Subgroups — #34030264

Order 18688.

718 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	#22626690	C₂	1	2	yes	maximal, complemented
2	#22626329	C₂	1	4	yes	maximal, complemented
3	D₁₆×#64408	C₂	1	4	yes	maximal, complemented
4	#22626282	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626240	C₂	1	2	yes	maximal, complemented
6	#22626139	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
8	#17989654	—	2	2	no	—
9	#17989643	C₂²	1	1	yes	characteristic
10	#17989628	—	2	4	no	—
11	D₈×#64408	C₂²	1	8	yes	complemented
12	#17989597	C₂²	1	2	yes	—
13	#17989579	—	2	2	no	—
14	#17989550	C₂²	1	1	yes	characteristic
15	#17989514	C₂²	1	2	yes	complemented
16	#17989513	C₂²	1	2	yes	complemented
17	#17989499	C₂²	1	8	yes	complemented
18	#17989474	C₂²	1	2	yes	complemented
19	C₁₆×#64408	C₂²	1	2	yes	complemented
20	C₂×D₈×D₇₃	C₂²	1	2	yes	—
21	C₂×D₁₁₆₈	C₄	1	1	yes	characteristic, complemented
22	C₂×#14556532	C₄	1	2	yes	complemented
23	D₁₆×D₇₃	C₄	1	2	yes	complemented
24	D₁₆×D₇₃	C₄	1	2	yes	complemented
25	D₁₆×D₇₃	C₂²	1	2	yes	—
26	D₁₆×D₇₃	C₂²	1	2	yes	—
27	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
28	C₂×D₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
29	D₇₃⋊M₁₆	D₄	1	1	yes	characteristic
30	D₄×#64408	—	2	8	no	—
31	D₄×#64408	—	4	4	no	—
32	#14556670	—	2	2	no	—
33	#14556670	D₄	1	1	yes	characteristic
34	C₂×C₂²⋊#64408	—	4	4	no	—
35	#14556666	—	2	1	no	—
36	#14556665	—	2	1	no	—
37	#14556664	C₂³	1	2	yes	—
38	#14556660	C₂³	1	4	yes	—
39	#14556659	—	2	2	no	—
40	#14556653	—	2	8	no	—
41	#14556645	D₄	1	2	yes	—
42	C₈×#64408	C₂³	1	2	yes	—
43	C₂×D₄×D₇₃	—	2	2	no	—
44	C₂×D₅₈₄	C₂×C₄	1	2	yes	complemented
45	C₂×#11924013	—	2	2	no	—
46	D₈×D₇₃	—	2	2	no	—
47	D₈×D₇₃	—	2	2	no	—
48	D₈×D₇₃	C₂×C₄	1	4	yes	complemented
49	D₈×D₇₃	C₂³	1	4	yes	—
50	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
51	D₁₁₆₈	C₂×C₄	1	2	yes	complemented
52	D₁₁₆₈	C₂×C₄	1	2	yes	complemented
53	#14556532	C₂×C₄	1	4	yes	complemented
54	#14556532	C₂×C₄	1	4	yes	complemented
55	C₂×#11924003	C₂×C₄	1	1	yes	characteristic, complemented
56	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic, complemented
57	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic, complemented
58	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
59	C₁₆×D₇₃	C₂³	1	1	yes	characteristic
60	C₂×D₈×C₇₃	C₂×C₄	1	2	yes	complemented
61	D₁₆×C₇₃	C₂×C₄	1	2	yes	complemented
62	D₁₆×C₇₃	C₂×C₄	1	2	yes	complemented
63	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic, complemented
64	C₂²×#64408	—	4	8	no	—
65	#11924038	—	2	1	no	—
66	#11924037	—	2	1	no	—
67	#11924036	C₂×D₄	1	2	yes	—
68	C₄⋊#64408	—	2	4	no	—
