Subgroups — C₄×#10597370 #14566170

Order 2432.

656 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597370	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926450	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926442	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926428	C₂	1	2	yes	maximal, complemented
6	#11926417	C₂	1	2	yes	maximal
7	#11926404	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₁₉⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₂²⋊Dic₁₉	C₂	1	2	yes	maximal
11	C₂²×C₄²×C₁₉	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₉	—	2	4	no	—
17	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
26	#10597370	C₄	1	8	yes	direct product factor
27	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
28	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
29	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
30	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
31	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
32	C₄×Dic₃₈	—	2	4	no	—
33	C₁₉⋊#95	C₂²	1	2	yes	—
34	C₁₉⋊#95	C₂²	1	2	yes	—
35	C₁₉⋊#95	C₂²	1	2	yes	—
36	C₁₉⋊#95	C₂²	1	2	yes	—
37	C₁₉⋊#95	C₂²	1	2	yes	—
38	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	—	2	2	no	—
40	C₂²×C₄×C₁₉	—	2	2	no	—
41	C₂²×C₄×C₁₉	—	2	2	no	—
42	C₂²×C₄×C₁₉	—	2	2	no	—
43	C₂²×C₄×C₁₉	—	2	2	no	—
44	C₂²×C₄×C₁₉	—	2	2	no	—
45	C₂²×C₄×C₁₉	—	2	2	no	—
46	C₂²×C₄×C₁₉	C₂×C₄	1	4	yes	—
47	C₂²×C₄×C₁₉	Q₈	1	4	yes	complemented
48	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₉	C₂³	1	2	yes	—
55	C₂²×Dic₁₉	C₂³	1	2	yes	—
56	C₂²×Dic₁₉	C₂³	1	2	yes	—
57	C₂²×Dic₁₉	C₂³	1	2	yes	—
58	C₂×Dic₃₈	—	2	2	no	—
59	C₂×Dic₃₈	—	2	2	no	—
60	C₂×Dic₃₈	C₂×C₄	1	4	yes	complemented
61	C₄²×C₁₉	—	2	2	no	—
62	C₄²×C₁₉	—	2	2	no	—
63	C₄²×C₁₉	—	2	2	no	—
64	C₄²×C₁₉	D₄	1	4	yes	—
65	C₂²⋊Dic₁₉	—	2	4	no	—
66	C₂²⋊Dic₁₉	C₂×C₄	1	8	yes	—
67	C₄⋊Dic₁₉	—	2	2	no	—
68	C₄⋊Dic₁₉	—	2	2	no	—
69	C₄⋊Dic₁₉	C₂×C₄	1	4	yes	complemented
70	Dic₁₉⋊C₄	—	2	4	no	—
71	Dic₁₉⋊C₄	—	2	4	no	—
72	Dic₁₉⋊C₄	C₂×C₄	1	8	yes	complemented
73	C₄×Dic₁₉	—	2	4	no	—
74	C₄×Dic₁₉	—	2	4	no	—
75	C₂³×C₁₉	—	2	1	no	—
76	C₂³×C₁₉	—	2	1	no	—
77	C₂³×C₁₉	—	2	1	no	—
78	C₂³×C₁₉	—	2	1	no	—
79	C₂³×C₁₉	C₂²×C₄	1	2	yes	—
80	C₂³×C₁₉	C₂×Q₈	1	2	yes	—
81	C₂³×C₁₉	#41	1	2	yes	—
82	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
83	C₂×C₄×C₁₉	—	2	8	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	—	2	1	no	—
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	2	8	no	—
93	C₂×C₄×C₁₉	—	2	8	no	—
94	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
100	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
101	C₂×C₄×C₁₉	#41	1	2	yes	—
102	C₂×C₄×C₁₉	#41	1	2	yes	—
103	C₂×C₄×C₁₉	#41	1	2	yes	—
104	C₂×Dic₁₉	—	2	2	no	—
105	C₂×Dic₁₉	—	2	2	no	—
106	C₂×Dic₁₉	—	2	2	no	—
107	C₂×Dic₁₉	—	2	2	no	—
108	C₂×Dic₁₉	—	2	2	no	—
109	C₂×Dic₁₉	—	2	2	no	—
110	C₂×Dic₁₉	—	2	2	no	—
111	C₂×Dic₁₉	—	2	2	no	—
112	C₂×Dic₁₉	—	2	2	no	—
