Subgroups — C₂²×#16837837 #29495180

Order 15104.

2334 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837837	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14273670	C₂	1	2	yes	maximal, complemented
4	C₂²×#14273668	C₂	1	2	yes	maximal, complemented
5	C₂²×#14273667	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#135×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₂₃₆	—	2	1	no	—
8	C₂²×D₄₇₂	—	2	1	no	—
9	C₂²×D₄₇₂	—	2	6	no	—
10	C₂²×D₄₇₂	C₂²	1	2	yes	—
11	C₂²×D₄₇₂	C₂²	1	12	yes	complemented
12	#16837837	C₂²	1	64	yes	direct product factor
13	C₂×#14273670	C₂²	1	24	yes	complemented
14	C₂×#14273668	C₂²	1	24	yes	complemented
15	C₂×#14273667	C₂²	1	12	yes	complemented
16	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	complemented
17	C₂²×#41×C₅₉	—	2	1	no	—
18	C₂×#135×C₅₉	—	2	6	no	—
19	C₂×#135×C₅₉	C₂²	1	12	yes	complemented
20	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic, complemented
21	C₂²×SD₁₆×C₅₉	—	2	1	no	—
22	C₂²×D₈×C₅₉	C₂²	1	1	yes	characteristic, complemented
23	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂⁴×D₅₉	—	4	1	no	—
25	C₂²×D₂₃₆	—	2	2	no	—
26	C₂²×D₂₃₆	—	2	12	no	—
27	C₂²×D₂₃₆	—	4	1	no	—
28	C₂²×D₂₃₆	—	4	6	no	—
29	C₂×D₄₇₂	—	2	6	no	—
30	C₂×D₄₇₂	—	2	24	no	—
31	C₂×D₄₇₂	—	2	12	no	—
32	C₂×D₄₇₂	C₂³	1	12	yes	—
33	C₂×D₄₇₂	C₂³	1	16	yes	complemented
34	#14273670	C₂³	1	32	yes	complemented
35	#14273668	C₂³	1	32	yes	complemented
36	#14273667	C₂³	1	16	yes	complemented
37	C₂×C₅₉⋊C₁₆	C₂³	1	12	yes	complemented
38	C₂×#41×C₅₉	—	2	12	no	—
39	C₂×#41×C₅₉	—	4	6	no	—
40	C₂²×Q₈×C₅₉	—	2	1	no	—
41	C₂²×D₄×C₅₉	—	2	1	no	—
42	C₂²×D₄×C₅₉	—	4	1	no	—
43	C₂³×C₄×C₅₉	—	4	1	no	—
44	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
45	#135×C₅₉	—	2	24	no	—
46	#135×C₅₉	C₂³	1	16	yes	complemented
47	C₂×Q₁₆×C₅₉	—	2	3	no	—
48	C₂×Q₁₆×C₅₉	C₂³	1	6	yes	complemented
49	C₂×SD₁₆×C₅₉	—	2	12	no	—
50	C₂×D₈×C₅₉	—	2	3	no	—
51	C₂×D₈×C₅₉	C₂³	1	6	yes	complemented
52	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₉	D₄	1	6	yes	—
54	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₅₉	C₂³	1	6	yes	—
56	C₂⁴×C₅₉	—	4	1	no	—
57	C₂⁴×C₅₉	D₈	1	1	yes	characteristic
58	C₂³×D₅₉	—	4	12	no	—
59	C₂³×D₅₉	—	4	2	no	—
60	C₂³×D₅₉	—	8	8	no	—
61	#41×C₅₉	—	2	16	no	—
62	#41×C₅₉	—	4	24	no	—
63	C₂×Q₈×C₅₉	—	2	6	no	—
64	C₂×Q₈×C₅₉	—	4	3	no	—
65	C₂×D₄×C₅₉	—	2	6	no	—
66	C₂×D₄×C₅₉	—	4	12	no	—
67	C₂×D₄×C₅₉	—	4	3	no	—
68	C₂²×C₄×C₅₉	—	4	1	no	—
69	C₂²×C₄×C₅₉	—	4	12	no	—
70	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	D₈	1	6	yes	—