69	C₄⋊#64408	D₈	1	2	yes	—
70	C₂²⋊#64408	—	4	2	no	—
71	C₂²⋊#64408	—	4	8	no	—
72	C₂²⋊#64408	D₈	1	2	yes	—
73	C₄×#64408	—	4	2	no	—
74	C₄×#64408	C₂×D₄	1	2	yes	—
75	#11924031	—	2	2	no	—
76	C₂³×D₇₃	—	4	2	no	—
77	C₂×D₂₉₂	—	2	2	no	—
78	C₂×C₂²⋊D₇₃	—	4	2	no	—
79	D₄×D₇₃	—	2	4	no	—
80	D₄×D₇₃	—	2	4	no	—
81	D₄×D₇₃	—	4	2	no	—
82	D₄×D₇₃	—	4	2	no	—
83	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
84	D₅₈₄	—	2	2	no	—
85	D₅₈₄	C₂²×C₄	1	4	yes	—
86	#11924013	—	2	4	no	—
87	#11924013	—	2	4	no	—
88	C₂×#10596831	#41	1	1	yes	characteristic
89	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
91	C₈×D₇₃	#41	1	1	yes	characteristic
92	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
93	#11924003	C₂²×C₄	1	1	yes	characteristic
94	#11924003	C₂²×C₄	1	1	yes	characteristic
95	C₂×D₄×C₇₃	—	2	2	no	—
96	D₈×C₇₃	—	2	2	no	—
97	D₈×C₇₃	C₂²×C₄	1	4	yes	—
98	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₇₃	—	4	2	no	—
102	C₂²×D₇₃	—	4	2	no	—
103	C₂²×D₇₃	—	4	4	no	—
104	C₂²×D₇₃	—	4	4	no	—
105	C₂²×D₇₃	—	16	4	no	—
106	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
107	C₂×#64408	—	4	4	no	—
108	C₂×#64408	—	16	8	no	—
109	C₂×#64408	C₂×D₈	1	4	yes	—
110	D₄×C₇₃	—	2	4	no	—
111	D₄×C₇₃	—	4	2	no	—
112	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
113	C₂²⋊D₇₃	—	4	4	no	—
114	C₂²⋊D₇₃	—	4	4	no	—
115	C₂×Dic₇₃	#135	1	1	yes	characteristic
116	D₂₉₂	—	2	4	no	—
117	D₂₉₂	—	4	2	no	—
118	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
119	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
120	C₄×D₇₃	#135	1	1	yes	characteristic
121	C₄×D₇₃	C₂²×D₄	1	1	yes	characteristic
122	#10596833	—	2	2	no	—
123	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
124	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
125	#10596831	C₄×D₄	1	1	yes	characteristic
126	#10596831	C₂×#41	1	1	yes	characteristic
127	C₂²×C₇₃	—	4	4	no	—
128	C₂²×C₇₃	—	16	2	no	—
129	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
130	C₂×D₇₃	—	2	2	no	—
131	C₂×D₇₃	—	4	4	no	—
132	C₂×D₇₃	—	8	8	no	—
133	C₂×D₇₃	—	16	4	no	—
134	C₂×D₇₃	—	16	2	no	—
135	C₂×D₇₃	C₄×D₈	1	1	yes	characteristic
136	C₂×D₇₃	C₂²×D₈	1	1	yes	characteristic
137	#64408	—	2	4	no	—
138	#64408	—	8	4	no	—
139	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
140	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
141	Dic₇₃	C₄×D₈	1	1	yes	characteristic
142	Dic₇₃	C₂×#135	1	1	yes	characteristic
143	#33456	—	73	1	no	maximal, Sylow
144	C₂×C₇₃	—	2	1	no	—
145	C₂×C₇₃	—	8	4	no	—
146	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
147	D₇₃	—	2	1	no	—
148	D₇₃	—	8	4	no	—
149	D₇₃	#3415	1	2	yes	—
150	C₂²×D₁₆	—	73	1	no	—
151	#2948	—	73	2	no	—
