Subgroups — #14577255

Order 2432.

1094 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927267	C₂	1	3	yes	maximal, complemented
2	#11927232	C₂	1	6	yes	maximal, complemented
3	Q₈×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#11927206	C₂	1	2	yes	maximal, complemented
5	#11927197	C₂	1	3	yes	maximal, complemented
6	#11927099	C₂	1	3	yes	maximal, complemented
7	#11927079	C₂	1	6	yes	maximal, complemented
8	C₄×#64545	C₂	1	3	yes	maximal, complemented
9	C₄×#64545	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×Q₈×D₁₉	C₂	1	2	yes	maximal, complemented
11	C₁₉×#557	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#64545	C₂²	1	3	yes	—
13	C₂×#64545	C₂²	1	1	yes	characteristic
14	C₂×Q₈×D₁₉	C₂²	1	2	yes	—
15	C₁₉×C₄⋊Q₈	C₂²	1	3	yes	—
16	C₁₉×#125	C₂²	1	6	yes	—
17	C₄×Q₈×C₁₉	C₂²	1	2	yes	—
18	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₉	C₂²	1	3	yes	—
20	#10597392	C₂²	1	6	yes	—
21	Q₈×Dic₁₉	C₂²	1	3	yes	—
22	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
23	#10597344	C₂²	1	12	yes	—
24	#10597343	C₂²	1	6	yes	—
25	#10597343	C₂²	1	6	yes	—
26	#10597341	C₂²	1	6	yes	—
27	#10597340	C₂²	1	12	yes	—
28	#10597339	C₂²	1	3	yes	—
29	#10597339	C₂²	1	6	yes	—
30	#10597339	C₂²	1	1	yes	characteristic
31	#10597338	C₂²	1	6	yes	—
32	#10597338	C₂²	1	6	yes	—
33	#10597338	C₂²	1	3	yes	—
34	#10597338	C₂²	1	3	yes	—
35	C₄⋊C₄×D₁₉	C₂²	1	6	yes	—
36	#10597336	C₂²	1	6	yes	—
37	#10597333	C₂²	1	6	yes	—
38	#10597321	C₂²	1	6	yes	—
39	C₄×D₇₆	C₂²	1	3	yes	—
40	C₄×D₇₆	C₂²	1	3	yes	—
41	C₄×D₇₆	C₂²	1	6	yes	—
42	C₄²×D₁₉	C₂²	1	6	yes	—
43	C₄²×D₁₉	C₂²	1	3	yes	—
44	#10597315	C₂²	1	3	yes	—
45	C₄×Dic₃₈	C₂²	1	6	yes	—
46	C₂×Q₈×C₁₉	C₂³	1	3	yes	—
47	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
48	#64545	—	2	4	no	—
49	#64545	—	2	12	no	—
50	Q₈×D₁₉	—	2	8	no	—
51	C₂×D₇₆	C₂³	1	3	yes	—
52	C₂×D₇₆	C₂³	1	6	yes	—
53	C₂×D₇₆	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₁₉	C₂³	1	3	yes	—
55	C₂×C₄×D₁₉	C₂³	1	6	yes	—
56	C₂×C₄×D₁₉	C₂³	1	3	yes	—
57	C₂×C₄×D₁₉	C₂³	1	6	yes	—
58	C₂×C₄×D₁₉	C₂³	1	2	yes	—
59	C₂×C₄×D₁₉	C₂³	1	3	yes	—
60	C₂×Dic₃₈	C₂³	1	6	yes	—
61	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
63	C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
64	C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
65	C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
66	C₄²×C₁₉	C₂³	1	6	yes	—
67	C₄²×C₁₉	C₂³	1	3	yes	—
68	(C₂×D₁₉)⋊C₄	C₂³	1	12	yes	—
69	(C₂×D₁₉)⋊C₄	C₂³	1	2	yes	—
70	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
71	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
72	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
73	C₄⋊Dic₁₉	C₂³	1	3	yes	—
74	C₄⋊Dic₁₉	C₂³	1	3	yes	—
75	C₄⋊Dic₁₉	C₂³	1	6	yes	—
76	C₄⋊Dic₁₉	C₂³	1	6	yes	—
77	Dic₁₉⋊C₄	C₂³	1	12	yes	—