72	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	C₂×D₄	1	6	yes	—
74	C₂×D₂₃₆	—	2	12	no	—
75	C₂×D₂₃₆	—	2	16	no	—
76	C₂×D₂₃₆	—	4	12	no	—
77	C₂×D₂₃₆	—	4	6	no	—
78	C₂×D₂₃₆	—	4	24	no	—
79	Q₁₆×C₅₉	—	2	6	no	—
80	Q₁₆×C₅₉	C₂⁴	1	4	yes	complemented
81	SD₁₆×C₅₉	—	2	16	no	—
82	D₈×C₅₉	—	2	6	no	—
83	D₈×C₅₉	C₂⁴	1	4	yes	complemented
84	C₂×C₈×C₅₉	C₂×D₄	1	3	yes	—
85	C₂×C₈×C₅₉	C₂×D₄	1	6	yes	—
86	C₂×C₈×C₅₉	C₂×D₄	1	12	yes	—
87	C₂×C₈×C₅₉	C₂⁴	1	3	yes	—
88	C₂×C₈×C₅₉	C₂⁴	1	4	yes	—
89	D₄₇₂	—	2	12	no	—
90	D₄₇₂	—	2	16	no	—
91	D₄₇₂	C₂⁴	1	8	yes	—
92	C₅₉⋊C₁₆	C₂⁴	1	8	yes	complemented
93	C₂³×C₅₉	—	2	4	no	—
94	C₂³×C₅₉	—	4	6	no	—
95	C₂³×C₅₉	—	8	4	no	—
96	C₂³×C₅₉	C₂×D₈	1	6	yes	—
97	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
98	C₂²×D₅₉	—	4	16	no	—
99	C₂²×D₅₉	—	4	12	no	—
100	C₂²×D₅₉	—	8	48	no	—
101	C₂²×D₅₉	—	8	8	no	—
102	Q₈×C₅₉	—	2	4	no	—
103	Q₈×C₅₉	—	4	6	no	—
104	D₄×C₅₉	—	2	4	no	—
105	D₄×C₅₉	—	4	16	no	—
106	D₄×C₅₉	—	4	6	no	—
107	C₂×C₄×C₅₉	—	4	6	no	—
108	C₂×C₄×C₅₉	—	4	16	no	—
109	C₂×C₄×C₅₉	C₂×D₈	1	6	yes	—
110	C₂×C₄×C₅₉	C₂×D₈	1	3	yes	—
111	C₂×C₄×C₅₉	C₂×D₈	1	12	yes	—
112	C₂×C₄×C₅₉	C₂²×D₄	1	3	yes	—
113	C₂×C₄×C₅₉	C₂²×D₄	1	4	yes	—
114	D₂₃₆	—	2	8	no	—
115	D₂₃₆	—	4	16	no	—
116	D₂₃₆	—	4	12	no	—
117	C₈×C₅₉	C₂²×D₄	1	3	yes	—
118	C₈×C₅₉	C₂²×D₄	1	4	yes	—
119	C₈×C₅₉	C₂⁵	1	1	yes	characteristic
120	C₂²×#509	—	59	1	no	maximal, Sylow
121	C₂²×C₅₉	—	2	12	no	—
122	C₂²×C₅₉	—	4	4	no	—
123	C₂²×C₅₉	—	8	12	no	—
124	C₂²×C₅₉	#509	1	4	yes	complemented
125	C₂²×C₅₉	C₂²×D₈	1	4	yes	—
126	C₂²×C₅₉	C₂²×D₈	1	3	yes	—
127	C₂×D₅₉	—	4	8	no	—
128	C₂×D₅₉	—	8	32	no	—
129	C₂×D₅₉	—	8	24	no	—
130	C₄×C₅₉	—	4	4	no	—
131	C₄×C₅₉	C₂²×D₈	1	4	yes	—
132	C₄×C₅₉	C₂²×D₈	1	3	yes	—
133	C₄×C₅₉	C₂³×D₄	1	1	yes	characteristic
134	C₂²×#135	D₅₉	1	1	yes	characteristic, complemented
135	C₂³×D₈	—	59	1	no	—
136	C₂×#509	—	59	24	no	—
137	C₂²×SD₃₂	—	59	2	no	—
138	C₂²×D₁₆	—	59	2	no	—
139	C₂²×M₃₂	—	59	1	no	—
140	C₂×C₅₉	—	2	4	no	—
141	C₂×C₅₉	—	8	4	no	—
142	C₂×C₅₉	C₂×#509	1	6	yes	complemented
143	C₂×C₅₉	C₂³×D₈	1	1	yes	characteristic
144	D₅₉	—	8	8	no	—
145	C₂²×#41	—	2	1	no	—
146	C₂³×D₄	—	118	1	no	—
147	C₂×#135	—	2	6	no	—
148	C₂×#135	C₂×D₅₉	1	12	yes	complemented
149	C₂²×Q₁₆	C₂×D₅₉	1	1	yes	characteristic, complemented
150	C₂²×SD₁₆	—	2	1	no	—
151	C₂²×D₈	—	59	2	no	—
152	C₂²×D₈	—	59	12	no	—
153	C₂²×D₈	—	118	1	no	—
154	C₂²×D₈	—	118	6	no	—
155	C₂²×D₈	C₂×D₅₉	1	1	yes	characteristic, complemented
156	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
157	#509	—	59	64	no	—
158	C₂×SD₃₂	—	59	24	no	—
