Subgroups — C₂²×#91171 #14411618

Order 1920.

2144 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#1200	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃⋊#170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₃×#170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#91171	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#4003	—	3	1	no	maximal, Hall
6	C₂⁴×D₁₅	C₄	1	1	yes	characteristic
7	C₂³×C₅×Dic₃	C₄	1	1	yes	characteristic
8	C₂³×C₃×Dic₅	C₂²	1	1	yes	characteristic
9	C₂²×#1200	—	2	4	no	—
10	C₂²×#1200	—	2	4	no	—
11	C₂²×#1200	C₄	1	6	yes	—
12	C₂²×#1200	C₂²	1	6	yes	—
13	C₂²×C₃⋊#170	—	2	4	no	—
14	C₂²×C₃⋊#170	C₂²	1	6	yes	complemented
15	C₂²×C₃×#170	—	2	4	no	—
16	C₂²×C₃×#170	C₂²	1	6	yes	complemented
17	#91171	C₂²	1	16	yes	direct product factor
18	C₂²×(C₂×D₃)⋊C₈	—	5	1	no	maximal, Hall
19	C₂³×C₄×D₅	—	3	1	no	—
20	C₂³×#170	—	3	1	no	—
21	C₂³×#170	D₃	1	1	yes	characteristic, complemented
22	C₂×#4003	—	3	12	no	—
23	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂³×D₁₅	—	2	4	no	—
25	C₂³×D₁₅	—	2	4	no	—
26	C₂³×D₁₅	C₈	1	8	yes	complemented
27	C₂³×D₁₅	C₂×C₄	1	6	yes	—
28	C₂²×C₅×Dic₃	—	2	4	no	—
29	C₂²×C₅×Dic₃	C₂×C₄	1	6	yes	—
30	C₂²×C₃×Dic₅	C₂×C₄	1	3	yes	—
31	C₂²×C₃×Dic₅	D₄	1	4	yes	—
32	C₂²×C₃×Dic₅	D₄	1	4	yes	—
33	C₂²×C₃×Dic₅	C₂³	1	3	yes	—
34	C₂×#1200	—	2	24	no	—
35	C₂×#1200	—	2	12	no	—
36	C₂×#1200	—	2	12	no	—
37	C₂×#1200	C₂×C₄	1	12	yes	—
38	C₂×#1200	C₂³	1	4	yes	—
39	C₂×C₃⋊#170	—	2	12	no	—
40	C₂×C₃⋊#170	C₂³	1	4	yes	complemented
41	C₂×C₃×#170	—	2	12	no	—
42	C₂×C₃×#170	C₂³	1	4	yes	complemented
43	C₂³×C₄×D₃	—	5	1	no	—
44	C₂³×C₃×C₈	—	5	1	no	—
45	C₂³×C₃⋊C₈	—	5	1	no	—
46	C₂×(C₂×D₃)⋊C₈	—	5	12	no	—
47	C₂⁴×D₅	—	3	1	no	—
48	C₂³×C₄×C₅	—	3	1	no	—
49	C₂³×Dic₅	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₄×D₅	—	3	6	no	—
51	C₂²×C₄×D₅	—	3	6	no	—
52	C₂²×C₄×D₅	—	6	4	no	—
53	C₂²×C₄×D₅	—	6	4	no	—
54	C₂²×#170	—	2	4	no	—
55	C₂²×#170	—	3	6	no	—
56	C₂²×#170	—	6	4	no	—
57	C₂²×#170	C₂×D₃	1	6	yes	complemented
58	#4003	—	3	16	no	—
59	C₂²×C₂²⋊C₈	—	15	1	no	Sylow
60	C₂³×C₃×C₅	C₂²⋊C₄	1	4	yes	—
61	C₂³×C₃×C₅	C₂×C₈	1	4	yes	—
62	C₂³×C₃×C₅	M₁₆	1	4	yes	—
