Subgroups — C₄×#16837217 #28503950

Order 15104.

434 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837217	C₂	1	2	yes	maximal
3	C₄×#14273667	C₂	1	1	yes	maximal, characteristic
4	#21422206	C₂	1	2	yes	maximal
5	C₄×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₃₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₁₈	—	2	1	no	—
8	C₂×#14273667	C₂²	1	2	yes	—
9	C₄×Dic₂₃₆	—	2	1	no	—
10	C₄×Dic₂₃₆	—	2	1	no	—
11	C₄×Dic₂₃₆	C₄	1	2	yes	—
12	C₄×Dic₂₃₆	C₄	1	2	yes	—
13	C₂×#14273661	C₂²	1	1	yes	characteristic
14	C₂×#14273652	—	2	1	no	—
15	#16837217	C₄	1	8	yes	direct product factor
16	#16837178	—	2	1	no	—
17	C₄×C₅₉⋊C₁₆	—	2	1	no	—
18	C₂×#108×C₅₉	C₂²	1	2	yes	—
19	C₄×M₁₆×C₅₉	—	2	1	no	—
20	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×#341	—	2	1	no	—
22	C₂²×Dic₁₁₈	—	2	1	no	—
23	C₂×Dic₂₃₆	—	2	1	no	—
24	C₂×Dic₂₃₆	—	2	2	no	—
25	C₂×Dic₂₃₆	C₂×C₄	1	4	yes	—
26	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
27	C₄×Dic₁₁₈	—	2	2	no	—
28	C₄×Dic₁₁₈	—	2	2	no	—
29	C₄×Dic₁₁₈	—	4	1	no	—
30	C₄×Dic₁₁₈	—	4	1	no	—
31	C₂×C₄⋊Dic₅₉	—	2	1	no	—
32	C₂×Dic₅₉⋊C₄	—	4	1	no	—
33	C₂×C₄×Dic₅₉	—	4	1	no	—
34	#14273667	—	2	2	no	—
35	#14273667	C₂×C₄	1	4	yes	—
36	C₂×C₅₉⋊C₁₆	—	2	2	no	—
37	#14273661	—	2	1	no	—
38	#14273661	C₂×C₄	1	2	yes	—
39	#14273652	—	2	2	no	—
40	#14273652	—	2	2	no	—
41	C₂×M₁₆×C₅₉	—	2	2	no	—
42	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
45	#108×C₅₉	—	2	2	no	—
46	#108×C₅₉	C₂×C₄	1	4	yes	—
47	#97×C₅₉	—	4	1	no	—
48	C₄×C₈×C₅₉	—	4	1	no	—
49	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₅₉	—	4	1	no	—
57	C₂²×Dic₅₉	—	4	1	no	—
58	C₂×Dic₁₁₈	—	2	2	no	—
59	C₂×Dic₁₁₈	—	2	4	no	—
60	C₂×Dic₁₁₈	—	4	1	no	—
61	C₂×Dic₁₁₈	—	4	2	no	—
62	M₁₆×C₅₉	—	2	4	no	—
63	M₁₆×C₅₉	—	4	2	no	—
64	C₂×C₈×C₅₉	—	2	2	no	—
65	C₂×C₈×C₅₉	—	4	2	no	—
66	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₉	#41	1	2	yes	—
72	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₅₉	D₈	1	1	yes	characteristic
75	C₄²×C₅₉	D₈	1	1	yes	characteristic
76	C₄⋊Dic₅₉	—	2	2	no	—
77	C₄⋊Dic₅₉	—	4	1	no	—
78	Dic₅₉⋊C₄	—	4	2	no	—
79	Dic₅₉⋊C₄	—	4	2	no	—
80	C₄×Dic₅₉	—	4	2	no	—
81	C₄×Dic₅₉	—	4	2	no	—
82	Dic₂₃₆	—	2	2	no	—
83	Dic₂₃₆	—	2	4	no	—
84	Dic₂₃₆	C₄²	1	4	yes	—
85	C₅₉⋊C₁₆	—	2	4	no	—
86	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₉	—	2	2	no	—
88	C₂×C₄×C₅₉	—	4	1	no	—
89	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
90	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
91	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
92	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
93	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅₉	#135	1	2	yes	—
97	C₂×Dic₅₉	—	4	2	no	—
