Subgroups — #14575378

Order 2432.

482 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	C₂×#10597350	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₇₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11927203	C₂	1	2	yes	maximal, complemented
4	#11926928	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926785	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926780	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926767	C₂	1	2	yes	maximal, complemented
8	#11926708	C₂	1	2	yes	maximal, complemented
9	#11926685	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₇₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#64541	C₂²	1	2	yes	—
13	C₂²×Dic₃₈	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
17	#10597385	—	2	2	no	—
18	#10597376	C₂²	1	2	yes	—
19	#10597370	C₂²	1	2	yes	—
20	C₂×Dic₇₆	—	2	4	no	—
21	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
22	C₂×Dic₇₆	C₂²	1	2	yes	—
23	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
24	#10597350	—	2	4	no	—
25	C₂×D₁₅₂	C₂²	1	1	yes	characteristic
26	C₂×#64516	C₂²	1	2	yes	—
27	C₂×#64516	C₂²	1	2	yes	—
28	#10597342	—	2	2	no	—
29	#10597331	—	2	2	no	—
30	(C₂²⋊D₁₉)⋊C₄	—	2	2	no	—
31	#10597321	C₂²	1	2	yes	—
32	C₄×Dic₃₈	C₂²	1	2	yes	—
33	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
36	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
37	#10597278	C₂²	1	2	yes	—
38	#10597275	C₂²	1	1	yes	characteristic
39	#10597273	C₂²	1	2	yes	—
40	#10597273	C₂²	1	2	yes	—
41	C₂×D₄×C₁₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₉	—	2	2	no	—
45	C₂×C₂²⋊D₁₉	—	2	2	no	—
46	C₂²×Dic₁₉	—	2	2	no	—
47	#64541	—	2	4	no	—
48	#64541	—	4	2	no	—
49	C₂×D₇₆	C₂³	1	2	yes	—
50	C₂×C₄×D₁₉	—	2	2	no	—
51	C₂×Dic₃₈	—	2	4	no	—
52	C₂×Dic₃₈	—	2	2	no	—
53	C₂×Dic₃₈	—	4	2	no	—
54	C₂×Dic₃₈	C₂³	1	2	yes	—
55	C₂×Dic₃₈	C₂³	1	2	yes	—
56	C₂×C₈×C₁₉	—	2	2	no	—
57	C₂×C₈×C₁₉	—	2	2	no	—
58	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
63	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
64	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
65	C₄²×C₁₉	C₂³	1	1	yes	characteristic
66	C₂²⋊Dic₁₉	—	2	2	no	—
67	(C₂×D₁₉)⋊C₄	—	2	2	no	—
68	(C₂×D₁₉)⋊C₄	—	2	2	no	—
69	(C₂×D₁₉)⋊C₄	—	2	2	no	—
70	C₄⋊Dic₁₉	C₂³	1	2	yes	—
71	Dic₁₉⋊C₄	—	2	2	no	—
72	Dic₁₉⋊C₄	—	2	2	no	—
73	C₄×Dic₁₉	—	2	2	no	—
74	Dic₇₆	—	2	4	no	—
75	Dic₇₆	—	2	2	no	—
76	Dic₇₆	D₄	1	4	yes	complemented
77	D₁₅₂	—	2	2	no	—
78	#64516	—	2	4	no	—
79	#64516	—	4	2	no	—
80	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₁₉	—	2	2	no	—
83	D₄×C₁₉	—	2	2	no	—
84	C₂×C₄×C₁₉	—	2	1	no	—
85	C₂×C₄×C₁₉	—	2	1	no	—
86	C₂×C₄×C₁₉	—	2	1	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₁₉	—	4	4	no	—
