Subgroups — C₂×#11927171 #14577032

Order 2432.

1080 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927171	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10597334	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10597390	C₂	1	2	yes	maximal, complemented
4	C₂×#10597316	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10597336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10597335	C₂	1	2	yes	maximal, complemented
7	C₂×#10597342	C₂	1	2	yes	maximal
8	C₂×#10597392	C₂	1	2	yes	maximal
9	C₂×#10597338	C₂	1	1	yes	maximal, characteristic
10	C₂×#10597318	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄⋊C₄×D₁₉	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₈×C₁₉	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₉	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₉	C₂²	1	1	yes	characteristic
16	C₂²×Dic₃₈	C₂²	1	2	yes	—
17	C₁₉×C₄⋊Q₈	C₂²	1	8	yes	complemented
18	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
22	#10597392	C₂²	1	16	yes	—
23	#10597390	C₂²	1	16	yes	complemented
24	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
26	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
28	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
30	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
31	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
32	C₂×Dic₁₉⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
34	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
35	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
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	#10597342	C₂²	1	16	yes	—
40	#10597338	C₂²	1	8	yes	—
41	C₄⋊C₄×D₁₉	C₂²	1	8	yes	—
42	#10597336	C₂²	1	8	yes	complemented
43	#10597335	C₂²	1	16	yes	complemented
44	#10597334	C₂²	1	8	yes	complemented
45	#10597318	C₂²	1	8	yes	—
46	#10597316	C₂²	1	8	yes	complemented
47	C₂³×D₁₉	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₁₉	—	4	4	no	—
49	C₂×Q₈×C₁₉	C₂³	1	8	yes	—
50	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
55	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₉	C₂³	1	2	yes	—
57	C₂²×Dic₁₉	C₂³	1	2	yes	—
58	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
59	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₁₉	—	4	4	no	—
61	C₂×C₄×D₁₉	Q₈	1	8	yes	complemented
62	C₂×C₄×D₁₉	C₂³	1	8	yes	—
63	C₂×C₄×D₁₉	C₂³	1	4	yes	—
64	C₂×Dic₃₈	—	4	4	no	—
65	C₂×Dic₃₈	C₂³	1	8	yes	—
66	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
67	C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
68	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
69	C₄²×C₁₉	C₂³	1	4	yes	—
70	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
71	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
72	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
73	C₄⋊Dic₁₉	C₂³	1	8	yes	—
74	C₄⋊Dic₁₉	C₂³	1	4	yes	—
75	C₄⋊Dic₁₉	C₂³	1	4	yes	—
76	Dic₁₉⋊C₄	C₂³	1	4	yes	—
77	Dic₁₉⋊C₄	C₂³	1	8	yes	—
78	Dic₁₉⋊C₄	C₂³	1	8	yes	—
79	Dic₁₉⋊C₄	C₂³	1	8	yes	—
80	Dic₁₉⋊C₄	C₂³	1	8	yes	—
81	Dic₁₉⋊C₄	C₂³	1	8	yes	—
82	Dic₁₉⋊C₄	C₂³	1	4	yes	—
83	C₄×Dic₁₉	C₂³	1	8	yes	—
84	C₄×Dic₁₉	C₂³	1	4	yes	—
85	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
