Subgroups — C₂×#11746779 #14313821

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#91223	C₂	1	1	yes	maximal, characteristic
3	C₂×D₅⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#11746779	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64908	—	3	1	no	maximal, Hall
6	C₂×C₃×Q₈×D₅	—	2	1	no	—
7	C₂×C₄⋊ΓL(1,𝔽₁₆)	—	2	1	no	—
8	C₂×C₃×C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
9	C₂×C₃×#689	—	2	1	no	—
10	C₃×D₅×Q₁₆	—	2	1	no	—
11	C₃×D₅×Q₁₆	—	2	1	no	—
12	C₃×D₅×Q₁₆	C₄	1	2	yes	complemented
13	C₃×D₅×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₃×Dic₂₀	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₃⋊#674	—	2	1	no	—
17	#91223	—	2	1	no	—
18	#91223	C₂²	1	2	yes	—
19	D₅⋊(C₃⋊C₁₆)	—	2	1	no	—
20	D₅⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
21	C₂×#4564	—	5	1	no	maximal, Hall
22	C₂×D₅×Q₁₆	D₃	1	1	yes	characteristic
23	C₂×#3994	—	3	1	no	—
24	C₂×D₅⋊C₁₆	—	3	1	no	—
25	#64908	—	3	4	no	—
26	C₂²×ΓL(1,𝔽₁₆)	—	4	1	no	—
27	C₂×C₃×C₅×Q₈	—	2	1	no	—
28	C₃×Q₈×D₅	—	2	2	no	—
29	C₃×Q₈×D₅	—	2	2	no	—
30	C₃×Q₈×D₅	—	4	1	no	—
31	C₃×Q₈×D₅	—	4	1	no	—
32	C₂×C₃×C₄×D₅	—	4	1	no	—
33	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
34	C₂×C₃×Dic₁₀	—	2	1	no	—
35	C₂×C₃×Dic₁₀	—	4	1	no	—
36	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
37	C₄⋊ΓL(1,𝔽₁₆)	—	4	1	no	—
38	C₃×C₅×Q₁₆	—	2	1	no	—
39	C₃×C₅×Q₁₆	C₂×C₄	1	2	yes	complemented
40	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₃×#689	—	2	2	no	—
42	C₃×#689	—	2	2	no	—
43	C₂×C₃×#168	D₄	1	1	yes	characteristic
44	C₃×Dic₂₀	—	2	1	no	—
45	C₃×Dic₂₀	C₂×C₄	1	2	yes	complemented
46	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
47	C₃×C₈×D₅	D₄	1	1	yes	characteristic
48	C₃×C₈×D₅	D₄	1	1	yes	characteristic
49	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
50	C₃⋊#674	—	2	2	no	—
51	C₂²×C₃×Q₁₆	—	5	1	no	—
52	C₂×#818	—	5	1	no	—
53	C₂²×C₃⋊C₁₆	—	5	1	no	—
54	#4564	—	5	4	no	—
55	C₂×Q₈×D₅	—	2	1	no	—
56	C₂×C₄⋊AGL(1,𝔽₅)	—	6	1	no	—
57	C₂×C₅×Q₁₆	Dic₃	1	1	yes	characteristic, complemented
58	C₂×#689	—	2	1	no	—
59	D₅×Q₁₆	—	2	1	no	—
60	D₅×Q₁₆	—	2	1	no	—
61	D₅×Q₁₆	Dic₃	1	2	yes	complemented
62	D₅×Q₁₆	C₂×D₃	1	2	yes	—
63	C₂×Dic₂₀	Dic₃	1	1	yes	characteristic, complemented
64	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
65	C₂×#674	—	6	1	no	—
66	#3994	—	3	2	no	—
67	#3994	—	6	1	no	—
68	D₅⋊C₁₆	—	3	2	no	—
69	D₅⋊C₁₆	—	6	1	no	—
70	C₂×#358	—	15	1	no	Sylow
