Subgroups — #14577652

Order 2432.

1010 conjugacy classes of subgroups.

The subgroup lattice has 252 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	1	yes	maximal, characteristic, complemented
2	#11927264	C₂	1	3	yes	maximal, complemented
3	#11927262	C₂	1	6	yes	maximal, complemented
4	#11927259	C₂	1	3	yes	maximal, complemented
5	Q₈×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#11927201	C₂	1	2	yes	maximal, complemented
7	#11927159	C₂	1	6	yes	maximal, complemented
8	#11927101	C₂	1	3	yes	maximal, complemented
9	#11927078	C₂	1	3	yes	maximal, complemented
10	#11927020	C₂	1	2	yes	maximal, complemented
11	C₄×#41×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₁₉	C₂²	1	2	yes	—
14	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₁₉	C₂²	1	3	yes	—
17	C₄×D₄×C₁₉	C₂²	1	3	yes	—
18	C₁₉×#117	C₂²	1	3	yes	—
19	C₂×C₄²×C₁₉	C₂²	1	3	yes	—
20	Q₈×Dic₁₉	C₂²	1	2	yes	—
21	#10597384	C₂²	1	6	yes	—
22	#10597382	C₂²	1	6	yes	—
23	#10597376	C₂²	1	3	yes	—
24	#10597375	C₂²	1	6	yes	—
25	C₄×C₂²⋊D₁₉	C₂²	1	6	yes	—
26	#10597372	C₂²	1	6	yes	—
27	#10597370	C₂²	1	3	yes	—
28	#10597370	C₂²	1	6	yes	—
29	#10597370	C₂²	1	6	yes	—
30	#10597343	C₂²	1	6	yes	—
31	#10597338	C₂²	1	6	yes	—
32	#10597334	C₂²	1	6	yes	—
33	#10597333	C₂²	1	6	yes	—
34	#10597331	C₂²	1	12	yes	—
35	#10597325	C₂²	1	12	yes	—
36	#10597322	C₂²	1	6	yes	—
37	#10597321	C₂²	1	1	yes	characteristic
38	C₄×D₇₆	C₂²	1	3	yes	—
39	#10597318	C₂²	1	6	yes	—
40	#10597316	C₂²	1	3	yes	—
41	#10597315	C₂²	1	3	yes	—
42	#10597315	C₂²	1	3	yes	—
43	C₄×Dic₃₈	C₂²	1	3	yes	—
44	C₄×Dic₃₈	C₂²	1	6	yes	—
45	C₄×Dic₃₈	C₂²	1	6	yes	—
46	#41×C₁₉	—	2	4	no	—
47	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₉	C₂³	1	3	yes	—
49	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
50	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
51	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
52	C₂×C₂²⋊D₁₉	C₂³	1	6	yes	—
53	Q₈×D₁₉	—	2	8	no	—
54	C₂×D₇₆	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₁₉	C₂³	1	6	yes	—
56	C₂×Dic₃₈	C₂³	1	6	yes	—
57	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
58	C₂×Dic₃₈	C₂³	1	6	yes	—
59	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
60	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
61	C₂²⋊C₄×C₁₉	C₂³	1	6	yes	—
62	C₄²×C₁₉	C₂³	1	1	yes	characteristic
63	C₄²×C₁₉	C₂³	1	3	yes	—
64	C₄²×C₁₉	C₂³	1	3	yes	—
65	C₄²×C₁₉	C₂³	1	3	yes	—
66	C₂²⋊Dic₁₉	C₂³	1	6	yes	—
67	C₂²⋊Dic₁₉	C₂³	1	12	yes	—
68	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
69	(C₂×D₁₉)⋊C₄	C₂³	1	2	yes	—
70	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
71	(C₂×D₁₉)⋊C₄	C₂³	1	2	yes	—
72	C₄⋊Dic₁₉	C₂³	1	3	yes	—
73	C₄⋊Dic₁₉	C₂³	1	6	yes	—
74	C₄⋊Dic₁₉	C₂³	1	6	yes	—
75	C₄⋊Dic₁₉	C₂³	1	3	yes	—
