Subgroups — C₂×#21426222 #28743253

Order 15104.

638 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837818	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837678	C₂	1	2	yes	maximal
4	#21426222	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16837217	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₉×#368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14273760	—	2	2	no	—
8	C₂²×Dic₂₃₆	C₂²	1	2	yes	—
9	C₂×#14273733	C₂²	1	1	yes	characteristic
10	C₂×Dic₄₇₂	—	2	1	no	—
11	C₂×Dic₄₇₂	—	2	1	no	—
12	C₂×Dic₄₇₂	—	2	2	no	—
13	C₂×Dic₄₇₂	C₂²	1	4	yes	complemented
14	#16837818	—	2	2	no	—
15	#16837818	C₂²	1	4	yes	—
16	C₂×#14273667	C₂²	1	1	yes	characteristic
17	C₂×#14273666	—	2	1	no	—
18	C₂×C₁₆×D₅₉	—	2	1	no	—
19	#16837678	—	2	4	no	—
20	#16837678	C₂²	1	8	yes	—
21	C₂×#14273645	—	2	2	no	—
22	#16837217	C₂²	1	8	yes	complemented
23	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×#108×C₅₉	C₂²	1	2	yes	—
25	C₅₉×#368	C₂²	1	4	yes	complemented
26	C₂×#11745685	D₄	1	1	yes	characteristic
27	C₂²×Dic₁₁₈	—	2	2	no	—
28	#14273760	—	2	8	no	—
29	#14273760	—	4	4	no	—
30	C₂×Dic₂₃₆	—	2	2	no	—
31	C₂×Dic₂₃₆	—	2	2	no	—
32	C₂×Dic₂₃₆	—	2	2	no	—
33	C₂×Dic₂₃₆	—	2	2	no	—
34	C₂×Dic₂₃₆	—	2	2	no	—
35	C₂×Dic₂₃₆	—	4	2	no	—
36	C₂×Dic₂₃₆	C₂³	1	4	yes	—
37	C₂×#11745660	—	4	2	no	—
38	#14273733	—	2	1	no	—
39	#14273733	—	2	1	no	—
40	#14273733	D₄	1	2	yes	—
41	#14273733	C₂³	1	2	yes	—
42	C₂×#11745664	D₄	1	1	yes	characteristic
43	C₂×#11745659	—	2	1	no	—
44	C₂×C₈×D₅₉	—	2	1	no	—
45	Dic₄₇₂	—	2	4	no	—
46	Dic₄₇₂	—	2	4	no	—
47	#14273667	—	2	1	no	—
48	#14273667	C₂³	1	2	yes	—
49	#14273666	—	2	4	no	—
50	C₂×C₅₉⋊C₁₆	—	2	1	no	—
51	C₁₆×D₅₉	—	2	4	no	—
52	#14273645	—	2	8	no	—
53	C₂×M₁₆×C₅₉	—	2	2	no	—
54	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
55	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
56	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
57	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
58	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
59	#108×C₅₉	—	2	2	no	—
60	#108×C₅₉	C₂³	1	4	yes	—
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₉	—	2	1	no	—
63	C₂²×Dic₅₉	—	4	2	no	—
64	#11745685	—	2	1	no	—
65	#11745685	—	2	1	no	—
66	#11745685	D₈	1	2	yes	—
67	#11745685	C₂×D₄	1	2	yes	—
68	C₂×D₂₃₆	D₈	1	1	yes	characteristic
69	C₂×C₄×D₅₉	—	2	1	no	—
70	C₂×Dic₁₁₈	—	2	4	no	—
71	C₂×Dic₁₁₈	—	4	2	no	—
72	C₂×Dic₁₁₈	—	4	2	no	—
73	C₂×Dic₁₁₈	—	4	2	no	—
74	C₂×Dic₁₁₈	—	4	2	no	—
75	C₂×Dic₁₁₈	—	4	2	no	—
76	C₂×Dic₁₁₈	D₈	1	1	yes	characteristic
77	M₁₆×C₅₉	—	2	4	no	—
78	M₁₆×C₅₉	—	4	2	no	—
79	C₂×C₈×C₅₉	—	4	2	no	—
80	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
84	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
85	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
86	#11745664	—	2	1	no	—
87	#11745664	C₂×D₄	1	2	yes	—
88	C₂×C₅₉⋊C₈	—	2	1	no	—
89	Dic₂₃₆	—	2	4	no	—
90	Dic₂₃₆	—	2	4	no	—
91	Dic₂₃₆	—	4	8	no	—
92	Dic₂₃₆	—	4	2	no	—
93	Dic₂₃₆	—	4	2	no	—
94	#11745660	—	4	8	no	—
95	#11745659	—	2	2	no	—
96	#11745659	—	2	2	no	—
97	C₈×D₅₉	—	2	2	no	—
98	C₈×D₅₉	—	2	2	no	—
99	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
100	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
101	C₅₉⋊C₁₆	—	2	2	no	—
102	C₂³×C₅₉	#135	1	1	yes	characteristic
