Subgroups — #14577436

Order 2432.

1100 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927255	C₂	1	2	yes	maximal, complemented
2	#11927254	C₂	1	4	yes	maximal, complemented
3	#11927254	C₂	1	2	yes	maximal, complemented
4	#11927250	C₂	1	4	yes	maximal, complemented
5	Q₈×D₇₆	C₂	1	2	yes	maximal, complemented
6	#11927209	C₂	1	2	yes	maximal, complemented
7	#11927172	C₂	1	2	yes	maximal, complemented
8	#11927169	C₂	1	4	yes	maximal, complemented
9	#11927101	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊Q₈×D₁₉	C₂	1	4	yes	maximal, complemented
11	#11927018	C₂	1	2	yes	maximal
12	C₂×C₄×D₇₆	C₂	1	1	yes	maximal, characteristic
13	C₁₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×Q₈×D₁₉	C₂²	1	4	yes	complemented
15	C₂²×D₇₆	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₉	C₂²	1	2	yes	—
17	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₁₉×#125	C₂²	1	2	yes	complemented
20	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₄×Q₈×C₁₉	C₂²	1	2	yes	complemented
23	C₄×Q₈×C₁₉	C₂²	1	2	yes	complemented
24	C₁₉×#117	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
26	#10597392	C₂²	1	8	yes	complemented
27	#10597392	C₂²	1	4	yes	complemented
28	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
29	#10597372	C₂²	1	2	yes	—
30	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
31	#10597370	C₂²	1	2	yes	—
32	#10597370	C₂²	1	2	yes	—
33	#10597343	C₂²	1	8	yes	complemented
34	#10597343	C₂²	1	4	yes	complemented
35	#10597342	C₂²	1	8	yes	complemented
36	#10597342	C₂²	1	8	yes	complemented
37	#10597342	C₂²	1	8	yes	complemented
38	#10597339	C₂²	1	4	yes	—
39	#10597339	C₂²	1	4	yes	—
40	C₄⋊C₄×D₁₉	C₂²	1	8	yes	complemented
41	C₄⋊C₄×D₁₉	C₂²	1	4	yes	complemented
42	#10597336	C₂²	1	4	yes	complemented
43	#10597334	C₂²	1	4	yes	complemented
44	C₂²⋊C₄×D₁₉	C₂²	1	8	yes	—
45	#10597324	C₂²	1	8	yes	—
46	C₄×D₇₆	C₂²	1	2	yes	—
47	C₄×D₇₆	C₂²	1	2	yes	—
48	C₄×D₇₆	C₂²	1	4	yes	—
49	C₄×D₇₆	C₂²	1	2	yes	—
50	C₄×D₇₆	C₂²	1	2	yes	—
51	C₄×D₇₆	C₂²	1	4	yes	—
52	#10597316	C₂²	1	2	yes	complemented
53	#10597316	C₂²	1	4	yes	complemented
54	#10597315	C₂²	1	1	yes	characteristic, complemented
55	#10597315	C₂²	1	1	yes	characteristic, complemented
56	#10597315	C₂²	1	4	yes	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	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
61	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
65	C₂²×Dic₁₉	C₂³	1	2	yes	—
66	Q₈×D₁₉	—	4	8	no	—
67	C₂×D₇₆	Q₈	1	8	yes	complemented
68	C₂×D₇₆	C₂³	1	2	yes	—
69	C₂×D₇₆	C₂³	1	2	yes	—
70	C₂×C₄×D₁₉	—	4	4	no	—
71	C₂×C₄×D₁₉	C₂³	1	4	yes	—
72	C₂×C₄×D₁₉	C₂³	1	8	yes	—
73	C₂×C₄×D₁₉	C₂³	1	4	yes	—
74	C₂×C₄×D₁₉	C₂³	1	8	yes	—
75	C₂×Dic₃₈	C₂³	1	2	yes	—
76	C₂×Dic₃₈	C₂³	1	8	yes	—
77	C₂×Dic₃₈	C₂³	1	2	yes	—
78	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
79	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
80	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
81	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
