Subgroups — #34034870

Order 18688.

602 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627155	C₂	1	2	yes	maximal, complemented
2	#22627140	C₂	1	2	yes	maximal, complemented
3	#22627139	C₂	1	4	yes	maximal, complemented
4	#22627130	C₂	1	2	yes	maximal, complemented
5	#22627128	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14556686	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×#11924040	C₂	1	1	yes	maximal, characteristic, complemented
8	#22626735	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556686	C₂²	1	4	yes	—
10	C₂²×#11924040	C₂²	1	2	yes	—
11	#17989690	C₂²	1	4	yes	—
12	C₄⋊#11924040	C₂²	1	2	yes	—
13	#17989684	C₂²	1	4	yes	—
14	C₂²⋊#11924040	C₂²	1	2	yes	—
15	C₄×#11924040	C₂²	1	1	yes	characteristic
16	C₄×#11924040	C₂²	1	1	yes	characteristic
17	C₄×#11924040	C₂²	1	2	yes	—
18	C₄×#11924040	C₂²	1	2	yes	—
19	C₄×#11924040	C₂²	1	4	yes	—
20	#17989636	C₄	1	1	yes	characteristic
21	#17989636	C₂²	1	1	yes	characteristic
22	#17989615	C₄	1	1	yes	characteristic
23	#17989615	C₂²	1	1	yes	characteristic
24	#17989614	C₄	1	2	yes	—
25	#17989614	C₂²	1	2	yes	—
26	#17989605	C₄	1	2	yes	—
27	#17989603	C₄	1	1	yes	characteristic
28	C₄×#11924036	C₂²	1	2	yes	—
29	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
30	#17989055	C₄	1	1	yes	characteristic
31	#14556686	—	2	8	no	—
32	C₂×#11924040	—	2	4	no	—
33	C₂×#11924040	C₂³	1	4	yes	—
34	C₂×#11924040	C₂³	1	2	yes	—
35	C₂×#11924040	C₂³	1	2	yes	—
36	C₂×#11924036	C₂×C₄	1	2	yes	—
37	C₂×#11924036	C₂³	1	2	yes	—
38	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
39	C₂²×#10596833	C₂³	1	1	yes	characteristic
40	#14556658	C₂×C₄	1	2	yes	—
41	#14556658	C₂×C₄	1	2	yes	—
42	C₄⋊#10596833	C₂×C₄	1	1	yes	characteristic
43	C₄⋊#10596833	C₂×C₄	1	1	yes	characteristic
44	#14556648	C₂×C₄	1	2	yes	—
45	#14556648	C₂×C₄	1	2	yes	—
46	C₂²⋊#10596833	C₂×C₄	1	2	yes	—
47	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
48	C₄×#10596833	C₂×C₄	1	2	yes	—
49	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
50	C₄×#10596833	C₂×C₄	1	2	yes	—
51	C₄×#10596833	C₂³	1	1	yes	characteristic
52	C₄×#10596833	C₂³	1	2	yes	—
53	C₄×#10596833	C₂³	1	1	yes	characteristic
54	C₂×#11924028	C₈	1	1	yes	characteristic
55	#14556556	C₈	1	1	yes	characteristic
56	#14556556	C₂×C₄	1	1	yes	characteristic
57	#14556553	C₈	1	1	yes	characteristic
58	#14556553	C₂×C₄	1	1	yes	characteristic
59	#14556551	C₈	1	2	yes	—
60	#14556551	C₂×C₄	1	2	yes	—
61	#14556548	C₈	1	2	yes	—
62	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
63	C₄²×D₇₃	C₂×C₄	1	2	yes	—
64	#117×C₇₃	C₈	1	1	yes	characteristic