78	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
79	C₄×Dic₁₉	C₂³	1	3	yes	—
80	C₄×Dic₁₉	C₂³	1	6	yes	—
81	C₄×Dic₁₉	C₂³	1	6	yes	—
82	C₄×Dic₁₉	C₂³	1	3	yes	—
83	C₂²×D₁₉	C₂⁴	1	3	yes	—
84	C₂²×D₁₉	C₂⁴	1	2	yes	—
85	Q₈×C₁₉	—	2	6	no	—
86	Q₈×C₁₉	#41	1	4	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×C₄×C₁₉	C₂⁴	1	3	yes	—
92	C₂×Dic₁₉	C₂⁴	1	3	yes	—
93	C₂×Dic₁₉	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₁₉	C₂⁴	1	6	yes	—
95	C₂×Dic₁₉	C₂⁴	1	1	yes	characteristic
96	D₇₆	—	2	12	no	—
97	D₇₆	—	2	6	no	—
98	D₇₆	#41	1	4	yes	complemented
99	C₄×D₁₉	—	2	12	no	—
100	C₄×D₁₉	—	2	12	no	—
101	C₄×D₁₉	—	2	6	no	—
102	C₄×D₁₉	—	2	6	no	—
103	C₄×D₁₉	—	2	4	no	—
104	C₄×D₁₉	—	2	6	no	—
105	Dic₃₈	—	2	12	no	—
106	#3482	—	19	1	no	maximal, Sylow
107	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
108	C₂×D₁₉	—	2	2	no	—
109	C₂×D₁₉	—	2	2	no	—
110	C₂×D₁₉	—	2	3	no	—
111	C₂×D₁₉	—	2	3	no	—
112	C₂×D₁₉	—	2	3	no	—
113	C₂×D₁₉	C₂×#41	1	4	yes	—
114	C₄×C₁₉	—	2	6	no	—
115	C₄×C₁₉	—	2	2	no	—
116	C₄×C₁₉	—	2	3	no	—
117	C₄×C₁₉	C₂×#41	1	6	yes	—
118	C₄×C₁₉	C₂×#41	1	2	yes	—
119	Dic₁₉	—	2	6	no	—
120	Dic₁₉	—	2	3	no	—
121	Dic₁₉	—	2	1	no	—
122	Dic₁₉	2₋¹⁺⁴	1	2	yes	—
123	#557	D₁₉	1	1	yes	characteristic, complemented
124	#556	—	19	3	no	—
125	#552	—	19	6	no	—
126	D₄×Q₈	—	19	1	no	—
127	#548	—	19	2	no	—
128	#547	—	19	3	no	—
129	#533	—	19	3	no	—
130	#529	—	19	6	no	—
131	C₄×#41	—	19	3	no	—
132	C₄×#41	—	19	1	no	—
133	C₂×C₄×Q₈	—	19	2	no	—
134	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
135	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
136	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
137	D₁₉	—	2	4	no	—
138	D₁₉	—	4	3	no	—
139	C₂×#41	—	19	3	no	—
140	C₂×#41	—	19	1	no	—
141	C₂²×Q₈	—	19	2	no	—
142	C₄⋊Q₈	—	19	3	no	—
143	C₄⋊Q₈	C₂×D₁₉	1	3	yes	—
144	#126	—	19	12	no	—
145	#125	—	19	6	no	—
146	#125	C₂×D₁₉	1	6	yes	—
147	#124	—	19	6	no	—
148	#123	—	19	12	no	—
149	C₂²⋊Q₈	—	19	6	no	—
150	C₂²⋊Q₈	—	19	6	no	—
151	C₂²⋊Q₈	—	19	6	no	—
152	#121	—	19	6	no	—
153	C₄×Q₈	—	19	1	no	—
154	C₄×Q₈	—	19	3	no	—
155	C₄×Q₈	—	19	6	no	—
156	C₄×Q₈	—	19	6	no	—
157	C₄×Q₈	C₂×D₁₉	1	2	yes	—
158	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
159	C₄×Q₈	C₂×D₁₉	1	3	yes	—
160	C₄×D₄	—	19	1	no	—
161	C₄×D₄	—	19	3	no	—
162	C₄×D₄	—	19	6	no	—
163	C₄×D₄	—	19	3	no	—
164	C₄×D₄	—	19	3	no	—
165	C₄×D₄	—	19	6	no	—
166	#117	—	19	3	no	—
167	#117	—	19	6	no	—
168	#117	—	19	3	no	—
169	#117	—	19	6	no	—
170	C₂×C₄⋊C₄	—	19	6	no	—
171	C₂×C₄²	—	19	6	no	—
172	C₂×C₄²	—	19	3	no	—
173	C₁₉	#3482	1	1	yes	characteristic, complemented, Sylow
174	#41	—	38	12	no	—
175	#41	—	38	4	no	—
176	C₂×Q₈	—	19	6	no	—
177	C₂×Q₈	—	38	8	no	—
178	C₂×Q₈	C₂²×D₁₉	1	3	yes	—
179	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
180	C₂×D₄	—	19	1	no	—
181	C₂×D₄	—	19	6	no	—