103	C₂²×D₅₉	—	2	1	no	—
104	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
105	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
110	C₂²⋊D₅₉	—	2	2	no	—
111	C₂²⋊D₅₉	—	2	2	no	—
112	C₂×Dic₅₉	—	2	1	no	—
113	C₂×Dic₅₉	—	4	4	no	—
114	C₂×Dic₅₉	—	8	2	no	—
115	C₂×Dic₅₉	—	8	2	no	—
116	D₂₃₆	—	2	1	no	—
117	D₂₃₆	C₂×D₈	1	2	yes	—
118	C₄×D₅₉	—	2	2	no	—
119	C₄×D₅₉	—	2	2	no	—
120	Dic₁₁₈	—	2	1	no	—
121	Dic₁₁₈	—	4	4	no	—
122	Dic₁₁₈	—	4	4	no	—
123	Dic₁₁₈	—	8	2	no	—
124	Dic₁₁₈	—	8	2	no	—
125	Dic₁₁₈	C₂×D₈	1	2	yes	—
126	C₈×C₅₉	—	4	4	no	—
127	C₈×C₅₉	#121	1	1	yes	characteristic
128	C₈×C₅₉	#121	1	1	yes	characteristic
129	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
131	C₅₉⋊C₈	—	2	2	no	—
132	C₂×#2222	—	59	1	no	maximal, Sylow
133	C₂²×C₅₉	—	2	2	no	—
134	C₂²×C₅₉	#466	1	1	yes	characteristic
135	C₂²×C₅₉	#466	1	1	yes	characteristic
136	C₂²×C₅₉	C₂×#135	1	1	yes	characteristic
137	C₂×D₅₉	—	2	2	no	—
138	C₂×D₅₉	—	4	2	no	—
139	C₄×C₅₉	#466	1	1	yes	characteristic
140	C₄×C₅₉	#466	1	1	yes	characteristic
141	C₄×C₅₉	C₂×#121	1	1	yes	characteristic
142	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
143	Dic₅₉	—	2	2	no	—
144	Dic₅₉	—	8	4	no	—
145	C₂²×Q₃₂	—	59	1	no	—
146	C₂×#504	—	59	1	no	—
147	C₂×#473	—	59	2	no	—
148	#2222	—	59	8	no	—
149	C₂×#368	D₅₉	1	1	yes	characteristic, complemented
150	C₂×#362	—	59	2	no	—
151	C₂×C₅₉	—	2	1	no	—
152	C₂×C₅₉	—	2	1	no	—
153	C₂×C₅₉	#2222	1	2	yes	complemented
154	C₂×C₅₉	C₂×#466	1	1	yes	characteristic
155	D₅₉	—	4	2	no	—
156	C₂×#137	—	118	2	no	—
157	C₂²×Q₁₆	—	59	2	no	—
158	C₂×#131	—	59	1	no	—
159	C₂×Q₃₂	—	59	4	no	—
160	C₂×Q₃₂	—	118	1	no	—
161	C₂×Q₃₂	—	118	1	no	—
162	C₂×Q₃₂	—	118	2	no	—
163	#504	—	59	4	no	—
164	#504	—	118	2	no	—
165	C₂×M₃₂	—	59	1	no	—
166	C₂×M₃₂	—	118	1	no	—
167	C₂²×C₁₆	—	118	1	no	—
168	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
169	#473	—	59	8	no	—
170	#473	—	118	4	no	—
171	C₂×#108	C₂×D₅₉	1	2	yes	—
172	C₂×#101	—	118	2	no	—
173	#368	C₂×D₅₉	1	4	yes	complemented
174	#362	—	59	8	no	—
175	C₅₉	C₂×#2222	1	1	yes	characteristic, complemented, Sylow
176	C₂×#41	—	59	1	no	—
177	C₂²×Q₈	—	118	2	no	—
178	#137	—	118	8	no	—
179	#137	—	236	4	no	—
180	C₂×Q₁₆	—	59	4	no	—
181	C₂×Q₁₆	—	118	2	no	—
182	C₂×Q₁₆	—	118	2	no	—
183	C₂×Q₁₆	—	118	2	no	—
184	C₂×Q₁₆	—	118	2	no	—
185	C₂×Q₁₆	—	118	2	no	—
186	C₂×Q₁₆	—	236	2	no	—
187	C₂×SD₁₆	—	236	2	no	—
188	#131	—	59	2	no	—
189	#131	—	59	2	no	—
190	#131	—	118	1	no	—
191	#131	—	118	1	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	59	1	no	—
194	C₂×M₁₆	—	118	1	no	—
195	C₂²×C₈	—	118	1	no	—
196	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
197	Q₃₂	—	118	4	no	—
198	Q₃₂	—	118	4	no	—
199	M₃₂	—	59	2	no	—
200	M₃₂	—	118	1	no	—
201	M₃₂	—	118	4	no	—
202	C₂×C₁₆	—	118	1	no	—
203	C₂×C₁₆	—	118	4	no	—
204	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
205	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
206	C₂×C₁₆	D₂₃₆	1	2	yes	—
207	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
208	#108	—	2	2	no	—
209	#108	C₂²×D₅₉	1	4	yes	—
210	#101	—	118	8	no	—
211	#41	—	59	2	no	—
212	#41	—	59	2	no	—
213	#41	—	118	1	no	—
214	#41	—	118	1	no	—
215	C₂×Q₈	—	59	1	no	—
216	C₂×Q₈	—	118	4	no	—
217	C₂×Q₈	—	236	2	no	—
218	C₂×Q₈	—	236	2	no	—
219	C₂×Q₈	—	236	2	no	—
