Subgroups — C₂²×#91786 #14479425

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	C₂³×C₄×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#6714	C₂	1	2	yes	maximal
5	C₂²×C₄⋊Dic₁₅	C₂	1	2	yes	maximal, complemented
6	C₂×#91786	C₂	1	24	yes	maximal, direct product factor
7	C₂²×#4043	—	3	1	no	maximal, Hall
8	C₂⁴×D₁₅	C₂²	1	1	yes	characteristic
9	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
10	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂³×Dic₁₅	C₂²	1	1	yes	characteristic
12	C₂³×Dic₁₅	C₂²	1	2	yes	—
13	C₂²×C₄×D₁₅	—	2	8	no	—
14	C₂²×C₄×D₁₅	C₂²	1	12	yes	—
15	C₂²×Dic₃₀	—	2	8	no	—
16	C₂²×Dic₃₀	C₂²	1	12	yes	complemented
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	12	yes	complemented
18	C₂×#6714	C₂²	1	24	yes	—
19	C₂×C₄⋊Dic₁₅	C₂²	1	24	yes	complemented
20	#91786	C₂²	1	64	yes	direct product factor
21	C₂²×#897	—	5	1	no	maximal, Hall
22	C₂³×C₄×D₅	—	3	1	no	—
23	C₂³×Dic₁₀	—	3	1	no	—
24	C₂²×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
25	C₂²×#685	—	3	2	no	—
26	C₂²×C₄⋊Dic₅	—	3	2	no	—
27	C₂×#4043	—	3	24	no	—
28	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
29	C₂³×D₁₅	—	2	4	no	—
30	C₂³×D₁₅	—	2	4	no	—
31	C₂³×D₁₅	Q₈	1	8	yes	complemented
32	C₂³×D₁₅	C₂³	1	6	yes	—
33	C₂²×C₃×C₄×C₅	—	2	8	no	—
34	C₂²×C₃×C₄×C₅	D₄	1	8	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	12	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
37	C₂²×Dic₁₅	—	2	8	no	—
38	C₂²×Dic₁₅	—	2	4	no	—
39	C₂²×Dic₁₅	C₂³	1	6	yes	—
40	C₂²×Dic₁₅	C₂³	1	12	yes	—
41	C₂×C₄×D₁₅	—	2	48	no	—
42	C₂×C₄×D₁₅	C₂³	1	16	yes	—
43	C₂×Dic₃₀	—	2	48	no	—
44	C₂×Dic₃₀	C₂³	1	16	yes	complemented
45	C₃×C₅×C₄⋊C₄	C₂³	1	16	yes	complemented
46	#6714	C₂³	1	32	yes	—
47	C₄⋊Dic₁₅	C₂³	1	32	yes	complemented
48	C₂³×C₄×D₃	—	5	1	no	—
49	C₂³×Dic₆	—	5	1	no	—
50	C₂²×C₃×C₄⋊C₄	D₅	1	1	yes	characteristic, complemented
51	C₂²×(C₂×D₃)⋊C₄	—	5	2	no	—
52	C₂²×C₄⋊Dic₃	—	5	2	no	—
53	C₂×#897	—	5	24	no	—
54	C₂⁴×D₅	—	3	1	no	—
55	C₂³×C₄×C₅	C₂×D₃	1	2	yes	—
56	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
57	C₂³×Dic₅	—	3	1	no	—
58	C₂³×Dic₅	—	3	2	no	—
59	C₂²×C₄×D₅	—	3	12	no	—
60	C₂²×C₄×D₅	—	6	8	no	—
61	C₂²×Dic₁₀	—	3	12	no	—