63	C₂³×C₃×C₅	C₂²×C₄	1	3	yes	—
64	C₂²×D₁₅	—	2	32	no	—
65	C₂²×D₁₅	—	2	12	no	—
66	C₂²×D₁₅	—	2	12	no	—
67	C₂²×D₁₅	C₂×C₈	1	24	yes	complemented
68	C₂²×D₁₅	C₂²×C₄	1	4	yes	—
69	C₂×C₅×Dic₃	—	2	12	no	—
70	C₂×C₅×Dic₃	C₂²×C₄	1	4	yes	—
71	C₂×C₃×Dic₅	C₂²⋊C₄	1	12	yes	—
72	C₂×C₃×Dic₅	C₂²×C₄	1	3	yes	—
73	C₂×C₃×Dic₅	C₂×D₄	1	6	yes	—
74	C₂×C₃×Dic₅	C₂×D₄	1	6	yes	—
75	C₂×C₃×Dic₅	C₂⁴	1	1	yes	characteristic
76	#1200	—	2	24	no	—
77	#1200	—	2	4	no	—
78	#1200	—	2	4	no	—
79	C₃⋊#170	—	2	4	no	—
80	C₃×#170	—	2	4	no	—
81	C₂⁴×D₃	—	5	1	no	—
82	C₂³×C₃×C₄	—	5	1	no	—
83	C₂³×Dic₃	AGL(1,𝔽₅)	1	1	yes	characteristic
84	C₂²×C₄×D₃	—	5	6	no	—
85	C₂²×C₄×D₃	—	5	6	no	—
86	C₂²×C₄×D₃	—	10	4	no	—
87	C₂²×C₄×D₃	—	10	4	no	—
88	C₂²×C₃×C₈	—	5	6	no	—
89	C₂²×C₃×C₈	—	10	4	no	—
90	C₂²×C₃⋊C₈	—	5	6	no	—
91	C₂²×C₃⋊C₈	—	10	4	no	—
92	(C₂×D₃)⋊C₈	—	5	16	no	—
93	C₂⁴×C₅	C₄×D₃	1	1	yes	characteristic
94	C₂³×D₅	—	3	8	no	—
95	C₂³×D₅	—	3	6	no	—
96	C₂³×D₅	—	6	4	no	—
97	C₂³×D₅	—	6	4	no	—
98	C₂²×C₄×C₅	—	3	6	no	—
99	C₂²×C₄×C₅	—	6	4	no	—
100	C₂²×Dic₅	C₄×D₃	1	3	yes	—
101	C₂²×Dic₅	D₁₂	1	4	yes	—
102	C₂²×Dic₅	#67	1	4	yes	—
103	C₂²×Dic₅	C₂²×D₃	1	3	yes	—
104	C₂×C₄×D₅	—	3	12	no	—
105	C₂×C₄×D₅	—	3	4	no	—
106	C₂×C₄×D₅	—	6	24	no	—
107	C₂×C₄×D₅	—	6	12	no	—
108	C₂×C₄×D₅	—	6	12	no	—
109	C₂×#170	—	2	12	no	—
110	C₂×#170	—	3	4	no	—
111	C₂×#170	—	6	12	no	—
112	C₂×#170	C₂²×D₃	1	4	yes	complemented
113	C₂⁴×C₄	—	15	1	no	—
114	C₂³×C₈	—	5	1	no	—
115	C₂³×C₈	—	15	1	no	—
116	C₂×C₂²⋊C₈	—	15	12	no	—
117	C₂²×C₃×C₅	C₂²⋊C₈	1	16	yes	complemented
118	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	6	yes	—
119	C₂²×C₃×C₅	C₂²×C₈	1	6	yes	—
120	C₂²×C₃×C₅	C₂×M₁₆	1	6	yes	—
121	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
122	C₂×D₁₅	—	2	4	no	—
123	C₂×D₁₅	—	2	4	no	—
124	C₂×D₁₅	—	2	48	no	—
125	C₂×D₁₅	C₂²×C₈	1	8	yes	complemented
126	C₃×Dic₅	C₂×C₂²⋊C₄	1	6	yes	—
127	C₃×Dic₅	C₂²×D₄	1	1	yes	characteristic
128	C₃×Dic₅	C₂²×D₄	1	1	yes	characteristic
129	C₅×Dic₃	—	2	4	no	—
130	C₂⁴×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₂³×D₃	—	5	8	no	—
132	C₂³×D₃	—	5	6	no	—
133	C₂³×D₃	—	10	4	no	—
134	C₂³×D₃	—	10	4	no	—
135	C₂²×C₃×C₄	—	5	4	no	—
136	C₂²×C₃×C₄	—	5	4	no	—