65	#11924040	—	2	2	no	—
66	#11924040	—	2	4	no	—
67	#11924040	#41	1	4	yes	complemented
68	#11924036	—	2	4	no	—
69	#11924036	—	2	4	no	—
70	C₂×#10596833	—	2	2	no	—
71	C₂×#10596833	—	2	2	no	—
72	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
73	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
74	C₂×#10596833	C₂²×C₄	1	2	yes	—
75	C₂×#10596833	C₂²×C₄	1	2	yes	—
76	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
77	C₂×#10596833	C₂⁴	1	1	yes	characteristic
78	#11924028	C₁₆	1	8	yes	complemented
79	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
80	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
81	C₂×C₂²⋊D₇₃	C₂×C₈	1	2	yes	—
82	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
85	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
87	#11924007	C₂×C₈	1	2	yes	—
88	#11924007	C₂×C₈	1	2	yes	—
89	#11924005	C₂×C₈	1	2	yes	—
90	#11924005	C₂×C₈	1	2	yes	—
91	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
92	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
93	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
94	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
96	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
98	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
99	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
100	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
101	C₄²×C₇₃	C₂×C₈	1	2	yes	—
102	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₇₃	C₂×C₁₆	1	4	yes	complemented
105	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
109	C₂²⋊D₇₃	C₂×C₁₆	1	8	yes	complemented
110	C₂×Dic₇₃	—	2	1	no	—
111	C₂×Dic₇₃	—	2	1	no	—
112	C₂×Dic₇₃	—	2	2	no	—
113	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
116	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
118	D₂₉₂	C₂×C₁₆	1	4	yes	complemented
119	C₄×D₇₃	—	2	2	no	—
120	C₄×D₇₃	—	2	2	no	—
121	C₄×D₇₃	—	2	4	no	—
122	C₄×D₇₃	C₂×C₁₆	1	8	yes	complemented
123	Dic₁₄₆	C₂×C₁₆	1	4	yes	complemented
124	#10596833	—	2	1	no	—
125	#10596833	—	2	2	no	—
126	#10596833	—	2	1	no	—
127	#10596833	—	2	2	no	—
128	#10596833	#131	1	2	yes	—
129	#10596833	C₂×#41	1	2	yes	—
130	C₂²×C₇₃	—	2	1	no	—
131	C₂²×C₇₃	—	2	1	no	—
132	C₂²×C₇₃	C₂²×C₁₆	1	2	yes	—
133	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
134	C₂×D₇₃	—	2	2	no	—
135	C₂×D₇₃	—	2	2	no	—
136	C₂×D₇₃	C₂²×C₁₆	1	4	yes	—
137	C₄×C₇₃	—	2	2	no	—
138	C₄×C₇₃	—	2	2	no	—
139	C₄×C₇₃	C₂²×C₁₆	1	2	yes	—
140	C₄×C₇₃	C₂²×C₁₆	1	2	yes	—
141	Dic₇₃	—	2	1	no	—
142	Dic₇₃	—	2	1	no	—
143	Dic₇₃	C₂²×C₁₆	1	4	yes	—
144	Dic₇₃	#504	1	2	yes	—
145	Dic₇₃	C₄×#41	1	1	yes	characteristic
