Subgroups — #14577484

Order 2432.

1140 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927276	C₂	1	4	yes	maximal, complemented
2	#11927268	C₂	1	2	yes	maximal, complemented
3	#11927259	C₂	1	2	yes	maximal, complemented
4	#11927240	C₂	1	4	yes	maximal, complemented
5	#11927214	C₂	1	2	yes	maximal, complemented
6	#11927198	C₂	1	2	yes	maximal, complemented
7	#11927131	C₂	1	4	yes	maximal, complemented
8	#11927108	C₂	1	2	yes	maximal, complemented
9	#11927101	C₂	1	1	yes	maximal, characteristic, complemented
10	#11927054	C₂	1	4	yes	maximal, complemented
11	#11927027	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
13	C₁₉×#532	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#41×C₁₉	C₂²	1	2	yes	—
15	C₂×#64543	C₂²	1	4	yes	complemented
16	C₂²×D₇₆	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₉	C₂²	1	2	yes	—
18	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₁₉×#127	C₂²	1	1	yes	characteristic, complemented
20	C₁₉×#124	C₂²	1	2	yes	complemented
21	C₁₉×#121	C₂²	1	2	yes	—
22	C₁₉×#121	C₂²	1	2	yes	—
23	C₄×D₄×C₁₉	C₂²	1	2	yes	complemented
24	C₄×D₄×C₁₉	C₂²	1	2	yes	complemented
25	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
26	#10597393	C₂²	1	4	yes	—
27	Q₈×Dic₁₉	C₂²	1	4	yes	complemented
28	#10597386	C₂²	1	4	yes	—
29	#10597385	C₂²	1	8	yes	complemented
30	#10597384	C₂²	1	4	yes	complemented
31	#10597383	C₂²	1	8	yes	—
32	D₄×Dic₁₉	C₂²	1	8	yes	—
33	D₄×Dic₁₉	C₂²	1	4	yes	complemented
34	#10597376	C₂²	1	4	yes	—
35	#10597376	C₂²	1	4	yes	—
36	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
37	#10597372	C₂²	1	2	yes	—
38	#10597372	C₂²	1	4	yes	complemented
39	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊Dic₁₉	C₂²	1	4	yes	complemented
41	#10597370	C₂²	1	2	yes	—
42	#10597370	C₂²	1	2	yes	—
43	#10597344	C₂²	1	8	yes	complemented
44	#10597343	C₂²	1	4	yes	complemented
45	#10597338	C₂²	1	4	yes	complemented
46	#10597332	C₂²	1	8	yes	complemented
47	#10597331	C₂²	1	8	yes	complemented
48	#10597329	C₂²	1	8	yes	complemented
49	(C₂²⋊D₁₉)⋊C₄	C₂²	1	8	yes	complemented
50	C₄×D₇₆	C₂²	1	2	yes	—
51	C₄×D₇₆	C₂²	1	2	yes	—
52	C₄×D₇₆	C₂²	1	2	yes	—
53	C₄×D₇₆	C₂²	1	2	yes	—
54	#10597316	C₂²	1	2	yes	complemented
55	#10597315	C₂²	1	1	yes	characteristic, complemented
56	#10597315	C₂²	1	1	yes	characteristic, complemented
57	C₄×Dic₃₈	C₂²	1	2	yes	complemented
58	C₄×Dic₃₈	C₂²	1	2	yes	complemented
59	C₂³×D₁₉	C₂³	1	2	yes	—
60	#41×C₁₉	—	2	8	no	—
61	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
62	C₂×D₄×C₁₉	C₂³	1	4	yes	—
63	C₂×D₄×C₁₉	C₂³	1	2	yes	—
64	C₂×D₄×C₁₉	C₂³	1	2	yes	—
65	C₂×D₄×C₁₉	C₂³	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
67	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
71	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
72	C₂²×Dic₁₉	C₂³	1	2	yes	—
73	C₂²×Dic₁₉	C₂³	1	8	yes	—
74	#64543	—	4	8	no	—
75	C₂×D₇₆	—	2	4	no	—
76	C₂×D₇₆	C₂³	1	2	yes	—
77	C₂×D₇₆	C₂³	1	2	yes	—
78	C₂×C₄×D₁₉	—	4	4	no	—
79	C₂×C₄×D₁₉	C₂³	1	4	yes	—
