Subgroups — C₄×#14273757 #21432913

Order 7552.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14273757	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837507	C₂	1	2	yes	maximal, complemented
4	#16837502	C₂	1	2	yes	maximal, complemented
5	C₄×#11745672	C₂	1	2	yes	maximal, complemented
6	C₄×#11745670	C₂	1	2	yes	maximal, complemented
7	#16837445	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837410	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837403	C₂	1	1	yes	maximal, characteristic
10	C₄×#11745664	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₅₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
13	#14273757	C₄	1	8	yes	direct product factor
14	C₂×#11745672	C₂²	1	2	yes	—
15	C₂×#11745670	C₂²	1	2	yes	—
16	C₂×#11745664	C₂²	1	1	yes	characteristic
17	C₂×#11745664	C₂²	1	1	yes	characteristic
18	C₄×Dic₁₁₈	—	2	1	no	—
19	C₄×Dic₁₁₈	—	2	1	no	—
20	C₄×Dic₁₁₈	C₂²	1	2	yes	—
21	C₄×Dic₁₁₈	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₅₉⋊C₄	—	2	1	no	—
24	C₂×C₄×Dic₅₉	—	2	1	no	—
25	#14273660	C₂²	1	2	yes	—
26	#14273658	C₂²	1	2	yes	—
27	#14273651	C₂²	1	2	yes	—
28	#14273650	C₂²	1	2	yes	—
29	#14273650	C₂²	1	2	yes	—
30	#14273648	C₂²	1	2	yes	—
31	#14273638	C₂²	1	2	yes	—
32	C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
33	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₅₉	—	2	1	no	—
37	C₄×D₄×C₅₉	—	2	1	no	—
38	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
40	#117×C₅₉	—	2	1	no	—
41	#117×C₅₉	C₂²	1	1	yes	characteristic
42	C₂×C₄²×C₅₉	—	2	1	no	—
43	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
44	#41×C₅₉	—	2	2	no	—
45	#41×C₅₉	C₂×C₄	1	4	yes	—
46	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₅₉	—	2	1	no	—
48	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	—	2	1	no	—
50	C₂²×C₄×C₅₉	—	2	1	no	—
51	C₂²×C₄×C₅₉	—	2	1	no	—
52	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
55	C₂²×Dic₅₉	—	2	1	no	—
56	C₂²×Dic₅₉	—	2	1	no	—
57	C₂×Dic₁₁₈	—	2	1	no	—
58	C₂×Dic₁₁₈	—	2	2	no	—
59	C₂×Dic₁₁₈	C₂×C₄	1	4	yes	complemented
60	C₂×Dic₁₁₈	C₂³	1	2	yes	—
61	C₄⋊C₄×C₅₉	—	2	1	no	—
62	C₄⋊C₄×C₅₉	—	2	1	no	—
63	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₅₉	—	2	1	no	—
66	C₂²⋊C₄×C₅₉	—	2	1	no	—
67	C₂²⋊C₄×C₅₉	—	2	1	no	—
68	C₄²×C₅₉	—	2	1	no	—
69	C₄²×C₅₉	—	2	1	no	—
70	C₄²×C₅₉	—	2	1	no	—
71	C₄²×C₅₉	D₄	1	1	yes	characteristic
72	C₄²×C₅₉	D₄	1	1	yes	characteristic
73	C₄²×C₅₉	C₂³	1	1	yes	characteristic
74	C₄²×C₅₉	C₂³	1	1	yes	characteristic
75	#11745672	C₂×C₄	1	8	yes	complemented
76	#11745670	C₂×C₄	1	8	yes	complemented
77	C₄⋊Dic₅₉	—	2	1	no	—
78	C₄⋊Dic₅₉	C₂³	1	2	yes	—
79	Dic₅₉⋊C₄	—	2	2	no	—
80	Dic₅₉⋊C₄	—	2	2	no	—
81	C₄×Dic₅₉	—	2	2	no	—
82	C₄×Dic₅₉	—	2	2	no	—
83	#11745664	—	2	2	no	—
84	#11745664	C₂×C₄	1	4	yes	complemented
85	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
