Subgroups — #14577489

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	#11927279	C₂	1	4	yes	maximal, complemented
2	#11927267	C₂	1	2	yes	maximal, complemented
3	#11927260	C₂	1	2	yes	maximal, complemented
4	#11927247	C₂	1	4	yes	maximal, complemented
5	#11927216	C₂	1	2	yes	maximal, complemented
6	#11927197	C₂	1	2	yes	maximal, complemented
7	#11927134	C₂	1	4	yes	maximal, complemented
8	#11927113	C₂	1	2	yes	maximal, complemented
9	#11927094	C₂	1	1	yes	maximal, characteristic, complemented
10	#11927057	C₂	1	4	yes	maximal, complemented
11	#11927024	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₇₆	C₂	1	1	yes	maximal, characteristic, complemented
13	C₁₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#64545	C₂²	1	4	yes	complemented
15	C₂×#64541	C₂²	1	2	yes	—
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×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₁₉×#125	C₂²	1	2	yes	complemented
21	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₉	C₂²	1	2	yes	complemented
24	C₄×Q₈×C₁₉	C₂²	1	2	yes	complemented
25	C₁₉×#117	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
27	#10597393	C₂²	1	8	yes	complemented
28	#10597392	C₂²	1	4	yes	complemented
29	#10597376	C₂²	1	2	yes	—
30	#10597376	C₂²	1	2	yes	—
31	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
32	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
33	#10597344	C₂²	1	8	yes	complemented
34	#10597341	C₂²	1	4	yes	—
35	#10597341	C₂²	1	4	yes	—
36	#10597341	C₂²	1	4	yes	complemented
37	#10597341	C₂²	1	8	yes	complemented
38	#10597340	C₂²	1	8	yes	complemented
39	#10597340	C₂²	1	8	yes	complemented
40	#10597339	C₂²	1	8	yes	complemented
41	#10597338	C₂²	1	4	yes	complemented
42	#10597338	C₂²	1	4	yes	complemented
43	C₄⋊C₄×D₁₉	C₂²	1	4	yes	complemented
44	#10597328	C₂²	1	8	yes	—
45	(C₂²⋊D₁₉)⋊C₄	C₂²	1	8	yes	—
46	#10597321	C₂²	1	4	yes	—
47	#10597321	C₂²	1	2	yes	complemented
48	#10597320	C₂²	1	4	yes	—
49	#10597320	C₂²	1	1	yes	characteristic, 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	C₄×D₇₆	C₂²	1	2	yes	complemented
55	C₄×D₇₆	C₂²	1	2	yes	complemented
56	C₄×D₇₆	C₂²	1	4	yes	complemented
57	#10597315	C₂²	1	1	yes	characteristic, complemented
58	C₄×Dic₃₈	C₂²	1	4	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	C₂×C₂²⋊D₁₉	C₂³	1	4	yes	—
66	C₂²×Dic₁₉	C₂³	1	2	yes	—
67	#64545	—	4	8	no	—
68	#64541	—	2	8	no	—
69	C₂×D₇₆	—	2	4	no	—
70	C₂×D₇₆	C₂³	1	2	yes	—
71	C₂×D₇₆	C₂³	1	2	yes	—
72	C₂×D₇₆	C₂³	1	2	yes	—
73	C₂×D₇₆	C₂³	1	2	yes	—
74	C₂×D₇₆	C₂³	1	2	yes	—
75	C₂×D₇₆	C₂³	1	8	yes	—
76	C₂×D₇₆	C₂³	1	8	yes	—
77	C₂×C₄×D₁₉	—	4	4	no	—
78	C₂×C₄×D₁₉	C₂³	1	4	yes	—
79	C₂×C₄×D₁₉	C₂³	1	4	yes	—
80	C₂×C₄×D₁₉	C₂³	1	4	yes	—
81	C₂×C₄×D₁₉	C₂³	1	8	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	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