71	C₂²×C₃×D₅	D₈	1	1	yes	characteristic
72	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
73	C₂×ΓL(1,𝔽₁₆)	—	8	2	no	—
74	C₃×C₅×Q₈	—	2	2	no	—
75	C₃×C₅×Q₈	—	4	1	no	—
76	C₂×C₃×C₄×C₅	—	4	1	no	—
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×Dic₅	—	4	1	no	—
79	C₂×C₃×Dic₅	SD₁₆	1	1	yes	characteristic
80	C₃×C₄×D₅	—	4	2	no	—
81	C₃×C₄×D₅	—	4	2	no	—
82	C₃×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₃×C₄×D₅	D₈	1	1	yes	characteristic
84	C₃×C₄×D₅	SD₁₆	1	1	yes	characteristic
85	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
86	C₃×Dic₁₀	—	2	2	no	—
87	C₃×Dic₁₀	—	4	1	no	—
88	C₃×Dic₁₀	—	4	2	no	—
89	C₃×Dic₁₀	—	4	2	no	—
90	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₃×#168	C₂²⋊C₄	1	1	yes	characteristic
93	C₃×#168	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₃×Q₈	—	10	1	no	—
95	C₂×C₃×Q₁₆	—	5	2	no	—
96	C₂×C₃×Q₁₆	—	5	2	no	—
97	C₂×C₃×Q₁₆	—	5	1	no	—
98	C₂×C₃×Q₁₆	—	10	1	no	—
99	C₂×C₃×Q₁₆	—	10	1	no	—
100	C₂×C₃×Q₁₆	—	10	1	no	—
101	C₂×C₃×Q₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
102	C₂²×C₃×C₈	—	5	1	no	—
103	C₂×C₄⋊Dic₃	—	10	1	no	—
104	#818	—	5	2	no	—
105	#818	—	10	1	no	—
106	C₂×C₃⋊C₁₆	—	5	2	no	—
107	C₂×C₃⋊C₁₆	—	10	1	no	—
108	C₂×C₃⋊C₁₆	—	10	1	no	—
109	C₂²×AGL(1,𝔽₅)	—	12	1	no	—
110	C₂×C₅×Q₈	—	2	1	no	—
111	Q₈×D₅	—	2	2	no	—
112	Q₈×D₅	—	2	2	no	—
113	Q₈×D₅	—	4	1	no	—
114	Q₈×D₅	—	4	1	no	—
115	C₂×C₄×D₅	—	4	1	no	—
116	C₂×C₄×D₅	#67	1	1	yes	characteristic
117	C₂×Dic₁₀	—	2	1	no	—
118	C₂×Dic₁₀	—	4	1	no	—
119	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
120	C₄⋊AGL(1,𝔽₅)	—	12	1	no	—
121	C₅×Q₁₆	—	2	1	no	—
122	C₅×Q₁₆	C₂×Dic₃	1	2	yes	complemented
123	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
124	#689	—	2	2	no	—
125	#689	—	2	2	no	—
126	C₂×#168	#67	1	1	yes	characteristic
127	Dic₂₀	—	2	1	no	—
128	Dic₂₀	C₂×Dic₃	1	2	yes	complemented
129	C₈×D₅	C₂×Dic₃	1	1	yes	characteristic
130	C₈×D₅	#67	1	1	yes	characteristic
131	C₈×D₅	#67	1	1	yes	characteristic
132	C₈×D₅	C₂²×D₃	1	1	yes	characteristic
133	#674	—	6	2	no	—
134	C₂²×Q₁₆	—	5	1	no	—
135	C₂²×C₁₆	—	15	1	no	—
136	C₂×#107	—	15	1	no	—
137	#358	—	15	4	no	—
138	C₂²×C₃×C₅	#102	1	1	yes	characteristic
139	C₂×C₃×D₅	#102	1	1	yes	characteristic
140	C₂×C₃×D₅	SD₃₂	1	2	yes	—
141	C₂×C₃×D₅	Q₃₂	1	2	yes	—
142	C₂×C₃×D₅	C₂×D₈	1	1	yes	characteristic
143	ΓL(1,𝔽₁₆)	—	8	2	no	—
144	C₃×C₄×C₅	—	4	2	no	—