86	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
87	C₂³×C₅₉	—	2	1	no	—
88	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₅₉	—	2	1	no	—
90	Q₈×C₅₉	C₂²×C₄	1	2	yes	—
91	D₄×C₅₉	—	2	4	no	—
92	D₄×C₅₉	—	2	1	no	—
93	D₄×C₅₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₅₉	—	2	1	no	—
95	C₂×C₄×C₅₉	—	2	4	no	—
96	C₂×C₄×C₅₉	—	2	2	no	—
97	C₂×C₄×C₅₉	—	2	1	no	—
98	C₂×C₄×C₅₉	—	2	1	no	—
99	C₂×C₄×C₅₉	—	4	1	no	—
100	C₂×C₄×C₅₉	—	4	2	no	—
101	C₂×C₄×C₅₉	—	4	2	no	—
102	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	#41	1	2	yes	—
109	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
110	C₂×Dic₅₉	—	2	4	no	—
111	C₂×Dic₅₉	—	2	2	no	—
112	C₂×Dic₅₉	—	2	2	no	—
113	C₂×Dic₅₉	—	4	2	no	—
114	Dic₁₁₈	—	2	4	no	—
115	Dic₁₁₈	—	2	2	no	—
116	Dic₁₁₈	C₂²×C₄	1	4	yes	—
117	C₅₉⋊C₈	—	2	2	no	—
118	C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
119	C₂²×C₅₉	—	2	1	no	—
120	C₂²×C₅₉	—	2	1	no	—
121	C₂²×C₅₉	—	2	2	no	—
122	C₂²×C₅₉	—	4	1	no	—
123	C₂²×C₅₉	—	4	1	no	—
124	C₂²×C₅₉	C₄×D₄	1	2	yes	—
125	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₅₉	—	2	2	no	—
127	C₄×C₅₉	—	2	1	no	—
128	C₄×C₅₉	—	2	1	no	—
129	C₄×C₅₉	—	2	1	no	—
130	C₄×C₅₉	—	4	1	no	—
131	C₄×C₅₉	—	4	1	no	—
132	C₄×C₅₉	C₄×D₄	1	2	yes	—
133	C₄×C₅₉	#137	1	2	yes	complemented
134	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
136	Dic₅₉	—	2	4	no	—
137	Dic₅₉	—	4	2	no	—
138	C₄×#137	—	59	1	no	maximal, Sylow
139	C₂×C₅₉	—	2	2	no	—
140	C₂×C₅₉	—	4	2	no	—
141	C₂×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
143	C₂×C₅₉	#575	1	1	yes	characteristic
144	C₂×#137	—	59	1	no	—
145	C₄×#41	D₅₉	1	1	yes	characteristic
146	C₂×C₄×Q₈	—	59	1	no	—
147	#441	—	59	2	no	—
148	#440	—	59	2	no	—
149	C₄×Q₁₆	—	59	2	no	—
150	C₄×SD₁₆	—	59	2	no	—
151	#428	—	59	1	no	—
152	#419	—	59	1	no	—
153	#417	—	59	1	no	—
154	C₄×M₁₆	—	59	1	no	—
155	C₅₉	C₄×#137	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
157	C₂²×Q₈	—	59	1	no	—
158	#137	—	59	8	no	—
159	C₂×Q₁₆	—	59	2	no	—
160	C₂×SD₁₆	—	59	2	no	—
161	C₂×M₁₆	—	59	1	no	—
162	C₂×M₁₆	—	59	1	no	—
163	C₄×Q₈	—	59	2	no	—
164	C₄×Q₈	—	59	2	no	—
165	C₄×Q₈	—	118	1	no	—
166	C₄×Q₈	—	118	1	no	—
167	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
168	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
172	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	#117	C₂×D₅₉	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	59	1	no	—
176	C₂×C₄⋊C₄	—	118	1	no	—
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	—	118	1	no	—
179	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
180	#107	—	59	2	no	—
181	#106	—	59	2	no	—
182	#103	—	59	2	no	—
183	#103	—	59	2	no	—
184	#103	—	59	2	no	—
185	#102	—	59	2	no	—
186	#97	—	59	2	no	—
187	C₄×C₈	—	59	2	no	—
188	#41	—	2	2	no	—
189	#41	C₄×D₅₉	1	4	yes	—
190	C₂×Q₈	—	59	4	no	—
191	C₂×Q₈	—	59	2	no	—