86	C₂²×D₁₉	—	2	2	no	—
87	C₂²×D₁₉	—	2	2	no	—
88	C₂²×D₁₉	C₂×Q₈	1	4	yes	—
89	C₂²×D₁₉	C₂⁴	1	2	yes	—
90	Q₈×C₁₉	—	4	8	no	—
91	C₂×C₄×C₁₉	—	2	4	no	—
92	C₂×C₄×C₁₉	—	2	4	no	—
93	C₂×C₄×C₁₉	—	2	2	no	—
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	C₂×Q₈	1	4	yes	—
96	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
97	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
98	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
99	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
100	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
101	C₂×Dic₁₉	—	2	2	no	—
102	C₂×Dic₁₉	—	2	4	no	—
103	C₂×Dic₁₉	—	2	4	no	—
104	C₂×Dic₁₉	—	2	2	no	—
105	C₂×Dic₁₉	C₂×Q₈	1	4	yes	—
106	C₂×Dic₁₉	C₂⁴	1	4	yes	—
107	C₂×Dic₁₉	C₂⁴	1	2	yes	—
108	C₂×Dic₁₉	C₂⁴	1	2	yes	—
109	C₂×Dic₁₉	C₂⁴	1	4	yes	—
110	C₂×Dic₁₉	C₂⁴	1	2	yes	—
111	C₄×D₁₉	—	4	8	no	—
112	C₄×D₁₉	C₂×Q₈	1	16	yes	complemented
113	Dic₃₈	—	4	8	no	—
114	C₂×#544	—	19	1	no	maximal, Sylow
115	C₂²×C₁₉	C₂²×Q₈	1	2	yes	—
116	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
117	C₂²×C₁₉	2₋¹⁺⁴	1	2	yes	—
118	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
119	C₂×D₁₉	—	2	8	no	—
120	C₂×D₁₉	—	2	2	no	—
121	C₂×D₁₉	—	2	2	no	—
122	C₂×D₁₉	C₂²×Q₈	1	4	yes	—
123	C₄×C₁₉	—	2	4	no	—
124	C₄×C₁₉	—	2	4	no	—
125	C₄×C₁₉	—	2	2	no	—
126	C₄×C₁₉	—	2	2	no	—
127	C₄×C₁₉	C₂²×Q₈	1	4	yes	—
128	Dic₁₉	—	2	2	no	—
129	Dic₁₉	—	2	4	no	—
130	Dic₁₉	—	2	4	no	—
131	Dic₁₉	—	2	2	no	—
132	Dic₁₉	C₂²×Q₈	1	4	yes	—
133	#544	—	19	16	no	—
134	C₂×C₄⋊Q₈	—	19	2	no	—
135	C₂×C₄⋊Q₈	—	19	1	no	—
136	C₂×C₄⋊Q₈	D₁₉	1	1	yes	characteristic, complemented
137	C₂×#125	—	19	2	no	—
138	C₂×#125	—	19	1	no	—
139	C₂×#125	—	19	1	no	—
140	C₂×C₂²⋊Q₈	—	19	2	no	—
141	C₂×C₂²⋊Q₈	—	19	2	no	—
142	C₂×#117	—	19	1	no	—
143	C₂×#117	—	19	1	no	—
144	C₂²×C₄⋊C₄	—	19	1	no	—
145	C₂×C₁₉	#544	1	4	yes	complemented
146	C₂×C₁₉	C₂³×Q₈	1	1	yes	characteristic
147	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
149	D₁₉	—	2	4	no	—
150	C₂²×Q₈	—	19	2	no	—
151	C₂²×Q₈	C₂×D₁₉	1	2	yes	—
152	C₂³×C₄	—	19	1	no	—
153	C₂³×C₄	—	19	2	no	—
154	C₄⋊Q₈	—	19	16	no	—
155	C₄⋊Q₈	—	19	8	no	—
156	C₄⋊Q₈	C₂×D₁₉	1	8	yes	complemented
157	#125	—	19	16	no	—
158	#125	—	19	8	no	—
159	#125	—	19	8	no	—
160	C₂²⋊Q₈	—	19	16	no	—
161	C₂²⋊Q₈	—	19	16	no	—
162	#117	—	19	8	no	—
163	#117	—	19	8	no	—
164	C₂×C₄⋊C₄	—	19	2	no	—
165	C₂×C₄⋊C₄	—	19	1	no	—
166	C₂×C₄⋊C₄	—	19	1	no	—
167	C₂×C₄⋊C₄	—	19	1	no	—
168	C₂×C₄⋊C₄	—	19	1	no	—
169	C₂×C₄⋊C₄	—	19	2	no	—
170	C₂×C₄⋊C₄	—	19	2	no	—
171	C₂×C₄⋊C₄	—	19	2	no	—
172	C₂×C₄⋊C₄	—	19	2	no	—
173	C₂×C₄⋊C₄	—	19	2	no	—
174	C₂×C₄⋊C₄	—	19	8	no	—
175	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
178	C₂×C₂²⋊C₄	—	19	2	no	—
179	C₂×C₂²⋊C₄	—	19	1	no	—
180	C₂×C₂²⋊C₄	—	19	1	no	—
181	C₂×C₄²	—	19	2	no	—
182	C₂×C₄²	—	19	1	no	—
183	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
184	C₁₉	C₂×#544	1	1	yes	characteristic, complemented, Sylow
185	C₂⁴	—	19	1	no	—
186	C₂×Q₈	—	4	4	no	—
187	C₂×Q₈	—	19	8	no	—
188	C₂×Q₈	—	76	4	no	—
189	C₂×Q₈	C₂²×D₁₉	1	8	yes	—
190	C₂²×C₄	—	19	1	no	—
191	C₂²×C₄	—	19	8	no	—
192	C₂²×C₄	—	19	4	no	—
193	C₂²×C₄	—	19	1	no	—
194	C₂²×C₄	—	19	2	no	—
