Subgroups — C₂×#11927168 #14577016

Order 2432.

1132 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927168	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10597344	C₂	1	2	yes	maximal, complemented
3	C₂×#10597393	C₂	1	2	yes	maximal, complemented
4	C₂×#10597321	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10597340	C₂	1	2	yes	maximal, complemented
6	C₂×#10597343	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10597392	C₂	1	2	yes	maximal, complemented
8	C₂×#10597341	C₂	1	1	yes	maximal, characteristic
9	C₂×#10597333	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10597339	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#10597318	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₈×C₁₉	C₂²	1	2	yes	—
14	C₂²×D₇₆	C₂²	1	1	yes	characteristic
15	C₂²×D₇₆	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₉	C₂²	1	2	yes	—
18	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
19	C₁₉×C₄⋊Q₈	C₂²	1	8	yes	complemented
20	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
24	#10597393	C₂²	1	16	yes	complemented
25	#10597392	C₂²	1	16	yes	complemented
26	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
27	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
29	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
30	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
32	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
33	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
34	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
35	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
36	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
37	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
38	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
39	#10597344	C₂²	1	16	yes	complemented
40	#10597343	C₂²	1	8	yes	complemented
41	#10597341	C₂²	1	8	yes	—
42	#10597340	C₂²	1	16	yes	complemented
43	#10597339	C₂²	1	8	yes	complemented
44	#10597333	C₂²	1	8	yes	complemented
45	#10597321	C₂²	1	8	yes	complemented
46	#10597318	C₂²	1	8	yes	complemented
47	C₂³×D₁₉	C₂³	1	1	yes	characteristic
48	C₂³×D₁₉	C₂³	1	2	yes	—
49	C₂×Q₈×C₁₉	—	4	4	no	—
50	C₂×Q₈×C₁₉	C₂³	1	8	yes	—
51	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
56	C₂²×Dic₁₉	C₂³	1	2	yes	—
57	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
59	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
60	C₂×D₇₆	—	2	4	no	—
61	C₂×D₇₆	—	4	4	no	—
62	C₂×D₇₆	C₂³	1	4	yes	—
63	C₂×D₇₆	C₂³	1	8	yes	—
64	C₂×C₄×D₁₉	—	2	4	no	—
65	C₂×C₄×D₁₉	—	4	4	no	—
66	C₂×C₄×D₁₉	C₂³	1	4	yes	—
67	C₂×C₄×D₁₉	C₂³	1	8	yes	—
68	C₂×Dic₃₈	—	2	4	no	—
69	C₂×Dic₃₈	C₂³	1	4	yes	—
70	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
71	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
72	C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
73	C₄²×C₁₉	C₂³	1	4	yes	—
74	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
75	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
76	(C₂×D₁₉)⋊C₄	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	4	yes	—
80	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
81	C₄⋊Dic₁₉	C₂³	1	8	yes	—
82	Dic₁₉⋊C₄	C₂³	1	4	yes	—
83	Dic₁₉⋊C₄	C₂³	1	8	yes	—
84	Dic₁₉⋊C₄	C₂³	1	4	yes	—
85	C₄×Dic₁₉	C₂³	1	8	yes	—
86	C₄×Dic₁₉	C₂³	1	4	yes	—
87	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
88	C₂²×D₁₉	—	2	2	no	—
89	C₂²×D₁₉	—	2	4	no	—
90	C₂²×D₁₉	—	2	4	no	—
91	C₂²×D₁₉	—	2	2	no	—
92	C₂²×D₁₉	—	2	2	no	—
93	C₂²×D₁₉	—	2	4	no	—