80	C₂×C₄×D₁₉	C₂³	1	4	yes	—
81	C₂×Dic₃₈	C₂³	1	2	yes	—
82	C₂×Dic₃₈	C₂³	1	2	yes	—
83	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
84	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
85	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
86	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
87	C₄²×C₁₉	C₂³	1	1	yes	characteristic
88	C₄²×C₁₉	C₂³	1	1	yes	characteristic
89	C₄²×C₁₉	C₂³	1	1	yes	characteristic
90	C₄²×C₁₉	C₂³	1	1	yes	characteristic
91	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
92	C₂²⋊Dic₁₉	C₂³	1	8	yes	—
93	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
94	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
95	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
96	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
97	C₄⋊Dic₁₉	C₂³	1	2	yes	—
98	C₄⋊Dic₁₉	C₂³	1	2	yes	—
99	C₄⋊Dic₁₉	C₂³	1	8	yes	—
100	C₄⋊Dic₁₉	C₂³	1	2	yes	—
101	C₄⋊Dic₁₉	C₂³	1	2	yes	—
102	C₄⋊Dic₁₉	C₂³	1	2	yes	—
103	C₄⋊Dic₁₉	C₂³	1	2	yes	—
104	Dic₁₉⋊C₄	C₂³	1	4	yes	—
105	Dic₁₉⋊C₄	C₂³	1	4	yes	—
106	C₄×Dic₁₉	C₂³	1	8	yes	—
107	C₄×Dic₁₉	C₂³	1	4	yes	—
108	C₂³×C₁₉	C₂⁴	1	4	yes	—
109	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
110	C₂²×D₁₉	—	2	4	no	—
111	C₂²×D₁₉	—	4	2	no	—
112	C₂²×D₁₉	—	4	2	no	—
113	C₂²×D₁₉	C₂⁴	1	4	yes	—
114	Q₈×C₁₉	—	2	4	no	—
115	D₄×C₁₉	—	2	8	no	—
116	D₄×C₁₉	—	2	4	no	—
117	D₄×C₁₉	—	2	4	no	—
118	D₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	2	8	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	4	1	no	—
122	C₂×C₄×C₁₉	#41	1	4	yes	complemented
123	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
124	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₁₉	—	4	8	no	—
131	C₂²⋊D₁₉	—	4	8	no	—
132	C₂×Dic₁₉	—	4	2	no	—
133	C₂×Dic₁₉	—	4	8	no	—
134	C₂×Dic₁₉	C₂⁴	1	4	yes	—
135	C₂×Dic₁₉	C₂⁴	1	4	yes	—
136	C₂×Dic₁₉	C₂⁴	1	4	yes	—
137	D₇₆	—	2	4	no	—
138	D₇₆	—	2	4	no	—
139	C₄×D₁₉	—	2	8	no	—
140	C₄×D₁₉	—	4	4	no	—
141	Dic₃₈	—	2	4	no	—
142	Dic₃₈	—	4	2	no	—
143	#3506	—	19	1	no	maximal, Sylow
144	C₂²×C₁₉	—	2	1	no	—
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	2	4	no	—
147	C₂²×C₁₉	—	2	4	no	—
148	C₂²×C₁₉	—	2	4	no	—
149	C₂²×C₁₉	C₂×#41	1	2	yes	—
150	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
151	C₂×D₁₉	—	2	4	no	—
152	C₂×D₁₉	—	2	4	no	—
153	C₂×D₁₉	—	2	4	no	—
154	C₂×D₁₉	—	4	8	no	—
155	C₄×C₁₉	—	2	4	no	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	2	1	no	—
158	C₄×C₁₉	—	2	1	no	—
159	C₄×C₁₉	C₂×#41	1	2	yes	—
160	C₄×C₁₉	C₂×#41	1	2	yes	—
161	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
162	Dic₁₉	—	2	4	no	—
163	Dic₁₉	—	2	4	no	—
164	Dic₁₉	—	2	4	no	—
165	#559	—	19	4	no	—
166	#556	—	19	2	no	—
167	#555	—	19	2	no	—
168	#553	—	19	4	no	—
169	#550	—	19	2	no	—
170	#547	—	19	2	no	—
171	#538	—	19	4	no	—
172	#535	—	19	2	no	—
173	#533	—	19	1	no	—
174	#532	D₁₉	1	1	yes	characteristic, complemented
175	#525	—	19	4	no	—
176	#520	—	19	2	no	—
177	C₂×C₄×D₄	—	19	1	no	—
178	C₂×C₁₉	—	2	2	no	—
179	C₂×C₁₉	—	4	4	no	—
