Subgroups — C₂×#11756493 #14483927

Order 1920.

2480 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#91836	C₂	1	4	yes	maximal, complemented
3	#11756493	C₂	1	16	yes	maximal, direct product factor
4	C₂×#91784	C₂	1	4	yes	maximal, complemented
5	C₂×#91781	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×D₆₀	C₂	1	2	yes	maximal, complemented
7	C₂×#66068	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
9	C₂²×D₆₀	C₂²	1	2	yes	—
10	C₂²×D₆₀	C₂²	1	4	yes	—
11	C₂²×C₄×D₁₅	C₂²	1	4	yes	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	8	yes	complemented
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	#91836	C₂²	1	32	yes	complemented
16	C₂×#6714	C₂²	1	4	yes	—
17	C₂×#6714	C₂²	1	8	yes	—
18	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
19	C₂×C₄×Dic₁₅	C₂²	1	4	yes	—
20	#91784	C₂²	1	32	yes	complemented
21	#91781	C₂²	1	32	yes	complemented
22	C₄×D₆₀	C₂²	1	16	yes	complemented
23	C₂×#5734	—	5	1	no	maximal, Hall
24	C₂×C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
25	C₂×#4093	—	3	4	no	—
26	#66068	—	3	16	no	—
27	C₂×#4041	—	3	4	no	—
28	C₂×#4038	—	3	4	no	—
29	C₂×C₄×D₂₀	—	3	2	no	—
30	C₂³×D₁₅	C₂³	1	4	yes	—
31	C₂×C₃×C₅×Q₈	—	4	4	no	—
32	C₂×C₃×C₅×Q₈	C₂³	1	8	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
36	C₂²×Dic₁₅	C₂³	1	4	yes	—
37	C₂×D₆₀	—	2	8	no	—
38	C₂×D₆₀	—	4	8	no	—
39	C₂×D₆₀	C₂³	1	8	yes	—
40	C₂×D₆₀	C₂³	1	16	yes	—
41	C₂×C₄×D₁₅	—	2	16	no	—
42	C₂×C₄×D₁₅	C₂³	1	16	yes	—
43	C₃×C₅×C₄⋊C₄	C₂³	1	16	yes	—
44	C₃×C₄²×C₅	C₂³	1	4	yes	—
45	#6714	C₂³	1	16	yes	—
46	#6714	C₂³	1	32	yes	—
47	C₄⋊Dic₁₅	C₂³	1	8	yes	—
48	C₄×Dic₁₅	C₂³	1	16	yes	—
49	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
50	C₂×#947	—	5	4	no	—
51	#5734	—	5	16	no	—
52	C₂×#895	—	5	4	no	—
53	C₂×#892	—	5	4	no	—
54	C₂×C₄×D₁₂	—	5	2	no	—
55	C₂²×C₅×Q₈	C₂×D₃	1	2	yes	—
56	C₂²×D₂₀	—	3	2	no	—
57	C₂²×D₂₀	—	3	4	no	—
58	C₂²×C₄×D₅	—	3	4	no	—
59	C₅×C₄⋊Q₈	C₂×D₃	1	8	yes	complemented
60	C₂×C₅×C₄⋊C₄	C₂×D₃	1	4	yes	—
61	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
62	#4093	—	3	32	no	—
63	C₂×#685	—	3	4	no	—
64	C₂×#685	—	3	8	no	—
65	C₂×C₄⋊Dic₅	—	3	2	no	—
66	C₂×C₄×Dic₅	—	3	4	no	—
67	#4041	—	3	32	no	—
68	#4038	—	3	32	no	—
69	C₄×D₂₀	—	3	16	no	—
70	C₂×#555	—	15	1	no	Sylow
71	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
72	C₂²×D₁₅	—	2	8	no	—
73	C₂²×D₁₅	—	2	8	no	—
74	C₂²×D₁₅	—	2	8	no	—
75	C₂²×D₁₅	C₂⁴	1	8	yes	—
76	C₃×C₅×Q₈	—	4	8	no	—
77	C₂×C₃×C₄×C₅	—	2	8	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	#41	1	8	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	8	yes	—
