Subgroups — C₂²×#91813 #14479422

Order 1920.

3680 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#6719	C₂	1	2	yes	maximal
4	C₂²×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#91813	C₂	1	24	yes	maximal, direct product factor
6	C₂²×#6712	C₂	1	2	yes	maximal, complemented
7	C₂²×#4070	—	3	1	no	maximal, Hall
8	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
9	C₂³×C₃×C₄×C₅	—	2	8	no	—
10	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄×C₅	C₂²	1	12	yes	—
12	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₅	C₂²	1	2	yes	—
14	C₂³×Dic₁₅	C₂²	1	2	yes	—
15	C₂²×Dic₃₀	—	2	8	no	—
16	C₂²×Dic₃₀	C₂²	1	12	yes	complemented
17	C₂×#6719	C₂²	1	24	yes	—
18	C₂×C₄⋊Dic₁₅	C₂²	1	12	yes	complemented
19	#91813	C₂²	1	64	yes	direct product factor
20	C₂×#6712	C₂²	1	24	yes	complemented
21	C₂²×#924	—	5	1	no	maximal, Hall
22	C₂⁴×C₄×C₅	D₃	1	1	yes	characteristic
23	C₂³×Dic₁₀	—	3	1	no	—
24	C₂²×C₂²⋊Dic₅	—	3	2	no	—
25	C₂²×C₄⋊Dic₅	—	3	1	no	—
26	C₂×#4070	—	3	24	no	—
27	C₂²×#683	—	3	2	no	—
28	C₂⁴×C₃×C₅	—	2	4	no	—
29	C₂⁴×C₃×C₅	—	2	4	no	—
30	C₂⁴×C₃×C₅	Q₈	1	8	yes	complemented
31	C₂⁴×C₃×C₅	C₂³	1	6	yes	—
32	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	—	2	4	no	—
34	C₂²×C₃×C₄×C₅	—	2	48	no	—
35	C₂²×C₃×C₄×C₅	D₄	1	8	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	16	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
39	C₂²×Dic₁₅	—	2	8	no	—
40	C₂²×Dic₁₅	—	2	8	no	—
41	C₂²×Dic₁₅	C₂³	1	12	yes	—
42	C₂²×Dic₁₅	C₂³	1	12	yes	—
43	C₂×Dic₃₀	—	2	48	no	—
44	C₂×Dic₃₀	C₂³	1	16	yes	complemented
45	#6719	C₂³	1	32	yes	—
46	C₄⋊Dic₁₅	C₂³	1	16	yes	complemented
47	#6712	C₂³	1	32	yes	complemented
48	C₂⁴×C₃×C₄	D₅	1	1	yes	characteristic
49	C₂³×Dic₆	—	5	1	no	—
50	C₂²×#217	—	5	2	no	—
51	C₂²×C₄⋊Dic₃	—	5	1	no	—
52	C₂×#924	—	5	24	no	—
53	C₂²×Dic₃⋊C₄	—	5	2	no	—
54	C₂⁵×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂³×C₄×C₅	—	2	8	no	—
56	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
57	C₂³×C₄×C₅	C₂×D₃	1	12	yes	—
58	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
59	C₂³×Dic₅	—	3	2	no	—
60	C₂³×Dic₅	—	3	2	no	—
61	C₂²×Dic₁₀	—	3	12	no	—
62	C₂²×Dic₁₀	—	6	8	no	—
63	C₂×C₂²⋊Dic₅	—	3	24	no	—
64	C₂×C₄⋊Dic₅	—	3	12	no	—
65	#4070	—	3	64	no	—
66	C₂×#683	—	3	24	no	—
67	C₂²×C₂²⋊Q₈	—	15	1	no	Sylow
68	C₂³×C₃×C₅	—	2	32	no	—
