Subgroups — #34007497

Order 18688.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625863	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10596833	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989451	C₂	1	2	yes	maximal
4	#22624761	C₂	1	2	yes	maximal
5	C₄⋊C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596833	—	2	1	no	—
7	C₂×C₄⋊#10596833	—	2	1	no	—
8	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
9	C₂×C₄⋊#10596833	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#10596833	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10596833	—	2	1	no	—
12	#17989453	—	2	1	no	—
13	#17989451	—	2	4	no	—
14	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
16	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
17	#17988341	C₄	1	2	yes	—
18	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
19	C₂²×#10596833	—	2	1	no	—
20	C₂²×#10596833	—	2	1	no	—
21	C₂²×#10596833	—	2	2	no	—
22	C₄⋊#10596833	—	2	2	no	—
23	C₄⋊#10596833	—	2	1	no	—
24	C₄⋊#10596833	—	2	1	no	—
25	C₄⋊#10596833	—	2	2	no	—
26	C₄⋊#10596833	—	4	2	no	—
27	C₄⋊#10596833	—	4	2	no	—
28	C₄⋊#10596833	D₄	1	2	yes	—
29	C₄⋊#10596833	Q₈	1	2	yes	—
30	C₄×#10596833	—	4	2	no	—
31	C₂²×#10596831	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
36	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
37	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
38	C₄⋊#10596831	C₈	1	4	yes	complemented
39	C₈×Dic₇₃	—	2	2	no	—
40	C₈×Dic₇₃	—	2	2	no	—
41	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₈×C₇₃	—	2	2	no	—
44	C₂×#10596833	—	2	2	no	—
45	C₂×#10596833	—	2	2	no	—
46	C₂×#10596833	—	4	1	no	—
47	C₂×#10596833	—	4	1	no	—
48	C₂×#10596833	—	4	1	no	—
49	C₂×#10596833	—	4	1	no	—
50	C₂×#10596833	—	4	2	no	—
51	C₂×#10596833	—	4	2	no	—
52	C₂×#10596833	—	4	2	no	—
53	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×#10596831	—	2	2	no	—
58	C₂×#10596831	—	2	2	no	—
59	C₂×#10596831	C₂×C₈	1	4	yes	—
60	C₄×Dic₇₃	—	2	2	no	—
61	C₄×Dic₇₃	—	2	1	no	—
62	C₄×Dic₇₃	—	2	1	no	—
63	C₄×Dic₇₃	—	2	1	no	—
64	C₄×Dic₇₃	—	2	1	no	—
65	C₄×Dic₇₃	—	4	1	no	—
66	C₄×Dic₇₃	D₈	1	2	yes	—
67	C₄×Dic₇₃	Q₁₆	1	2	yes	—
68	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
69	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
70	C₄×Dic₇₃	#41	1	1	yes	characteristic
71	C₄×Dic₇₃	#41	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₇₃	—	2	2	no	—
76	C₂×C₈×C₇₃	—	2	2	no	—
77	C₂×C₈×C₇₃	—	2	2	no	—
78	C₄²×C₇₃	C₂×C₈	1	2	yes	—
79	C₄²×C₇₃	M₁₆	1	2	yes	—
80	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₃	—	2	1	no	—
82	C₂×C₄×C₇₃	—	2	1	no	—
83	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
84	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₇₃	#107	1	2	yes	—
86	C₂×C₄×C₇₃	#108	1	2	yes	—
87	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