81	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
83	C₂×Dic₁₅	—	2	8	no	—
84	C₂×Dic₁₅	C₂⁴	1	8	yes	—
85	D₆₀	—	2	16	no	—
86	D₆₀	—	4	16	no	—
87	C₄×D₁₅	—	2	32	no	—
88	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
89	C₂²×D₁₂	—	5	2	no	—
90	C₂²×D₁₂	—	5	4	no	—
91	C₂²×C₄×D₃	—	5	4	no	—
92	C₃×C₄⋊Q₈	C₂×D₅	1	8	yes	complemented
93	C₂×C₃×C₄⋊C₄	C₂×D₅	1	4	yes	—
94	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
95	#947	—	5	32	no	—
96	C₂×(C₂×D₃)⋊C₄	—	5	4	no	—
97	C₂×(C₂×D₃)⋊C₄	—	5	8	no	—
98	C₂×C₄⋊Dic₃	—	5	2	no	—
99	C₂×C₄×Dic₃	—	5	4	no	—
100	#895	—	5	32	no	—
101	#892	—	5	32	no	—
102	C₄×D₁₂	—	5	16	no	—
103	C₂³×D₅	—	3	4	no	—
104	C₂×C₅×Q₈	—	4	4	no	—
105	C₂×C₅×Q₈	C₂²×D₃	1	8	yes	—
106	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
107	C₂²×C₄×C₅	C₂²×D₃	1	4	yes	—
108	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
109	C₂²×Dic₅	—	3	4	no	—
110	C₂×D₂₀	—	3	8	no	—
111	C₂×D₂₀	—	3	16	no	—
112	C₂×D₂₀	—	6	8	no	—
113	C₂×D₂₀	—	12	8	no	—
114	C₂×C₄×D₅	—	3	16	no	—
115	C₂×C₄×D₅	—	6	16	no	—
116	C₅×C₄⋊C₄	C₂²×D₃	1	16	yes	—
117	C₄²×C₅	C₂²×D₃	1	4	yes	—
118	#685	—	3	16	no	—
119	#685	—	3	32	no	—
120	C₄⋊Dic₅	—	3	8	no	—
121	C₄×Dic₅	—	3	16	no	—
122	#555	—	15	16	no	—
123	C₂×C₄⋊Q₈	D₁₅	1	1	yes	characteristic, complemented
124	C₂×#124	—	15	4	no	—
125	C₂×#121	—	15	4	no	—
126	C₂×C₄×D₄	—	15	2	no	—
127	C₂×#117	—	15	4	no	—
128	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
129	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
130	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
131	C₂×D₁₅	—	2	8	no	—
132	C₂×D₁₅	—	2	8	no	—
133	C₂×D₁₅	—	2	8	no	—
134	C₂×D₁₅	—	4	16	no	—
135	C₃×C₄×C₅	—	2	8	no	—
136	C₃×C₄×C₅	—	2	2	no	—
137	C₃×C₄×C₅	C₂×#41	1	8	yes	—
138	Dic₁₅	—	2	8	no	—
139	C₂³×D₃	—	5	4	no	—
140	C₂×C₃×Q₈	—	4	4	no	—
141	C₂×C₃×Q₈	C₂²×D₅	1	8	yes	—
142	C₂²×C₃×C₄	C₂²×D₅	1	4	yes	—
143	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
145	C₂²×Dic₃	—	5	4	no	—
146	C₂×D₁₂	—	5	8	no	—
147	C₂×D₁₂	—	5	16	no	—
148	C₂×D₁₂	—	10	8	no	—
149	C₂×D₁₂	—	20	8	no	—
150	C₂×C₄×D₃	—	5	16	no	—
151	C₂×C₄×D₃	—	10	16	no	—
152	C₃×C₄⋊C₄	C₂²×D₅	1	16	yes	—
153	C₃×C₄²	C₂²×D₅	1	4	yes	—
154	(C₂×D₃)⋊C₄	—	5	16	no	—
155	(C₂×D₃)⋊C₄	—	5	32	no	—
156	C₄⋊Dic₃	—	5	8	no	—
157	C₄×Dic₃	—	5	16	no	—
158	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
159	C₂²×D₅	—	3	8	no	—
160	C₂²×D₅	—	6	8	no	—
161	C₂²×D₅	—	6	8	no	—
162	C₂²×D₅	—	6	8	no	—
163	C₅×Q₈	—	4	8	no	—
