Subgroups — C₂×#11927203 #14577101

Order 2432.

1620 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#64541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11927203	C₂	1	16	yes	maximal, direct product factor
4	C₂×#10597321	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10597331	C₂	1	2	yes	maximal, complemented
6	C₂×#10597370	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10597342	C₂	1	2	yes	maximal, complemented
8	C₂×#10597376	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10597385	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×(C₂²⋊D₁₉)⋊C₄	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊D₁₉	C₂²	1	2	yes	—
17	C₂²×C₂²⋊D₁₉	C₂²	1	2	yes	—
18	C₂³×Dic₁₉	C₂²	1	2	yes	—
19	C₂×#64541	—	2	8	no	—
20	C₂×#64541	C₂²	1	8	yes	complemented
21	C₂²×D₇₆	C₂²	1	1	yes	characteristic
22	C₂²×C₄×D₁₉	C₂²	1	2	yes	—
23	C₂²×Dic₃₈	—	2	8	no	—
24	C₂²×Dic₃₈	C₂²	1	2	yes	—
25	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
26	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
27	C₂²×Dic₃₈	C₂²	1	8	yes	complemented
28	C₄×D₄×C₁₉	C₂²	1	8	yes	complemented
29	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊Dic₁₉	C₂²	1	2	yes	—
34	#10597385	C₂²	1	16	yes	complemented
35	#10597376	C₂²	1	8	yes	complemented
36	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
37	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
38	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
39	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
40	#10597370	C₂²	1	8	yes	complemented
41	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
42	C₂×Dic₁₉⋊C₄	C₂²	1	2	yes	—
43	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
44	#10597342	C₂²	1	16	yes	complemented
45	#10597331	C₂²	1	16	yes	complemented
46	(C₂²⋊D₁₉)⋊C₄	C₂²	1	16	yes	complemented
47	#10597321	C₂²	1	8	yes	complemented
48	C₄×Dic₃₈	C₂²	1	8	yes	complemented
49	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
50	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
51	C₂³×D₁₉	C₂³	1	2	yes	—
52	C₂×D₄×C₁₉	—	2	4	no	—
53	C₂×D₄×C₁₉	C₂³	1	4	yes	—
54	C₂²×C₄×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	—	2	4	no	—
56	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
57	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
62	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₉	—	2	8	no	—
64	C₂×C₂²⋊D₁₉	—	2	8	no	—
65	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
66	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
67	C₂²×Dic₁₉	—	2	8	no	—
68	C₂²×Dic₁₉	C₂³	1	8	yes	—
69	C₂²×Dic₁₉	C₂³	1	2	yes	—
70	C₂²×Dic₁₉	C₂³	1	2	yes	—
71	C₂²×Dic₁₉	C₂³	1	2	yes	—
72	#64541	—	2	32	no	—
73	C₂×D₇₆	—	2	4	no	—
74	C₂×D₇₆	C₂³	1	4	yes	—
75	C₂×C₄×D₁₉	—	2	8	no	—
76	C₂×C₄×D₁₉	C₂³	1	8	yes	—
77	C₂×Dic₃₈	—	2	4	no	—
78	C₂×Dic₃₈	—	2	8	no	—
79	C₂×Dic₃₈	—	2	32	no	—
80	C₂×Dic₃₈	D₄	1	8	yes	complemented
81	C₂×Dic₃₈	C₂³	1	4	yes	—