195	C₂²×C₄	—	19	1	no	—
196	C₂²×C₄	—	19	2	no	—
197	C₂²×C₄	—	19	8	no	—
198	C₂²×C₄	—	76	4	no	—
199	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
200	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
203	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
204	C₄⋊C₄	—	19	8	no	—
205	C₄⋊C₄	—	19	4	no	—
206	C₄⋊C₄	—	19	4	no	—
207	C₄⋊C₄	—	19	8	no	—
208	C₄⋊C₄	—	19	8	no	—
209	C₄⋊C₄	—	19	8	no	—
210	C₄⋊C₄	—	19	8	no	—
211	C₄⋊C₄	—	19	4	no	—
212	C₄⋊C₄	—	19	4	no	—
213	C₄⋊C₄	—	19	8	no	—
214	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
215	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
216	C₄⋊C₄	C₂²×D₁₉	1	8	yes	—
217	C₂²⋊C₄	—	19	8	no	—
218	C₂²⋊C₄	—	19	4	no	—
219	C₂²⋊C₄	—	19	4	no	—
220	C₄²	—	19	8	no	—
221	C₄²	—	19	4	no	—
222	C₄²	C₂²×D₁₉	1	4	yes	—
223	C₂³	—	19	4	no	—
224	C₂³	—	19	2	no	—
225	C₂³	—	38	2	no	—
226	C₂³	—	38	2	no	—
227	C₂³	C₂³×D₁₉	1	1	yes	characteristic
228	Q₈	—	4	8	no	—
229	Q₈	—	76	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	19	4	no	—
235	C₂×C₄	—	19	2	no	—
236	C₂×C₄	—	19	16	no	—
237	C₂×C₄	—	19	2	no	—
238	C₂×C₄	—	19	4	no	—
239	C₂×C₄	—	19	2	no	—
240	C₂×C₄	—	19	4	no	—
241	C₂×C₄	—	38	4	no	—
242	C₂×C₄	—	38	2	no	—
243	C₂×C₄	—	38	2	no	—
244	C₂×C₄	—	38	4	no	—
245	C₂×C₄	—	76	8	no	—
246	C₂×C₄	Q₈×D₁₉	1	4	yes	—
247	C₂×C₄	C₂³×D₁₉	1	4	yes	—
248	C₂×C₄	C₂³×D₁₉	1	2	yes	—
249	C₂×C₄	C₂³×D₁₉	1	2	yes	—
250	C₂×C₄	C₂³×D₁₉	1	4	yes	—
251	C₂×C₄	C₂³×D₁₉	1	2	yes	—
252	C₂²	—	19	4	no	—
253	C₂²	—	38	2	no	—
254	C₂²	—	38	2	no	—
255	C₂²	—	38	8	no	—
256	C₂²	#10597430	1	2	yes	—
257	C₂²	C₂×Q₈×D₁₉	1	2	yes	—
258	C₂²	#10597433	1	2	yes	—
259	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	19	4	no	—
265	C₄	—	38	4	no	—
266	C₄	—	38	2	no	—
267	C₄	—	38	2	no	—
268	C₄	—	38	4	no	—
269	C₄	C₂×Q₈×D₁₉	1	4	yes	—
270	C₂	—	38	4	no	—
271	C₂	#11927171	1	4	yes	direct product factor
272	C₂	C₂²×Q₈×D₁₉	1	1	yes	characteristic
273	C₂	C₂×#10597430	1	1	yes	characteristic
274	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	C₂³×D₁₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₉
hypocenter	C₁₉	C₂×#544
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. #11927171	C₂	direct (N × H)
2. C₂×#10597334	C₂	semidirect
3. C₂×#10597390	C₂	semidirect
4. C₂×#10597316	C₂	semidirect
5. C₂×#10597336	C₂	semidirect
6. C₂×#10597335	C₂	semidirect
12. C₂×C₁₉×C₄⋊Q₈	C₂	semidirect
17. C₁₉×C₄⋊Q₈	C₂²	semidirect
23. #10597390	C₂²	semidirect
42. #10597336	C₂²	semidirect
43. #10597335	C₂²	semidirect
44. #10597334	C₂²	semidirect
46. #10597316	C₂²	semidirect
61. C₂×C₄×D₁₉	Q₈	semidirect
112. C₄×D₁₉	C₂×Q₈	semidirect
136. C₂×C₄⋊Q₈	D₁₉	semidirect
145. C₂×C₁₉	#544	semidirect
156. C₄⋊Q₈	C₂×D₁₉	semidirect
184. C₁₉	C₂×#544	semidirect
271. C₂	#11927171	direct (N × H)

Maximal subgroups

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

subgroup	index
#11927171	2
C₂×#10597334	2
C₂×#10597390	2
C₂×#10597316	2
C₂×#10597336	2
C₂×#10597335	2
C₂×#10597342	2
C₂×#10597392	2
C₂×#10597338	2
C₂×#10597318	2
C₂×C₄⋊C₄×D₁₉	2
C₂×C₁₉×C₄⋊Q₈	2
C₂×#544	19

Subgroups — C2×#11927171 #14577032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11927171 #14577032