69	C₂³×C₃×C₅	—	2	12	no	—
70	C₂³×C₃×C₅	—	2	12	no	—
71	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
72	C₂³×C₃×C₅	C₂×Q₈	1	4	yes	—
73	C₂³×C₃×C₅	C₂×Q₈	1	24	yes	complemented
74	C₂³×C₃×C₅	#41	1	4	yes	—
75	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
76	C₂³×C₃×C₅	C₂⁴	1	3	yes	—
77	C₂×C₃×C₄×C₅	—	2	12	no	—
78	C₂×C₃×C₄×C₅	—	2	32	no	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	24	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
81	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
82	C₂×Dic₁₅	—	2	24	no	—
83	C₂×Dic₁₅	—	2	24	no	—
84	C₂×Dic₁₅	C₂⁴	1	8	yes	—
85	C₂×Dic₁₅	C₂⁴	1	8	yes	—
86	Dic₃₀	—	2	32	no	—
87	C₂⁵×C₃	C₂×D₅	1	1	yes	characteristic
88	C₂³×C₃×C₄	—	2	8	no	—
89	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
90	C₂³×C₃×C₄	C₂×D₅	1	12	yes	—
91	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
92	C₂³×Dic₃	—	5	2	no	—
93	C₂³×Dic₃	—	5	2	no	—
94	C₂²×Dic₆	—	5	12	no	—
95	C₂²×Dic₆	—	10	8	no	—
96	C₂×#217	—	5	24	no	—
97	C₂×C₄⋊Dic₃	—	5	12	no	—
98	#924	—	5	64	no	—
99	C₂×Dic₃⋊C₄	—	5	24	no	—
100	C₂⁴×C₅	—	2	4	no	—
101	C₂⁴×C₅	—	2	4	no	—
102	C₂⁴×C₅	Dic₆	1	8	yes	complemented
103	C₂⁴×C₅	C₂²×D₃	1	6	yes	—
104	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
105	C₂²×C₄×C₅	—	2	4	no	—
106	C₂²×C₄×C₅	—	2	48	no	—
107	C₂²×C₄×C₅	#67	1	8	yes	—
108	C₂²×C₄×C₅	C₂²×D₃	1	6	yes	—
109	C₂²×C₄×C₅	C₂²×D₃	1	16	yes	—
110	C₂²×C₄×C₅	C₂²×D₃	1	6	yes	—
111	C₂²×Dic₅	—	3	12	no	—
112	C₂²×Dic₅	—	3	12	no	—
113	C₂²×Dic₅	—	6	8	no	—
114	C₂²×Dic₅	—	6	8	no	—
115	C₂×Dic₁₀	—	3	16	no	—
116	C₂×Dic₁₀	—	6	48	no	—
117	C₂²⋊Dic₅	—	3	32	no	—
118	C₄⋊Dic₅	—	3	16	no	—
119	#683	—	3	32	no	—
120	C₂³×Q₈	—	15	1	no	—
121	C₂⁴×C₄	D₁₅	1	1	yes	characteristic
122	C₂×C₂²⋊Q₈	—	15	24	no	—
123	C₂²×C₄⋊C₄	—	15	2	no	—
124	C₂²×C₄⋊C₄	—	15	1	no	—
125	C₂²×C₂²⋊C₄	—	15	2	no	—
126	C₂²×C₃×C₅	—	2	48	no	—
127	C₂²×C₃×C₅	—	2	4	no	—
128	C₂²×C₃×C₅	—	2	4	no	—
129	C₂²×C₃×C₅	C₂²⋊Q₈	1	16	yes	complemented
130	C₂²×C₃×C₅	C₂²×D₄	1	6	yes	—
131	C₂²×C₃×C₅	C₂²×Q₈	1	6	yes	—
132	C₂²×C₃×C₅	C₂²×Q₈	1	8	yes	complemented
133	C₂²×C₃×C₅	C₂×#41	1	6	yes	—
134	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
135	C₃×C₄×C₅	—	2	4	no	—
136	C₃×C₄×C₅	C₂²×D₄	1	8	yes	—
137	Dic₁₅	—	2	8	no	—
138	Dic₁₅	—	2	8	no	—
139	C₂⁴×C₃	—	2	4	no	—
140	C₂⁴×C₃	—	2	4	no	—
141	C₂⁴×C₃	Dic₁₀	1	8	yes	complemented