146	Dic₇₃	C₂×#131	1	1	yes	characteristic
147	C₁₆×#41	—	73	1	no	maximal, Sylow
148	C₂×C₇₃	—	2	2	no	—
149	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
150	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
151	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
152	D₇₃	—	2	4	no	—
153	C₈×#41	—	73	1	no	—
154	Q₈×C₁₆	—	73	2	no	—
155	D₄×C₁₆	—	73	4	no	—
156	D₄×C₁₆	—	73	2	no	—
157	#2162	—	73	1	no	—
158	#2162	—	73	2	no	—
159	C₂×C₄×C₁₆	—	73	1	no	—
160	C₂×C₄×C₁₆	—	73	2	no	—
161	C₇₃	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
162	C₄×#41	—	73	1	no	—
163	C₂²×C₁₆	—	73	2	no	—
164	C₂²×C₁₆	—	73	4	no	—
165	C₈×Q₈	—	73	1	no	—
166	C₈×Q₈	—	73	1	no	—
167	C₈×D₄	—	73	2	no	—
168	C₈×D₄	—	73	1	no	—
169	C₈×D₄	—	73	2	no	—
170	C₈×D₄	—	73	1	no	—
171	#431	—	73	1	no	—
172	#431	—	73	2	no	—
173	C₂×C₄×C₈	—	73	1	no	—
174	C₂×C₄×C₈	—	73	2	no	—
175	C₄⋊C₁₆	—	73	2	no	—
176	C₄⋊C₁₆	—	73	4	no	—
177	C₂²⋊C₁₆	—	73	2	no	—
178	C₂²⋊C₁₆	—	73	4	no	—
179	C₄×C₁₆	—	73	1	no	—
180	C₄×C₁₆	—	73	1	no	—
181	C₄×C₁₆	—	73	2	no	—
182	C₄×C₁₆	—	73	2	no	—
183	C₄×C₁₆	—	73	4	no	—
184	C₂×#41	—	73	1	no	—
185	C₂²×C₈	—	73	1	no	—
186	C₂²×C₈	—	73	1	no	—
187	C₂²×C₈	—	73	2	no	—
188	C₂²×C₈	—	73	2	no	—
189	C₄×Q₈	—	73	1	no	—
190	C₄×Q₈	—	73	1	no	—
191	C₄×D₄	—	73	1	no	—
192	C₄×D₄	—	73	1	no	—
193	C₄×D₄	—	73	2	no	—
194	C₄×D₄	—	73	2	no	—
195	#117	—	73	2	no	—
196	#117	#10596842	1	1	yes	characteristic
197	C₂×C₄²	—	73	1	no	—
198	C₂×C₄²	—	73	2	no	—
199	C₂×C₁₆	—	73	2	no	—
200	C₂×C₁₆	—	73	4	no	—
201	C₂×C₁₆	—	73	2	no	—
202	C₂×C₁₆	—	146	4	no	—
203	C₂×C₁₆	—	146	8	no	—
204	C₄⋊C₈	—	73	1	no	—
205	C₄⋊C₈	—	73	1	no	—
206	C₄⋊C₈	—	73	2	no	—
207	C₄⋊C₈	—	73	2	no	—
208	C₂²⋊C₈	—	73	2	no	—
209	C₂²⋊C₈	—	73	2	no	—
210	C₂²⋊C₈	—	73	2	no	—
211	C₄×C₈	—	73	1	no	—
212	C₄×C₈	—	73	1	no	—
213	C₄×C₈	—	73	2	no	—
214	C₄×C₈	—	73	1	no	—
215	C₄×C₈	—	73	2	no	—
216	C₄×C₈	—	73	1	no	—
217	C₄×C₈	—	73	2	no	—
218	#41	—	73	8	no	—
219	C₂×Q₈	—	73	1	no	—
220	C₂×D₄	—	73	2	no	—
221	C₂×D₄	—	73	1	no	—
222	C₂²×C₄	—	73	2	no	—
223	C₂²×C₄	—	73	1	no	—
224	C₂²×C₄	—	73	1	no	—
225	C₂²×C₄	—	73	2	no	—
226	C₂²×C₄	—	73	2	no	—
227	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
228	C₂×C₈	—	73	1	no	—
229	C₂×C₈	—	73	1	no	—
230	C₂×C₈	—	73	1	no	—
231	C₂×C₈	—	73	1	no	—
232	C₂×C₈	—	73	2	no	—
233	C₂×C₈	—	73	2	no	—
234	C₂×C₈	—	146	2	no	—
235	C₂×C₈	—	146	2	no	—
236	C₂×C₈	—	146	4	no	—
237	C₂×C₈	—	146	4	no	—
238	C₄⋊C₄	—	73	2	no	—
239	C₄⋊C₄	—	73	2	no	—
240	C₄⋊C₄	C₂×#10596842	1	1	yes	characteristic
241	C₄⋊C₄	C₂×#10596842	1	1	yes	characteristic
242	C₂²⋊C₄	—	73	2	no	—
243	C₂²⋊C₄	—	73	2	no	—
