Subgroups — #14577025

Order 2432.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927165	C₂	1	4	yes	maximal, complemented
2	#11927156	C₂	1	2	yes	maximal, complemented
3	#11927149	C₂	1	2	yes	maximal, complemented
4	#11927141	C₂	1	4	yes	maximal, complemented
5	#11927132	C₂	1	4	yes	maximal, complemented
6	#11927101	C₂	1	1	yes	maximal, characteristic, complemented
7	#11927094	C₂	1	1	yes	maximal, characteristic, complemented
8	#11927072	C₂	1	8	yes	maximal, complemented
9	#11927055	C₂	1	2	yes	maximal, complemented
10	C₄×#64541	C₂	1	2	yes	maximal, complemented
11	C₂×C₁₉×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×#64541	C₂²	1	2	yes	—
14	C₁₉×#117	C₂²	1	4	yes	—
15	C₁₉×#117	C₂²	1	2	yes	—
16	C₁₉×#117	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
21	#10597399	C₂²	1	4	yes	—
22	#10597376	C₂²	1	4	yes	—
23	#10597376	C₂²	1	4	yes	—
24	#10597375	C₂²	1	4	yes	—
25	C₄×C₂²⋊D₁₉	C₂²	1	8	yes	—
26	C₄×C₂²⋊D₁₉	C₂²	1	8	yes	—
27	#10597372	C₂²	1	2	yes	—
28	#10597370	C₂²	1	4	yes	—
29	#10597370	C₂²	1	4	yes	—
30	#10597344	C₂²	1	8	yes	complemented
31	#10597342	C₂²	1	8	yes	complemented
32	#10597340	C₂²	1	8	yes	complemented
33	#10597335	C₂²	1	8	yes	complemented
34	#10597331	C₂²	1	8	yes	—
35	#10597330	C₂²	1	8	yes	—
36	#10597329	C₂²	1	8	yes	—
37	#10597325	C₂²	1	8	yes	—
38	#10597321	C₂²	1	2	yes	complemented
39	#10597320	C₂²	1	1	yes	characteristic, complemented
40	C₄×D₇₆	C₂²	1	4	yes	complemented
41	C₄×D₇₆	C₂²	1	4	yes	complemented
42	#10597318	C₂²	1	4	yes	complemented
43	C₄²×D₁₉	C₂²	1	4	yes	complemented
44	#10597316	C₂²	1	2	yes	complemented
45	#10597315	C₂²	1	1	yes	characteristic, complemented
46	#10597315	C₂²	1	2	yes	complemented
47	C₄×Dic₃₈	C₂²	1	4	yes	complemented
48	C₄×Dic₃₈	C₂²	1	4	yes	complemented
49	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	—	2	4	no	—
51	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₉	C₂³	1	4	yes	—
58	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
59	#64541	—	2	8	no	—
60	C₂×D₇₆	C₂³	1	2	yes	—
61	C₂×D₇₆	C₂³	1	4	yes	—
62	C₂×C₄×D₁₉	C₂³	1	4	yes	—
63	C₂×C₄×D₁₉	C₂³	1	8	yes	—
64	C₂×Dic₃₈	C₂³	1	2	yes	—
65	C₂×Dic₃₈	C₂³	1	4	yes	—
66	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
67	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
68	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
69	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
70	C₄²×C₁₉	C₂³	1	2	yes	—
71	C₄²×C₁₉	C₂³	1	2	yes	—
72	C₄²×C₁₉	C₂³	1	1	yes	characteristic
73	C₄²×C₁₉	C₂³	1	1	yes	characteristic
74	C₄²×C₁₉	C₂³	1	2	yes	—
75	C₂²⋊Dic₁₉	C₂³	1	8	yes	—
76	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
77	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
78	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
79	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
80	C₄⋊Dic₁₉	C₂³	1	4	yes	—
81	C₄⋊Dic₁₉	C₂³	1	4	yes	—
82	C₄⋊Dic₁₉	C₂³	1	4	yes	—
83	Dic₁₉⋊C₄	C₂³	1	8	yes	—
84	Dic₁₉⋊C₄	C₂³	1	8	yes	—
85	Dic₁₉⋊C₄	C₂³	1	8	yes	—
86	C₄×Dic₁₉	C₂³	1	4	yes	—
87	C₄×Dic₁₉	C₂³	1	8	yes	—
88	C₂³×C₁₉	—	2	2	no	—
