Subgroups — #14577448

Order 2432.

1060 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₃₈	C₂	1	2	yes	maximal, complemented
2	#11927270	C₂	1	2	yes	maximal, complemented
3	#11927253	C₂	1	6	yes	maximal, complemented
4	D₄×Dic₃₈	C₂	1	6	yes	maximal, complemented
5	#11927175	C₂	1	3	yes	maximal, complemented
6	#11927172	C₂	1	3	yes	maximal, complemented
7	#11927097	C₂	1	3	yes	maximal, complemented
8	C₂×#10597334	C₂	1	3	yes	maximal, complemented
9	#11927027	C₂	1	1	yes	maximal, characteristic
10	#41×Dic₁₉	C₂	1	1	yes	maximal, characteristic
11	C₁₉×#520	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₃₈	C₂²	1	6	yes	—
14	C₄×Q₈×C₁₉	C₂²	1	2	yes	—
15	C₄×D₄×C₁₉	C₂²	1	6	yes	—
16	C₁₉×#117	C₂²	1	3	yes	—
17	C₂×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
18	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
19	Q₈×Dic₁₉	C₂²	1	2	yes	—
20	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
21	D₄×Dic₁₉	C₂²	1	3	yes	—
22	D₄×Dic₁₉	C₂²	1	6	yes	—
23	D₄×Dic₁₉	C₂²	1	3	yes	—
24	#10597372	C₂²	1	3	yes	—
25	#10597372	C₂²	1	3	yes	—
26	C₂×C₄⋊Dic₁₉	C₂²	1	3	yes	—
27	#10597370	C₂²	1	12	yes	—
28	C₂×Dic₁₉⋊C₄	C₂²	1	6	yes	—
29	C₂×C₄×Dic₁₉	C₂²	1	3	yes	—
30	#10597336	C₂²	1	3	yes	—
31	#10597336	C₂²	1	3	yes	—
32	#10597336	C₂²	1	6	yes	—
33	#10597334	C₂²	1	3	yes	—
34	#10597334	C₂²	1	6	yes	—
35	#10597334	C₂²	1	6	yes	—
36	#10597334	C₂²	1	6	yes	—
37	#10597334	C₂²	1	1	yes	characteristic
38	#10597334	C₂²	1	6	yes	—
39	#10597324	C₂²	1	6	yes	—
40	#10597324	C₂²	1	6	yes	—
41	#10597324	C₂²	1	12	yes	—
42	#10597316	C₂²	1	6	yes	—
43	#10597315	C₂²	1	6	yes	—
44	C₄×Dic₃₈	C₂²	1	6	yes	—
45	C₄×Dic₃₈	C₂²	1	6	yes	—
46	#41×C₁₉	Q₈	1	8	yes	complemented
47	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₉	C₂³	1	3	yes	—
49	C₂²×C₄×C₁₉	C₂³	1	6	yes	—
50	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
51	C₂²×Dic₁₉	C₂³	1	3	yes	—
52	C₂²×Dic₁₉	C₂³	1	3	yes	—
53	C₂²×Dic₁₉	C₂³	1	6	yes	—
54	C₂×Dic₃₈	—	4	12	no	—
55	C₂×Dic₃₈	C₂³	1	6	yes	—
56	C₂×Dic₃₈	C₂³	1	2	yes	—
57	C₂×Dic₃₈	C₂³	1	12	yes	—
58	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
59	C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
60	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂²⋊C₄×C₁₉	C₂³	1	6	yes	—
62	C₄²×C₁₉	C₂³	1	6	yes	—
63	C₂²⋊Dic₁₉	C₂³	1	6	yes	—
64	C₂²⋊Dic₁₉	C₂³	1	6	yes	—
65	C₄⋊Dic₁₉	C₂³	1	3	yes	—
66	C₄⋊Dic₁₉	C₂³	1	6	yes	—
67	C₄⋊Dic₁₉	C₂³	1	3	yes	—
68	C₄⋊Dic₁₉	C₂³	1	6	yes	—
69	C₄⋊Dic₁₉	C₂³	1	6	yes	—
70	C₄⋊Dic₁₉	C₂³	1	3	yes	—
71	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
72	Dic₁₉⋊C₄	C₂³	1	2	yes	—
73	Dic₁₉⋊C₄	C₂³	1	6	yes	—
74	Dic₁₉⋊C₄	C₂³	1	6	yes	—
75	Dic₁₉⋊C₄	C₂³	1	6	yes	—
76	Dic₁₉⋊C₄	C₂³	1	12	yes	—
77	C₄×Dic₁₉	C₂³	1	3	yes	—
78	C₄×Dic₁₉	C₂³	1	3	yes	—