86	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
87	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
88	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
89	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
90	C₄²×C₁₉	C₂³	1	1	yes	characteristic
91	C₄²×C₁₉	C₂³	1	1	yes	characteristic
92	C₄²×C₁₉	C₂³	1	1	yes	characteristic
93	C₄²×C₁₉	C₂³	1	1	yes	characteristic
94	C₄²×C₁₉	C₂³	1	4	yes	—
95	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
96	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
97	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
98	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
99	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
100	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
101	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
102	C₄⋊Dic₁₉	C₂³	1	2	yes	—
103	C₄⋊Dic₁₉	C₂³	1	2	yes	—
104	C₄⋊Dic₁₉	C₂³	1	2	yes	—
105	C₄⋊Dic₁₉	C₂³	1	2	yes	—
106	Dic₁₉⋊C₄	C₂³	1	8	yes	—
107	C₄×Dic₁₉	C₂³	1	8	yes	—
108	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
109	C₂²×D₁₉	—	2	4	no	—
110	C₂²×D₁₉	—	4	2	no	—
111	C₂²×D₁₉	—	4	2	no	—
112	C₂²×D₁₉	C₂⁴	1	4	yes	—
113	C₂²×D₁₉	C₂⁴	1	4	yes	—
114	Q₈×C₁₉	—	2	4	no	—
115	Q₈×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	2	2	no	—
117	C₂×C₄×C₁₉	—	4	1	no	—
118	C₂×C₄×C₁₉	#41	1	4	yes	complemented
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×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₁₉	—	2	8	no	—
128	C₂×Dic₁₉	—	4	2	no	—
129	C₂×Dic₁₉	C₂⁴	1	4	yes	—
130	C₂×Dic₁₉	C₂⁴	1	4	yes	—
131	D₇₆	—	2	4	no	—
132	D₇₆	—	2	4	no	—
133	D₇₆	—	2	4	no	—
134	D₇₆	—	2	4	no	—
135	D₇₆	—	4	2	no	—
136	D₇₆	—	4	8	no	—
137	D₇₆	—	4	8	no	—
138	C₄×D₁₉	—	2	8	no	—
139	C₄×D₁₉	—	2	8	no	—
140	C₄×D₁₉	—	4	4	no	—
141	C₄×D₁₉	—	4	8	no	—
142	Dic₃₈	—	2	4	no	—
143	#3506	—	19	1	no	maximal, Sylow
144	C₂²×C₁₉	—	2	1	no	—
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	C₂×#41	1	2	yes	—
147	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
148	C₂×D₁₉	—	2	4	no	—
149	C₂×D₁₉	—	2	4	no	—
150	C₂×D₁₉	—	2	4	no	—
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	1	no	—
156	C₄×C₁₉	—	2	4	no	—
157	C₄×C₁₉	—	2	1	no	—
158	C₄×C₁₉	—	2	4	no	—
159	C₄×C₁₉	—	2	1	no	—
160	C₄×C₁₉	C₂×#41	1	2	yes	—
161	C₄×C₁₉	C₂×#41	1	2	yes	—
162	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
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	D₁₉	1	1	yes	characteristic, complemented
174	#532	—	19	1	no	—
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	C₂²×#41	1	1	yes	characteristic
180	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
181	C₂×C₁₉	#585	1	1	yes	characteristic
182	D₁₉	—	4	4	no	—
183	D₁₉	—	4	4	no	—
184	C₂×#41	—	19	4	no	—
185	C₂×#41	—	19	2	no	—
186	C₂²×D₄	—	19	1	no	—
187	C₂³×C₄	—	19	2	no	—
188	C₄⋊Q₈	—	19	1	no	—