89	C₂³×C₁₉	—	4	2	no	—
90	C₂³×C₁₉	C₂⁴	1	2	yes	—
91	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
92	C₂²×D₁₉	C₂⁴	1	4	yes	—
93	C₂×C₄×C₁₉	—	2	2	no	—
94	C₂×C₄×C₁₉	—	2	4	no	—
95	C₂×C₄×C₁₉	—	2	4	no	—
96	C₂×C₄×C₁₉	—	4	2	no	—
97	C₂×C₄×C₁₉	—	4	2	no	—
98	C₂×C₄×C₁₉	#41	1	4	yes	—
99	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
100	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
101	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
102	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
103	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₁₉	—	2	8	no	—
106	C₂²⋊D₁₉	—	4	8	no	—
107	C₂×Dic₁₉	C₂⁴	1	4	yes	—
108	C₂×Dic₁₉	C₂⁴	1	8	yes	—
109	D₇₆	—	2	4	no	—
110	D₇₆	—	4	4	no	—
111	C₄×D₁₉	—	2	8	no	—
112	C₄×D₁₉	—	4	8	no	—
113	Dic₃₈	—	2	4	no	—
114	Dic₃₈	—	4	4	no	—
115	#3455	—	19	1	no	maximal, Sylow
116	C₂²×C₁₉	—	2	2	no	—
117	C₂²×C₁₉	—	2	2	no	—
118	C₂²×C₁₉	—	2	4	no	—
119	C₂²×C₁₉	—	4	4	no	—
120	C₂²×C₁₉	C₂×#41	1	2	yes	—
121	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
122	C₂×D₁₉	—	2	4	no	—
123	C₂×D₁₉	—	2	8	no	—
124	C₄×C₁₉	—	2	2	no	—
125	C₄×C₁₉	—	2	2	no	—
126	C₄×C₁₉	—	2	4	no	—
127	C₄×C₁₉	—	2	2	no	—
128	C₄×C₁₉	C₂×#41	1	2	yes	—
129	C₄×C₁₉	C₂×#41	1	2	yes	—
130	Dic₁₉	—	2	4	no	—
131	Dic₁₉	—	2	8	no	—
132	#542	—	19	4	no	—
133	#541	—	19	2	no	—
134	#540	—	19	2	no	—
135	#539	—	19	4	no	—
136	#538	—	19	4	no	—
137	#533	—	19	1	no	—
138	#532	—	19	1	no	—
139	#529	—	19	8	no	—
140	#525	—	19	2	no	—
141	C₄×#41	—	19	2	no	—
142	C₂×#117	D₁₉	1	1	yes	characteristic, complemented
143	C₂×C₁₉	—	2	2	no	—
144	C₂×C₁₉	—	4	2	no	—
145	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
146	C₂×C₁₉	#585	1	2	yes	—
147	D₁₉	—	4	4	no	—
148	C₂×#41	—	19	2	no	—
149	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
150	C₄⋊Q₈	—	19	1	no	—
151	C₄⋊Q₈	—	19	2	no	—
152	#127	—	19	1	no	—
153	#126	—	19	8	no	—
154	#126	—	19	8	no	—
155	#125	—	19	8	no	—
156	#125	—	19	2	no	—
157	#124	—	19	8	no	—
158	#124	—	19	2	no	—
159	#123	—	19	4	no	—
160	#123	—	19	8	no	—
161	#123	—	19	8	no	—
162	C₂²⋊Q₈	—	19	4	no	—
163	C₂²⋊Q₈	—	19	4	no	—
164	C₂²⋊Q₈	—	19	8	no	—
165	#121	—	19	4	no	—
166	#121	—	19	4	no	—
167	#121	—	19	8	no	—
168	#120	—	19	4	no	—
169	C₄×Q₈	—	19	4	no	—
170	C₄×Q₈	—	19	4	no	—
171	C₄×D₄	—	19	8	no	—
172	C₄×D₄	—	19	8	no	—
173	C₄×D₄	—	19	4	no	—
174	C₄×D₄	—	19	4	no	—
175	#117	—	19	2	no	—
176	#117	—	19	4	no	—
177	#117	C₂×D₁₉	1	4	yes	—
178	#117	C₂×D₁₉	1	2	yes	—
179	#117	C₂×D₁₉	1	2	yes	—
180	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂×D₁₉	1	2	yes	—
182	C₂×C₄²	—	19	4	no	—
183	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
184	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
185	C₁₉	#3455	1	1	yes	characteristic, complemented, Sylow
186	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
187	#41	—	38	8	no	—
188	C₂×Q₈	—	19	2	no	—
189	C₂×Q₈	—	19	4	no	—
190	C₂×D₄	—	19	4	no	—
191	C₂×D₄	—	19	8	no	—
192	C₂×D₄	—	19	2	no	—
193	C₂×D₄	—	19	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	19	4	no	—
196	C₂²×C₄	—	19	8	no	—