137	C₂²×C₃×C₄	—	5	3	no	—
138	C₂²×C₃×C₄	—	5	3	no	—
139	C₂²×Dic₃	—	2	4	no	—
140	C₂²×Dic₃	C₂×AGL(1,𝔽₅)	1	6	yes	—
141	C₂×C₄×D₃	—	5	4	no	—
142	C₂×C₄×D₃	—	5	12	no	—
143	C₂×C₄×D₃	—	10	24	no	—
144	C₂×C₄×D₃	—	10	12	no	—
145	C₂×C₄×D₃	—	10	12	no	—
146	C₂×C₃×C₈	—	5	4	no	—
147	C₂×C₃×C₈	—	10	12	no	—
148	C₂×C₃⋊C₈	—	5	4	no	—
149	C₂×C₃⋊C₈	—	10	12	no	—
150	C₂³×C₅	D₃×C₈	1	4	yes	—
151	C₂³×C₅	ΓL(1,𝔽₂₅)	1	4	yes	—
152	C₂³×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
153	C₂³×C₅	C₂×C₄×D₃	1	3	yes	—
154	C₂²×D₅	—	3	24	no	—
155	C₂²×D₅	—	3	4	no	—
156	C₂²×D₅	—	6	32	no	—
157	C₂²×D₅	—	6	12	no	—
158	C₂²×D₅	—	6	12	no	—
159	C₂×C₄×C₅	—	3	4	no	—
160	C₂×C₄×C₅	—	6	12	no	—
161	C₂×Dic₅	(C₂×D₃)⋊C₄	1	12	yes	—
162	C₂×Dic₅	C₂×C₄×D₃	1	3	yes	—
163	C₂×Dic₅	C₂×D₁₂	1	6	yes	—
164	C₂×Dic₅	C₂×#67	1	6	yes	—
165	C₂×Dic₅	C₂³×D₃	1	1	yes	characteristic
166	C₄×D₅	—	6	24	no	—
167	C₄×D₅	—	6	4	no	—
168	C₄×D₅	—	6	4	no	—
169	#170	—	2	4	no	—
170	#170	—	6	4	no	—
171	C₂⁵	—	15	1	no	—
172	C₂³×C₄	—	3	1	no	—
173	C₂³×C₄	—	5	1	no	—
174	C₂³×C₄	—	15	6	no	—
175	C₂³×C₄	—	15	6	no	—
176	C₂³×C₄	—	30	4	no	—
177	C₂³×C₄	—	30	4	no	—
178	C₂²×C₈	—	5	6	no	—
179	C₂²×C₈	—	10	4	no	—
180	C₂²×C₈	—	15	6	no	—
181	C₂²×C₈	—	30	4	no	—
182	C₂²⋊C₈	—	15	16	no	—
183	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	12	yes	complemented
184	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
186	C₂×C₃×C₅	C₂²×M₁₆	1	1	yes	characteristic
187	D₁₅	—	2	8	no	—
188	C₂³×C₃	D₅⋊C₈	1	4	yes	—
189	C₂³×C₃	#700	1	4	yes	—
190	C₂³×C₃	C₂²⋊AGL(1,𝔽₅)	1	4	yes	—
191	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	3	yes	—
192	C₂²×D₃	—	5	24	no	—
193	C₂²×D₃	—	5	4	no	—
194	C₂²×D₃	—	10	12	no	—
195	C₂²×D₃	—	10	12	no	—
196	C₂²×D₃	—	10	32	no	—
197	C₂×C₃×C₄	—	5	12	no	—
198	C₂×C₃×C₄	—	5	6	no	—
199	C₂×C₃×C₄	—	5	6	no	—
200	C₂×C₃×C₄	—	5	3	no	—
201	C₂×C₃×C₄	—	5	1	no	—
202	C₂×Dic₃	—	2	12	no	—
203	C₂×Dic₃	C₂²×AGL(1,𝔽₅)	1	4	yes	—
204	C₄×D₃	—	10	24	no	—
205	C₄×D₃	—	10	4	no	—
206	C₄×D₃	—	10	4	no	—
207	C₃×C₈	—	10	4	no	—
208	C₃⋊C₈	—	10	4	no	—
209	C₂²×C₅	(C₂×D₃)⋊C₈	1	16	yes	complemented
210	C₂²×C₅	C₂×D₃×C₈	1	6	yes	—
211	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	6	yes	—