98	C₂×Dic₅₉	—	4	4	no	—
99	C₂×Dic₅₉	—	4	2	no	—
100	C₂×Dic₅₉	—	8	2	no	—
101	Dic₁₁₈	—	2	4	no	—
102	Dic₁₁₈	—	4	2	no	—
103	Dic₁₁₈	—	4	4	no	—
104	C₈×C₅₉	—	2	1	no	—
105	C₈×C₅₉	—	2	1	no	—
106	C₈×C₅₉	—	4	4	no	—
107	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
108	C₈×C₅₉	#117	1	1	yes	characteristic
109	C₈×C₅₉	C₄×D₄	1	2	yes	—
110	C₄×#362	—	59	1	no	maximal, Sylow
111	C₂²×C₅₉	—	2	1	no	—
112	C₂²×C₅₉	—	2	1	no	—
113	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
114	C₂²×C₅₉	C₄×SD₁₆	1	2	yes	—
115	C₄×C₅₉	—	2	1	no	—
116	C₄×C₅₉	—	2	1	no	—
117	C₄×C₅₉	—	2	1	no	—
118	C₄×C₅₉	#362	1	2	yes	complemented
119	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₅₉	#416	1	1	yes	characteristic
121	C₄×C₅₉	C₄×D₈	1	2	yes	—
122	Dic₅₉	—	4	4	no	—
123	Dic₅₉	—	8	2	no	—
124	C₂×C₄×Q₁₆	—	59	1	no	—
125	C₂×#362	—	59	2	no	—
126	C₄×M₃₂	—	59	1	no	—
127	C₄×#108	D₅₉	1	1	yes	characteristic
128	#1387	—	59	2	no	—
129	C₂×C₅₉	—	2	2	no	—
130	C₂×C₅₉	C₄×#102	1	1	yes	characteristic
131	C₂×C₅₉	C₂×#362	1	1	yes	characteristic
132	C₂×C₅₉	#2148	1	1	yes	characteristic
133	C₂²×Q₁₆	—	59	1	no	—
134	C₂×C₄×Q₈	—	118	1	no	—
135	C₂×M₃₂	—	59	2	no	—
136	C₄×Q₁₆	—	59	2	no	—
137	C₄×Q₁₆	—	59	2	no	—
138	C₄×Q₁₆	—	118	1	no	—
139	C₄×Q₁₆	—	118	1	no	—
140	C₂×#108	C₂×D₅₉	1	2	yes	—
141	C₂×#107	—	59	1	no	—
142	C₂×#103	—	118	1	no	—
143	C₄×M₁₆	—	2	1	no	—
144	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
145	#362	—	59	8	no	—
146	#346	—	118	1	no	—
147	C₄×C₁₆	—	118	1	no	—
148	#341	—	2	1	no	—
149	C₅₉	C₄×#362	1	1	yes	characteristic, complemented, Sylow
150	C₂²×Q₈	—	118	1	no	—
151	C₂×Q₁₆	—	59	4	no	—
152	C₂×Q₁₆	—	59	2	no	—
153	C₂×Q₁₆	—	118	1	no	—
154	C₂×Q₁₆	—	118	2	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
158	C₄×Q₈	—	118	2	no	—
159	C₄×Q₈	—	118	2	no	—
160	C₄×Q₈	—	236	1	no	—
161	C₄×Q₈	—	236	1	no	—
162	C₂×C₄⋊C₄	—	118	1	no	—
163	C₂×C₄⋊C₄	—	236	1	no	—
164	C₂×C₄²	—	236	1	no	—
165	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
166	M₃₂	—	59	4	no	—
167	M₃₂	—	118	2	no	—
168	C₂×C₁₆	—	118	2	no	—
169	#108	—	2	2	no	—
170	#108	C₄×D₅₉	1	4	yes	—
171	#107	—	59	2	no	—
172	#107	—	118	1	no	—
173	#103	—	118	2	no	—
174	#103	—	118	2	no	—
175	#97	—	4	1	no	—
176	C₄×C₈	—	4	1	no	—
177	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
178	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
179	C₄×C₈	D₂₃₆	1	1	yes	characteristic
180	C₄×C₈	D₂₃₆	1	1	yes	characteristic
181	C₂×Q₈	—	118	2	no	—
182	C₂×Q₈	—	118	4	no	—
183	C₂×Q₈	—	236	1	no	—
184	C₂×Q₈	—	236	2	no	—
185	C₂²×C₄	—	236	1	no	—
186	C₂²×C₄	—	236	1	no	—
187	C₂²×C₄	#11745671	1	1	yes	characteristic
188	C₂²×C₄	#11745671	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