182	C₂×D₄	—	19	3	no	—
183	C₂²×C₄	—	19	2	no	—
184	C₂²×C₄	—	19	6	no	—
185	C₂²×C₄	—	19	6	no	—
186	C₂²×C₄	—	19	3	no	—
187	C₂²×C₄	—	19	3	no	—
188	C₂²×C₄	—	19	3	no	—
189	C₄⋊C₄	—	19	12	no	—
190	C₄⋊C₄	—	19	3	no	—
191	C₄⋊C₄	—	19	6	no	—
192	C₄⋊C₄	—	19	3	no	—
193	C₄⋊C₄	—	19	6	no	—
194	C₄⋊C₄	C₂²×D₁₉	1	6	yes	—
195	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
196	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
197	C₄⋊C₄	C₂²×D₁₉	1	6	yes	—
198	C₄⋊C₄	C₂²×D₁₉	1	6	yes	—
199	C₂²⋊C₄	—	19	2	no	—
200	C₂²⋊C₄	—	19	6	no	—
201	C₂²⋊C₄	—	19	6	no	—
202	C₂²⋊C₄	—	19	6	no	—
203	C₂²⋊C₄	—	19	12	no	—
204	C₄²	—	19	3	no	—
205	C₄²	—	19	1	no	—
206	C₄²	—	19	3	no	—
207	C₄²	—	19	6	no	—
208	C₄²	—	19	6	no	—
209	C₄²	C₂²×D₁₉	1	3	yes	—
210	C₄²	C₂²×D₁₉	1	6	yes	—
211	C₂³	—	19	2	no	—
212	C₂³	—	19	3	no	—
213	Q₈	—	2	6	no	—
214	Q₈	—	38	12	no	—
215	Q₈	#64545	1	4	yes	complemented
216	D₄	—	19	4	no	—
217	D₄	—	38	12	no	—
218	D₄	—	38	6	no	—
219	C₂×C₄	—	19	1	no	—
220	C₂×C₄	—	19	3	no	—
221	C₂×C₄	—	19	1	no	—
222	C₂×C₄	—	19	6	no	—
223	C₂×C₄	—	38	4	no	—
224	C₂×C₄	—	38	6	no	—
225	C₂×C₄	—	38	12	no	—
226	C₂×C₄	—	38	12	no	—
227	C₂×C₄	—	38	6	no	—
228	C₂×C₄	—	38	6	no	—
229	C₂×C₄	C₂³×D₁₉	1	2	yes	—
230	C₂×C₄	C₂³×D₁₉	1	3	yes	—
231	C₂×C₄	C₂³×D₁₉	1	6	yes	—
232	C₂×C₄	C₂³×D₁₉	1	3	yes	—
233	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
234	C₂²	—	19	4	no	—
235	C₂²	—	38	2	no	—
236	C₂²	—	38	2	no	—
237	C₂²	—	38	3	no	—
238	C₂²	—	38	3	no	—
239	C₂²	—	38	3	no	—
240	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
241	C₄	—	2	6	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	3	no	—
244	C₄	—	19	2	no	—
245	C₄	—	38	3	no	—
246	C₄	—	38	1	no	—
247	C₄	—	38	6	no	—
248	C₄	C₂×#64545	1	6	yes	—
249	C₄	#41×D₁₉	1	2	yes	—
250	C₂	—	38	4	no	—
251	C₂	—	76	3	no	—
252	C₂	C₂×#41×D₁₉	1	1	yes	characteristic
253	C₂	C₂²×#64545	1	1	yes	characteristic
254	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×#557	C₂
hypercenter	#557	D₁₉
hypocenter	C₁₉	#3482
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #557

Complements

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

N	H ≅ G/N	type
1. #11927267	C₂	semidirect
2. #11927232	C₂	semidirect
3. Q₈×D₇₆	C₂	semidirect
4. #11927206	C₂	semidirect
5. #11927197	C₂	semidirect
6. #11927099	C₂	semidirect
7. #11927079	C₂	semidirect
8. C₄×#64545	C₂	semidirect
9. C₄×#64545	C₂	semidirect
10. C₄×Q₈×D₁₉	C₂	semidirect
11. C₁₉×#557	C₂	semidirect
86. Q₈×C₁₉	#41	semidirect
98. D₇₆	#41	semidirect
123. #557	D₁₉	semidirect
173. C₁₉	#3482	semidirect
215. Q₈	#64545	semidirect

Maximal subgroups

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

subgroup	index
#11927267	2
#11927232	2
Q₈×D₇₆	2
#11927206	2
#11927197	2
#11927099	2
#11927079	2
C₄×#64545	2
C₄×Q₈×D₁₉	2
C₁₉×#557	2
#3482	19