180	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
181	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
182	C₂×C₁₉	#585	1	1	yes	characteristic
183	D₁₉	—	4	4	no	—
184	C₂×#41	—	19	4	no	—
185	C₂×#41	C₂×D₁₉	1	2	yes	—
186	C₂²×D₄	—	19	1	no	—
187	C₂³×C₄	—	19	2	no	—
188	C₄⋊Q₈	—	19	1	no	—
189	C₄⋊Q₈	—	19	1	no	—
190	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic, complemented
191	#127	—	19	4	no	—
192	#127	C₂×D₁₉	1	1	yes	characteristic, complemented
193	#126	—	19	8	no	—
194	#125	—	19	2	no	—
195	#124	—	19	4	no	—
196	#124	—	19	8	no	—
197	#124	C₂×D₁₉	1	2	yes	complemented
198	#123	—	19	8	no	—
199	#123	—	19	8	no	—
200	C₂²⋊Q₈	—	19	2	no	—
201	C₂²⋊Q₈	—	19	2	no	—
202	C₂²⋊Q₈	—	19	4	no	—
203	#121	—	19	4	no	—
204	#121	—	19	8	no	—
205	#121	—	19	4	no	—
206	#121	—	19	4	no	—
207	#121	C₂×D₁₉	1	2	yes	—
208	#121	C₂×D₁₉	1	2	yes	—
209	#120	—	19	8	no	—
210	C₄×Q₈	—	19	4	no	—
211	C₄×Q₈	—	19	2	no	—
212	C₄×Q₈	—	19	2	no	—
213	C₄×D₄	—	19	8	no	—
214	C₄×D₄	—	19	4	no	—
215	C₄×D₄	—	19	8	no	—
216	C₄×D₄	—	19	2	no	—
217	C₄×D₄	—	19	2	no	—
218	C₄×D₄	—	19	2	no	—
219	C₄×D₄	—	19	2	no	—
220	C₄×D₄	C₂×D₁₉	1	2	yes	complemented
221	C₄×D₄	C₂×D₁₉	1	2	yes	complemented
222	#117	—	19	2	no	—
223	#117	—	19	4	no	—
224	#117	—	19	4	no	—
225	C₂×C₄⋊C₄	—	19	1	no	—
226	C₂×C₄⋊C₄	—	19	4	no	—
227	C₂×C₂²⋊C₄	—	19	2	no	—
228	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
229	C₁₉	#3506	1	1	yes	characteristic, complemented, Sylow
230	C₂⁴	—	19	2	no	—
231	#41	—	2	8	no	—
232	#41	—	76	8	no	—
233	C₂×Q₈	—	19	2	no	—
234	C₂×Q₈	—	19	2	no	—
235	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
236	C₂×D₄	—	19	8	no	—
237	C₂×D₄	—	19	8	no	—
238	C₂×D₄	—	19	2	no	—
239	C₂×D₄	—	19	2	no	—
240	C₂×D₄	—	38	4	no	—
241	C₂×D₄	C₂²×D₁₉	1	4	yes	—
242	C₂×D₄	C₂²×D₁₉	1	2	yes	—
243	C₂×D₄	C₂²×D₁₉	1	2	yes	—
244	C₂×D₄	C₂²×D₁₉	1	2	yes	—
245	C₂²×C₄	—	19	2	no	—
246	C₂²×C₄	—	19	8	no	—
247	C₂²×C₄	—	19	4	no	—
248	C₂²×C₄	—	19	4	no	—
249	C₂²×C₄	—	76	4	no	—
250	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
252	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
254	C₄⋊C₄	—	19	4	no	—
255	C₄⋊C₄	—	19	4	no	—
256	C₄⋊C₄	—	19	2	no	—
257	C₄⋊C₄	—	19	2	no	—
258	C₄⋊C₄	—	19	2	no	—
259	C₄⋊C₄	—	19	8	no	—
260	C₄⋊C₄	—	19	2	no	—
261	C₄⋊C₄	—	19	2	no	—
262	C₄⋊C₄	—	19	2	no	—
263	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
264	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
265	C₂²⋊C₄	—	19	4	no	—
266	C₂²⋊C₄	—	19	8	no	—
267	C₂²⋊C₄	—	19	4	no	—
268	C₂²⋊C₄	—	19	4	no	—
269	C₂²⋊C₄	—	19	8	no	—
270	C₂²⋊C₄	—	19	8	no	—
271	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
272	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
273	C₄²	—	19	8	no	—
274	C₄²	—	19	4	no	—
275	C₄²	C₂²×D₁₉	1	1	yes	characteristic
276	C₄²	C₂²×D₁₉	1	1	yes	characteristic
277	C₄²	C₂²×D₁₉	1	1	yes	characteristic
278	C₄²	C₂²×D₁₉	1	1	yes	characteristic
279	C₂³	—	19	4	no	—
280	C₂³	—	38	4	no	—
281	C₂³	—	76	2	no	—
282	C₂³	—	76	2	no	—
283	C₂³	C₂³×D₁₉	1	4	yes	—