113	C₂×Dic₁₉	—	2	2	no	—
114	C₂×Dic₁₉	C₂²×C₄	1	4	yes	—
115	C₂×Dic₁₉	C₂²×C₄	1	4	yes	—
116	Dic₃₈	—	2	8	no	—
117	C₄×C₂²⋊Q₈	—	19	1	no	maximal, Sylow
118	C₂²×C₁₉	—	2	2	no	—
119	C₂²×C₁₉	—	2	2	no	—
120	C₂²×C₁₉	—	2	2	no	—
121	C₂²×C₁₉	—	2	2	no	—
122	C₂²×C₁₉	—	2	2	no	—
123	C₂²×C₁₉	—	2	2	no	—
124	C₂²×C₁₉	C₄×Q₈	1	4	yes	complemented
125	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
126	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
128	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
129	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
130	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
131	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
132	C₄×C₁₉	—	2	2	no	—
133	C₄×C₁₉	—	2	2	no	—
134	C₄×C₁₉	—	2	2	no	—
135	C₄×C₁₉	—	2	2	no	—
136	C₄×C₁₉	C₄×D₄	1	4	yes	—
137	C₄×C₁₉	C₂²⋊Q₈	1	4	yes	complemented
138	Dic₁₉	—	2	4	no	—
139	Dic₁₉	—	2	4	no	—
140	Dic₁₉	—	4	2	no	—
141	Dic₁₉	—	4	2	no	—
142	C₂×C₂²⋊Q₈	—	19	1	no	—
143	C₂×C₄×Q₈	—	19	1	no	—
144	C₂²×C₄²	D₁₉	1	1	yes	characteristic
145	#391	—	19	1	no	—
146	#389	—	19	1	no	—
147	#387	—	19	2	no	—
148	#385	—	19	2	no	—
149	#380	—	19	1	no	—
150	C₄×C₄⋊C₄	—	19	1	no	—
151	C₄×C₄⋊C₄	—	19	2	no	—
152	C₄×C₂²⋊C₄	—	19	2	no	—
153	C₂×C₁₉	—	2	4	no	—
154	C₂×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₂×C₁₉	C₄×#41	1	1	yes	characteristic
157	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₁₉	#525	1	1	yes	characteristic
159	C₂×C₁₉	#529	1	1	yes	characteristic
160	C₂×C₁₉	#533	1	1	yes	characteristic
161	C₂²×Q₈	—	19	1	no	—
162	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
163	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
164	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
165	C₂²⋊Q₈	—	19	8	no	—
166	C₄×Q₈	—	38	4	no	—
167	C₂×C₄⋊C₄	—	19	2	no	—
168	C₂×C₄⋊C₄	—	19	1	no	—
169	C₂×C₄⋊C₄	—	19	1	no	—
170	C₂×C₄⋊C₄	—	19	2	no	—
171	C₂×C₄⋊C₄	—	19	1	no	—
172	C₂×C₄⋊C₄	—	19	2	no	—
173	C₂×C₂²⋊C₄	—	19	2	no	—
174	C₂×C₂²⋊C₄	—	19	2	no	—
175	C₂×C₄²	—	2	4	no	—
176	C₂×C₄²	—	19	2	no	—
177	C₂×C₄²	—	19	2	no	—
178	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
179	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
180	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
181	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
182	#95	—	19	2	no	—
183	#95	—	19	2	no	—
184	#95	—	19	2	no	—
185	#95	—	19	2	no	—
186	#95	—	19	2	no	—
187	C₁₉	C₄×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
188	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
189	C₂×Q₈	—	19	4	no	—
190	C₂×Q₈	—	38	2	no	—
191	C₂×Q₈	—	38	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	19	2	no	—
200	C₂²×C₄	—	19	2	no	—
201	C₂²×C₄	—	19	2	no	—
202	C₂²×C₄	—	19	2	no	—
203	C₂²×C₄	Dic₃₈	1	4	yes	complemented
204	C₂²×C₄	C₄×D₁₉	1	4	yes	—
205	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
211	C₄⋊C₄	—	19	8	no	—
212	C₄⋊C₄	—	19	4	no	—
213	C₄⋊C₄	—	38	4	no	—
