	characteristic
185	C₂²⋊C₄	D₇₆	1	1	yes	characteristic
186	C₄²	—	38	2	no	—
187	C₄²	C₂²×D₁₉	1	1	yes	characteristic
188	C₄²	C₂²×D₁₉	1	1	yes	characteristic
189	C₂³	—	38	2	no	—
190	C₂³	C₂×D₇₆	1	1	yes	characteristic
191	Q₈	—	38	2	no	—
192	Q₈	—	38	2	no	—
193	Q₈	—	152	2	no	—
194	D₄	—	38	2	no	—
195	D₄	—	38	2	no	—
196	D₄	—	152	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	38	2	no	—
200	C₂×C₄	—	38	2	no	—
201	C₂×C₄	—	38	2	no	—
202	C₂×C₄	—	152	2	no	—
203	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
204	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
205	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
206	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
207	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	#64541	1	2	yes	—
212	C₈	#64541	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	76	2	no	—
217	C₂²	—	76	2	no	—
218	C₂²	—	76	2	no	—
219	C₂²	C₂²×D₇₆	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	76	2	no	—
226	C₄	—	76	2	no	—
227	C₄	—	76	2	no	—
228	C₄	C₂×#64541	1	1	yes	characteristic
229	C₄	C₂×#64541	1	1	yes	characteristic
230	C₂	—	4	1	no	—
231	C₂	—	152	2	no	—
232	C₂	#11927094	1	1	yes	characteristic
233	C₂	#11927349	1	1	yes	characteristic
234	C₂	#11927361	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₆
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#405	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	#11926780	C₂
hypercenter	#405	D₁₉
hypocenter	C₁₉	#3175
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
1. #11927165	C₂	semidirect
2. #11926949	C₂	semidirect
3. #11926928	C₂	semidirect
4. #11926904	C₂	semidirect
5. #11926790	C₂	semidirect
6. #11926785	C₂	semidirect
8. #11926646	C₂	semidirect
9. C₄×Dic₇₆	C₂	semidirect
10. C₄×D₁₅₂	C₂	semidirect
11. C₁₉×#405	C₂	semidirect
123. #405	D₁₉	semidirect
156. C₁₉	#3175	semidirect

Maximal subgroups

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

subgroup	index
#11927165	2
#11926949	2
#11926928	2
#11926904	2
#11926790	2
#11926785	2
#11926780	2
#11926646	2
C₄×Dic₇₆	2
C₄×D₁₅₂	2
C₁₉×#405	2
#3175	19