Subgroups — #14575539

Order 2432.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927240	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927203	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926899	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926886	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926863	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926852	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926780	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926773	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926760	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926691	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926685	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#64516	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926541	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#10597273	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#64541	C₂²	1	1	yes	characteristic
17	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
21	#10597385	—	2	1	no	—
22	#10597385	—	2	1	no	—
23	#10597376	C₂²	1	1	yes	characteristic
24	#10597376	C₂²	1	1	yes	characteristic
25	#10597376	C₂²	1	1	yes	characteristic
26	#10597372	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
28	#10597370	C₂²	1	1	yes	characteristic
29	C₂×D₁₅₂	C₂²	1	1	yes	characteristic
30	C₂×#64516	C₂²	1	1	yes	characteristic
31	C₂×#64516	C₂²	1	1	yes	characteristic
32	#10597344	—	2	1	no	—
33	#10597342	—	2	1	no	—
34	#10597331	—	2	1	no	—
35	#10597329	—	2	1	no	—
36	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
37	(C₂²⋊D₁₉)⋊C₄	—	2	1	no	—
38	#10597321	C₂²	1	1	yes	characteristic
39	C₄×D₇₆	C₂²	1	1	yes	characteristic
40	#10597316	C₂²	1	1	yes	characteristic
41	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
42	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
46	#10597278	C₂²	1	1	yes	characteristic
47	#10597278	C₂²	1	1	yes	characteristic
48	#10597278	C₂²	1	1	yes	characteristic
49	#10597278	C₂²	1	1	yes	characteristic
50	#10597275	C₂²	1	1	yes	characteristic
51	#10597274	C₂²	1	1	yes	characteristic
52	#10597274	C₂²	1	1	yes	characteristic
53	#10597273	C₂²	1	1	yes	characteristic
54	#10597273	C₂²	1	1	yes	characteristic
55	#10597273	C₂²	1	1	yes	characteristic
56	#10597273	C₂²	1	1	yes	characteristic
57	#10597273	C₂²	1	1	yes	characteristic
58	#10597273	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₉	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₉	—	2	1	no	—
63	C₂×C₂²⋊D₁₉	—	2	1	no	—
64	C₂×C₂²⋊D₁₉	—	2	1	no	—
65	C₂×C₂²⋊D₁₉	—	2	1	no	—
66	C₂²×Dic₁₉	—	2	1	no	—
67	C₂²×Dic₁₉	—	2	1	no	—
68	#64541	—	2	2	no	—
69	#64541	—	4	1	no	—
70	C₂×D₇₆	C₂³	1	1	yes	characteristic
71	C₂×D₇₆	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₁₉	—	2	1	no	—
73	C₂×C₄×D₁₉	—	2	1	no	—
74	C₂×Dic₃₈	—	2	2	no	—
75	C₂×Dic₃₈	—	2	1	no	—
76	C₂×Dic₃₈	—	4	1	no	—
77	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
78	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₉	—	4	1	no	—
80	C₂×C₈×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
84	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
87	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂³	1	1	yes	characteristic
89	C₂²⋊Dic₁₉	—	2	1	no	—
90	C₂²⋊Dic₁₉	—	2	1	no	—
91	(C₂×D₁₉)⋊C₄	—	2	1	no	—
92	(C₂×D₁₉)⋊C₄	—	2	1	no	—
93	(C₂×D₁₉)⋊C₄	—	2	1	no	—
94	(C₂×D₁₉)⋊C₄	—	2	1	no	—
95	(C₂×D₁₉)⋊C₄	—	2	1	no	—
96	(C₂×D₁₉)⋊C₄	—	2	1	no	—
97	C₄⋊Dic₁₉	—	2	1	no	—
98	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
99	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
100	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