159	C₂×D₁₆	—	59	24	no	—
160	C₂×M₃₂	—	59	12	no	—
161	C₂²×C₁₆	—	59	2	no	—
162	C₅₉	C₂²×#509	1	1	yes	characteristic, complemented, Sylow
163	C₂⁵	—	236	1	no	—
164	C₂×#41	—	2	12	no	—
165	C₂×#41	—	4	6	no	—
166	C₂²×Q₈	—	2	1	no	—
167	C₂²×D₄	—	2	1	no	—
168	C₂²×D₄	—	4	1	no	—
169	C₂²×D₄	—	118	12	no	—
170	C₂²×D₄	—	118	2	no	—
171	C₂²×D₄	—	236	6	no	—
172	C₂²×D₄	—	236	1	no	—
173	C₂³×C₄	—	4	1	no	—
174	C₂³×C₄	C₂²⋊D₅₉	1	1	yes	characteristic
175	#135	—	2	24	no	—
176	#135	C₂²×D₅₉	1	16	yes	complemented
177	C₂×Q₁₆	—	2	3	no	—
178	C₂×Q₁₆	C₂²×D₅₉	1	6	yes	complemented
179	C₂×SD₁₆	—	2	12	no	—
180	C₂×D₈	—	2	3	no	—
181	C₂×D₈	—	59	12	no	—
182	C₂×D₈	—	59	16	no	—
183	C₂×D₈	—	118	6	no	—
184	C₂×D₈	—	118	12	no	—
185	C₂×D₈	—	118	24	no	—
186	C₂×D₈	C₂²×D₅₉	1	6	yes	complemented
187	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
188	C₂²×C₈	C₂²⋊D₅₉	1	6	yes	—
189	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₅₉	1	6	yes	—
191	SD₃₂	—	59	32	no	—
192	D₁₆	—	59	32	no	—
193	M₃₂	—	59	16	no	—
194	C₂×C₁₆	—	59	12	no	—
195	C₂⁴	—	4	1	no	—
196	C₂⁴	—	472	8	no	—
197	C₂⁴	—	236	12	no	—
198	C₂⁴	—	236	2	no	—
199	C₂⁴	#11745669	1	1	yes	characteristic
200	#41	—	2	16	no	—
201	#41	—	4	24	no	—
202	C₂×Q₈	—	2	6	no	—
203	C₂×Q₈	—	4	3	no	—
204	C₂×D₄	—	2	6	no	—
205	C₂×D₄	—	4	12	no	—
206	C₂×D₄	—	4	3	no	—
207	C₂×D₄	—	118	12	no	—
208	C₂×D₄	—	118	16	no	—
209	C₂×D₄	—	236	12	no	—
210	C₂×D₄	—	236	6	no	—
211	C₂×D₄	—	236	24	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	12	no	—
214	C₂²×C₄	#11745669	1	6	yes	—
215	C₂²×C₄	#11745669	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₅₉	1	6	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
218	Q₁₆	—	2	6	no	—
219	Q₁₆	C₂³×D₅₉	1	4	yes	complemented
220	SD₁₆	—	2	16	no	—
221	D₈	—	2	6	no	—
222	D₈	—	59	8	no	—
223	D₈	—	118	12	no	—
224	D₈	—	118	16	no	—
225	D₈	C₂³×D₅₉	1	4	yes	complemented
226	C₂×C₈	C₂×C₂²⋊D₅₉	1	3	yes	—
227	C₂×C₈	C₂×C₂²⋊D₅₉	1	6	yes	—
228	C₂×C₈	C₂×C₂²⋊D₅₉	1	12	yes	—
229	C₂×C₈	C₂³×D₅₉	1	3	yes	—
230	C₂×C₈	C₂³×D₅₉	1	4	yes	—
231	C₁₆	—	59	8	no	—
232	C₂³	—	2	4	no	—
233	C₂³	—	4	6	no	—
234	C₂³	—	8	4	no	—
235	C₂³	—	472	48	no	—
236	C₂³	—	472	8	no	—
237	C₂³	—	236	16	no	—
238	C₂³	—	236	12	no	—
239	C₂³	C₂×#11745669	1	6	yes	—
240	C₂³	C₂×#11745669	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	4	6	no	—
243	D₄	—	2	4	no	—
244	D₄	—	4	16	no	—
245	D₄	—	4	6	no	—
246	D₄	—	118	8	no	—
247	D₄	—	236	16	no	—
248	D₄	—	236	12	no	—
249	C₂×C₄	—	4	6	no	—
250	C₂×C₄	—	4	16	no	—
251	C₂×C₄	C₂×#11745669	1	6	yes	—
252	C₂×C₄	C₂×#11745669	1	3	yes	—