189	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic, complemented
190	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic, complemented
191	#127	—	19	1	no	—
192	#127	—	19	4	no	—
193	#126	—	19	8	no	—
194	#125	C₂×D₁₉	1	2	yes	complemented
195	#124	—	19	8	no	—
196	#124	—	19	2	no	—
197	#124	—	19	4	no	—
198	#123	—	19	8	no	—
199	#123	—	19	8	no	—
200	C₂²⋊Q₈	—	19	4	no	—
201	C₂²⋊Q₈	C₂×D₁₉	1	2	yes	—
202	C₂²⋊Q₈	C₂×D₁₉	1	2	yes	—
203	#121	—	19	2	no	—
204	#121	—	19	2	no	—
205	#121	—	19	4	no	—
206	#121	—	19	4	no	—
207	#121	—	19	4	no	—
208	#121	—	19	8	no	—
209	#120	—	19	8	no	—
210	C₄×Q₈	—	19	4	no	—
211	C₄×Q₈	C₂×D₁₉	1	2	yes	complemented
212	C₄×Q₈	C₂×D₁₉	1	2	yes	complemented
213	C₄×D₄	—	19	8	no	—
214	C₄×D₄	—	19	8	no	—
215	C₄×D₄	—	19	2	no	—
216	C₄×D₄	—	19	2	no	—
217	C₄×D₄	—	19	2	no	—
218	C₄×D₄	—	19	2	no	—
219	C₄×D₄	—	19	4	no	—
220	C₄×D₄	—	19	2	no	—
221	C₄×D₄	—	19	2	no	—
222	#117	—	19	4	no	—
223	#117	—	19	4	no	—
224	#117	C₂×D₁₉	1	2	yes	—
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	—	38	8	no	—
232	#41	—	76	8	no	—
233	C₂×Q₈	—	19	2	no	—
234	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
235	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
236	C₂×D₄	—	19	4	no	—
237	C₂×D₄	—	19	2	no	—
238	C₂×D₄	—	19	2	no	—
239	C₂×D₄	—	19	2	no	—
240	C₂×D₄	—	19	2	no	—
241	C₂×D₄	—	19	8	no	—
242	C₂×D₄	—	19	2	no	—
243	C₂×D₄	—	19	8	no	—
244	C₂×D₄	—	38	4	no	—
245	C₂²×C₄	—	19	2	no	—
246	C₂²×C₄	—	19	4	no	—
247	C₂²×C₄	—	19	4	no	—
248	C₂²×C₄	—	19	8	no	—
249	C₂²×C₄	—	19	4	no	—
250	C₂²×C₄	—	76	4	no	—
251	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
252	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
253	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
254	C₄⋊C₄	—	19	8	no	—
255	C₄⋊C₄	—	19	2	no	—
256	C₄⋊C₄	—	19	2	no	—
257	C₄⋊C₄	—	19	2	no	—
258	C₄⋊C₄	—	19	2	no	—
259	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
260	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
261	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
262	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
263	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
264	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
265	C₂²⋊C₄	—	19	4	no	—
266	C₂²⋊C₄	—	19	4	no	—
267	C₂²⋊C₄	—	19	8	no	—
268	C₂²⋊C₄	—	19	8	no	—
269	C₂²⋊C₄	—	19	4	no	—
270	C₂²⋊C₄	—	19	8	no	—
271	C₂²⋊C₄	—	19	4	no	—
272	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
273	C₄²	—	19	8	no	—
274	C₄²	C₂²×D₁₉	1	1	yes	characteristic
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	4	yes	—
279	C₂³	—	19	4	no	—
280	C₂³	—	19	4	no	—
281	C₂³	—	38	4	no	—
282	C₂³	—	76	2	no	—
283	C₂³	—	76	2	no	—
284	C₂³	C₂³×D₁₉	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	38	4	no	—
288	D₄	—	38	8	no	—
289	D₄	—	38	4	no	—
290	D₄	—	38	4	no	—