164	C₂×C₄×C₅	—	2	8	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	#239	1	8	yes	—
167	C₂×C₄×C₅	C₂³×D₃	1	8	yes	—
168	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
169	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
170	C₂×Dic₅	—	3	8	no	—
171	C₂×Dic₅	—	6	8	no	—
172	D₂₀	—	6	16	no	—
173	D₂₀	—	12	16	no	—
174	C₄×D₅	—	6	32	no	—
175	C₂²×Q₈	C₂×D₁₅	1	2	yes	—
176	C₂²×D₄	—	15	2	no	—
177	C₂²×D₄	—	15	4	no	—
178	C₂³×C₄	—	15	4	no	—
179	C₄⋊Q₈	C₂×D₁₅	1	8	yes	complemented
180	#124	—	15	32	no	—
181	#121	—	15	32	no	—
182	C₄×D₄	—	15	16	no	—
183	#117	—	15	32	no	—
184	C₂×C₄⋊C₄	—	15	2	no	—
185	C₂×C₄⋊C₄	C₂×D₁₅	1	4	yes	—
186	C₂×C₂²⋊C₄	—	15	8	no	—
187	C₂×C₂²⋊C₄	—	15	4	no	—
188	C₂×C₄²	—	15	4	no	—
189	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
190	C₂×C₃×C₅	#555	1	4	yes	complemented
191	C₂×C₃×C₅	C₂²×#41	1	2	yes	—
192	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
193	D₁₅	—	4	8	no	—
194	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
195	C₂²×D₃	—	5	8	no	—
196	C₂²×D₃	—	10	8	no	—
197	C₂²×D₃	—	10	8	no	—
198	C₂²×D₃	—	10	8	no	—
199	C₃×Q₈	—	4	8	no	—
200	C₂×C₃×C₄	—	2	8	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	#713	1	8	yes	—
203	C₂×C₃×C₄	C₂³×D₅	1	8	yes	—
204	C₂×C₃×C₄	C₂³×D₅	1	2	yes	—
205	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
206	C₂×Dic₃	—	5	8	no	—
207	C₂×Dic₃	—	10	8	no	—
208	D₁₂	—	10	16	no	—
209	D₁₂	—	20	16	no	—
210	C₄×D₃	—	10	32	no	—
211	C₂²×C₅	#1004	1	2	yes	—
212	C₂²×C₅	C₂×#239	1	4	yes	—
213	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
214	C₂×D₅	—	6	8	no	—
215	C₂×D₅	—	6	8	no	—
216	C₂×D₅	—	6	8	no	—
217	C₂×D₅	—	12	16	no	—
218	C₄×C₅	—	2	8	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	C₂×#239	1	8	yes	—
221	Dic₅	—	6	8	no	—
222	C₂⁴	—	15	4	no	—
223	C₂×Q₈	—	4	4	no	—
224	C₂×Q₈	C₂²×D₁₅	1	8	yes	—
225	C₂×D₄	—	15	8	no	—
226	C₂×D₄	—	15	16	no	—
227	C₂×D₄	—	30	8	no	—
228	C₂×D₄	—	60	8	no	—
229	C₂²×C₄	—	15	4	no	—
230	C₂²×C₄	—	15	16	no	—
231	C₂²×C₄	—	30	16	no	—
232	C₂²×C₄	C₂²×D₁₅	1	4	yes	—
233	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
234	C₂²×C₄	C₂²×D₁₅	1	2	yes	—
235	C₄⋊C₄	—	15	8	no	—
236	C₄⋊C₄	C₂²×D₁₅	1	16	yes	—
237	C₂²⋊C₄	—	15	32	no	—
238	C₂²⋊C₄	—	15	16	no	—
239	C₄²	—	15	16	no	—
240	C₄²	C₂²×D₁₅	1	4	yes	—
241	C₃×C₅	C₂×#555	1	1	yes	characteristic, complemented, Hall
242	C₂²×C₃	#4144	1	2	yes	—
243	C₂²×C₃	C₂×#713	1	4	yes	—
244	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
245	C₂×D₃	—	10	8	no	—
246	C₂×D₃	—	10	8	no	—
247	C₂×D₃	—	10	8	no	—
248	C₂×D₃	—	20	16	no	—