284	C₂³	C₂³×D₁₉	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	38	4	no	—
287	Q₈	—	76	2	no	—
288	D₄	—	2	8	no	—
289	D₄	—	2	4	no	—
290	D₄	—	2	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	38	4	no	—
293	D₄	—	38	4	no	—
294	D₄	—	76	8	no	—
295	D₄	—	76	8	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	19	4	no	—
300	C₂×C₄	—	19	4	no	—
301	C₂×C₄	—	19	4	no	—
302	C₂×C₄	—	38	8	no	—
303	C₂×C₄	—	76	4	no	—
304	C₂×C₄	—	76	2	no	—
305	C₂×C₄	—	76	8	no	—
306	C₂×C₄	#64543	1	4	yes	complemented
307	C₂×C₄	C₂³×D₁₉	1	4	yes	—
308	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
309	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
313	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	38	4	no	—
320	C₂²	—	38	4	no	—
321	C₂²	—	38	4	no	—
322	C₂²	—	76	8	no	—
323	C₂²	C₂×#64543	1	2	yes	—
324	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
325	C₄	—	2	4	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	38	4	no	—
330	C₄	—	38	4	no	—
331	C₄	—	38	4	no	—
332	C₄	C₂×#64543	1	2	yes	—
333	C₄	C₂×#64543	1	2	yes	—
334	C₄	#10597435	1	2	yes	—
335	C₂	—	2	2	no	—
336	C₂	—	4	4	no	—
337	C₂	—	76	4	no	—
338	C₂	C₂²×#64543	1	1	yes	characteristic
339	C₂	C₂×#10597435	1	1	yes	characteristic
340	C₂	#11927380	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₁₉×#532	C₂
hypercenter	#532	D₁₉
hypocenter	C₁₉	#3506
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. #11927276	C₂	semidirect
2. #11927268	C₂	semidirect
3. #11927259	C₂	semidirect
4. #11927240	C₂	semidirect
5. #11927214	C₂	semidirect
6. #11927198	C₂	semidirect
7. #11927131	C₂	semidirect
8. #11927108	C₂	semidirect
9. #11927101	C₂	semidirect
10. #11927054	C₂	semidirect
11. #11927027	C₂	semidirect
12. C₂×C₄×D₇₆	C₂	semidirect
13. C₁₉×#532	C₂	semidirect
15. C₂×#64543	C₂²	semidirect
18. C₁₉×C₄⋊Q₈	C₂²	semidirect
19. C₁₉×#127	C₂²	semidirect
20. C₁₉×#124	C₂²	semidirect
23. C₄×D₄×C₁₉	C₂²	semidirect
24. C₄×D₄×C₁₉	C₂²	semidirect
27. Q₈×Dic₁₉	C₂²	semidirect
29. #10597385	C₂²	semidirect
30. #10597384	C₂²	semidirect
33. D₄×Dic₁₉	C₂²	semidirect
38. #10597372	C₂²	semidirect
40. C₂×C₄⋊Dic₁₉	C₂²	semidirect
43. #10597344	C₂²	semidirect
44. #10597343	C₂²	semidirect
45. #10597338	C₂²	semidirect
46. #10597332	C₂²	semidirect
47. #10597331	C₂²	semidirect
48. #10597329	C₂²	semidirect
49. (C₂²⋊D₁₉)⋊C₄	C₂²	semidirect
54. #10597316	C₂²	semidirect
55. #10597315	C₂²	semidirect
56. #10597315	C₂²	semidirect
57. C₄×Dic₃₈	C₂²	semidirect
58. C₄×Dic₃₈	C₂²	semidirect
122. C₂×C₄×C₁₉	#41	semidirect
174. #532	D₁₉	semidirect
190. C₄⋊Q₈	C₂×D₁₉	semidirect
192. #127	C₂×D₁₉	semidirect
197. #124	C₂×D₁₉	semidirect
220. C₄×D₄	C₂×D₁₉	semidirect
221. C₄×D₄	C₂×D₁₉	semidirect
229. C₁₉	#3506	semidirect
306. C₂×C₄	#64543	semidirect

Maximal subgroups

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

subgroup	index
#11927276	2
#11927268	2
#11927259	2
#11927240	2
#11927214	2
#11927198	2
#11927131	2
#11927108	2
#11927101	2
#11927054	2
#11927027	2
C₂×C₄×D₇₆	2
C₁₉×#532	2
#3506	19