212	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	6	yes	—
213	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
214	C₂×D₅	—	3	8	no	—
215	C₂×D₅	—	6	4	no	—
216	C₂×D₅	—	6	4	no	—
217	C₂×D₅	—	6	48	no	—
218	C₄×C₅	—	6	4	no	—
219	Dic₅	C₂×(C₂×D₃)⋊C₄	1	6	yes	—
220	Dic₅	C₂²×D₁₂	1	1	yes	characteristic
221	Dic₅	C₂²×#67	1	1	yes	characteristic
222	C₂⁴	—	15	8	no	—
223	C₂⁴	—	15	6	no	—
224	C₂⁴	—	30	4	no	—
225	C₂⁴	—	30	4	no	—
226	C₂⁴	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
227	C₂²×C₄	—	3	6	no	—
228	C₂²×C₄	—	5	4	no	—
229	C₂²×C₄	—	5	4	no	—
230	C₂²×C₄	—	5	3	no	—
231	C₂²×C₄	—	5	3	no	—
232	C₂²×C₄	—	6	4	no	—
233	C₂²×C₄	—	15	4	no	—
234	C₂²×C₄	—	15	12	no	—
235	C₂²×C₄	—	30	12	no	—
236	C₂²×C₄	—	30	12	no	—
237	C₂²×C₄	—	30	24	no	—
238	C₂×C₈	—	5	4	no	—
239	C₂×C₈	—	10	12	no	—
240	C₂×C₈	—	15	4	no	—
241	C₂×C₈	—	30	12	no	—
242	C₃×C₅	C₂²×C₂²⋊C₈	1	1	yes	characteristic, complemented, Hall
243	C₂²×C₃	#4003	1	16	yes	complemented
244	C₂²×C₃	C₂×D₅⋊C₈	1	6	yes	—
245	C₂²×C₃	C₂×#700	1	6	yes	—
246	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	6	yes	—
247	C₂²×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
248	C₂×D₃	—	5	8	no	—
249	C₂×D₃	—	10	4	no	—
250	C₂×D₃	—	10	48	no	—
251	C₂×D₃	—	10	4	no	—
252	C₃×C₄	—	5	6	no	—
253	C₃×C₄	—	5	1	no	—
254	C₃×C₄	—	5	1	no	—
255	Dic₃	—	2	4	no	—
256	C₂×C₅	C₂×(C₂×D₃)⋊C₈	1	12	yes	complemented
257	C₂×C₅	C₂²×D₃×C₈	1	1	yes	characteristic
258	C₂×C₅	C₂²×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
259	C₂×C₅	C₂²×(C₂×D₃)⋊C₄	1	1	yes	characteristic
260	D₅	—	6	8	no	—
261	C₂³	—	15	24	no	—
262	C₂³	—	15	4	no	—
263	C₂³	—	30	12	no	—
264	C₂³	—	30	12	no	—
265	C₂³	—	30	32	no	—
266	C₂³	(C₂×D₃)⋊AGL(1,𝔽₅)	1	4	yes	—
267	C₂³	#6738	1	4	yes	—
268	C₂³	#6740	1	4	yes	—
269	C₂³	C₂×D₃×AGL(1,𝔽₅)	1	3	yes	—
270	C₂×C₄	—	3	4	no	—
271	C₂×C₄	—	5	6	no	—
272	C₂×C₄	—	5	6	no	—
273	C₂×C₄	—	5	12	no	—
274	C₂×C₄	—	5	1	no	—
275	C₂×C₄	—	5	3	no	—
276	C₂×C₄	—	6	12	no	—
277	C₂×C₄	—	30	4	no	—
278	C₂×C₄	—	30	4	no	—
279	C₂×C₄	—	30	24	no	—
280	C₈	—	10	4	no	—
281	C₈	—	30	4	no	—
282	C₂×C₃	C₂×#4003	1	12	yes	complemented
283	C₂×C₃	C₂²×D₅⋊C₈	1	1	yes	characteristic