190	Q₁₆	—	59	4	no	—
191	Q₁₆	—	118	2	no	—
192	Q₁₆	—	118	4	no	—
193	M₁₆	—	2	4	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
198	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
201	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
202	C₂×C₈	#11745685	1	2	yes	—
203	C₄⋊C₄	—	118	2	no	—
204	C₄⋊C₄	—	236	2	no	—
205	C₄⋊C₄	—	236	2	no	—
206	C₄⋊C₄	—	236	1	no	—
207	C₄²	—	236	2	no	—
208	C₄²	—	236	2	no	—
209	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
210	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
211	C₄²	#11745669	1	1	yes	characteristic
212	C₄²	#11745669	1	1	yes	characteristic
213	C₁₆	—	118	4	no	—
214	C₂³	C₂×#11745671	1	1	yes	characteristic
215	Q₈	—	118	4	no	—
216	Q₈	—	236	2	no	—
217	Q₈	—	236	4	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	472	2	no	—
221	C₂×C₄	—	236	2	no	—
222	C₂×C₄	—	236	2	no	—
223	C₂×C₄	—	236	4	no	—
224	C₂×C₄	#14273647	1	1	yes	characteristic
225	C₂×C₄	#14273647	1	1	yes	characteristic
226	C₂×C₄	#14273647	1	1	yes	characteristic
227	C₂×C₄	#14273647	1	1	yes	characteristic
228	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
229	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
230	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
231	C₂×C₄	#14273747	1	2	yes	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	4	4	no	—
235	C₈	C₄²×D₅₉	1	1	yes	characteristic
236	C₈	#14273684	1	1	yes	characteristic
237	C₈	C₄×D₂₃₆	1	2	yes	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	C₂×#14273647	1	1	yes	characteristic
241	C₂²	C₄×#11745671	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	472	2	no	—
246	C₄	—	236	4	no	—
247	C₄	#16837217	1	2	yes	direct product factor
248	C₄	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
249	C₄	#16837396	1	1	yes	characteristic
250	C₄	C₄×#11745669	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	C₄×#14273647	1	1	yes	characteristic
253	C₂	C₂×#16837217	1	1	yes	characteristic
254	C₂	#21425843	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273647
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×#108×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂³×C₅₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×#108	D₅₉
hypocenter	C₅₉	C₄×#362
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
15. #16837217	C₄	direct (N × H)
118. C₄×C₅₉	#362	semidirect
149. C₅₉	C₄×#362	semidirect
247. C₄	#16837217	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×C₄×Dic₂₃₆	2
C₂×#16837217	2
C₄×#14273667	2
#21422206	2
C₄×#108×C₅₉	2
C₄×#362	59

Subgroups — C4×#16837217 #28503950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16837217 #28503950