145	C₃×C₄×C₅	#102	1	1	yes	characteristic
146	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₃×Dic₅	—	4	2	no	—
148	C₃×Dic₅	#102	1	1	yes	characteristic
149	C₃×Dic₅	C₂×SD₁₆	1	1	yes	characteristic
150	C₂×C₃×Q₈	—	2	1	no	—
151	C₂×C₃×Q₈	—	10	2	no	—
152	C₂×C₃×Q₈	—	10	2	no	—
153	C₂×C₃×Q₈	—	10	1	no	—
154	C₂×C₃×Q₈	—	20	1	no	—
155	C₂×C₃×Q₈	—	20	1	no	—
156	C₂×C₃×Q₈	—	20	1	no	—
157	C₂²×C₃×C₄	—	5	1	no	—
158	C₂²×C₃×C₄	—	20	1	no	—
159	C₂²×Dic₃	—	20	1	no	—
160	C₃×Q₁₆	—	2	1	no	—
161	C₃×Q₁₆	—	5	2	no	—
162	C₃×Q₁₆	—	10	2	no	—
163	C₃×Q₁₆	—	10	2	no	—
164	C₃×Q₁₆	—	10	1	no	—
165	C₃×Q₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
166	C₂×C₃×C₈	—	5	1	no	—
167	C₂×C₃×C₈	—	5	1	no	—
168	C₂×C₃×C₈	—	5	1	no	—
169	C₂×C₃×C₈	—	5	1	no	—
170	C₂×C₃×C₈	—	5	1	no	—
171	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
172	C₄⋊Dic₃	—	10	2	no	—
173	C₄⋊Dic₃	—	20	1	no	—
174	C₃⋊C₁₆	—	10	2	no	—
175	C₂²×D₅	#213	1	1	yes	characteristic
176	C₂×AGL(1,𝔽₅)	—	12	2	no	—
177	C₂×AGL(1,𝔽₅)	—	24	2	no	—
178	C₅×Q₈	—	2	2	no	—
179	C₅×Q₈	—	4	1	no	—
180	C₂×C₄×C₅	—	4	1	no	—
181	C₂×C₄×C₅	#217	1	1	yes	characteristic
182	C₂×Dic₅	—	4	1	no	—
183	C₂×Dic₅	#214	1	1	yes	characteristic
184	C₄×D₅	—	4	2	no	—
185	C₄×D₅	—	4	2	no	—
186	C₄×D₅	#213	1	1	yes	characteristic
187	C₄×D₅	#214	1	1	yes	characteristic
188	C₄×D₅	#217	1	1	yes	characteristic
189	C₄×D₅	C₂×#67	1	1	yes	characteristic
190	Dic₁₀	—	2	2	no	—
191	Dic₁₀	—	4	1	no	—
192	Dic₁₀	—	4	2	no	—
193	Dic₁₀	—	4	2	no	—
194	C₅×C₈	#217	1	1	yes	characteristic
195	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
196	#168	#217	1	1	yes	characteristic
197	#168	C₂×#67	1	1	yes	characteristic
198	C₂²×Q₈	—	10	1	no	—
199	C₂×Q₁₆	—	5	2	no	—
200	C₂×Q₁₆	—	5	2	no	—
201	C₂×Q₁₆	—	5	1	no	—
202	C₂×Q₁₆	—	10	1	no	—
203	C₂×Q₁₆	—	10	1	no	—
204	C₂×Q₁₆	—	10	1	no	—
205	C₂×Q₁₆	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
206	C₂²×C₈	—	5	1	no	—
207	C₂×C₄⋊C₄	—	30	1	no	—
208	C₂×C₁₆	—	15	2	no	—
209	C₂×C₁₆	—	30	1	no	—
210	C₂×C₁₆	—	30	1	no	—
211	#107	—	15	2	no	—
212	#107	—	30	1	no	—
213	C₂×C₃×C₅	#358	1	2	yes	complemented
214	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
215	C₃×D₅	#358	1	2	yes	complemented
216	C₃×D₅	C₂×SD₃₂	1	1	yes	characteristic
217	C₃×D₅	C₂×Q₃₂	1	1	yes	characteristic
218	C₂³×C₃	—	5	1	no	—
219	C₃×Q₈	—	2	2	no	—
220	C₃×Q₈	—	4	1	no	—
221	C₃×Q₈	—	10	2	no	—
222	C₃×Q₈	—	20	2	no	—
223	C₃×Q₈	—	20	2	no	—
224	C₃×Q₈	—	20	1	no	—