62	C₂²×Dic₁₀	—	6	8	no	—
63	C₂×C₅×C₄⋊C₄	C₂×D₃	1	12	yes	complemented
64	C₂×#685	—	3	24	no	—
65	C₂×C₄⋊Dic₅	—	3	24	no	—
66	#4043	—	3	64	no	—
67	C₂²×C₂²⋊Q₈	—	15	1	no	Sylow
68	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
69	C₂³×C₃×C₅	C₂×Q₈	1	4	yes	—
70	C₂³×C₃×C₅	#41	1	4	yes	—
71	C₂³×C₃×C₅	C₂⁴	1	3	yes	—
72	C₂²×D₁₅	—	2	32	no	—
73	C₂²×D₁₅	—	2	12	no	—
74	C₂²×D₁₅	—	2	12	no	—
75	C₂²×D₁₅	C₂×Q₈	1	24	yes	complemented
76	C₂²×D₁₅	C₂⁴	1	4	yes	—
77	C₂×C₃×C₄×C₅	—	2	24	no	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	24	yes	—
79	C₂×C₃×C₄×C₅	C₂⁴	1	8	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
81	C₂×Dic₁₅	—	2	24	no	—
82	C₂×Dic₁₅	—	2	12	no	—
83	C₂×Dic₁₅	C₂⁴	1	4	yes	—
84	C₂×Dic₁₅	C₂⁴	1	8	yes	—
85	C₄×D₁₅	—	2	32	no	—
86	Dic₃₀	—	2	32	no	—
87	C₂⁴×D₃	—	5	1	no	—
88	C₂³×C₃×C₄	C₂×D₅	1	2	yes	—
89	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
90	C₂³×Dic₃	—	5	2	no	—
91	C₂³×Dic₃	—	5	1	no	—
92	C₂²×C₄×D₃	—	5	12	no	—
93	C₂²×C₄×D₃	—	10	8	no	—
94	C₂²×Dic₆	—	5	12	no	—
95	C₂²×Dic₆	—	10	8	no	—
96	C₂×C₃×C₄⋊C₄	C₂×D₅	1	12	yes	complemented
97	C₂×(C₂×D₃)⋊C₄	—	5	24	no	—
98	C₂×C₄⋊Dic₃	—	5	24	no	—
99	#897	—	5	64	no	—
100	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
101	C₂³×D₅	—	3	8	no	—
102	C₂³×D₅	—	3	6	no	—
103	C₂³×D₅	—	6	4	no	—
104	C₂³×D₅	—	6	4	no	—
105	C₂²×C₄×C₅	—	2	8	no	—
106	C₂²×C₄×C₅	D₁₂	1	8	yes	—
107	C₂²×C₄×C₅	C₂²×D₃	1	12	yes	—
108	C₂²×C₄×C₅	C₂²×D₃	1	6	yes	—
109	C₂²×Dic₅	—	3	12	no	—
110	C₂²×Dic₅	—	3	6	no	—
111	C₂²×Dic₅	—	6	8	no	—
112	C₂²×Dic₅	—	6	4	no	—
113	C₂×C₄×D₅	—	3	16	no	—
114	C₂×C₄×D₅	—	6	48	no	—
115	C₂×Dic₁₀	—	3	16	no	—
116	C₂×Dic₁₀	—	6	48	no	—
117	C₅×C₄⋊C₄	C₂²×D₃	1	16	yes	complemented
118	#685	—	3	32	no	—
119	C₄⋊Dic₅	—	3	32	no	—
120	C₂³×Q₈	—	15	1	no	—
121	C₂⁴×C₄	—	15	1	no	—
122	C₂×C₂²⋊Q₈	—	15	24	no	—
123	C₂²×C₄⋊C₄	—	15	2	no	—
124	C₂²×C₄⋊C₄	D₁₅	1	1	yes	characteristic, complemented
125	C₂²×C₂²⋊C₄	—	15	2	no	—
126	C₂²×C₃×C₅	C₂²⋊Q₈	1	16	yes	complemented
127	C₂²×C₃×C₅	C₂²×D₄	1	6	yes	—
128	C₂²×C₃×C₅	C₂²×Q₈	1	6	yes	—
129	C₂²×C₃×C₅	C₂×#41	1	6	yes	—
130	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
131	C₂×D₁₅	—	2	48	no	—
132	C₂×D₁₅	—	2	4	no	—
133	C₂×D₁₅	—	2	4	no	—