82	C₂×Dic₃₈	C₂³	1	8	yes	—
83	C₂×Dic₃₈	C₂³	1	4	yes	—
84	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
85	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
86	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
87	C₄²×C₁₉	C₂³	1	4	yes	—
88	C₂²⋊Dic₁₉	C₂³	1	8	yes	—
89	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
90	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
91	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
92	C₄⋊Dic₁₉	C₂³	1	4	yes	—
93	Dic₁₉⋊C₄	C₂³	1	8	yes	—
94	Dic₁₉⋊C₄	C₂³	1	8	yes	—
95	C₄×Dic₁₉	C₂³	1	8	yes	—
96	C₂³×C₁₉	—	2	2	no	—
97	C₂³×C₁₉	—	2	2	no	—
98	C₂³×C₁₉	—	2	2	no	—
99	C₂³×C₁₉	—	2	2	no	—
100	C₂³×C₁₉	—	2	2	no	—
101	C₂³×C₁₉	#41	1	4	yes	complemented
102	C₂³×C₁₉	C₂⁴	1	2	yes	—
103	C₂³×C₁₉	C₂⁴	1	2	yes	—
104	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
105	C₂²×D₁₉	—	2	4	no	—
106	C₂²×D₁₉	—	2	4	no	—
107	C₂²×D₁₉	—	2	4	no	—
108	C₂²×D₁₉	C₂⁴	1	4	yes	—
109	D₄×C₁₉	—	2	8	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	2	2	no	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	8	no	—
115	C₂×C₄×C₁₉	—	2	8	no	—
116	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	—
117	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
118	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
119	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
120	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
121	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
122	C₂²⋊D₁₉	—	2	16	no	—
123	C₂²⋊D₁₉	—	2	16	no	—
124	C₂×Dic₁₉	—	2	4	no	—
125	C₂×Dic₁₉	—	2	4	no	—
126	C₂×Dic₁₉	—	2	16	no	—
127	C₂×Dic₁₉	C₂×D₄	1	8	yes	—
128	C₂×Dic₁₉	C₂⁴	1	4	yes	—
129	C₂×Dic₁₉	C₂⁴	1	4	yes	—
130	C₂×Dic₁₉	C₂⁴	1	4	yes	—
131	D₇₆	—	2	8	no	—
132	C₄×D₁₉	—	2	16	no	—
133	Dic₃₈	—	2	8	no	—
134	Dic₃₈	—	2	16	no	—
135	Dic₃₈	C₂×D₄	1	16	yes	complemented
136	C₂×#548	—	19	1	no	maximal, Sylow
137	C₂²×C₁₉	—	2	8	no	—
138	C₂²×C₁₉	—	2	2	no	—
139	C₂²×C₁₉	—	2	2	no	—
140	C₂²×C₁₉	—	2	2	no	—
141	C₂²×C₁₉	—	2	2	no	—
142	C₂²×C₁₉	—	2	2	no	—
143	C₂²×C₁₉	—	4	4	no	—
144	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
145	C₂²×C₁₉	C₂×#41	1	2	yes	—
146	C₂²×C₁₉	C₂×#41	1	4	yes	complemented
147	C₂²×C₁₉	2₋¹⁺⁴	1	2	yes	—
148	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
149	C₂×D₁₉	—	2	4	no	—
150	C₂×D₁₉	—	2	4	no	—
151	C₂×D₁₉	—	2	4	no	—
152	C₂×D₁₉	—	4	8	no	—
153	C₄×C₁₉	—	2	2	no	—
154	C₄×C₁₉	—	2	2	no	—
155	C₄×C₁₉	—	2	2	no	—
156	C₄×C₁₉	—	2	2	no	—
157	C₄×C₁₉	C₂²×D₄	1	4	yes	—
158	Dic₁₉	—	2	4	no	—
159	Dic₁₉	—	2	4	no	—
160	Dic₁₉	C₂²×D₄	1	8	yes	—
161	C₂²×#41	—	19	1	no	—
162	C₂³×Q₈	—	19	1	no	—
163	#548	—	19	16	no	—
164	C₂×#124	—	19	2	no	—
165	C₂×#124	—	19	1	no	—
166	C₂×C₂²⋊Q₈	—	19	1	no	—
167	C₂×C₂²⋊Q₈	—	19	2	no	—
168	C₂×#121	—	19	2	no	—
169	C₂×#121	—	19	1	no	—
170	C₂×C₄×Q₈	—	19	1	no	—
171	C₂×C₄×D₄	—	19	2	no	—
172	C₂×C₄×D₄	D₁₉	1	1	yes	characteristic, complemented
173	C₂×C₁₉	—	2	4	no	—
174	C₂×C₁₉	—	4	2	no	—