152	#2177	—	73	4	no	—
153	C₄×D₁₆	—	73	4	no	—
154	#2158	—	73	1	no	—
155	#2140	—	73	2	no	—
156	#2108	—	73	1	no	—
157	C₇₃	#33456	1	1	yes	characteristic, complemented, Sylow
158	C₂²×D₈	—	73	2	no	—
159	#518	—	146	2	no	—
160	C₂×D₁₆	—	73	2	no	—
161	C₂×D₁₆	—	73	2	no	—
162	C₂×D₁₆	—	73	2	no	—
163	C₂×D₁₆	—	73	2	no	—
164	C₂×D₁₆	—	73	2	no	—
165	C₂×D₁₆	—	73	1	no	—
166	C₂×D₁₆	#64408	1	1	yes	characteristic, complemented
167	C₂×M₃₂	—	73	1	no	—
168	C₂²×C₁₆	—	73	1	no	—
169	#442	—	146	4	no	—
170	C₄×D₈	—	73	8	no	—
171	#427	—	73	2	no	—
172	#418	—	146	2	no	—
173	#405	—	73	1	no	—
174	#367	—	73	2	no	—
175	#366	—	73	2	no	—
176	#357	—	73	8	no	—
177	#346	—	73	2	no	—
178	C₄×C₁₆	—	73	2	no	—
179	C₂²×D₄	—	146	2	no	—
180	C₂×D₈	—	73	4	no	—
181	C₂×D₈	—	73	4	no	—
182	C₂×D₈	—	73	2	no	—
183	C₂×D₈	—	146	2	no	—
184	C₂×D₈	—	146	2	no	—
185	C₂×D₈	—	146	2	no	—
186	C₂×D₈	C₂×#64408	1	2	yes	complemented
187	C₂×M₁₆	—	73	1	no	—
188	C₂²×C₈	—	73	1	no	—
189	C₄×D₄	—	292	4	no	—
190	C₄×D₄	—	146	8	no	—
191	#117	—	73	1	no	—
192	#117	—	146	2	no	—
193	C₂×C₂²⋊C₄	—	292	4	no	—
194	D₁₆	—	73	4	no	—
195	D₁₆	—	73	4	no	—
196	D₁₆	—	73	2	no	—
197	D₁₆	—	73	2	no	—
198	D₁₆	C₂×#64408	1	2	yes	complemented
199	D₁₆	C₂×#64408	1	2	yes	complemented
200	M₃₂	—	146	1	no	—
201	M₃₂	—	146	1	no	—
202	C₂×C₁₆	—	73	1	no	—
203	C₂×C₁₆	—	73	1	no	—
204	C₂×C₁₆	—	73	1	no	—
205	C₂×C₁₆	—	73	1	no	—
206	C₂×C₁₆	—	73	1	no	—
207	C₂×C₁₆	—	73	2	no	—
208	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic, complemented
209	#107	—	73	4	no	—
210	#106	—	146	2	no	—
211	#102	—	146	8	no	—
212	#97	—	73	2	no	—
213	C₄×C₈	—	73	2	no	—
214	C₂⁴	—	292	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	292	2	no	—
217	C₂×D₄	—	292	2	no	—
218	C₂×D₄	—	292	2	no	—
219	C₂×D₄	—	146	4	no	—
220	C₂×D₄	—	146	4	no	—
221	C₂×D₄	—	146	2	no	—
222	C₂²×C₄	—	292	8	no	—
223	C₂²×C₄	—	73	1	no	—
224	D₈	—	2	2	no	—
225	D₈	—	73	4	no	—
226	D₈	—	146	4	no	—
227	D₈	—	146	4	no	—
228	D₈	—	146	2	no	—
229	D₈	C₂²×#64408	1	4	yes	—
230	M₁₆	—	146	1	no	—
231	M₁₆	—	146	1	no	—
232	C₂×C₈	—	73	1	no	—
233	C₂×C₈	—	73	1	no	—
234	C₂×C₈	—	73	1	no	—
235	C₂×C₈	—	73	1	no	—
236	C₂×C₈	—	73	1	no	—
237	C₂×C₈	—	73	2	no	—
238	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
239	C₄⋊C₄	—	73	2	no	—
240	C₄⋊C₄	—	146	4	no	—
241	C₂²⋊C₄	—	292	2	no	—
242	C₂²⋊C₄	—	292	8	no	—
243	C₂²⋊C₄	—	73	2	no	—
244	C₄²	—	292	2	no	—
245	C₄²	—	73	2	no	—
246	C₁₆	—	73	1	no	—
247	C₁₆	—	73	1	no	—
248	C₁₆	—	146	2	no	—
249	C₁₆	C₂²×#64408	1	1	yes	characteristic
250	C₁₆	C₂²×#64408	1	1	yes	characteristic
251	C₂³	—	4	2	no	—
252	C₂³	—	292	4	no	—
253	C₂³	—	292	4	no	—
254	C₂³	—	292	2	no	—
255	C₂³	—	73	1	no	—