94	C₂²×D₁₉	C₂⁴	1	2	yes	—
95	C₂²×D₁₉	C₂⁴	1	4	yes	—
96	Q₈×C₁₉	—	4	8	no	—
97	C₂×C₄×C₁₉	—	2	4	no	—
98	C₂×C₄×C₁₉	—	2	4	no	—
99	C₂×C₄×C₁₉	—	2	2	no	—
100	C₂×C₄×C₁₉	—	2	2	no	—
101	C₂×C₄×C₁₉	#41	1	4	yes	—
102	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
103	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
104	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
105	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
106	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
107	C₂×Dic₁₉	—	2	4	no	—
108	C₂×Dic₁₉	—	2	2	no	—
109	C₂×Dic₁₉	—	2	2	no	—
110	C₂×Dic₁₉	—	2	2	no	—
111	C₂×Dic₁₉	C₂⁴	1	2	yes	—
112	C₂×Dic₁₉	C₂⁴	1	2	yes	—
113	C₂×Dic₁₉	C₂⁴	1	4	yes	—
114	C₂×Dic₁₉	C₂⁴	1	2	yes	—
115	D₇₆	—	2	8	no	—
116	D₇₆	—	4	8	no	—
117	C₄×D₁₉	—	2	8	no	—
118	C₄×D₁₉	—	4	8	no	—
119	Dic₃₈	—	2	8	no	—
120	C₂×#542	—	19	1	no	maximal, Sylow
121	C₂²×C₁₉	C₂×#41	1	2	yes	—
122	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
123	C₂²×C₁₉	2₋¹⁺⁴	1	2	yes	—
124	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
125	C₂×D₁₉	—	2	2	no	—
126	C₂×D₁₉	—	2	4	no	—
127	C₂×D₁₉	—	2	4	no	—
128	C₂×D₁₉	—	2	2	no	—
129	C₂×D₁₉	—	2	2	no	—
130	C₂×D₁₉	—	2	4	no	—
131	C₂×D₁₉	—	4	4	no	—
132	C₂×D₁₉	—	4	8	no	—
133	C₄×C₁₉	—	2	4	no	—
134	C₄×C₁₉	—	2	4	no	—
135	C₄×C₁₉	—	2	2	no	—
136	C₄×C₁₉	—	2	2	no	—
137	C₄×C₁₉	C₂×#41	1	4	yes	—
138	Dic₁₉	—	2	4	no	—
139	Dic₁₉	—	2	2	no	—
140	Dic₁₉	—	2	2	no	—
141	Dic₁₉	—	2	2	no	—
142	#542	—	19	16	no	—
143	C₂×C₄⋊Q₈	D₁₉	1	1	yes	characteristic, complemented
144	C₂×#126	—	19	2	no	—
145	C₂×#124	—	19	2	no	—
146	C₂×#124	—	19	1	no	—
147	C₂×#123	—	19	2	no	—
148	C₂×C₂²⋊Q₈	—	19	2	no	—
149	C₂×C₂²⋊Q₈	—	19	1	no	—
150	C₂×#121	—	19	1	no	—
151	C₂×C₄×Q₈	—	19	1	no	—
152	C₂×C₄×D₄	—	19	1	no	—
153	C₂×#117	—	19	1	no	—
154	C₂×C₁₉	#542	1	4	yes	complemented
155	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
156	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
158	D₁₉	—	4	2	no	—
159	D₁₉	—	4	4	no	—
160	C₂²×Q₈	—	19	1	no	—
161	C₂²×Q₈	C₂×D₁₉	1	2	yes	—
162	C₂²×D₄	—	19	1	no	—
163	C₂²×D₄	—	19	2	no	—
164	C₂³×C₄	—	19	1	no	—
165	C₂³×C₄	—	19	2	no	—
166	C₄⋊Q₈	C₂×D₁₉	1	8	yes	complemented
167	#126	—	19	16	no	—
168	#124	—	19	16	no	—
169	#124	—	19	8	no	—
170	#123	—	19	16	no	—
171	C₂²⋊Q₈	—	19	16	no	—
172	C₂²⋊Q₈	—	19	8	no	—
173	#121	—	19	8	no	—
174	C₄×Q₈	—	19	8	no	—
175	C₄×D₄	—	19	8	no	—
176	#117	—	19	8	no	—
177	C₂×C₄⋊C₄	—	19	1	no	—
178	C₂×C₄⋊C₄	—	19	1	no	—
179	C₂×C₄⋊C₄	—	19	2	no	—
180	C₂×C₄⋊C₄	—	19	2	no	—
181	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
184	C₂×C₂²⋊C₄	—	19	1	no	—
185	C₂×C₂²⋊C₄	—	19	1	no	—
186	C₂×C₂²⋊C₄	—	19	2	no	—
187	C₂×C₂²⋊C₄	—	19	2	no	—
188	C₂×C₂²⋊C₄	—	19	2	no	—
189	C₂×C₂²⋊C₄	—	19	2	no	—
190	C₂×C₂²⋊C₄	—	19	2	no	—
191	C₂×C₄²	—	19	1	no	—
192	C₂×C₄²	—	19	2	no	—
193	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
194	C₁₉	C₂×#542	1	1	yes	characteristic, complemented, Sylow
195	C₂⁴	—	19	2	no	—
196	C₂⁴	—	19	1	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×Q₈	—	19	4	no	—
199	C₂×Q₈	—	38	4	no	—
200	C₂×Q₈	C₂²×D₁₉	1	8	yes	—
201	C₂×D₄	—	19	4	no	—
202	C₂×D₄	—	19	8	no	—
203	C₂×D₄	—	38	4	no	—
204	C₂×D₄	—	76	4	no	—
205	C₂²×C₄	—	19	1	no	—
206	C₂²×C₄	—	19	2	no	—
207	C₂²×C₄	—	19	1	no	—
208	C₂²×C₄	—	19	1	no	—
209	C₂²×C₄	—	19	4	no	—
210	C₂²×C₄	—	19	8	no	—
211	C₂²×C₄	—	38	4	no	—
212	C₂²×C₄	—	76	4	no	—
213	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
214	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