175	C₂×C₁₉	#548	1	4	yes	complemented
176	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
177	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
178	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
179	D₁₉	—	4	4	no	—
180	C₂×#41	—	19	8	no	—
181	C₂×#41	—	38	8	no	—
182	C₂²×Q₈	—	19	1	no	—
183	C₂²×Q₈	—	19	2	no	—
184	C₂²×Q₈	—	19	1	no	—
185	C₂²×Q₈	—	19	8	no	—
186	C₂²×Q₈	—	38	8	no	—
187	C₂²×D₄	—	19	2	no	—
188	C₂²×D₄	—	19	2	no	—
189	C₂²×D₄	—	19	1	no	—
190	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
191	C₂³×C₄	—	19	2	no	—
192	C₂³×C₄	—	19	2	no	—
193	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
194	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
195	#124	—	19	16	no	—
196	#124	—	19	8	no	—
197	C₂²⋊Q₈	—	19	8	no	—
198	C₂²⋊Q₈	—	19	16	no	—
199	#121	—	19	16	no	—
200	#121	—	19	8	no	—
201	C₄×Q₈	—	19	8	no	—
202	C₄×D₄	—	19	16	no	—
203	C₄×D₄	C₂×D₁₉	1	8	yes	complemented
204	C₂×C₄⋊C₄	—	19	2	no	—
205	C₂×C₄⋊C₄	—	19	2	no	—
206	C₂×C₄⋊C₄	—	19	1	no	—
207	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
208	C₂×C₂²⋊C₄	—	19	2	no	—
209	C₂×C₂²⋊C₄	—	19	2	no	—
210	C₂×C₂²⋊C₄	—	19	2	no	—
211	C₂×C₂²⋊C₄	—	19	2	no	—
212	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
214	C₂×C₄²	—	19	2	no	—
215	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
216	C₁₉	C₂×#548	1	1	yes	characteristic, complemented, Sylow
217	C₂⁴	—	19	2	no	—
218	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
219	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
220	#41	—	38	32	no	—
221	C₂×Q₈	—	19	8	no	—
222	C₂×Q₈	—	19	4	no	—
223	C₂×Q₈	—	19	8	no	—
224	C₂×Q₈	—	19	4	no	—
225	C₂×Q₈	—	38	8	no	—
226	C₂×Q₈	—	38	32	no	—
227	C₂×Q₈	—	38	4	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	19	8	no	—
230	C₂×D₄	—	19	8	no	—
231	C₂×D₄	—	19	4	no	—
232	C₂×D₄	—	38	8	no	—
233	C₂×D₄	—	38	8	no	—
234	C₂×D₄	—	38	4	no	—
235	C₂×D₄	C₂²×D₁₉	1	4	yes	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	19	2	no	—
239	C₂²×C₄	—	19	8	no	—
240	C₂²×C₄	—	19	2	no	—
241	C₂²×C₄	—	19	2	no	—
242	C₂²×C₄	—	19	8	no	—
243	C₂²×C₄	—	38	8	no	—
244	C₂²×C₄	—	38	8	no	—
245	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
246	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
247	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
252	C₄⋊C₄	—	19	8	no	—
253	C₄⋊C₄	—	19	8	no	—
254	C₄⋊C₄	—	19	4	no	—
255	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
256	C₂²⋊C₄	—	19	8	no	—
257	C₂²⋊C₄	—	19	8	no	—
258	C₂²⋊C₄	—	19	8	no	—
259	C₂²⋊C₄	—	19	8	no	—
260	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
261	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
262	C₄²	—	19	8	no	—
263	C₄²	C₂²×D₁₉	1	4	yes	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	19	4	no	—
270	C₂³	—	38	4	no	—
271	C₂³	—	38	4	no	—
272	C₂³	—	38	4	no	—
273	C₂³	#64541	1	4	yes	complemented
274	C₂³	C₂³×D₁₉	1	2	yes	—
275	C₂³	C₂³×D₁₉	1	2	yes	—