225	C₂×C₃×C₄	—	4	1	no	—
226	C₂×C₃×C₄	—	5	1	no	—
227	C₂×C₃×C₄	—	5	1	no	—
228	C₂×C₃×C₄	—	5	1	no	—
229	C₂×C₃×C₄	—	5	1	no	—
230	C₂×C₃×C₄	—	5	1	no	—
231	C₂×C₃×C₄	—	20	2	no	—
232	C₂×C₃×C₄	—	20	2	no	—
233	C₂×C₃×C₄	—	20	1	no	—
234	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
235	C₂×Dic₃	—	20	2	no	—
236	C₂×Dic₃	—	40	2	no	—
237	C₃×C₈	—	5	1	no	—
238	C₃×C₈	—	5	1	no	—
239	C₃×C₈	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
240	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
241	C₂²×C₅	#832	1	1	yes	characteristic
242	C₂×D₅	#828	1	2	yes	—
243	C₂×D₅	#829	1	2	yes	—
244	C₂×D₅	#832	1	1	yes	characteristic
245	C₂×D₅	C₂×#213	1	1	yes	characteristic
246	AGL(1,𝔽₅)	—	24	2	no	—
247	C₄×C₅	—	4	2	no	—
248	C₄×C₅	#832	1	1	yes	characteristic
249	C₄×C₅	C₂×#217	1	1	yes	characteristic
250	Dic₅	—	4	2	no	—
251	Dic₅	#832	1	1	yes	characteristic
252	Dic₅	C₂×#214	1	1	yes	characteristic
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	10	2	no	—
255	C₂×Q₈	—	10	2	no	—
256	C₂×Q₈	—	10	1	no	—
257	C₂×Q₈	—	20	1	no	—
258	C₂×Q₈	—	20	1	no	—
259	C₂×Q₈	—	20	1	no	—
260	C₂²×C₄	—	5	1	no	—
261	C₂²×C₄	—	20	1	no	—
262	C₂²×C₄	—	60	1	no	—
263	Q₁₆	—	2	1	no	—
264	Q₁₆	—	5	2	no	—
265	Q₁₆	—	10	2	no	—
266	Q₁₆	—	10	2	no	—
267	Q₁₆	—	10	1	no	—
268	Q₁₆	C₂×ΓL(1,𝔽₁₆)	1	2	yes	complemented
269	C₂×C₈	—	5	1	no	—
270	C₂×C₈	—	5	1	no	—
271	C₂×C₈	—	5	1	no	—
272	C₂×C₈	—	5	1	no	—
273	C₂×C₈	—	5	1	no	—
274	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
275	C₄⋊C₄	—	30	2	no	—
276	C₄⋊C₄	—	60	1	no	—
277	C₁₆	—	30	2	no	—
278	C₃×C₅	C₂×#358	1	1	yes	characteristic, complemented, Hall
279	C₂²×C₃	—	5	2	no	—
280	C₂²×C₃	—	5	2	no	—
281	C₂²×C₃	—	5	1	no	—
282	C₂²×C₃	—	5	1	no	—
283	C₂²×C₃	#4007	1	1	yes	characteristic
284	C₃×C₄	—	4	2	no	—
285	C₃×C₄	—	5	1	no	—
286	C₃×C₄	—	5	1	no	—
287	C₃×C₄	—	20	2	no	—
288	C₃×C₄	#4007	1	1	yes	characteristic
289	C₃×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
290	Dic₃	—	40	2	no	—
291	C₂×C₅	#4564	1	2	yes	complemented
292	C₂×C₅	C₂×#832	1	1	yes	characteristic
293	D₅	#4564	1	2	yes	complemented
294	D₅	C₂×#828	1	1	yes	characteristic
295	D₅	C₂×#829	1	1	yes	characteristic
296	C₂³	—	5	1	no	—
297	Q₈	—	2	2	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	10	2	no	—
300	Q₈	—	20	2	no	—
301	Q₈	—	20	2	no	—
302	Q₈	—	20	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	5	1	no	—
305	C₂×C₄	—	5	1	no	—
306	C₂×C₄	—	5	1	no	—