215	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
217	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
218	C₄⋊C₄	—	19	4	no	—
219	C₄⋊C₄	—	19	4	no	—
220	C₄⋊C₄	—	19	8	no	—
221	C₄⋊C₄	—	19	8	no	—
222	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
223	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
224	C₄⋊C₄	C₂²×D₁₉	1	8	yes	—
225	C₂²⋊C₄	—	19	4	no	—
226	C₂²⋊C₄	—	19	4	no	—
227	C₂²⋊C₄	—	19	8	no	—
228	C₂²⋊C₄	—	19	8	no	—
229	C₂²⋊C₄	—	19	8	no	—
230	C₂²⋊C₄	—	19	8	no	—
231	C₂²⋊C₄	—	19	8	no	—
232	C₄²	—	19	4	no	—
233	C₄²	—	19	8	no	—
234	C₄²	C₂²×D₁₉	1	4	yes	—
235	C₂³	—	19	2	no	—
236	C₂³	—	19	4	no	—
237	C₂³	—	38	4	no	—
238	C₂³	—	38	2	no	—
239	C₂³	—	38	2	no	—
240	C₂³	—	38	4	no	—
241	C₂³	—	38	4	no	—
242	C₂³	—	38	2	no	—
243	C₂³	C₂³×D₁₉	1	1	yes	characteristic
244	Q₈	—	4	8	no	—
245	Q₈	—	38	8	no	—
246	D₄	—	38	8	no	—
247	D₄	—	76	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	19	2	no	—
253	C₂×C₄	—	19	4	no	—
254	C₂×C₄	—	19	2	no	—
255	C₂×C₄	—	19	2	no	—
256	C₂×C₄	—	38	2	no	—
257	C₂×C₄	—	38	2	no	—
258	C₂×C₄	—	38	4	no	—
259	C₂×C₄	—	38	2	no	—
260	C₂×C₄	—	38	8	no	—
261	C₂×C₄	—	76	8	no	—
262	C₂×C₄	#64545	1	4	yes	—
263	C₂×C₄	C₂³×D₁₉	1	4	yes	—
264	C₂×C₄	C₂³×D₁₉	1	2	yes	—
265	C₂×C₄	C₂³×D₁₉	1	2	yes	—
266	C₂×C₄	C₂³×D₁₉	1	4	yes	—
267	C₂×C₄	C₂³×D₁₉	1	2	yes	—
268	C₂²	—	38	2	no	—
269	C₂²	—	38	4	no	—
270	C₂²	—	38	4	no	—
271	C₂²	—	38	2	no	—
272	C₂²	—	38	2	no	—
273	C₂²	—	38	4	no	—
274	C₂²	—	76	8	no	—
275	C₂²	—	76	4	no	—
276	C₂²	#10597430	1	2	yes	—
277	C₂²	C₂×#64545	1	2	yes	—
278	C₂²	#10597433	1	2	yes	—
279	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	38	2	no	—
285	C₄	—	38	4	no	—
286	C₄	—	38	2	no	—
287	C₄	—	38	2	no	—
288	C₄	C₂×#64545	1	4	yes	—
289	C₂	—	76	2	no	—
290	C₂	—	76	4	no	—
291	C₂	#11927168	1	4	yes	direct product factor
292	C₂	C₂²×#64545	1	1	yes	characteristic
293	C₂	C₂×#10597430	1	1	yes	characteristic
294	C₂	C₂×#10597433	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₉
commutator, cosocle	C₂²×C₁₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉
Fitting	C₂×C₁₉×C₄⋊Q₈	C₂
socle	C₂³×C₁₉	C₂⁴
prime radical	C₂×#10597341	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₉
hypocenter	C₁₉	C₂×#542
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #11927168	C₂	direct (N × H)
2. C₂×#10597344	C₂	semidirect
3. C₂×#10597393	C₂	semidirect
4. C₂×#10597321	C₂	semidirect
5. C₂×#10597340	C₂	semidirect
6. C₂×#10597343	C₂	semidirect
7. C₂×#10597392	C₂	semidirect
9. C₂×#10597333	C₂	semidirect
10. C₂×#10597339	C₂	semidirect
11. C₂×#10597318	C₂	semidirect
12. C₂×C₁₉×C₄⋊Q₈	C₂	semidirect
19. C₁₉×C₄⋊Q₈	C₂²	semidirect
24. #10597393	C₂²	semidirect
25. #10597392	C₂²	semidirect
39. #10597344	C₂²	semidirect
40. #10597343	C₂²	semidirect
42. #10597340	C₂²	semidirect
43. #10597339	C₂²	semidirect
44. #10597333	C₂²	semidirect
45. #10597321	C₂²	semidirect
46. #10597318	C₂²	semidirect
143. C₂×C₄⋊Q₈	D₁₉	semidirect
154. C₂×C₁₉	#542	semidirect
166. C₄⋊Q₈	C₂×D₁₉	semidirect
194. C₁₉	C₂×#542	semidirect
291. C₂	#11927168	direct (N × H)

Maximal subgroups

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

subgroup	index
#11927168	2
C₂×#10597344	2
C₂×#10597393	2
C₂×#10597321	2
C₂×#10597340	2
C₂×#10597343	2
C₂×#10597392	2
C₂×#10597341	2
C₂×#10597333	2
C₂×#10597339	2
C₂×#10597318	2
C₂×C₁₉×C₄⋊Q₈	2
C₂×#542	19

Subgroups — C2×#11927168 #14577016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11927168 #14577016