101	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
102	Dic₁₉⋊C₄	—	2	1	no	—
103	Dic₁₉⋊C₄	—	2	1	no	—
104	Dic₁₉⋊C₄	—	2	1	no	—
105	Dic₁₉⋊C₄	—	2	1	no	—
106	C₄×Dic₁₉	—	2	1	no	—
107	C₄×Dic₁₉	—	2	1	no	—
108	D₁₅₂	—	4	1	no	—
109	#64516	—	2	2	no	—
110	#64516	—	4	1	no	—
111	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₉	—	2	1	no	—
114	C₂²×D₁₉	—	2	1	no	—
115	D₄×C₁₉	—	2	2	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₁₉	—	4	2	no	—
126	C₂²⋊D₁₉	—	4	2	no	—
127	C₂²⋊D₁₉	—	4	2	no	—
128	C₂²⋊D₁₉	—	8	1	no	—
129	C₂×Dic₁₉	—	2	1	no	—
130	C₂×Dic₁₉	—	2	1	no	—
131	C₂×Dic₁₉	—	2	1	no	—
132	C₂×Dic₁₉	—	2	1	no	—
133	C₂×Dic₁₉	—	2	1	no	—
134	C₂×Dic₁₉	—	2	1	no	—
135	C₂×Dic₁₉	—	4	2	no	—
136	C₂×Dic₁₉	—	8	1	no	—
137	D₇₆	—	2	1	no	—
138	D₇₆	—	2	1	no	—
139	D₇₆	—	2	1	no	—
140	D₇₆	—	2	1	no	—
141	C₄×D₁₉	—	4	2	no	—
142	C₄×D₁₉	—	8	1	no	—
143	Dic₃₈	—	2	1	no	—
144	Dic₃₈	—	2	1	no	—
145	Dic₃₈	—	2	1	no	—
146	Dic₃₈	—	4	2	no	—
147	Dic₃₈	#41	1	2	yes	—
148	C₈×C₁₉	—	2	1	no	—
149	C₈×C₁₉	—	2	1	no	—
150	C₈×C₁₉	—	2	1	no	—
151	C₈×C₁₉	—	2	1	no	—
152	#3301	—	19	1	no	maximal, Sylow
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	#135	1	2	yes	—
159	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
160	C₂×D₁₉	—	4	1	no	—
161	C₂×D₁₉	—	4	1	no	—
162	C₂×D₁₉	—	4	1	no	—
163	C₂×D₁₉	—	4	1	no	—
164	C₂×D₁₉	—	4	1	no	—
165	C₂×D₁₉	—	4	1	no	—
166	C₄×C₁₉	—	2	1	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	2	1	no	—
169	C₄×C₁₉	—	2	1	no	—
170	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
171	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
172	Dic₁₉	—	2	2	no	—
173	Dic₁₉	—	4	1	no	—
174	Dic₁₉	—	4	1	no	—
175	Dic₁₉	—	4	1	no	—
176	Dic₁₉	—	4	1	no	—
177	Dic₁₉	—	4	1	no	—
178	#553	—	19	1	no	—
179	#548	—	19	1	no	—
180	#488	—	19	1	no	—
181	#485	—	19	1	no	—
182	#481	—	19	1	no	—
183	#479	—	19	1	no	—
184	#466	—	19	1	no	—
185	#465	—	19	1	no	—
186	#463	—	19	1	no	—
187	#449	—	19	1	no	—
188	#448	—	19	1	no	—
189	C₄×SD₁₆	—	19	1	no	—
190	C₈×D₄	D₁₉	1	1	yes	characteristic, complemented
191	#416	—	19	1	no	—
192	C₂×#103	—	19	1	no	—
193	C₂×C₁₉	—	2	2	no	—
194	C₂×C₁₉	—	4	1	no	—
195	C₂×C₁₉	#546	1	1	yes	characteristic
196	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
197	C₂×C₁₉	#577	1	1	yes	characteristic
198	D₁₉	—	8	1	no	—
199	D₁₉	—	8	1	no	—
200	C₂×#41	—	19	1	no	—
201	C₂²×Q₈	—	19	1	no	—
202	C₂×SD₁₆	—	19	1	no	—
203	C₂×SD₁₆	—	19	1	no	—
204	C₂×D₈	—	19	1	no	—
205	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
206	C₂²×C₈	C₂×D₁₉	1	1	yes	characteristic
207	#126	—	38	1	no	—
208	#125	—	19	1	no	—
209	#124	—	19	1	no	—
210	#124	—	38	1	no	—
211	#123	—	38	1	no	—
212	C₂²⋊Q₈	—	19	1	no	—
213	C₂²⋊Q₈	—	38	1	no	—
214	#121	—	19	1	no	—
215	#121	—	19	1	no	—
216	#121	—	19	1	no	—
217	#121	—	38	1	no	—
218	#121	—	38	1	no	—
219	C₄×Q₈	—	19	1	no	—
220	C₄×D₄	—	19	1	no	—
221	C₄×D₄	—	38	1	no	—
222	C₄×D₄	—	38	1	no	—
223	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
224	#117	—	19	1	no	—
225	C₂×C₄⋊C₄	—	19	1	no	—
226	#107	—	19	1	no	—
227	#106	—	19	1	no	—
228	#106	—	19	1	no	—
229	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
230	#103	—	19	1	no	—
231	#103	—	19	1	no	—
232	#103	—	19	1	no	—
233	#103	—	19	1	no	—
234	#103	—	19	1	no	—
235	#103	—	19	1	no	—
236	#102	—	19	1	no	—
237	#102	—	19	1	no	—
238	#102	—	19	1	no	—
239	#102	—	19	1	no	—
240	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
241	C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
242	C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
243	C₁₉	#3301	1	1	yes	characteristic, complemented, Sylow
244	#41	—	38	2	no	—
245	#41	—	76	1	no	—
246	C₂×Q₈	—	19	1	no	—