276	C₂³	C₂³×D₁₉	1	1	yes	characteristic
277	Q₈	—	19	16	no	—
278	Q₈	—	38	16	no	—
279	Q₈	—	38	8	no	—
280	D₄	—	2	8	no	—
281	D₄	—	38	16	no	—
282	D₄	—	38	16	no	—
283	D₄	—	38	8	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	19	8	no	—
291	C₂×C₄	—	19	4	no	—
292	C₂×C₄	—	19	4	no	—
293	C₂×C₄	—	19	4	no	—
294	C₂×C₄	—	38	16	no	—
295	C₂×C₄	—	38	16	no	—
296	C₂×C₄	—	38	4	no	—
297	C₂×C₄	—	38	4	no	—
298	C₂×C₄	D₄×D₁₉	1	4	yes	—
299	C₂×C₄	C₂³×D₁₉	1	2	yes	—
300	C₂×C₄	C₂³×D₁₉	1	2	yes	—
301	C₂×C₄	C₂³×D₁₉	1	2	yes	—
302	C₂×C₄	C₂³×D₁₉	1	2	yes	—
303	C₂×C₄	C₂³×D₁₉	1	2	yes	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	8	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	38	4	no	—
312	C₂²	—	38	4	no	—
313	C₂²	—	38	4	no	—
314	C₂²	—	76	8	no	—
315	C₂²	C₂×#64541	1	2	yes	—
316	C₂²	C₂×#64541	1	4	yes	complemented
317	C₂²	C₂×D₄×D₁₉	1	2	yes	—
318	C₂²	#10597436	1	2	yes	—
319	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	19	8	no	—
325	C₄	—	38	4	no	—
326	C₄	—	38	4	no	—
327	C₄	C₂×D₄×D₁₉	1	4	yes	—
328	C₂	—	2	4	no	—
329	C₂	—	4	2	no	—
330	C₂	—	76	4	no	—
331	C₂	#11927203	1	4	yes	direct product factor
332	C₂	C₂²×D₄×D₁₉	1	1	yes	characteristic
333	C₂	C₂²×#64541	1	1	yes	characteristic
334	C₂	C₂×#10597436	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₉
commutator, cosocle	C₂²×C₁₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₉
Fitting	C₂×C₄×D₄×C₁₉	C₂
socle	C₂³×C₁₉	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₉
hypocenter	C₁₉	C₂×#548
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#64541	C₂	semidirect
2. C₂³×Dic₃₈	C₂	semidirect
3. #11927203	C₂	direct (N × H)
4. C₂×#10597321	C₂	semidirect
5. C₂×#10597331	C₂	semidirect
6. C₂×#10597370	C₂	semidirect
7. C₂×#10597342	C₂	semidirect
8. C₂×#10597376	C₂	semidirect
9. C₂×#10597385	C₂	semidirect
10. C₂×C₄×Dic₃₈	C₂	semidirect
11. C₂×(C₂²⋊D₁₉)⋊C₄	C₂	semidirect
12. C₂×C₄×D₄×C₁₉	C₂	semidirect
20. C₂×#64541	C₂²	semidirect
27. C₂²×Dic₃₈	C₂²	semidirect
28. C₄×D₄×C₁₉	C₂²	semidirect
34. #10597385	C₂²	semidirect
35. #10597376	C₂²	semidirect
40. #10597370	C₂²	semidirect
44. #10597342	C₂²	semidirect
45. #10597331	C₂²	semidirect
46. (C₂²⋊D₁₉)⋊C₄	C₂²	semidirect
47. #10597321	C₂²	semidirect
48. C₄×Dic₃₈	C₂²	semidirect
80. C₂×Dic₃₈	D₄	semidirect
101. C₂³×C₁₉	#41	semidirect
135. Dic₃₈	C₂×D₄	semidirect
146. C₂²×C₁₉	C₂×#41	semidirect
172. C₂×C₄×D₄	D₁₉	semidirect
175. C₂×C₁₉	#548	semidirect
203. C₄×D₄	C₂×D₁₉	semidirect
216. C₁₉	C₂×#548	semidirect
273. C₂³	#64541	semidirect
316. C₂²	C₂×#64541	semidirect
331. C₂	#11927203	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#64541	2
C₂³×Dic₃₈	2
#11927203	2
C₂×#10597321	2
C₂×#10597331	2
C₂×#10597370	2
C₂×#10597342	2
C₂×#10597376	2
C₂×#10597385	2
C₂×C₄×Dic₃₈	2
C₂×(C₂²⋊D₁₉)⋊C₄	2
C₂×C₄×D₄×C₁₉	2
C₂×#548	19

Subgroups — C2×#11927203 #14577101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11927203 #14577101