142	C₂⁴×C₃	C₂²×D₅	1	6	yes	—
143	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
144	C₂²×C₃×C₄	—	2	4	no	—
145	C₂²×C₃×C₄	—	2	48	no	—
146	C₂²×C₃×C₄	C₂²⋊D₅	1	8	yes	—
147	C₂²×C₃×C₄	C₂²×D₅	1	6	yes	—
148	C₂²×C₃×C₄	C₂²×D₅	1	16	yes	—
149	C₂²×C₃×C₄	C₂²×D₅	1	6	yes	—
150	C₂²×Dic₃	—	5	12	no	—
151	C₂²×Dic₃	—	5	12	no	—
152	C₂²×Dic₃	—	10	8	no	—
153	C₂²×Dic₃	—	10	8	no	—
154	C₂×Dic₆	—	5	16	no	—
155	C₂×Dic₆	—	10	48	no	—
156	#217	—	5	32	no	—
157	C₄⋊Dic₃	—	5	16	no	—
158	Dic₃⋊C₄	—	5	32	no	—
159	C₂³×C₅	—	2	32	no	—
160	C₂³×C₅	—	2	12	no	—
161	C₂³×C₅	—	2	12	no	—
162	C₂³×C₅	C₂×Dic₆	1	4	yes	—
163	C₂³×C₅	C₂×Dic₆	1	24	yes	complemented
164	C₂³×C₅	#235	1	4	yes	—
165	C₂³×C₅	C₂×#67	1	4	yes	—
166	C₂³×C₅	C₂³×D₃	1	4	yes	—
167	C₂³×C₅	C₂³×D₃	1	3	yes	—
168	C₂×C₄×C₅	—	2	12	no	—
169	C₂×C₄×C₅	—	2	32	no	—
170	C₂×C₄×C₅	C₂×#67	1	24	yes	—
171	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
172	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
173	C₂×Dic₅	—	3	8	no	—
174	C₂×Dic₅	—	3	8	no	—
175	C₂×Dic₅	—	6	24	no	—
176	C₂×Dic₅	—	6	24	no	—
177	Dic₁₀	—	6	32	no	—
178	C₂⁵	C₂×D₁₅	1	1	yes	characteristic
179	C₂²×Q₈	—	15	12	no	—
180	C₂²×Q₈	—	30	8	no	—
181	C₂³×C₄	—	2	8	no	—
182	C₂³×C₄	—	15	2	no	—
183	C₂³×C₄	—	15	2	no	—
184	C₂³×C₄	C₂×D₁₅	1	12	yes	—
185	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
186	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
187	C₂²⋊Q₈	—	15	64	no	—
188	C₂×C₄⋊C₄	—	15	24	no	—
189	C₂×C₄⋊C₄	—	15	12	no	—
190	C₂×C₂²⋊C₄	—	15	24	no	—
191	C₂×C₃×C₅	—	2	8	no	—
192	C₂×C₃×C₅	C₂×C₂²⋊Q₈	1	12	yes	complemented
193	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
194	C₂×C₃×C₅	C₂³×Q₈	1	1	yes	characteristic
195	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
196	C₂³×C₃	—	2	12	no	—
197	C₂³×C₃	—	2	12	no	—
198	C₂³×C₃	—	2	32	no	—
199	C₂³×C₃	C₂×Dic₁₀	1	4	yes	—
200	C₂³×C₃	C₂×Dic₁₀	1	24	yes	complemented
201	C₂³×C₃	#709	1	4	yes	—
202	C₂³×C₃	C₂×C₂²⋊D₅	1	4	yes	—
203	C₂³×C₃	C₂³×D₅	1	4	yes	—
204	C₂³×C₃	C₂³×D₅	1	3	yes	—
205	C₂×C₃×C₄	—	2	12	no	—
206	C₂×C₃×C₄	—	2	32	no	—
207	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	24	yes	—
208	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
209	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
210	C₂×Dic₃	—	5	8	no	—
211	C₂×Dic₃	—	5	8	no	—