307	C₂×C₄	—	5	1	no	—
308	C₂×C₄	—	5	1	no	—
309	C₂×C₄	—	20	2	no	—
310	C₂×C₄	—	20	2	no	—
311	C₂×C₄	—	20	1	no	—
312	C₂×C₄	—	60	2	no	—
313	C₂×C₄	—	120	2	no	—
314	C₂×C₄	#6763	1	1	yes	characteristic
315	C₈	—	5	1	no	—
316	C₈	—	5	1	no	—
317	C₈	#6763	1	1	yes	characteristic
318	C₈	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
319	C₂×C₃	—	5	1	no	—
320	C₂×C₃	—	5	2	no	—
321	C₂×C₃	—	5	1	no	—
322	C₂×C₃	#64908	1	2	yes	complemented
323	C₂×C₃	C₂×#4007	1	1	yes	characteristic
324	C₅	C₂×#4564	1	1	yes	characteristic, complemented, Sylow
325	C₂²	—	5	2	no	—
326	C₂²	—	5	2	no	—
327	C₂²	—	5	1	no	—
328	C₂²	—	5	1	no	—
329	C₂²	#91236	1	1	yes	characteristic
330	C₄	—	4	2	no	—
331	C₄	—	5	1	no	—
332	C₄	—	5	1	no	—
333	C₄	—	20	2	no	—
334	C₄	—	120	2	no	—
335	C₄	#91236	1	1	yes	characteristic
336	C₄	C₂×#6763	1	1	yes	characteristic
337	C₃	C₂×#64908	1	1	yes	characteristic, complemented, Sylow
338	C₂	—	5	1	no	—
339	C₂	—	5	1	no	—
340	C₂	—	5	2	no	—
341	C₂	#11746779	1	2	yes	direct product factor
342	C₂	C₂×#91236	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#91236
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₈	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₅×Q₁₆	C₄
socle	C₂²×C₃×C₅	#102
prime radical	C₂²×C₃×D₅	D₈
cosocle	C₃×C₈×D₅	C₂³
hypercenter	C₂×Q₁₆	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₂×#358
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
4. #11746779	C₂	direct (N × H)
8. C₂×C₃×C₅×Q₁₆	C₄	semidirect
12. C₃×D₅×Q₁₆	C₄	semidirect
14. C₂×C₃×Dic₂₀	C₄	semidirect
39. C₃×C₅×Q₁₆	C₂×C₄	semidirect
45. C₃×Dic₂₀	C₂×C₄	semidirect
57. C₂×C₅×Q₁₆	Dic₃	semidirect
61. D₅×Q₁₆	Dic₃	semidirect
63. C₂×Dic₂₀	Dic₃	semidirect
101. C₂×C₃×Q₁₆	AGL(1,𝔽₅)	semidirect
122. C₅×Q₁₆	C₂×Dic₃	semidirect
128. Dic₂₀	C₂×Dic₃	semidirect
165. C₃×Q₁₆	C₂×AGL(1,𝔽₅)	semidirect
205. C₂×Q₁₆	ΓL(1,𝔽₁₆)	semidirect
213. C₂×C₃×C₅	#358	semidirect
215. C₃×D₅	#358	semidirect
268. Q₁₆	C₂×ΓL(1,𝔽₁₆)	semidirect
278. C₃×C₅	C₂×#358	semidirect
291. C₂×C₅	#4564	semidirect
293. D₅	#4564	semidirect
322. C₂×C₃	#64908	semidirect
324. C₅	C₂×#4564	semidirect
337. C₃	C₂×#64908	semidirect
341. C₂	#11746779	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×D₅×Q₁₆	2
C₂×#91223	2
C₂×D₅⋊(C₃⋊C₁₆)	2
#11746779	2
C₂×#64908	3
C₂×#4564	5

Subgroups — C2×#11746779 #14313821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746779 #14313821