82	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
83	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
84	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
85	C₄²×C₁₉	C₂³	1	1	yes	characteristic
86	C₄²×C₁₉	C₂³	1	1	yes	characteristic
87	C₄²×C₁₉	C₂³	1	1	yes	characteristic
88	C₄²×C₁₉	C₂³	1	1	yes	characteristic
89	C₄²×C₁₉	C₂³	1	4	yes	—
90	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
91	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
92	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
93	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
94	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
95	C₄⋊Dic₁₉	C₂³	1	2	yes	—
96	C₄⋊Dic₁₉	C₂³	1	8	yes	—
97	C₄⋊Dic₁₉	C₂³	1	2	yes	—
98	C₄⋊Dic₁₉	C₂³	1	2	yes	—
99	C₄⋊Dic₁₉	C₂³	1	2	yes	—
100	C₄⋊Dic₁₉	C₂³	1	2	yes	—
101	C₄⋊Dic₁₉	C₂³	1	2	yes	—
102	Dic₁₉⋊C₄	C₂³	1	4	yes	—
103	Dic₁₉⋊C₄	C₂³	1	8	yes	—
104	Dic₁₉⋊C₄	C₂³	1	8	yes	—
105	Dic₁₉⋊C₄	C₂³	1	4	yes	—
106	C₄×Dic₁₉	C₂³	1	4	yes	—
107	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
108	C₂²×D₁₉	—	4	2	no	—
109	C₂²×D₁₉	—	4	2	no	—
110	C₂²×D₁₉	C₂×Q₈	1	8	yes	complemented
111	C₂²×D₁₉	C₂⁴	1	4	yes	—
112	Q₈×C₁₉	—	2	4	no	—
113	Q₈×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	4	1	no	—
116	C₂×C₄×C₁₉	C₂×Q₈	1	4	yes	complemented
117	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
120	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
121	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₁₉	—	4	2	no	—
126	C₂×Dic₁₉	C₂⁴	1	4	yes	—
127	C₂×Dic₁₉	C₂⁴	1	4	yes	—
128	C₂×Dic₁₉	C₂⁴	1	4	yes	—
129	D₇₆	C₂×Q₈	1	8	yes	complemented
130	D₇₆	C₂×Q₈	1	8	yes	complemented
131	C₄×D₁₉	—	2	8	no	—
132	C₄×D₁₉	—	4	4	no	—
133	C₄×D₁₉	—	4	8	no	—
134	C₄×D₁₉	—	4	8	no	—
135	Dic₃₈	—	2	4	no	—
136	Dic₃₈	—	4	2	no	—
137	Dic₃₈	—	4	8	no	—
138	#3501	—	19	1	no	maximal, Sylow
139	C₂²×C₁₉	—	2	1	no	—
140	C₂²×C₁₉	—	2	1	no	—
141	C₂²×C₁₉	C₂²×Q₈	1	2	yes	—
142	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
143	C₂×D₁₉	—	2	4	no	—
144	C₂×D₁₉	—	2	4	no	—
145	C₂×D₁₉	—	4	8	no	—
146	C₂×D₁₉	C₂²×Q₈	1	8	yes	—
147	C₄×C₁₉	—	2	4	no	—
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	—	2	4	no	—
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	C₂²×Q₈	1	2	yes	—
153	C₄×C₁₉	C₂²×Q₈	1	2	yes	—
154	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
155	Dic₁₉	—	2	4	no	—
156	Dic₁₉	—	2	4	no	—
157	Dic₁₉	—	2	4	no	—
158	#554	—	19	2	no	—
159	#554	—	19	4	no	—
160	#554	—	19	2	no	—
161	#554	—	19	4	no	—
162	D₄×Q₈	—	19	2	no	—
163	D₄×Q₈	—	19	2	no	—
164	#544	—	19	2	no	—
165	#543	—	19	4	no	—
166	#533	—	19	1	no	—
167	#533	D₁₉	1	1	yes	characteristic, complemented
168	C₂×C₂²⋊Q₈	—	19	4	no	—
169	#518	—	19	2	no	—
170	C₂×C₄×D₄	—	19	1	no	—
171	C₂×C₁₉	—	2	2	no	—
172	C₂×C₁₉	C₂³×Q₈	1	1	yes	characteristic
173	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
174	C₂×C₁₉	#585	1	1	yes	characteristic
175	D₁₉	—	2	8	no	—
176	C₂²×Q₈	—	19	4	no	—
177	C₂²×D₄	—	19	1	no	—
178	C₂³×C₄	—	19	2	no	—
179	C₄⋊Q₈	—	19	4	no	—
180	C₄⋊Q₈	—	19	1	no	—
181	C₄⋊Q₈	—	19	1	no	—