79	C₄×Dic₁₉	C₂³	1	6	yes	—
80	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
81	C₄×Dic₁₉	C₂³	1	3	yes	—
82	C₂³×C₁₉	C₂⁴	1	3	yes	—
83	Q₈×C₁₉	C₂×Q₈	1	4	yes	complemented
84	D₄×C₁₉	C₂×Q₈	1	12	yes	complemented
85	C₂×C₄×C₁₉	—	4	6	no	—
86	C₂×C₄×C₁₉	C₂×Q₈	1	12	yes	complemented
87	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
88	C₂×C₄×C₁₉	C₂⁴	1	6	yes	—
89	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
90	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
91	C₂×Dic₁₉	—	2	6	no	—
92	C₂×Dic₁₉	—	2	6	no	—
93	C₂×Dic₁₉	—	4	6	no	—
94	C₂×Dic₁₉	C₂⁴	1	2	yes	—
95	C₂×Dic₁₉	C₂⁴	1	3	yes	—
96	C₂×Dic₁₉	C₂⁴	1	3	yes	—
97	C₂×Dic₁₉	C₂⁴	1	6	yes	—
98	C₂×Dic₁₉	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₁₉	C₂⁴	1	1	yes	characteristic
100	Dic₃₈	—	2	12	no	—
101	Dic₃₈	—	2	4	no	—
102	Dic₃₈	—	4	12	no	—
103	#3503	—	19	1	no	maximal, Sylow
104	C₂²×C₁₉	—	2	3	no	—
105	C₂²×C₁₉	—	2	3	no	—
106	C₂²×C₁₉	C₂²×Q₈	1	6	yes	—
107	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
108	C₄×C₁₉	—	2	6	no	—
109	C₄×C₁₉	—	2	2	no	—
110	C₄×C₁₉	C₂²×Q₈	1	2	yes	—
111	C₄×C₁₉	C₂²×Q₈	1	6	yes	—
112	Dic₁₉	—	2	6	no	—
113	Dic₁₉	—	2	3	no	—
114	Dic₁₉	—	2	2	no	—
115	Dic₁₉	—	2	3	no	—
116	Dic₁₉	H(5,𝔽₂)	1	2	yes	—
117	Dic₁₉	2₋¹⁺⁴	1	2	yes	—
118	Q₈²	—	19	2	no	—
119	#557	—	19	2	no	—
120	#554	—	19	6	no	—
121	D₄×Q₈	—	19	6	no	—
122	#544	—	19	3	no	—
123	#544	—	19	3	no	—
124	#533	—	19	3	no	—
125	C₂×C₄⋊Q₈	—	19	3	no	—
126	#520	—	19	1	no	—
127	#520	D₁₉	1	1	yes	characteristic
128	C₄×#41	—	19	1	no	—
129	C₂×C₁₉	—	2	6	no	—
130	C₂×C₁₉	C₂³×Q₈	1	1	yes	characteristic
131	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
134	C₂²×Q₈	—	19	6	no	—
135	C₄⋊Q₈	—	19	3	no	—
136	C₄⋊Q₈	—	19	6	no	—
137	C₄⋊Q₈	—	19	6	no	—
138	C₄⋊Q₈	—	19	6	no	—
139	C₄⋊Q₈	—	19	1	no	—
140	C₄⋊Q₈	—	19	6	no	—
141	C₄⋊Q₈	—	19	6	no	—
142	#125	—	19	3	no	—
143	#125	—	19	3	no	—
144	#125	—	19	6	no	—
145	#125	—	19	6	no	—
146	C₂²⋊Q₈	—	19	12	no	—
147	C₂²⋊Q₈	—	19	6	no	—
148	C₂²⋊Q₈	—	19	6	no	—
149	C₂²⋊Q₈	—	19	12	no	—
150	C₄×Q₈	—	19	1	no	—
151	C₄×Q₈	—	19	1	no	—
152	C₄×Q₈	—	19	2	no	—
153	C₄×Q₈	—	19	6	no	—
154	C₄×Q₈	—	19	6	no	—
155	C₄×Q₈	C₂×D₁₉	1	2	yes	—
156	C₄×D₄	—	19	3	no	—
157	C₄×D₄	—	19	3	no	—
158	C₄×D₄	—	19	6	no	—
159	C₄×D₄	C₂×D₁₉	1	6	yes	—
160	#117	—	19	3	no	—
161	#117	—	19	3	no	—
162	#117	C₂×D₁₉	1	3	yes	—
163	C₂×C₄⋊C₄	—	19	3	no	—
164	C₂×C₄⋊C₄	—	19	6	no	—
165	C₂×C₄⋊C₄	C₂×D₁₉	1	3	yes	—
166	C₂×C₄²	—	19	3	no	—
167	C₁₉	#3503	1	1	yes	characteristic, complemented, Sylow
168	#41	Dic₃₈	1	8	yes	complemented
169	C₂×Q₈	—	19	2	no	—
170	C₂×Q₈	—	19	6	no	—
171	C₂×Q₈	—	19	12	no	—
172	C₂×Q₈	—	76	12	no	—
173	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
174	C₂×D₄	C₂²×D₁₉	1	3	yes	—
175	C₂²×C₄	—	19	3	no	—