134	C₂×D₁₅	C₂²×Q₈	1	8	yes	complemented
135	C₃×C₄×C₅	—	2	8	no	—
136	C₃×C₄×C₅	C₂²×D₄	1	8	yes	—
137	Dic₁₅	—	2	8	no	—
138	Dic₁₅	—	2	4	no	—
139	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
140	C₂³×D₃	—	5	8	no	—
141	C₂³×D₃	—	5	6	no	—
142	C₂³×D₃	—	10	4	no	—
143	C₂³×D₃	—	10	4	no	—
144	C₂²×C₃×C₄	—	2	8	no	—
145	C₂²×C₃×C₄	D₂₀	1	8	yes	—
146	C₂²×C₃×C₄	C₂²×D₅	1	12	yes	—
147	C₂²×C₃×C₄	C₂²×D₅	1	6	yes	—
148	C₂²×Dic₃	—	5	6	no	—
149	C₂²×Dic₃	—	5	12	no	—
150	C₂²×Dic₃	—	10	8	no	—
151	C₂²×Dic₃	—	10	4	no	—
152	C₂×C₄×D₃	—	5	16	no	—
153	C₂×C₄×D₃	—	10	48	no	—
154	C₂×Dic₆	—	5	16	no	—
155	C₂×Dic₆	—	10	48	no	—
156	C₃×C₄⋊C₄	C₂²×D₅	1	16	yes	complemented
157	(C₂×D₃)⋊C₄	—	5	32	no	—
158	C₄⋊Dic₃	—	5	32	no	—
159	C₂³×C₅	C₂×D₁₂	1	4	yes	—
160	C₂³×C₅	#237	1	4	yes	—
161	C₂³×C₅	D₃×Q₈	1	4	yes	—
162	C₂³×C₅	C₂³×D₃	1	3	yes	—
163	C₂²×D₅	—	3	24	no	—
164	C₂²×D₅	—	3	4	no	—
165	C₂²×D₅	—	6	32	no	—
166	C₂²×D₅	—	6	12	no	—
167	C₂²×D₅	—	6	12	no	—
168	C₂×C₄×C₅	—	2	24	no	—
169	C₂×C₄×C₅	C₂×D₁₂	1	24	yes	—
170	C₂×C₄×C₅	C₂³×D₃	1	8	yes	—
171	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
172	C₂×Dic₅	—	3	4	no	—
173	C₂×Dic₅	—	3	8	no	—
174	C₂×Dic₅	—	6	24	no	—
175	C₂×Dic₅	—	6	12	no	—
176	C₄×D₅	—	6	32	no	—
177	Dic₁₀	—	6	32	no	—
178	C₂⁵	—	15	1	no	—
179	C₂²×Q₈	—	15	12	no	—
180	C₂²×Q₈	—	30	8	no	—
181	C₂³×C₄	—	15	2	no	—
182	C₂³×C₄	—	15	1	no	—
183	C₂³×C₄	—	15	12	no	—
184	C₂³×C₄	—	30	8	no	—
185	C₂³×C₄	C₂×D₁₅	1	2	yes	—
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	C₂×D₁₅	1	12	yes	complemented
190	C₂×C₂²⋊C₄	—	15	24	no	—
191	C₂×C₃×C₅	C₂×C₂²⋊Q₈	1	12	yes	complemented
192	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂³×Q₈	1	1	yes	characteristic
194	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
195	D₁₅	—	2	8	no	—
196	C₂³×C₃	C₂×D₂₀	1	4	yes	—
197	C₂³×C₃	#711	1	4	yes	—
198	C₂³×C₃	Q₈×D₅	1	4	yes	—
199	C₂³×C₃	C₂³×D₅	1	3	yes	—
200	C₂²×D₃	—	5	24	no	—
201	C₂²×D₃	—	5	4	no	—
202	C₂²×D₃	—	10	12	no	—
203	C₂²×D₃	—	10	12	no	—
204	C₂²×D₃	—	10	32	no	—
205	C₂×C₃×C₄	—	2	24	no	—
206	C₂×C₃×C₄	C₂×D₂₀	1	24	yes	—
207	C₂×C₃×C₄	C₂³×D₅	1	8	yes	—
208	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
209	C₂×Dic₃	—	5	4	no	—