220	C₂×Q₈	—	236	2	no	—
221	C₂×Q₈	—	236	2	no	—
222	C₂×D₄	—	59	1	no	—
223	C₂×D₄	—	118	1	no	—
224	C₂²×C₄	—	118	1	no	—
225	C₂²×C₄	—	236	2	no	—
226	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
227	Q₁₆	—	118	4	no	—
228	Q₁₆	—	118	4	no	—
229	Q₁₆	—	236	8	no	—
230	Q₁₆	—	236	2	no	—
231	Q₁₆	—	236	2	no	—
232	SD₁₆	—	236	8	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	59	2	no	—
236	M₁₆	—	118	1	no	—
237	M₁₆	—	118	2	no	—
238	M₁₆	—	118	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	118	1	no	—
241	C₂×C₈	—	118	2	no	—
242	C₂×C₈	—	118	2	no	—
243	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
244	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
245	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
246	C₂×C₈	#11745689	1	1	yes	characteristic
247	C₂×C₈	#11745689	1	1	yes	characteristic
248	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
249	C₁₆	—	118	2	no	—
250	C₁₆	C₂×D₂₃₆	1	2	yes	—
251	C₁₆	C₂×D₂₃₆	1	2	yes	—
252	C₂³	—	118	1	no	—
253	C₂³	#14273750	1	1	yes	characteristic
254	Q₈	—	59	2	no	—
255	Q₈	—	118	1	no	—
256	Q₈	—	472	2	no	—
257	Q₈	—	472	2	no	—
258	Q₈	—	236	4	no	—
259	Q₈	—	236	4	no	—
260	D₄	—	59	2	no	—
261	D₄	—	118	2	no	—
262	D₄	—	118	2	no	—
263	D₄	—	118	1	no	—
264	C₂×C₄	—	118	1	no	—
265	C₂×C₄	—	118	2	no	—
266	C₂×C₄	—	118	2	no	—
267	C₂×C₄	—	472	2	no	—
268	C₂×C₄	—	472	2	no	—
269	C₂×C₄	—	236	4	no	—
270	C₂×C₄	#14273702	1	1	yes	characteristic
271	C₂×C₄	#14273702	1	1	yes	characteristic
272	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
273	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
274	C₂×C₄	#14273750	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
276	C₈	—	4	4	no	—
277	C₈	—	118	2	no	—
278	C₈	#14273702	1	1	yes	characteristic
279	C₈	#14273702	1	1	yes	characteristic
280	C₈	C₂²×D₂₃₆	1	1	yes	characteristic
281	C₈	C₂×#11745689	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	118	2	no	—
284	C₂²	—	236	2	no	—
285	C₂²	#16837639	1	1	yes	characteristic
286	C₂²	#16837639	1	1	yes	characteristic
287	C₂²	C₂×#14273750	1	1	yes	characteristic
288	C₄	—	118	2	no	—
289	C₄	—	472	4	no	—
290	C₄	#16837639	1	1	yes	characteristic
291	C₄	#16837639	1	1	yes	characteristic
292	C₄	C₂×#14273702	1	1	yes	characteristic
293	C₄	C₂×D₈×D₅₉	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	236	2	no	—
297	C₂	#21426222	1	2	yes	direct product factor
298	C₂	C₂×#16837639	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837639
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#368	C₂
socle	C₂²×C₅₉	#466
prime radical	C₂×#11745685	D₄
hypercenter	C₂×#368	D₅₉
hypocenter	C₅₉	C₂×#2222
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₄₇₂	C₂	semidirect
4. #21426222	C₂	direct (N × H)
5. C₂×#16837217	C₂	semidirect
6. C₂×C₅₉×#368	C₂	semidirect
13. C₂×Dic₄₇₂	C₂²	semidirect
22. #16837217	C₂²	semidirect
25. C₅₉×#368	C₂²	semidirect
149. C₂×#368	D₅₉	semidirect
153. C₂×C₅₉	#2222	semidirect
173. #368	C₂×D₅₉	semidirect
175. C₅₉	C₂×#2222	semidirect
297. C₂	#21426222	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₇₂	2
C₂×#16837818	2
C₂×#16837678	2
#21426222	2
C₂×#16837217	2
C₂×C₅₉×#368	2
C₂×#2222	59

Subgroups — C2×#21426222 #28743253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426222 #28743253