291	D₄	—	38	4	no	—
292	D₄	—	38	4	no	—
293	D₄	—	76	8	no	—
294	D₄	—	76	2	no	—
295	D₄	—	76	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	19	4	no	—
299	C₂×C₄	—	19	4	no	—
300	C₂×C₄	—	38	8	no	—
301	C₂×C₄	—	38	8	no	—
302	C₂×C₄	—	76	4	no	—
303	C₂×C₄	—	76	8	no	—
304	C₂×C₄	—	76	2	no	—
305	C₂×C₄	#64545	1	4	yes	complemented
306	C₂×C₄	C₂³×D₁₉	1	4	yes	—
307	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
308	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
309	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₉	1	4	yes	—
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	—	38	4	no	—
317	C₂²	—	38	4	no	—
318	C₂²	—	38	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×#64545	1	2	yes	—
324	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	4	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	4	no	—
330	C₄	—	38	4	no	—
331	C₄	—	38	4	no	—
332	C₄	C₂×#64545	1	2	yes	—
333	C₄	C₂×#64545	1	2	yes	—
334	C₄	#10597435	1	2	yes	—
335	C₂	—	2	2	no	—
336	C₂	—	76	4	no	—
337	C₂	—	76	4	no	—
338	C₂	C₂²×#64545	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×#533	C₂
hypercenter	#533	D₁₉
hypocenter	C₁₉	#3506
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. #11927279	C₂	semidirect
2. #11927267	C₂	semidirect
3. #11927260	C₂	semidirect
4. #11927247	C₂	semidirect
5. #11927216	C₂	semidirect
6. #11927197	C₂	semidirect
7. #11927134	C₂	semidirect
8. #11927113	C₂	semidirect
9. #11927094	C₂	semidirect
10. #11927057	C₂	semidirect
11. #11927024	C₂	semidirect
12. C₂×C₄×D₇₆	C₂	semidirect
13. C₁₉×#533	C₂	semidirect
14. C₂×#64545	C₂²	semidirect
18. C₁₉×C₄⋊Q₈	C₂²	semidirect
19. C₁₉×C₄⋊Q₈	C₂²	semidirect
20. C₁₉×#125	C₂²	semidirect
23. C₄×Q₈×C₁₉	C₂²	semidirect
24. C₄×Q₈×C₁₉	C₂²	semidirect
27. #10597393	C₂²	semidirect
28. #10597392	C₂²	semidirect
33. #10597344	C₂²	semidirect
36. #10597341	C₂²	semidirect
37. #10597341	C₂²	semidirect
38. #10597340	C₂²	semidirect
39. #10597340	C₂²	semidirect
40. #10597339	C₂²	semidirect
41. #10597338	C₂²	semidirect
42. #10597338	C₂²	semidirect
43. C₄⋊C₄×D₁₉	C₂²	semidirect
47. #10597321	C₂²	semidirect
49. #10597320	C₂²	semidirect
54. C₄×D₇₆	C₂²	semidirect
55. C₄×D₇₆	C₂²	semidirect
56. C₄×D₇₆	C₂²	semidirect
57. #10597315	C₂²	semidirect
58. C₄×Dic₃₈	C₂²	semidirect
118. C₂×C₄×C₁₉	#41	semidirect
173. #533	D₁₉	semidirect
189. C₄⋊Q₈	C₂×D₁₉	semidirect
190. C₄⋊Q₈	C₂×D₁₉	semidirect
194. #125	C₂×D₁₉	semidirect
211. C₄×Q₈	C₂×D₁₉	semidirect
212. C₄×Q₈	C₂×D₁₉	semidirect
229. C₁₉	#3506	semidirect
305. C₂×C₄	#64545	semidirect

Maximal subgroups

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

subgroup	index
#11927279	2
#11927267	2
#11927260	2
#11927247	2
#11927216	2
#11927197	2
#11927134	2
#11927113	2
#11927094	2
#11927057	2
#11927024	2
C₂×C₄×D₇₆	2
C₁₉×#533	2
#3506	19