247	C₂×Q₈	—	19	1	no	—
248	C₂×Q₈	—	38	2	no	—
249	C₂×Q₈	—	38	1	no	—
250	C₂×Q₈	—	76	1	no	—
251	C₂×D₄	—	19	1	no	—
252	C₂×D₄	—	19	1	no	—
253	C₂×D₄	—	38	1	no	—
254	C₂×D₄	—	38	1	no	—
255	C₂×D₄	—	38	1	no	—
256	C₂×D₄	—	38	1	no	—
257	C₂×D₄	D₇₆	1	1	yes	characteristic
258	C₂²×C₄	—	38	1	no	—
259	C₂²×C₄	—	38	1	no	—
260	C₂²×C₄	—	38	1	no	—
261	C₂²×C₄	—	38	1	no	—
262	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
264	SD₁₆	—	38	2	no	—
265	SD₁₆	—	76	1	no	—
266	D₈	—	76	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
273	C₄⋊C₄	—	19	1	no	—
274	C₄⋊C₄	—	19	1	no	—
275	C₄⋊C₄	—	19	1	no	—
276	C₄⋊C₄	—	19	1	no	—
277	C₄⋊C₄	—	38	1	no	—
278	C₄⋊C₄	—	38	1	no	—
279	C₄⋊C₄	—	38	1	no	—
280	C₄⋊C₄	—	38	1	no	—
281	C₄⋊C₄	—	38	1	no	—
282	C₄⋊C₄	D₇₆	1	1	yes	characteristic
283	C₂²⋊C₄	—	38	1	no	—
284	C₂²⋊C₄	—	38	1	no	—
285	C₂²⋊C₄	—	38	1	no	—
286	C₂²⋊C₄	—	38	1	no	—
287	C₂²⋊C₄	—	38	1	no	—
288	C₂²⋊C₄	—	38	1	no	—
289	C₂²⋊C₄	—	38	1	no	—
290	C₂²⋊C₄	—	38	1	no	—
291	C₂²⋊C₄	D₇₆	1	1	yes	characteristic
292	C₂²⋊C₄	D₇₆	1	1	yes	characteristic
293	C₄²	—	38	1	no	—
294	C₄²	—	38	1	no	—
295	C₄²	C₂²×D₁₉	1	1	yes	characteristic
296	C₂³	—	38	1	no	—
297	C₂³	—	38	1	no	—
298	C₂³	C₂×D₇₆	1	1	yes	characteristic
299	C₂³	C₂×D₇₆	1	1	yes	characteristic
300	Q₈	—	19	2	no	—
301	Q₈	—	38	1	no	—
302	Q₈	—	38	1	no	—
303	Q₈	—	38	1	no	—
304	Q₈	—	76	2	no	—
305	D₄	—	2	2	no	—
306	D₄	—	38	1	no	—
307	D₄	—	38	1	no	—
308	D₄	—	38	1	no	—
309	D₄	—	38	1	no	—
310	D₄	—	76	2	no	—
311	D₄	—	76	2	no	—
312	D₄	—	76	2	no	—
313	D₄	—	152	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	38	1	no	—
319	C₂×C₄	—	38	1	no	—
320	C₂×C₄	—	38	1	no	—
321	C₂×C₄	—	38	1	no	—
322	C₂×C₄	—	38	1	no	—
323	C₂×C₄	—	38	1	no	—
324	C₂×C₄	—	76	2	no	—
325	C₂×C₄	—	76	2	no	—
326	C₂×C₄	—	152	1	no	—
327	C₂×C₄	—	152	1	no	—
328	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
329	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
330	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
331	C₂×C₄	C₂×D₇₆	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	76	1	no	—
343	C₂²	—	76	1	no	—
344	C₂²	—	76	1	no	—
345	C₂²	—	76	1	no	—
346	C₂²	—	76	1	no	—
347	C₂²	—	76	1	no	—
348	C₂²	#10597350	1	2	yes	—
349	C₂²	C₂²×D₇₆	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	38	2	no	—
355	C₄	—	76	1	no	—
356	C₄	—	76	1	no	—
357	C₄	—	76	1	no	—
358	C₄	—	76	1	no	—
359	C₄	—	76	1	no	—
360	C₄	#10597430	1	1	yes	characteristic
361	C₄	#41×D₁₉	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	—	152	1	no	—
365	C₂	—	152	1	no	—
366	C₂	#11927189	1	1	yes	characteristic
367	C₂	C₂×#10597350	1	1	yes	characteristic
368	C₂	#11927356	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₆
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₈×D₄×C₁₉	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	C₈×D₄	D₁₉
hypocenter	C₁₉	#3301
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #11927240	C₂	semidirect
2. #11927203	C₂	semidirect
3. #11926899	C₂	semidirect
4. #11926886	C₂	semidirect
5. #11926863	C₂	semidirect
6. #11926852	C₂	semidirect
7. #11926780	C₂	semidirect
8. #11926773	C₂	semidirect
9. #11926760	C₂	semidirect
10. #11926691	C₂	semidirect
11. #11926685	C₂	semidirect
12. C₄×#64516	C₂	semidirect
13. #11926541	C₂	semidirect
14. C₂×#10597273	C₂	semidirect
15. C₈×D₄×C₁₉	C₂	semidirect
190. C₈×D₄	D₁₉	semidirect
243. C₁₉	#3301	semidirect

Maximal subgroups

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

subgroup	index
#11927240	2
#11927203	2
#11926899	2
#11926886	2
#11926863	2
#11926852	2
#11926780	2
#11926773	2
#11926760	2
#11926691	2
#11926685	2
C₄×#64516	2
#11926541	2
C₂×#10597273	2
C₈×D₄×C₁₉	2
#3301	19