94	C₂²⋊D₁₉	—	4	4	no	—
95	C₂×Dic₁₉	—	2	2	no	—
96	C₂×Dic₁₉	—	2	2	no	—
97	C₂×Dic₁₉	—	2	2	no	—
98	C₂×Dic₁₉	—	4	4	no	—
99	D₇₆	—	2	2	no	—
100	D₇₆	—	2	2	no	—
101	C₄×D₁₉	—	4	4	no	—
102	Dic₃₈	—	2	2	no	—
103	Dic₃₈	—	2	2	no	—
104	Dic₃₈	—	2	2	no	—
105	Dic₃₈	—	4	4	no	—
106	Dic₃₈	C₂×D₄	1	4	yes	—
107	C₈×C₁₉	—	2	1	no	—
108	C₈×C₁₉	—	2	1	no	—
109	C₈×C₁₉	—	2	1	no	—
110	C₈×C₁₉	C₂×D₄	1	2	yes	—
111	#3285	—	19	1	no	maximal, Sylow
112	C₂²×C₁₉	—	2	1	no	—
113	C₂²×C₁₉	—	2	1	no	—
114	C₂²×C₁₉	—	2	1	no	—
115	C₂²×C₁₉	—	2	1	no	—
116	C₂²×C₁₉	—	2	1	no	—
117	C₂²×C₁₉	#135	1	2	yes	complemented
118	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₁₉	—	4	2	no	—
120	C₂×D₁₉	—	4	2	no	—
121	C₂×D₁₉	—	4	2	no	—
122	C₄×C₁₉	—	2	1	no	—
123	C₄×C₁₉	—	2	1	no	—
124	C₄×C₁₉	—	2	1	no	—
125	C₄×C₁₉	—	2	1	no	—
126	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
128	Dic₁₉	—	2	4	no	—
129	Dic₁₉	—	4	2	no	—
130	Dic₁₉	—	4	2	no	—
131	C₂×#135	—	19	1	no	—
132	C₂²×Q₁₆	—	19	1	no	—
133	#548	—	19	2	no	—
134	#495	—	19	1	no	—
135	#467	—	19	1	no	—
136	#466	—	19	1	no	—
137	#464	—	19	2	no	—
138	#452	—	19	2	no	—
139	#448	—	19	2	no	—
140	C₄×Q₁₆	—	19	1	no	—
141	C₈×D₄	D₁₉	1	1	yes	characteristic, complemented
142	C₂×C₁₉	—	2	2	no	—
143	C₂×C₁₉	—	4	1	no	—
144	C₂×C₁₉	D₄²	1	1	yes	characteristic
145	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
146	C₂×C₁₉	#578	1	1	yes	characteristic
147	D₁₉	—	8	2	no	—
148	C₂×#41	—	19	2	no	—
149	C₂²×Q₈	—	19	2	no	—
150	#135	—	38	4	no	—
151	C₂×Q₁₆	—	19	1	no	—
152	C₂×Q₁₆	—	19	2	no	—
153	C₂×Q₁₆	—	19	1	no	—
154	C₂×Q₁₆	—	38	4	no	—
155	C₂×SD₁₆	—	19	2	no	—
156	C₂×SD₁₆	—	19	2	no	—
157	C₂×D₈	—	19	1	no	—
158	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
159	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
160	#124	—	19	2	no	—
161	#124	—	38	2	no	—
162	C₂²⋊Q₈	—	19	2	no	—
163	C₂²⋊Q₈	—	38	2	no	—
164	#121	—	19	2	no	—
165	#121	—	38	2	no	—
166	C₄×Q₈	—	19	2	no	—
167	C₄×D₄	—	38	2	no	—
168	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
169	#107	—	19	1	no	—
170	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
171	#103	—	19	2	no	—
172	#103	—	19	2	no	—
173	#102	—	19	2	no	—
174	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
175	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
176	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
177	C₁₉	#3285	1	1	yes	characteristic, complemented, Sylow
178	#41	—	38	4	no	—
179	#41	—	76	2	no	—
180	C₂×Q₈	—	19	2	no	—
181	C₂×Q₈	—	19	2	no	—
182	C₂×Q₈	—	38	4	no	—
183	C₂×Q₈	—	38	2	no	—
184	C₂×Q₈	—	76	2	no	—
185	C₂×D₄	—	19	2	no	—
186	C₂×D₄	—	38	2	no	—
187	C₂×D₄	—	38	2	no	—
188	C₂×D₄	D₇₆	1	1	yes	characteristic
189	C₂²×C₄	—	38	2	no	—