284	C₂×C₃	C₂²×#700	1	1	yes	characteristic
285	C₂×C₃	C₂²×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
286	D₃	—	10	8	no	—
287	C₅	C₂²×(C₂×D₃)⋊C₈	1	1	yes	characteristic, complemented, Sylow
288	C₂²	—	15	8	no	—
289	C₂²	—	30	4	no	—
290	C₂²	—	30	48	no	—
291	C₂²	—	30	4	no	—
292	C₂²	#91171	1	16	yes	direct product factor
293	C₂²	C₂×(C₂×D₃)⋊AGL(1,𝔽₅)	1	6	yes	—
294	C₂²	C₂×#6738	1	6	yes	—
295	C₂²	C₂×#6740	1	6	yes	—
296	C₂²	C₂²×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
297	C₄	—	5	1	no	—
298	C₄	—	5	1	no	—
299	C₄	—	5	6	no	—
300	C₄	—	6	4	no	—
301	C₃	C₂²×#4003	1	1	yes	characteristic, complemented, Sylow
302	C₂	—	30	8	no	—
303	C₂	C₂×#91171	1	12	yes	direct product factor
304	C₂	C₂²×(C₂×D₃)⋊AGL(1,𝔽₅)	1	1	yes	characteristic
305	C₂	C₂²×#6738	1	1	yes	characteristic
306	C₂	C₂²×#6740	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴	D₃×AGL(1,𝔽₅)
commutator	C₂×C₃×C₅	C₂³×C₈
Frattini	C₂²	C₂²×D₃×AGL(1,𝔽₅)
Fitting, socle	C₂⁴×C₃×C₅	C₂×C₄
prime radical	C₂⁴×D₁₅	C₄
cosocle	C₂×C₃×Dic₅	C₂⁴
hypocenter	C₃×C₅	C₂²×C₂²⋊C₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃⋊#170	C₂	semidirect
3. C₂³×C₃×#170	C₂	semidirect
4. C₂×#91171	C₂	direct (N × H)
14. C₂²×C₃⋊#170	C₂²	semidirect
16. C₂²×C₃×#170	C₂²	semidirect
17. #91171	C₂²	direct (N × H)
21. C₂³×#170	D₃	semidirect
26. C₂³×D₁₅	C₈	semidirect
40. C₂×C₃⋊#170	C₂³	semidirect
42. C₂×C₃×#170	C₂³	semidirect
57. C₂²×#170	C₂×D₃	semidirect
67. C₂²×D₁₅	C₂×C₈	semidirect
112. C₂×#170	C₂²×D₃	semidirect
117. C₂²×C₃×C₅	C₂²⋊C₈	semidirect
125. C₂×D₁₅	C₂²×C₈	semidirect
183. C₂×C₃×C₅	C₂×C₂²⋊C₈	semidirect
209. C₂²×C₅	(C₂×D₃)⋊C₈	semidirect
242. C₃×C₅	C₂²×C₂²⋊C₈	semidirect
243. C₂²×C₃	#4003	semidirect
256. C₂×C₅	C₂×(C₂×D₃)⋊C₈	semidirect
282. C₂×C₃	C₂×#4003	semidirect
287. C₅	C₂²×(C₂×D₃)⋊C₈	semidirect
292. C₂²	#91171	direct (N × H)
301. C₃	C₂²×#4003	semidirect
303. C₂	C₂×#91171	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#1200	2
C₂³×C₃⋊#170	2
C₂³×C₃×#170	2
C₂×#91171	2
C₂²×#4003	3
C₂²×(C₂×D₃)⋊C₈	5

Subgroups — C22×#91171 #14411618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#91171 #14411618