192	C₂×Q₈	—	118	2	no	—
193	C₂×Q₈	—	118	1	no	—
194	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	118	1	no	—
201	C₂²×C₄	—	118	1	no	—
202	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
205	Q₁₆	—	59	8	no	—
206	SD₁₆	—	59	8	no	—
207	M₁₆	—	59	4	no	—
208	M₁₆	—	118	2	no	—
209	C₂×C₈	—	59	2	no	—
210	C₂×C₈	—	59	2	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	59	2	no	—
214	C₄⋊C₄	—	118	1	no	—
215	C₄⋊C₄	—	118	2	no	—
216	C₄⋊C₄	—	118	2	no	—
217	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
218	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	118	2	no	—
226	C₄²	—	118	2	no	—
227	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
228	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
229	C₄²	C₂²×D₅₉	1	1	yes	characteristic
230	C₄²	C₂²×D₅₉	1	1	yes	characteristic
231	C₂³	—	2	1	no	—
232	C₂³	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
233	Q₈	—	2	1	no	—
234	Q₈	—	59	4	no	—
235	Q₈	—	118	4	no	—
236	Q₈	—	118	2	no	—
237	Q₈	C₂×C₄×D₅₉	1	2	yes	—
238	D₄	—	2	4	no	—
239	D₄	—	2	1	no	—
240	D₄	C₂×C₄×D₅₉	1	2	yes	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	118	4	no	—
250	C₂×C₄	—	118	2	no	—
251	C₂×C₄	—	118	2	no	—
252	C₂×C₄	—	236	2	no	—
253	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
254	C₂×C₄	#11745685	1	2	yes	—
255	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
259	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
260	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
261	C₈	—	59	4	no	—
262	C₈	—	118	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	C₄×C₂²⋊D₅₉	1	2	yes	—
269	C₂²	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	118	4	no	—
277	C₄	—	236	2	no	—
278	C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
279	C₄	#14273757	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
281	C₄	C₂×#11745685	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	4	2	no	—
284	C₂	C₂×C₄×C₂²⋊D₅₉	1	1	yes	characteristic
285	C₂	C₂×#14273757	1	1	yes	characteristic
286	C₂	#16838200	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₄×#41×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₄×#41	D₅₉
hypocenter	C₅₉	C₄×#137
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₁₁₈	C₂	semidirect
3. #16837507	C₂	semidirect
4. #16837502	C₂	semidirect
5. C₄×#11745672	C₂	semidirect
6. C₄×#11745670	C₂	semidirect
7. #16837445	C₂	semidirect
8. #16837410	C₂	semidirect
10. C₄×#11745664	C₂	semidirect
13. #14273757	C₄	direct (N × H)
59. C₂×Dic₁₁₈	C₂×C₄	semidirect
75. #11745672	C₂×C₄	semidirect
76. #11745670	C₂×C₄	semidirect
84. #11745664	C₂×C₄	semidirect
133. C₄×C₅₉	#137	semidirect
155. C₅₉	C₄×#137	semidirect
279. C₄	#14273757	direct (N × H)

Subgroups — C4×#14273757 #21432913

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#14273757 #21432913