256	C₂³	—	1168	4	no	—
257	D₄	—	2	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	292	4	no	—
260	D₄	—	292	4	no	—
261	D₄	—	292	2	no	—
262	D₄	—	146	4	no	—
263	C₂×C₄	—	292	4	no	—
264	C₂×C₄	—	73	1	no	—
265	C₂×C₄	—	73	1	no	—
266	C₂×C₄	—	73	1	no	—
267	C₂×C₄	—	73	1	no	—
268	C₂×C₄	—	73	1	no	—
269	C₂×C₄	—	73	4	no	—
270	C₂×C₄	—	1168	8	no	—
271	C₂×C₄	D₄×#64408	1	1	yes	characteristic
272	C₈	—	73	1	no	—
273	C₈	—	73	1	no	—
274	C₈	—	146	2	no	—
275	C₈	D₄×#64408	1	1	yes	characteristic
276	C₈	C₂³×#64408	1	1	yes	characteristic
277	C₂²	—	4	4	no	—
278	C₂²	—	16	2	no	—
279	C₂²	—	292	4	no	—
280	C₂²	—	584	8	no	—
281	C₂²	—	73	1	no	—
282	C₂²	—	73	1	no	—
283	C₂²	—	1168	2	no	—
284	C₂²	—	1168	4	no	—
285	C₂²	—	146	2	no	—
286	C₂²	D₈×#64408	1	1	yes	characteristic
287	C₄	—	584	4	no	—
288	C₄	—	73	1	no	—
289	C₄	—	73	1	no	—
290	C₄	—	146	4	no	—
291	C₄	D₈×#64408	1	1	yes	characteristic
292	C₄	C₂×D₄×#64408	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	8	4	no	—
295	C₂	—	584	4	no	—
296	C₂	—	73	2	no	—
297	C₂	—	146	1	no	—
298	C₂	C₂×D₈×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#64408
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₈	C₂³×#64408
Fitting	C₂×D₁₆×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₄×D₈
prime radical	C₂×D₁₆×D₇₃	C₂
cosocle	C₈×D₇₃	C₂⁴
hypercenter	C₂×D₁₆	#64408
hypocenter	C₇₃	#33456
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. #22626690	C₂	semidirect
2. #22626329	C₂	semidirect
3. D₁₆×#64408	C₂	semidirect
4. #22626282	C₂	semidirect
5. #22626240	C₂	semidirect
6. #22626139	C₂	semidirect
11. D₈×#64408	C₂²	semidirect
15. #17989514	C₂²	semidirect
16. #17989513	C₂²	semidirect
17. #17989499	C₂²	semidirect
18. #17989474	C₂²	semidirect
19. C₁₆×#64408	C₂²	semidirect
21. C₂×D₁₁₆₈	C₄	semidirect
22. C₂×#14556532	C₄	semidirect
23. D₁₆×D₇₃	C₄	semidirect
24. D₁₆×D₇₃	C₄	semidirect
28. C₂×D₁₆×C₇₃	C₄	semidirect
44. C₂×D₅₈₄	C₂×C₄	semidirect
48. D₈×D₇₃	C₂×C₄	semidirect
51. D₁₁₆₈	C₂×C₄	semidirect
52. D₁₁₆₈	C₂×C₄	semidirect
53. #14556532	C₂×C₄	semidirect
54. #14556532	C₂×C₄	semidirect
55. C₂×#11924003	C₂×C₄	semidirect
56. C₁₆×D₇₃	C₂×C₄	semidirect
57. C₁₆×D₇₃	C₂×C₄	semidirect
60. C₂×D₈×C₇₃	C₂×C₄	semidirect
61. D₁₆×C₇₃	C₂×C₄	semidirect
62. D₁₆×C₇₃	C₂×C₄	semidirect
63. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
157. C₇₃	#33456	semidirect
166. C₂×D₁₆	#64408	semidirect
186. C₂×D₈	C₂×#64408	semidirect
198. D₁₆	C₂×#64408	semidirect
199. D₁₆	C₂×#64408	semidirect
208. C₂×C₁₆	C₂×#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626690	2
#22626329	2
D₁₆×#64408	2
#22626282	2
#22626240	2
#22626139	2
C₂×D₁₆×D₇₃	2
#33456	73