182	C₄⋊Q₈	—	19	4	no	—
183	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic, complemented
184	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic, complemented
185	#125	—	19	4	no	—
186	#125	—	19	2	no	—
187	#125	—	19	4	no	—
188	#125	C₂×D₁₉	1	2	yes	complemented
189	C₂²⋊Q₈	—	19	4	no	—
190	C₂²⋊Q₈	—	19	8	no	—
191	C₂²⋊Q₈	—	19	2	no	—
192	C₂²⋊Q₈	—	19	2	no	—
193	C₂²⋊Q₈	—	19	4	no	—
194	C₂²⋊Q₈	—	19	8	no	—
195	C₂²⋊Q₈	—	19	8	no	—
196	C₂²⋊Q₈	—	19	8	no	—
197	C₂²⋊Q₈	—	19	8	no	—
198	C₂²⋊Q₈	—	19	8	no	—
199	C₂²⋊Q₈	C₂×D₁₉	1	2	yes	—
200	C₂²⋊Q₈	C₂×D₁₉	1	2	yes	—
201	C₄×Q₈	—	19	2	no	—
202	C₄×Q₈	—	19	2	no	—
203	C₄×Q₈	C₂×D₁₉	1	2	yes	complemented
204	C₄×Q₈	C₂×D₁₉	1	2	yes	complemented
205	C₄×D₄	—	19	4	no	—
206	C₄×D₄	—	19	4	no	—
207	C₄×D₄	—	19	2	no	—
208	C₄×D₄	—	19	2	no	—
209	C₄×D₄	—	19	2	no	—
210	C₄×D₄	—	19	2	no	—
211	C₄×D₄	—	19	4	no	—
212	C₄×D₄	—	19	4	no	—
213	#117	—	19	2	no	—
214	#117	C₂×D₁₉	1	2	yes	—
215	C₂×C₄⋊C₄	—	19	1	no	—
216	C₂×C₄⋊C₄	—	19	8	no	—
217	C₂×C₄⋊C₄	—	19	4	no	—
218	C₂×C₂²⋊C₄	—	19	2	no	—
219	C₂×C₂²⋊C₄	—	19	8	no	—
220	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
221	C₁₉	#3501	1	1	yes	characteristic, complemented, Sylow
222	C₂⁴	—	19	2	no	—
223	C₂×Q₈	—	19	2	no	—
224	C₂×Q₈	—	19	2	no	—
225	C₂×Q₈	—	19	8	no	—
226	C₂×Q₈	—	76	8	no	—
227	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
228	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
229	C₂×D₄	—	19	8	no	—
230	C₂×D₄	—	19	2	no	—
231	C₂×D₄	—	19	2	no	—
232	C₂²×C₄	—	19	2	no	—
233	C₂²×C₄	—	19	4	no	—
234	C₂²×C₄	—	19	8	no	—
235	C₂²×C₄	—	19	8	no	—
236	C₂²×C₄	—	19	4	no	—
237	C₂²×C₄	—	76	4	no	—
238	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
239	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
240	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
241	C₄⋊C₄	—	19	4	no	—
242	C₄⋊C₄	—	19	8	no	—
243	C₄⋊C₄	—	19	8	no	—
244	C₄⋊C₄	—	19	4	no	—
245	C₄⋊C₄	—	19	2	no	—
246	C₄⋊C₄	—	19	2	no	—
247	C₄⋊C₄	—	19	2	no	—
248	C₄⋊C₄	—	19	8	no	—
249	C₄⋊C₄	—	19	2	no	—
250	C₄⋊C₄	—	19	2	no	—
251	C₄⋊C₄	—	19	2	no	—
252	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
253	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
254	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
255	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
256	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
257	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
258	C₂²⋊C₄	—	19	4	no	—
259	C₂²⋊C₄	—	19	8	no	—
260	C₂²⋊C₄	—	19	8	no	—
261	C₂²⋊C₄	—	19	4	no	—
262	C₂²⋊C₄	—	19	4	no	—
263	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
264	C₄²	—	19	4	no	—
265	C₄²	C₂²×D₁₉	1	1	yes	characteristic
266	C₄²	C₂²×D₁₉	1	1	yes	characteristic
267	C₄²	C₂²×D₁₉	1	1	yes	characteristic
268	C₄²	C₂²×D₁₉	1	1	yes	characteristic
269	C₄²	C₂²×D₁₉	1	4	yes	—
270	C₂³	—	19	4	no	—
271	C₂³	—	19	8	no	—
272	C₂³	—	76	2	no	—
273	C₂³	—	76	2	no	—
274	C₂³	C₂³×D₁₉	1	1	yes	characteristic
275	Q₈	—	2	4	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	38	4	no	—
278	Q₈	—	76	8	no	—