190	C₂²×C₄	—	38	2	no	—
191	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
193	Q₁₆	—	19	4	no	—
194	Q₁₆	—	38	4	no	—
195	Q₁₆	—	38	2	no	—
196	SD₁₆	—	38	4	no	—
197	SD₁₆	—	76	2	no	—
198	D₈	—	38	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
204	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
205	C₄⋊C₄	—	19	2	no	—
206	C₄⋊C₄	—	38	2	no	—
207	C₄⋊C₄	—	38	2	no	—
208	C₄⋊C₄	D₇₆	1	1	yes	characteristic
209	C₂²⋊C₄	—	38	2	no	—
210	C₂²⋊C₄	—	38	2	no	—
211	C₂²⋊C₄	—	38	2	no	—
212	C₂²⋊C₄	—	38	2	no	—
213	C₂²⋊C₄	D₇₆	1	1	yes	characteristic
214	C₂²⋊C₄	D₇₆	1	1	yes	characteristic
215	C₄²	—	38	2	no	—
216	C₄²	C₂²×D₁₉	1	1	yes	characteristic
217	C₂³	—	38	2	no	—
218	C₂³	C₂×D₇₆	1	1	yes	characteristic
219	C₂³	C₂×D₇₆	1	1	yes	characteristic
220	Q₈	—	19	4	no	—
221	Q₈	—	38	2	no	—
222	Q₈	—	38	2	no	—
223	Q₈	—	38	2	no	—
224	Q₈	—	76	4	no	—
225	D₄	—	2	2	no	—
226	D₄	—	38	2	no	—
227	D₄	—	38	2	no	—
228	D₄	—	76	4	no	—
229	D₄	—	76	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	38	2	no	—
235	C₂×C₄	—	38	2	no	—
236	C₂×C₄	—	38	2	no	—
237	C₂×C₄	—	76	4	no	—
238	C₂×C₄	—	76	4	no	—
239	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
240	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
241	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
242	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	D₄×D₁₉	1	2	yes	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	76	2	no	—
254	C₂²	—	76	2	no	—
255	C₂²	—	76	2	no	—
256	C₂²	#10597350	1	2	yes	complemented
257	C₂²	C₂²×D₇₆	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	38	4	no	—
263	C₄	—	76	2	no	—
264	C₄	—	76	2	no	—
265	C₄	C₂×D₄×D₁₉	1	1	yes	characteristic
266	C₄	#10597435	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	4	1	no	—
269	C₂	—	152	2	no	—
270	C₂	D₄×D₇₆	1	1	yes	characteristic
271	C₂	C₂×#10597350	1	1	yes	characteristic
272	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	C₈×D₄×C₁₉	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	C₈×D₄	D₁₉
hypocenter	C₁₉	#3285
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#10597350	C₂	semidirect
2. C₂²×Dic₇₆	C₂	semidirect
3. #11927203	C₂	semidirect
4. #11926928	C₂	semidirect
5. #11926785	C₂	semidirect
6. #11926780	C₂	semidirect
7. #11926767	C₂	semidirect
8. #11926708	C₂	semidirect
9. #11926685	C₂	semidirect
10. C₄×Dic₇₆	C₂	semidirect
11. C₈×D₄×C₁₉	C₂	semidirect
76. Dic₇₆	D₄	semidirect
117. C₂²×C₁₉	#135	semidirect
141. C₈×D₄	D₁₉	semidirect
177. C₁₉	#3285	semidirect
256. C₂²	#10597350	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597350	2
C₂²×Dic₇₆	2
#11927203	2
#11926928	2
#11926785	2
#11926780	2
#11926767	2
#11926708	2
#11926685	2
C₄×Dic₇₆	2
C₈×D₄×C₁₉	2
#3285	19