244	C₂²⋊C₄	C₂×#10596842	1	2	yes	—
245	C₄²	—	73	2	no	—
246	C₄²	—	73	1	no	—
247	C₄²	—	73	2	no	—
248	C₄²	—	73	1	no	—
249	C₄²	—	73	1	no	—
250	C₄²	—	73	1	no	—
251	C₄²	C₂×#10596842	1	2	yes	—
252	C₁₆	—	73	4	no	—
253	C₁₆	—	146	2	no	—
254	C₁₆	—	146	4	no	—
255	C₂³	—	73	2	no	—
256	C₂³	C₂²×#10596842	1	1	yes	characteristic
257	Q₈	—	73	4	no	—
258	D₄	—	73	8	no	—
259	D₄	—	73	4	no	—
260	C₂×C₄	—	73	1	no	—
261	C₂×C₄	—	73	1	no	—
262	C₂×C₄	—	73	2	no	—
263	C₂×C₄	—	73	8	no	—
264	C₂×C₄	—	73	1	no	—
265	C₂×C₄	—	73	1	no	—
266	C₂×C₄	—	146	2	no	—
267	C₂×C₄	—	146	2	no	—
268	C₂×C₄	—	146	4	no	—
269	C₂×C₄	—	146	1	no	—
270	C₂×C₄	—	146	1	no	—
271	C₂×C₄	—	146	2	no	—
272	C₂×C₄	#14556686	1	4	yes	complemented
273	C₂×C₄	C₂²×#10596842	1	2	yes	—
274	C₂×C₄	C₂²×#10596842	1	2	yes	—
275	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
276	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
277	C₈	—	73	2	no	—
278	C₈	—	73	2	no	—
279	C₈	—	146	1	no	—
280	C₈	—	146	2	no	—
281	C₈	—	146	1	no	—
282	C₈	—	146	2	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	73	4	no	—
286	C₂²	—	146	2	no	—
287	C₂²	—	146	2	no	—
288	C₂²	C₂×#14556686	1	2	yes	—
289	C₂²	C₂³×#10596842	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	73	1	no	—
293	C₄	—	73	1	no	—
294	C₄	—	73	4	no	—
295	C₄	—	73	2	no	—
296	C₄	—	146	1	no	—
297	C₄	—	146	1	no	—
298	C₄	C₂×#14556686	1	2	yes	—
299	C₄	C₂×#14556686	1	2	yes	—
300	C₂	—	2	2	no	—
301	C₂	—	146	4	no	—
302	C₂	#41×#10596842	1	1	yes	characteristic
303	C₂	C₂²×#14556686	1	1	yes	characteristic
304	C₂	#22627182	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#10596842
commutator	C₂×C₇₃	C₂³×C₁₆
Frattini	C₂²	C₂³×#10596842
Fitting	#117×C₇₃	C₈
socle	C₂²×C₇₃	C₂³×C₈
prime radical	C₂×#11924028	C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	#117	#10596842
hypocenter	C₇₃	C₁₆×#41
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
1. #22627155	C₂	semidirect
2. #22627140	C₂	semidirect
3. #22627139	C₂	semidirect
4. #22627130	C₂	semidirect
5. #22627128	C₂	semidirect
6. C₄×#14556686	C₂	semidirect
7. C₂×C₄×#11924040	C₂	semidirect
67. #11924040	#41	semidirect
78. #11924028	C₁₆	semidirect
104. C₂×C₄×C₇₃	C₂×C₁₆	semidirect
109. C₂²⋊D₇₃	C₂×C₁₆	semidirect
118. D₂₉₂	C₂×C₁₆	semidirect
122. C₄×D₇₃	C₂×C₁₆	semidirect
123. Dic₁₄₆	C₂×C₁₆	semidirect
161. C₇₃	C₁₆×#41	semidirect
272. C₂×C₄	#14556686	semidirect

Maximal subgroups

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

subgroup	index
#22627155	2
#22627140	2
#22627139	2
#22627130	2
#22627128	2
C₄×#14556686	2
C₂×C₄×#11924040	2
#22626735	2
C₁₆×#41	73