210	C₂×Dic₃	—	5	8	no	—
211	C₂×Dic₃	—	10	24	no	—
212	C₂×Dic₃	—	10	12	no	—
213	C₄×D₃	—	10	32	no	—
214	Dic₆	—	10	32	no	—
215	C₂²×C₅	#897	1	16	yes	complemented
216	C₂²×C₅	C₂²×D₁₂	1	6	yes	—
217	C₂²×C₅	C₂×#237	1	6	yes	—
218	C₂²×C₅	C₂×D₃×Q₈	1	6	yes	—
219	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
220	C₂×D₅	—	3	8	no	—
221	C₂×D₅	—	6	4	no	—
222	C₂×D₅	—	6	4	no	—
223	C₂×D₅	—	6	48	no	—
224	C₄×C₅	—	2	8	no	—
225	C₄×C₅	C₂²×D₁₂	1	8	yes	—
226	Dic₅	—	6	8	no	—
227	Dic₅	—	6	4	no	—
228	C₂⁴	—	15	8	no	—
229	C₂⁴	—	15	6	no	—
230	C₂⁴	—	30	4	no	—
231	C₂⁴	—	30	4	no	—
232	C₂⁴	C₂²×D₁₅	1	1	yes	characteristic
233	C₂×Q₈	—	15	16	no	—
234	C₂×Q₈	—	30	48	no	—
235	C₂²×C₄	—	2	8	no	—
236	C₂²×C₄	—	15	12	no	—
237	C₂²×C₄	—	15	6	no	—
238	C₂²×C₄	—	15	16	no	—
239	C₂²×C₄	—	30	4	no	—
240	C₂²×C₄	—	30	8	no	—
241	C₂²×C₄	—	30	48	no	—
242	C₂²×C₄	D₆₀	1	8	yes	—
243	C₂²×C₄	C₂²×D₁₅	1	12	yes	—
244	C₂²×C₄	C₂²×D₁₅	1	6	yes	—
245	C₄⋊C₄	—	15	32	no	—
246	C₄⋊C₄	C₂²×D₁₅	1	16	yes	complemented
247	C₂²⋊C₄	—	15	32	no	—
248	C₃×C₅	C₂²×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
249	C₂²×C₃	#4043	1	16	yes	complemented
250	C₂²×C₃	C₂²×D₂₀	1	6	yes	—
251	C₂²×C₃	C₂×#711	1	6	yes	—
252	C₂²×C₃	C₂×Q₈×D₅	1	6	yes	—
253	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
254	C₂×D₃	—	5	8	no	—
255	C₂×D₃	—	10	48	no	—
256	C₂×D₃	—	10	4	no	—
257	C₂×D₃	—	10	4	no	—
258	C₃×C₄	—	2	8	no	—
259	C₃×C₄	C₂²×D₂₀	1	8	yes	—
260	Dic₃	—	10	8	no	—
261	Dic₃	—	10	4	no	—
262	C₂×C₅	C₂×#897	1	12	yes	complemented
263	C₂×C₅	C₂³×D₁₂	1	1	yes	characteristic
264	C₂×C₅	C₂²×#237	1	1	yes	characteristic
265	C₂×C₅	C₂²×D₃×Q₈	1	1	yes	characteristic
266	D₅	—	6	8	no	—
267	C₂³	—	15	24	no	—
268	C₂³	—	15	4	no	—
269	C₂³	—	30	32	no	—
270	C₂³	—	30	12	no	—
271	C₂³	—	30	12	no	—
272	C₂³	C₂×D₆₀	1	4	yes	—
273	C₂³	#6819	1	4	yes	—
274	C₂³	Q₈×D₁₅	1	4	yes	—
275	C₂³	C₂³×D₁₅	1	3	yes	—
276	Q₈	—	30	32	no	—
277	C₂×C₄	—	2	24	no	—
278	C₂×C₄	—	15	8	no	—
279	C₂×C₄	—	15	4	no	—
280	C₂×C₄	—	30	12	no	—
281	C₂×C₄	—	30	24	no	—
282	C₂×C₄	—	30	32	no	—
283	C₂×C₄	C₂×D₆₀	1	24	yes	—
284	C₂×C₄	C₂³×D₁₅	1	8	yes	—
285	C₂×C₄	C₂³×D₁₅	1	4	yes	—
286	C₂×C₃	C₂×#4043	1	12	yes	complemented