249	C₃×C₄	—	2	8	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	C₂×#713	1	8	yes	—
252	Dic₃	—	10	8	no	—
253	C₂×C₅	#5734	1	4	yes	complemented
254	C₂×C₅	C₂×#1004	1	1	yes	characteristic
255	C₂×C₅	C₂²×#239	1	2	yes	—
256	D₅	—	12	8	no	—
257	C₂³	—	15	8	no	—
258	C₂³	—	30	8	no	—
259	C₂³	—	30	8	no	—
260	C₂³	—	30	8	no	—
261	C₂³	C₂³×D₁₅	1	1	yes	characteristic
262	Q₈	—	4	8	no	—
263	D₄	—	30	16	no	—
264	D₄	—	60	16	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	15	8	no	—
268	C₂×C₄	—	30	8	no	—
269	C₂×C₄	—	30	32	no	—
270	C₂×C₄	#6821	1	8	yes	—
271	C₂×C₄	C₂³×D₁₅	1	8	yes	—
272	C₂×C₄	C₂³×D₁₅	1	2	yes	—
273	C₂×C₄	C₂³×D₁₅	1	4	yes	—
274	C₂×C₃	#66068	1	4	yes	complemented
275	C₂×C₃	C₂×#4144	1	1	yes	characteristic
276	C₂×C₃	C₂²×#713	1	2	yes	—
277	D₃	—	20	8	no	—
278	C₅	C₂×#5734	1	1	yes	characteristic, complemented, Sylow
279	C₂²	—	30	8	no	—
280	C₂²	—	30	8	no	—
281	C₂²	—	30	8	no	—
282	C₂²	—	60	16	no	—
283	C₂²	#92095	1	2	yes	—
284	C₂²	C₂×#6821	1	4	yes	—
285	C₂²	C₂⁴×D₁₅	1	1	yes	characteristic
286	C₄	—	2	8	no	—
287	C₄	—	2	2	no	—
288	C₄	—	30	8	no	—
289	C₄	C₂×#6821	1	8	yes	—
290	C₃	C₂×#66068	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	60	8	no	—
292	C₂	#11756493	1	4	yes	direct product factor
293	C₂	C₂×#92095	1	1	yes	characteristic
294	C₂	C₂²×#6821	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₅
commutator, cosocle	C₂²×C₃×C₅	C₂⁵
Frattini	C₂²	C₂⁴×D₁₅
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₁₅
hypocenter	C₃×C₅	C₂×#555
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₄⋊Q₈	C₂	semidirect
2. C₂×#91836	C₂	semidirect
3. #11756493	C₂	direct (N × H)
4. C₂×#91784	C₂	semidirect
5. C₂×#91781	C₂	semidirect
6. C₂×C₄×D₆₀	C₂	semidirect
12. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
15. #91836	C₂²	semidirect
20. #91784	C₂²	semidirect
21. #91781	C₂²	semidirect
22. C₄×D₆₀	C₂²	semidirect
24. C₂×C₅×C₄⋊Q₈	D₃	semidirect
49. C₂×C₃×C₄⋊Q₈	D₅	semidirect
59. C₅×C₄⋊Q₈	C₂×D₃	semidirect
92. C₃×C₄⋊Q₈	C₂×D₅	semidirect
123. C₂×C₄⋊Q₈	D₁₅	semidirect
179. C₄⋊Q₈	C₂×D₁₅	semidirect
190. C₂×C₃×C₅	#555	semidirect
241. C₃×C₅	C₂×#555	semidirect
253. C₂×C₅	#5734	semidirect
274. C₂×C₃	#66068	semidirect
278. C₅	C₂×#5734	semidirect
290. C₃	C₂×#66068	semidirect
292. C₂	#11756493	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×C₅×C₄⋊Q₈	2
C₂×#91836	2
#11756493	2
C₂×#91784	2
C₂×#91781	2
C₂×C₄×D₆₀	2
C₂×#66068	3
C₂×#5734	5

Subgroups — C2×#11756493 #14483927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11756493 #14483927