279	Q₈	—	76	2	no	—
280	D₄	—	19	8	no	—
281	D₄	—	19	8	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	19	4	no	—
285	C₂×C₄	—	19	4	no	—
286	C₂×C₄	—	19	4	no	—
287	C₂×C₄	—	38	8	no	—
288	C₂×C₄	—	76	4	no	—
289	C₂×C₄	—	76	8	no	—
290	C₂×C₄	—	76	8	no	—
291	C₂×C₄	—	76	2	no	—
292	C₂×C₄	Q₈×D₁₉	1	4	yes	complemented
293	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
294	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
295	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₁₉	1	4	yes	—
297	C₂×C₄	C₂³×D₁₉	1	4	yes	—
298	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
299	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	19	8	no	—
304	C₂²	—	38	4	no	—
305	C₂²	—	38	4	no	—
306	C₂²	—	76	8	no	—
307	C₂²	C₂×Q₈×D₁₉	1	2	yes	—
308	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	4	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	4	no	—
314	C₄	—	38	4	no	—
315	C₄	—	38	4	no	—
316	C₄	—	38	4	no	—
317	C₄	C₂×Q₈×D₁₉	1	2	yes	—
318	C₄	C₂×Q₈×D₁₉	1	2	yes	—
319	C₄	#10597435	1	2	yes	—
320	C₂	—	2	2	no	—
321	C₂	—	38	8	no	—
322	C₂	C₂²×Q₈×D₁₉	1	1	yes	characteristic
323	C₂	C₂×#10597435	1	1	yes	characteristic
324	C₂	#11927380	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₁₉×#533	C₂
prime radical	C₂²×D₇₆	C₂²
hypercenter	#533	D₁₉
hypocenter	C₁₉	#3501
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. #11927255	C₂	semidirect
2. #11927254	C₂	semidirect
3. #11927254	C₂	semidirect
4. #11927250	C₂	semidirect
5. Q₈×D₇₆	C₂	semidirect
6. #11927209	C₂	semidirect
7. #11927172	C₂	semidirect
8. #11927169	C₂	semidirect
9. #11927101	C₂	semidirect
10. C₂²⋊Q₈×D₁₉	C₂	semidirect
13. C₁₉×#533	C₂	semidirect
14. C₂×Q₈×D₁₉	C₂²	semidirect
17. C₁₉×C₄⋊Q₈	C₂²	semidirect
18. C₁₉×C₄⋊Q₈	C₂²	semidirect
19. C₁₉×#125	C₂²	semidirect
22. C₄×Q₈×C₁₉	C₂²	semidirect
23. C₄×Q₈×C₁₉	C₂²	semidirect
26. #10597392	C₂²	semidirect
27. #10597392	C₂²	semidirect
33. #10597343	C₂²	semidirect
34. #10597343	C₂²	semidirect
35. #10597342	C₂²	semidirect
36. #10597342	C₂²	semidirect
37. #10597342	C₂²	semidirect
40. C₄⋊C₄×D₁₉	C₂²	semidirect
41. C₄⋊C₄×D₁₉	C₂²	semidirect
42. #10597336	C₂²	semidirect
43. #10597334	C₂²	semidirect
52. #10597316	C₂²	semidirect
53. #10597316	C₂²	semidirect
54. #10597315	C₂²	semidirect
55. #10597315	C₂²	semidirect
56. #10597315	C₂²	semidirect
57. C₄×Dic₃₈	C₂²	semidirect
58. C₄×Dic₃₈	C₂²	semidirect
67. C₂×D₇₆	Q₈	semidirect
110. C₂²×D₁₉	C₂×Q₈	semidirect
116. C₂×C₄×C₁₉	C₂×Q₈	semidirect
129. D₇₆	C₂×Q₈	semidirect
130. D₇₆	C₂×Q₈	semidirect
167. #533	D₁₉	semidirect
183. C₄⋊Q₈	C₂×D₁₉	semidirect
184. C₄⋊Q₈	C₂×D₁₉	semidirect
188. #125	C₂×D₁₉	semidirect
203. C₄×Q₈	C₂×D₁₉	semidirect
204. C₄×Q₈	C₂×D₁₉	semidirect
221. C₁₉	#3501	semidirect
292. C₂×C₄	Q₈×D₁₉	semidirect

Maximal subgroups

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

subgroup	index
#11927255	2
#11927254	2
#11927250	2
Q₈×D₇₆	2
#11927209	2
#11927172	2
#11927169	2
#11927101	2
C₂²⋊Q₈×D₁₉	2
#11927018	2
C₂×C₄×D₇₆	2
C₁₉×#533	2
#3501	19