76	Dic₁₉⋊C₄	C₂³	1	6	yes	—
77	Dic₁₉⋊C₄	C₂³	1	6	yes	—
78	Dic₁₉⋊C₄	C₂³	1	6	yes	—
79	Dic₁₉⋊C₄	C₂³	1	6	yes	—
80	C₄×Dic₁₉	C₂³	1	6	yes	—
81	C₄×Dic₁₉	C₂³	1	12	yes	—
82	C₂³×C₁₉	C₂⁴	1	3	yes	—
83	C₂²×D₁₉	C₂⁴	1	2	yes	—
84	Q₈×C₁₉	#41	1	4	yes	—
85	D₄×C₁₉	—	2	6	no	—
86	C₂×C₄×C₁₉	—	2	6	no	—
87	C₂×C₄×C₁₉	—	2	6	no	—
88	C₂×C₄×C₁₉	—	4	3	no	—
89	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
90	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
93	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
94	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₁₉	—	4	6	no	—
96	C₂×Dic₁₉	C₂⁴	1	2	yes	—
97	C₂×Dic₁₉	C₂⁴	1	6	yes	—
98	C₂×Dic₁₉	C₂⁴	1	6	yes	—
99	D₇₆	—	2	2	no	—
100	C₄×D₁₉	—	2	12	no	—
101	Dic₃₈	—	2	12	no	—
102	Dic₃₈	—	2	2	no	—
103	Dic₃₈	—	4	6	no	—
104	#3518	—	19	1	no	maximal, Sylow
105	C₂²×C₁₉	—	2	3	no	—
106	C₂²×C₁₉	—	2	3	no	—
107	C₂²×C₁₉	—	2	3	no	—
108	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
109	C₂×D₁₉	—	2	2	no	—
110	C₂×D₁₉	—	2	2	no	—
111	C₂×D₁₉	—	2	2	no	—
112	C₄×C₁₉	—	2	1	no	—
113	C₄×C₁₉	—	2	3	no	—
114	C₄×C₁₉	—	2	3	no	—
115	C₄×C₁₉	—	2	1	no	—
116	C₄×C₁₉	C₂×#41	1	6	yes	—
117	C₄×C₁₉	2₋¹⁺⁴	1	2	yes	—
118	Dic₁₉	—	2	6	no	—
119	Dic₁₉	—	2	2	no	—
120	Dic₁₉	—	2	6	no	—
121	Q₈²	—	19	1	no	—
122	#556	—	19	3	no	—
123	#556	—	19	6	no	—
124	#555	—	19	3	no	—
125	D₄×Q₈	—	19	1	no	—
126	#548	—	19	2	no	—
127	#542	—	19	6	no	—
128	#533	—	19	3	no	—
129	#529	—	19	3	no	—
130	#519	—	19	2	no	—
131	C₄×#41	D₁₉	1	1	yes	characteristic, complemented
132	C₂×C₁₉	—	4	3	no	—
133	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
134	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
135	C₂×C₁₉	#585	1	1	yes	characteristic
136	D₁₉	—	4	2	no	—
137	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
138	C₂²×Q₈	—	19	2	no	—
139	C₄⋊Q₈	—	19	6	no	—
140	C₄⋊Q₈	—	19	3	no	—
141	C₄⋊Q₈	—	19	3	no	—
142	#126	—	19	12	no	—
143	#126	—	19	6	no	—
144	#125	—	19	3	no	—
145	#124	—	19	6	no	—
146	#124	—	19	12	no	—
147	#124	—	19	1	no	—
148	#123	—	19	6	no	—
149	C₂²⋊Q₈	—	19	3	no	—
150	C₂²⋊Q₈	—	19	6	no	—
151	C₂²⋊Q₈	—	19	6	no	—
152	C₂²⋊Q₈	—	19	6	no	—
153	#121	—	19	6	no	—
154	#121	—	19	3	no	—
155	C₄×Q₈	—	19	2	no	—
156	C₄×Q₈	—	19	6	no	—
157	C₄×Q₈	—	19	6	no	—
158	C₄×Q₈	—	19	3	no	—
159	C₄×Q₈	—	19	6	no	—
160	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
161	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
162	C₄×D₄	—	19	6	no	—
163	C₄×D₄	—	19	3	no	—
164	C₄×D₄	C₂×D₁₉	1	3	yes	—
165	C₄×D₄	C₂×D₁₉	1	3	yes	—
166	#117	—	19	6	no	—
167	#117	—	19	6	no	—
168	#117	—	19	6	no	—
169	#117	C₂×D₁₉	1	3	yes	—
170	C₂×C₄²	C₂×D₁₉	1	3	yes	—
171	C₁₉	#3518	1	1	yes	characteristic, complemented, Sylow
172	#41	—	2	4	no	—
173	C₂×Q₈	—	19	1	no	—
174	C₂×Q₈	—	19	6	no	—
175	C₂×Q₈	—	19	6	no	—