176	C₂²×C₄	—	19	3	no	—
177	C₂²×C₄	—	19	6	no	—
178	C₂²×C₄	C₂²×D₁₉	1	6	yes	—
179	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
180	C₄⋊C₄	—	19	6	no	—
181	C₄⋊C₄	—	19	12	no	—
182	C₄⋊C₄	—	19	2	no	—
183	C₄⋊C₄	—	19	6	no	—
184	C₄⋊C₄	—	19	6	no	—
185	C₄⋊C₄	—	19	3	no	—
186	C₄⋊C₄	—	19	6	no	—
187	C₄⋊C₄	—	19	1	no	—
188	C₄⋊C₄	—	19	3	no	—
189	C₄⋊C₄	—	19	3	no	—
190	C₄⋊C₄	—	19	6	no	—
191	C₄⋊C₄	—	19	6	no	—
192	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
193	C₄⋊C₄	C₂²×D₁₉	1	6	yes	—
194	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
195	C₂²⋊C₄	—	19	6	no	—
196	C₂²⋊C₄	—	19	6	no	—
197	C₂²⋊C₄	C₂²×D₁₉	1	6	yes	—
198	C₄²	—	19	1	no	—
199	C₄²	—	19	3	no	—
200	C₄²	—	19	3	no	—
201	C₄²	—	19	3	no	—
202	C₄²	—	19	6	no	—
203	C₄²	C₂²×D₁₉	1	6	yes	—
204	C₂³	C₂³×D₁₉	1	3	yes	—
205	Q₈	—	38	12	no	—
206	Q₈	—	38	4	no	—
207	Q₈	—	76	12	no	—
208	Q₈	C₂×Dic₃₈	1	4	yes	complemented
209	D₄	C₂×Dic₃₈	1	12	yes	complemented
210	C₂×C₄	—	4	6	no	—
211	C₂×C₄	—	19	1	no	—
212	C₂×C₄	—	19	1	no	—
213	C₂×C₄	—	19	2	no	—
214	C₂×C₄	—	19	3	no	—
215	C₂×C₄	—	19	3	no	—
216	C₂×C₄	—	19	6	no	—
217	C₂×C₄	—	38	6	no	—
218	C₂×C₄	—	38	6	no	—
219	C₂×C₄	—	76	6	no	—
220	C₂×C₄	C₂×Dic₃₈	1	12	yes	complemented
221	C₂×C₄	C₂³×D₁₉	1	2	yes	—
222	C₂×C₄	C₂³×D₁₉	1	6	yes	—
223	C₂×C₄	C₂³×D₁₉	1	3	yes	—
224	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
225	C₂²	—	2	3	no	—
226	C₂²	—	2	3	no	—
227	C₂²	C₂²×Dic₃₈	1	6	yes	—
228	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
229	C₄	—	2	6	no	—
230	C₄	—	2	2	no	—
231	C₄	—	19	2	no	—
232	C₄	—	19	2	no	—
233	C₄	—	38	3	no	—
234	C₄	—	38	6	no	—
235	C₄	—	38	3	no	—
236	C₄	—	38	2	no	—
237	C₄	C₂²×Dic₃₈	1	2	yes	—
238	C₄	C₂²×Dic₃₈	1	6	yes	—
239	C₂	—	2	6	no	—
240	C₂	C₂³×Dic₃₈	1	1	yes	characteristic
241	C₂	H(5,𝔽₂)×D₁₉	1	1	yes	characteristic
242	C₂	2₋¹⁺⁴×D₁₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₁₉×#520	C₂
prime radical	C₂×#41×C₁₉	C₂²
hypercenter	#520	D₁₉
hypocenter	C₁₉	#3503
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #520

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₃₈	C₂	semidirect
2. #11927270	C₂	semidirect
3. #11927253	C₂	semidirect
4. D₄×Dic₃₈	C₂	semidirect
5. #11927175	C₂	semidirect
6. #11927172	C₂	semidirect
7. #11927097	C₂	semidirect
8. C₂×#10597334	C₂	semidirect
46. #41×C₁₉	Q₈	semidirect
83. Q₈×C₁₉	C₂×Q₈	semidirect
84. D₄×C₁₉	C₂×Q₈	semidirect
86. C₂×C₄×C₁₉	C₂×Q₈	semidirect
167. C₁₉	#3503	semidirect
168. #41	Dic₃₈	semidirect
208. Q₈	C₂×Dic₃₈	semidirect
209. D₄	C₂×Dic₃₈	semidirect
220. C₂×C₄	C₂×Dic₃₈	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₃₈	2
#11927270	2
#11927253	2
D₄×Dic₃₈	2
#11927175	2
#11927172	2
#11927097	2
C₂×#10597334	2
#11927027	2
#41×Dic₁₉	2
C₁₉×#520	2
#3503	19