197	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
199	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
200	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
203	C₄⋊C₄	—	19	8	no	—
204	C₄⋊C₄	—	19	8	no	—
205	C₄⋊C₄	—	19	8	no	—
206	C₄⋊C₄	—	19	4	no	—
207	C₄⋊C₄	—	19	4	no	—
208	C₄⋊C₄	—	19	4	no	—
209	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
210	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
211	C₂²⋊C₄	—	19	4	no	—
212	C₂²⋊C₄	—	19	8	no	—
213	C₂²⋊C₄	—	19	8	no	—
214	C₂²⋊C₄	—	19	8	no	—
215	C₂²⋊C₄	—	19	8	no	—
216	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
217	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
218	C₄²	—	19	8	no	—
219	C₄²	—	19	4	no	—
220	C₄²	C₂²×D₁₉	1	2	yes	—
221	C₄²	C₂²×D₁₉	1	1	yes	characteristic
222	C₄²	C₂²×D₁₉	1	1	yes	characteristic
223	C₄²	C₂²×D₁₉	1	2	yes	—
224	C₄²	C₂²×D₁₉	1	2	yes	—
225	C₂³	—	2	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	19	4	no	—
228	C₂³	C₂³×D₁₉	1	2	yes	—
229	C₂³	C₂³×D₁₉	1	1	yes	characteristic
230	Q₈	—	38	4	no	—
231	Q₈	—	76	4	no	—
232	D₄	—	38	8	no	—
233	D₄	—	38	4	no	—
234	D₄	—	76	4	no	—
235	D₄	—	76	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	19	4	no	—
242	C₂×C₄	—	19	8	no	—
243	C₂×C₄	—	38	8	no	—
244	C₂×C₄	—	76	8	no	—
245	C₂×C₄	#64541	1	4	yes	—
246	C₂×C₄	C₂³×D₁₉	1	4	yes	—
247	C₂×C₄	C₂³×D₁₉	1	2	yes	—
248	C₂×C₄	C₂³×D₁₉	1	2	yes	—
249	C₂×C₄	C₂³×D₁₉	1	2	yes	—
250	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	38	4	no	—
257	C₂²	—	38	8	no	—
258	C₂²	C₂×#64541	1	2	yes	—
259	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	38	4	no	—
265	C₄	—	38	8	no	—
266	C₄	C₂×#64541	1	2	yes	—
267	C₄	C₂×#64541	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	2	no	—
270	C₂	—	76	4	no	—
271	C₂	C₂²×#64541	1	1	yes	characteristic
272	C₂	#11927380	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉
Fitting	C₂×C₁₉×#117	C₂
hypercenter	C₂×#117	D₁₉
hypocenter	C₁₉	#3455
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #11927165	C₂	semidirect
2. #11927156	C₂	semidirect
3. #11927149	C₂	semidirect
4. #11927141	C₂	semidirect
5. #11927132	C₂	semidirect
6. #11927101	C₂	semidirect
7. #11927094	C₂	semidirect
8. #11927072	C₂	semidirect
9. #11927055	C₂	semidirect
10. C₄×#64541	C₂	semidirect
11. C₂×C₁₉×#117	C₂	semidirect
30. #10597344	C₂²	semidirect
31. #10597342	C₂²	semidirect
32. #10597340	C₂²	semidirect
33. #10597335	C₂²	semidirect
38. #10597321	C₂²	semidirect
39. #10597320	C₂²	semidirect
40. C₄×D₇₆	C₂²	semidirect
41. C₄×D₇₆	C₂²	semidirect
42. #10597318	C₂²	semidirect
43. C₄²×D₁₉	C₂²	semidirect
44. #10597316	C₂²	semidirect
45. #10597315	C₂²	semidirect
46. #10597315	C₂²	semidirect
47. C₄×Dic₃₈	C₂²	semidirect
48. C₄×Dic₃₈	C₂²	semidirect
142. C₂×#117	D₁₉	semidirect
185. C₁₉	#3455	semidirect

Maximal subgroups

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

subgroup	index
#11927165	2
#11927156	2
#11927149	2
#11927141	2
#11927132	2
#11927101	2
#11927094	2
#11927072	2
#11927055	2
C₄×#64541	2
C₂×C₁₉×#117	2
#3455	19