214	C₄⋊C₄	—	38	2	no	—
215	C₄⋊C₄	—	38	4	no	—
216	C₄⋊C₄	—	38	2	no	—
217	C₂²⋊C₄	—	19	8	no	—
218	C₂²⋊C₄	—	38	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	38	4	no	—
223	C₄²	—	38	4	no	—
224	C₄²	C₂²⋊D₁₉	1	4	yes	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	C₂×Dic₃₈	1	2	yes	—
230	C₂³	C₂×C₄×D₁₉	1	2	yes	—
231	C₂³	#64541	1	2	yes	—
232	C₂³	C₂³×D₁₉	1	1	yes	characteristic
233	Q₈	—	38	8	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	19	4	no	—
246	C₂×C₄	—	19	4	no	—
247	C₂×C₄	—	38	2	no	—
248	C₂×C₄	—	38	2	no	—
249	C₂×C₄	—	38	2	no	—
250	C₂×C₄	—	38	2	no	—
251	C₂×C₄	—	38	2	no	—
252	C₂×C₄	—	38	2	no	—
253	C₂×C₄	—	38	2	no	—
254	C₂×C₄	—	38	2	no	—
255	C₂×C₄	—	38	2	no	—
256	C₂×C₄	—	38	2	no	—
257	C₂×C₄	C₂×Dic₃₈	1	2	yes	—
258	C₂×C₄	C₂×Dic₃₈	1	2	yes	—
259	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
260	C₂×C₄	C₂×C₄×D₁₉	1	2	yes	—
261	C₂×C₄	#64541	1	2	yes	—
262	C₂×C₄	#64541	1	2	yes	—
263	C₂×C₄	#64541	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
265	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	C₄×Dic₃₈	1	4	yes	complemented
274	C₂²	C₂²×Dic₃₈	1	1	yes	characteristic
275	C₂²	C₂²×C₄×D₁₉	1	1	yes	characteristic
276	C₂²	C₂×#64541	1	1	yes	characteristic
277	C₂²	C₂×#64541	1	1	yes	characteristic
278	C₂²	C₂×#64541	1	1	yes	characteristic
279	C₂²	C₂×#64541	1	1	yes	characteristic
280	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	38	4	no	—
286	C₄	—	38	4	no	—
287	C₄	—	76	2	no	—
288	C₄	—	76	2	no	—
289	C₄	#10597370	1	4	yes	direct product factor
290	C₄	C₄×C₂²⋊D₁₉	1	4	yes	—
291	C₂	—	2	4	no	—
292	C₂	C₂×C₄×C₂²⋊D₁₉	1	1	yes	characteristic
293	C₂	C₂×C₄×Dic₃₈	1	1	yes	characteristic
294	C₂	C₄×#64541	1	1	yes	characteristic
295	C₂	C₂×#10597370	1	1	yes	characteristic
296	C₂	#11927055	1	1	yes	characteristic
297	C₂	#11927078	1	1	yes	characteristic
298	C₂	#11927101	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂³×C₄
Frattini	C₂³	C₂³×D₁₉
Fitting	C₂²×C₄²×C₁₉	C₂
socle, cosocle	C₂³×C₁₉	C₂⁴
prime radical	C₂⁴×C₁₉	C₂³
hypercenter	C₂²×C₄²	D₁₉
hypocenter	C₁₉	C₄×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₃₈	C₂	semidirect
3. #11926450	C₂	semidirect
4. #11926442	C₂	semidirect
5. #11926428	C₂	semidirect
8. C₄×Dic₁₉⋊C₄	C₂	semidirect
9. C₄×C₄⋊Dic₁₉	C₂	semidirect
26. #10597370	C₄	direct (N × H)
47. C₂²×C₄×C₁₉	Q₈	semidirect
60. C₂×Dic₃₈	C₂×C₄	semidirect
69. C₄⋊Dic₁₉	C₂×C₄	semidirect
72. Dic₁₉⋊C₄	C₂×C₄	semidirect
124. C₂²×C₁₉	C₄×Q₈	semidirect
137. C₄×C₁₉	C₂²⋊Q₈	semidirect
187. C₁₉	C₄×C₂²⋊Q₈	semidirect
203. C₂²×C₄	Dic₃₈	semidirect
273. C₂²	C₄×Dic₃₈	semidirect
289. C₄	#10597370	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#10597370	2
C₂×C₄×Dic₃₈	2
#11926450	2
#11926442	2
#11926428	2
#11926417	2
#11926404	2
C₄×Dic₁₉⋊C₄	2
C₄×C₄⋊Dic₁₉	2
C₄×C₂²⋊Dic₁₉	2
C₂²×C₄²×C₁₉	2
C₄×C₂²⋊Q₈	19

Subgroups — C4×#10597370 #14566170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#10597370 #14566170