212	C₂×Dic₃	—	10	24	no	—
213	C₂×Dic₃	—	10	24	no	—
214	Dic₆	—	10	32	no	—
215	C₂²×C₅	—	2	4	no	—
216	C₂²×C₅	—	2	4	no	—
217	C₂²×C₅	—	2	48	no	—
218	C₂²×C₅	#924	1	16	yes	complemented
219	C₂²×C₅	C₂²×Dic₆	1	6	yes	—
220	C₂²×C₅	C₂²×Dic₆	1	8	yes	complemented
221	C₂²×C₅	C₂×#235	1	6	yes	—
222	C₂²×C₅	C₂²×#67	1	6	yes	—
223	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
224	C₄×C₅	—	2	4	no	—
225	C₄×C₅	C₂²×#67	1	8	yes	—
226	Dic₅	—	6	8	no	—
227	Dic₅	—	6	8	no	—
228	C₂⁴	—	2	4	no	—
229	C₂⁴	—	2	4	no	—
230	C₂⁴	Dic₃₀	1	8	yes	complemented
231	C₂⁴	C₂²×D₁₅	1	6	yes	—
232	C₂⁴	C₂²×D₁₅	1	1	yes	characteristic
233	C₂×Q₈	—	15	16	no	—
234	C₂×Q₈	—	30	48	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	48	no	—
237	C₂²×C₄	—	15	12	no	—
238	C₂²×C₄	—	15	12	no	—
239	C₂²×C₄	—	30	8	no	—
240	C₂²×C₄	—	30	8	no	—
241	C₂²×C₄	#1220	1	8	yes	—
242	C₂²×C₄	C₂²×D₁₅	1	6	yes	—
243	C₂²×C₄	C₂²×D₁₅	1	16	yes	—
244	C₂²×C₄	C₂²×D₁₅	1	6	yes	—
245	C₄⋊C₄	—	15	32	no	—
246	C₄⋊C₄	—	15	16	no	—
247	C₂²⋊C₄	—	15	32	no	—
248	C₃×C₅	C₂²×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
249	C₂²×C₃	—	2	48	no	—
250	C₂²×C₃	—	2	4	no	—
251	C₂²×C₃	—	2	4	no	—
252	C₂²×C₃	#4070	1	16	yes	complemented
253	C₂²×C₃	C₂²×Dic₁₀	1	6	yes	—
254	C₂²×C₃	C₂²×Dic₁₀	1	8	yes	complemented
255	C₂²×C₃	C₂×#709	1	6	yes	—
256	C₂²×C₃	C₂²×C₂²⋊D₅	1	6	yes	—
257	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
258	C₃×C₄	—	2	4	no	—
259	C₃×C₄	C₂²×C₂²⋊D₅	1	8	yes	—
260	Dic₃	—	10	8	no	—
261	Dic₃	—	10	8	no	—
262	C₂×C₅	—	2	8	no	—
263	C₂×C₅	C₂×#924	1	12	yes	complemented
264	C₂×C₅	C₂³×Dic₆	1	1	yes	characteristic
265	C₂×C₅	C₂²×#235	1	1	yes	characteristic
266	C₂×C₅	C₂³×#67	1	1	yes	characteristic
267	C₂³	—	2	32	no	—
268	C₂³	—	2	12	no	—
269	C₂³	—	2	12	no	—
270	C₂³	C₂×Dic₃₀	1	4	yes	—
271	C₂³	C₂×Dic₃₀	1	24	yes	complemented
272	C₂³	#6817	1	4	yes	—
273	C₂³	C₂×#1220	1	4	yes	—
274	C₂³	C₂³×D₁₅	1	4	yes	—
275	C₂³	C₂³×D₁₅	1	3	yes	—
276	Q₈	—	30	32	no	—
277	C₂×C₄	—	2	12	no	—
278	C₂×C₄	—	2	32	no	—
279	C₂×C₄	—	15	8	no	—
280	C₂×C₄	—	15	8	no	—
281	C₂×C₄	—	30	24	no	—
282	C₂×C₄	—	30	24	no	—
283	C₂×C₄	C₂×#1220	1	24	yes	—
284	C₂×C₄	C₂³×D₁₅	1	4	yes	—
285	C₂×C₄	C₂³×D₁₅	1	4	yes	—
286	C₂×C₃	—	2	8	no	—
287	C₂×C₃	C₂×#4070	1	12	yes	complemented
288	C₂×C₃	C₂³×Dic₁₀	1	1	yes	characteristic
289	C₂×C₃	C₂²×#709	1	1	yes	characteristic