176	C₂×Q₈	—	38	8	no	—
177	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
178	C₂×D₄	—	19	6	no	—
179	C₂×D₄	—	19	1	no	—
180	C₂×D₄	C₂²×D₁₉	1	3	yes	—
181	C₂²×C₄	—	19	6	no	—
182	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
183	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
184	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
185	C₄⋊C₄	—	19	6	no	—
186	C₄⋊C₄	—	19	6	no	—
187	C₄⋊C₄	—	19	6	no	—
188	C₄⋊C₄	—	19	6	no	—
189	C₄⋊C₄	—	19	3	no	—
190	C₄⋊C₄	—	19	3	no	—
191	C₄⋊C₄	—	19	6	no	—
192	C₄⋊C₄	—	19	6	no	—
193	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
194	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
195	C₂²⋊C₄	—	19	12	no	—
196	C₂²⋊C₄	—	19	6	no	—
197	C₂²⋊C₄	—	19	6	no	—
198	C₂²⋊C₄	—	19	2	no	—
199	C₂²⋊C₄	—	19	2	no	—
200	C₂²⋊C₄	—	19	6	no	—
201	C₂²⋊C₄	C₂²×D₁₉	1	6	yes	—
202	C₄²	—	19	12	no	—
203	C₄²	—	19	6	no	—
204	C₄²	C₂²×D₁₉	1	1	yes	characteristic
205	C₄²	C₂²×D₁₉	1	3	yes	—
206	C₄²	C₂²×D₁₉	1	3	yes	—
207	C₄²	C₂²×D₁₉	1	3	yes	—
208	C₂³	—	19	2	no	—
209	C₂³	C₂³×D₁₉	1	3	yes	—
210	Q₈	—	38	12	no	—
211	Q₈	—	38	2	no	—
212	Q₈	—	76	6	no	—
213	Q₈	#64541	1	4	yes	—
214	D₄	—	2	6	no	—
215	D₄	—	38	2	no	—
216	D₄	—	76	6	no	—
217	C₂×C₄	—	2	6	no	—
218	C₂×C₄	—	2	6	no	—
219	C₂×C₄	—	4	3	no	—
220	C₂×C₄	—	19	2	no	—
221	C₂×C₄	—	19	6	no	—
222	C₂×C₄	—	19	6	no	—
223	C₂×C₄	—	38	12	no	—
224	C₂×C₄	C₂³×D₁₉	1	3	yes	—
225	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
227	C₂×C₄	C₂³×D₁₉	1	3	yes	—
228	C₂×C₄	C₂³×D₁₉	1	3	yes	—
229	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
230	C₂²	—	2	3	no	—
231	C₂²	—	2	3	no	—
232	C₂²	—	2	3	no	—
233	C₂²	—	38	2	no	—
234	C₂²	—	38	2	no	—
235	C₂²	—	38	2	no	—
236	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	3	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	3	no	—
241	C₄	—	38	6	no	—
242	C₄	—	38	2	no	—
243	C₄	—	38	6	no	—
244	C₄	C₂×#64541	1	6	yes	—
245	C₄	#10597436	1	2	yes	—
246	C₂	—	4	3	no	—
247	C₂	—	76	2	no	—
248	C₂	C₂²×#64541	1	1	yes	characteristic
249	C₂	C₂×#10597436	1	1	yes	characteristic
250	C₂	#11927380	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₄×#41×C₁₉	C₂
hypercenter	C₄×#41	D₁₉
hypocenter	C₁₉	#3518
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₃₈	C₂	semidirect
2. #11927264	C₂	semidirect
3. #11927262	C₂	semidirect
4. #11927259	C₂	semidirect
5. Q₈×D₇₆	C₂	semidirect
6. #11927201	C₂	semidirect
7. #11927159	C₂	semidirect
8. #11927101	C₂	semidirect
9. #11927078	C₂	semidirect
10. #11927020	C₂	semidirect
11. C₄×#41×C₁₉	C₂	semidirect
131. C₄×#41	D₁₉	semidirect
171. C₁₉	#3518	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
#11927264	2
#11927262	2
#11927259	2
Q₈×D₇₆	2
#11927201	2
#11927159	2
#11927101	2
#11927078	2
#11927020	2
C₄×#41×C₁₉	2
#3518	19