253	C₂×C₄	C₂×#11745669	1	12	yes	—
254	C₂×C₄	C₂²×C₂²⋊D₅₉	1	3	yes	—
255	C₂×C₄	C₂²×C₂²⋊D₅₉	1	4	yes	—
256	C₈	C₂²×C₂²⋊D₅₉	1	3	yes	—
257	C₈	C₂²×C₂²⋊D₅₉	1	4	yes	—
258	C₈	C₂⁴×D₅₉	1	1	yes	characteristic
259	C₂²	—	2	12	no	—
260	C₂²	—	4	4	no	—
261	C₂²	—	8	12	no	—
262	C₂²	—	472	32	no	—
263	C₂²	—	472	24	no	—
264	C₂²	—	236	8	no	—
265	C₂²	#16837837	1	4	yes	direct product factor
266	C₂²	C₂²×#11745669	1	4	yes	—
267	C₂²	C₂²×#11745669	1	3	yes	—
268	C₄	—	4	4	no	—
269	C₄	C₂²×#11745669	1	4	yes	—
270	C₄	C₂²×#11745669	1	3	yes	—
271	C₄	C₂³×C₂²⋊D₅₉	1	1	yes	characteristic
272	C₂	—	2	4	no	—
273	C₂	—	8	4	no	—
274	C₂	—	472	8	no	—
275	C₂	C₂×#16837837	1	6	yes	direct product factor
276	C₂	C₂³×#11745669	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator, cosocle	C₈×C₅₉	C₂⁵
Frattini	C₈	C₂⁴×D₅₉
Fitting	C₂²×#135×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₈
hypercenter	C₂²×#135	D₅₉
hypocenter	C₅₉	C₂²×#509
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#135

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄₇₂	C₂	semidirect
2. C₂×#16837837	C₂	direct (N × H)
3. C₂²×#14273670	C₂	semidirect
4. C₂²×#14273668	C₂	semidirect
5. C₂²×#14273667	C₂	semidirect
6. C₂²×#135×C₅₉	C₂	semidirect
11. C₂²×D₄₇₂	C₂²	semidirect
12. #16837837	C₂²	direct (N × H)
13. C₂×#14273670	C₂²	semidirect
14. C₂×#14273668	C₂²	semidirect
15. C₂×#14273667	C₂²	semidirect
16. C₂²×C₅₉⋊C₁₆	C₂²	semidirect
19. C₂×#135×C₅₉	C₂²	semidirect
20. C₂²×Q₁₆×C₅₉	C₂²	semidirect
22. C₂²×D₈×C₅₉	C₂²	semidirect
33. C₂×D₄₇₂	C₂³	semidirect
34. #14273670	C₂³	semidirect
35. #14273668	C₂³	semidirect
36. #14273667	C₂³	semidirect
37. C₂×C₅₉⋊C₁₆	C₂³	semidirect
46. #135×C₅₉	C₂³	semidirect
48. C₂×Q₁₆×C₅₉	C₂³	semidirect
51. C₂×D₈×C₅₉	C₂³	semidirect
80. Q₁₆×C₅₉	C₂⁴	semidirect
83. D₈×C₅₉	C₂⁴	semidirect
92. C₅₉⋊C₁₆	C₂⁴	semidirect
124. C₂²×C₅₉	#509	semidirect
134. C₂²×#135	D₅₉	semidirect
142. C₂×C₅₉	C₂×#509	semidirect
148. C₂×#135	C₂×D₅₉	semidirect
149. C₂²×Q₁₆	C₂×D₅₉	semidirect
155. C₂²×D₈	C₂×D₅₉	semidirect
162. C₅₉	C₂²×#509	semidirect
176. #135	C₂²×D₅₉	semidirect
178. C₂×Q₁₆	C₂²×D₅₉	semidirect
186. C₂×D₈	C₂²×D₅₉	semidirect
219. Q₁₆	C₂³×D₅₉	semidirect
225. D₈	C₂³×D₅₉	semidirect
265. C₂²	#16837837	direct (N × H)
275. C₂	C₂×#16837837	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×D₄₇₂	2
C₂×#16837837	2
C₂²×#14273670	2
C₂²×#14273668	2
C₂²×#14273667	2
C₂²×#135×C₅₉	2
C₂²×#509	59

Subgroups — C22×#16837837 #29495180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16837837 #29495180