290	C₂×C₃	C₂³×C₂²⋊D₅	1	1	yes	characteristic
291	C₅	C₂²×#924	1	1	yes	characteristic, complemented, Sylow
292	C₂²	—	2	4	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	48	no	—
295	C₂²	#91813	1	16	yes	direct product factor
296	C₂²	C₂²×Dic₃₀	1	6	yes	—
297	C₂²	C₂²×Dic₃₀	1	8	yes	complemented
298	C₂²	C₂×#6817	1	6	yes	—
299	C₂²	C₂²×#1220	1	6	yes	—
300	C₂²	C₂⁴×D₁₅	1	1	yes	characteristic
301	C₄	—	2	4	no	—
302	C₄	—	30	8	no	—
303	C₄	—	30	8	no	—
304	C₄	C₂²×#1220	1	8	yes	—
305	C₃	C₂²×#4070	1	1	yes	characteristic, complemented, Sylow
306	C₂	—	2	8	no	—
307	C₂	C₂×#91813	1	12	yes	direct product factor
308	C₂	C₂³×Dic₃₀	1	1	yes	characteristic
309	C₂	C₂²×#6817	1	1	yes	characteristic
310	C₂	C₂³×#1220	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₅
commutator, cosocle	C₂²×C₃×C₅	C₂⁵
Frattini	C₂²	C₂⁴×D₁₅
Fitting	C₂⁴×C₃×C₄×C₅	C₂
socle	C₂⁴×C₃×C₅	C₂³
prime radical	C₂⁵×C₃×C₅	C₂²
hypercenter	C₂⁴×C₄	D₁₅
hypocenter	C₃×C₅	C₂²×C₂²⋊Q₈
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×Dic₃₀	C₂	semidirect
4. C₂²×C₄⋊Dic₁₅	C₂	semidirect
5. C₂×#91813	C₂	direct (N × H)
6. C₂²×#6712	C₂	semidirect
16. C₂²×Dic₃₀	C₂²	semidirect
18. C₂×C₄⋊Dic₁₅	C₂²	semidirect
19. #91813	C₂²	direct (N × H)
20. C₂×#6712	C₂²	semidirect
30. C₂⁴×C₃×C₅	Q₈	semidirect
44. C₂×Dic₃₀	C₂³	semidirect
46. C₄⋊Dic₁₅	C₂³	semidirect
47. #6712	C₂³	semidirect
73. C₂³×C₃×C₅	C₂×Q₈	semidirect
102. C₂⁴×C₅	Dic₆	semidirect
129. C₂²×C₃×C₅	C₂²⋊Q₈	semidirect
132. C₂²×C₃×C₅	C₂²×Q₈	semidirect
141. C₂⁴×C₃	Dic₁₀	semidirect
163. C₂³×C₅	C₂×Dic₆	semidirect
192. C₂×C₃×C₅	C₂×C₂²⋊Q₈	semidirect
200. C₂³×C₃	C₂×Dic₁₀	semidirect
218. C₂²×C₅	#924	semidirect
220. C₂²×C₅	C₂²×Dic₆	semidirect
230. C₂⁴	Dic₃₀	semidirect
248. C₃×C₅	C₂²×C₂²⋊Q₈	semidirect
252. C₂²×C₃	#4070	semidirect
254. C₂²×C₃	C₂²×Dic₁₀	semidirect
263. C₂×C₅	C₂×#924	semidirect
271. C₂³	C₂×Dic₃₀	semidirect
287. C₂×C₃	C₂×#4070	semidirect
291. C₅	C₂²×#924	semidirect
295. C₂²	#91813	direct (N × H)
297. C₂²	C₂²×Dic₃₀	semidirect
305. C₃	C₂²×#4070	semidirect
307. C₂	C₂×#91813	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₄×C₅	2
C₂³×Dic₃₀	2
C₂²×#6719	2
C₂²×C₄⋊Dic₁₅	2
C₂×#91813	2
C₂²×#6712	2
C₂²×#4070	3
C₂²×#924	5

Subgroups — C22×#91813 #14479422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#91813 #14479422