287	C₂×C₃	C₂³×D₂₀	1	1	yes	characteristic
288	C₂×C₃	C₂²×#711	1	1	yes	characteristic
289	C₂×C₃	C₂²×Q₈×D₅	1	1	yes	characteristic
290	D₃	—	10	8	no	—
291	C₅	C₂²×#897	1	1	yes	characteristic, complemented, Sylow
292	C₂²	—	15	8	no	—
293	C₂²	—	30	4	no	—
294	C₂²	—	30	4	no	—
295	C₂²	—	30	48	no	—
296	C₂²	#91786	1	16	yes	direct product factor
297	C₂²	C₂²×D₆₀	1	6	yes	—
298	C₂²	C₂×#6819	1	6	yes	—
299	C₂²	C₂×Q₈×D₁₅	1	6	yes	—
300	C₂²	C₂⁴×D₁₅	1	1	yes	characteristic
301	C₄	—	2	8	no	—
302	C₄	—	30	4	no	—
303	C₄	—	30	8	no	—
304	C₄	C₂²×D₆₀	1	8	yes	—
305	C₃	C₂²×#4043	1	1	yes	characteristic, complemented, Sylow
306	C₂	—	30	8	no	—
307	C₂	C₂×#91786	1	12	yes	direct product factor
308	C₂	C₂³×D₆₀	1	1	yes	characteristic
309	C₂	C₂²×#6819	1	1	yes	characteristic
310	C₂	C₂²×Q₈×D₁₅	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂³×Dic₃₀	C₂	semidirect
3. C₂²×C₃×C₅×C₄⋊C₄	C₂	semidirect
5. C₂²×C₄⋊Dic₁₅	C₂	semidirect
6. C₂×#91786	C₂	direct (N × H)
16. C₂²×Dic₃₀	C₂²	semidirect
17. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
19. C₂×C₄⋊Dic₁₅	C₂²	semidirect
20. #91786	C₂²	direct (N × H)
24. C₂²×C₅×C₄⋊C₄	D₃	semidirect
31. C₂³×D₁₅	Q₈	semidirect
44. C₂×Dic₃₀	C₂³	semidirect
45. C₃×C₅×C₄⋊C₄	C₂³	semidirect
47. C₄⋊Dic₁₅	C₂³	semidirect
50. C₂²×C₃×C₄⋊C₄	D₅	semidirect
63. C₂×C₅×C₄⋊C₄	C₂×D₃	semidirect
75. C₂²×D₁₅	C₂×Q₈	semidirect
96. C₂×C₃×C₄⋊C₄	C₂×D₅	semidirect
117. C₅×C₄⋊C₄	C₂²×D₃	semidirect
124. C₂²×C₄⋊C₄	D₁₅	semidirect
126. C₂²×C₃×C₅	C₂²⋊Q₈	semidirect
134. C₂×D₁₅	C₂²×Q₈	semidirect
156. C₃×C₄⋊C₄	C₂²×D₅	semidirect
189. C₂×C₄⋊C₄	C₂×D₁₅	semidirect
191. C₂×C₃×C₅	C₂×C₂²⋊Q₈	semidirect
215. C₂²×C₅	#897	semidirect
246. C₄⋊C₄	C₂²×D₁₅	semidirect
248. C₃×C₅	C₂²×C₂²⋊Q₈	semidirect
249. C₂²×C₃	#4043	semidirect
262. C₂×C₅	C₂×#897	semidirect
286. C₂×C₃	C₂×#4043	semidirect
291. C₅	C₂²×#897	semidirect
296. C₂²	#91786	direct (N × H)
305. C₃	C₂²×#4043	semidirect
307. C₂	C₂×#91786	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×C₄×D₁₅	2
C₂³×Dic₃₀	2
C₂²×C₃×C₅×C₄⋊C₄	2
C₂²×#6714	2
C₂²×C₄⋊Dic₁₅	2
C₂×#91786	2
C₂²×#4043	3
C₂²×#897	5

Subgroups — C22×#91786 #14479425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#91786 #14479425