92	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
93	C₂×Dic₇₃	—	2	1	no	—
94	C₂×Dic₇₃	—	2	1	no	—
95	C₂×Dic₇₃	—	2	1	no	—
96	C₂×Dic₇₃	—	2	1	no	—
97	C₂×Dic₇₃	—	2	1	no	—
98	C₂×Dic₇₃	—	2	1	no	—
99	C₂×Dic₇₃	—	2	1	no	—
100	C₂×Dic₇₃	#107	1	2	yes	—
101	C₂×Dic₇₃	#121	1	1	yes	characteristic
102	C₂×Dic₇₃	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×Dic₇₃	#125	1	1	yes	characteristic
104	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
106	C₂×Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×Dic₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×Dic₇₃	#137	1	1	yes	characteristic
109	#10596833	—	4	2	no	—
110	#10596833	—	4	2	no	—
111	#10596833	—	8	2	no	—
112	C₈×C₇₃	—	4	2	no	—
113	#10596831	—	2	4	no	—
114	C₂²×C₇₃	#389	1	1	yes	characteristic
115	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂²×C₇₃	#424	1	1	yes	characteristic
117	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
118	C₂²×C₇₃	#428	1	1	yes	characteristic
119	C₂²×C₇₃	C₂×#108	1	1	yes	characteristic
120	C₂²×C₇₃	#430	1	1	yes	characteristic
121	C₄×C₇₃	—	2	2	no	—
122	C₄×C₇₃	#335	1	4	yes	complemented
123	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
124	C₄×C₇₃	#436	1	1	yes	characteristic
125	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
126	C₄×C₇₃	#446	1	1	yes	characteristic
127	Dic₇₃	—	2	2	no	—
128	Dic₇₃	—	2	1	no	—
129	Dic₇₃	—	2	1	no	—
130	Dic₇₃	—	4	1	no	—
131	Dic₇₃	#459	1	1	yes	characteristic
132	Dic₇₃	#462	1	1	yes	characteristic
133	Dic₇₃	#474	1	1	yes	characteristic
134	Dic₇₃	#477	1	1	yes	characteristic
135	#9601	—	73	1	no	maximal, Sylow
136	C₂×C₇₃	C₂×#335	1	1	yes	characteristic
137	C₂×C₇₃	#1565	1	1	yes	characteristic
138	C₂×C₇₃	#1571	1	1	yes	characteristic
139	C₂×C₇₃	#1574	1	1	yes	characteristic
140	C₂×C₇₃	#1939	1	1	yes	characteristic
141	C₂×C₇₃	#1947	1	1	yes	characteristic
142	C₂×C₇₃	#1950	1	1	yes	characteristic
143	#1842	—	73	1	no	—
144	C₄×C₄⋊C₈	—	73	1	no	—
145	C₄×C₄⋊C₈	—	73	1	no	—
146	C₂×#335	—	73	2	no	—
147	#1277	—	73	2	no	—
148	C₇₃	#9601	1	1	yes	characteristic, complemented, Sylow
149	C₂×C₄⋊C₈	—	73	1	no	—
150	C₂×C₄⋊C₈	—	73	1	no	—
151	C₂×C₄⋊C₈	—	73	1	no	—
152	C₂×C₄⋊C₈	—	73	2	no	—
153	C₂×C₄⋊C₈	—	146	1	no	—
154	C₂×C₄⋊C₈	—	146	1	no	—
155	C₂×C₄⋊C₈	#64408	1	1	yes	characteristic
156	C₂×C₄×C₈	—	73	2	no	—
157	C₂×C₄×C₈	—	146	1	no	—
158	C₄³	—	73	1	no	—
159	#336	—	73	2	no	—
160	#336	—	146	1	no	—
161	#335	—	146	4	no	—
162	C₂²×C₈	—	73	2	no	—
163	C₂²×C₈	—	146	2	no	—
164	C₂²×C₈	—	146	1	no	—
165	C₂²×C₈	—	146	1	no	—
166	C₂²×C₈	C₂×#64408	1	2	yes	—
167	C₂×C₄²	—	73	1	no	—
168	C₂×C₄²	—	73	1	no	—
169	C₂×C₄²	—	73	1	no	—
170	C₂×C₄²	—	73	1	no	—
171	C₂×C₄²	—	73	1	no	—
172	C₂×C₄²	—	73	1	no	—
173	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	292	2	no	—
176	C₄⋊C₈	—	292	2	no	—
177	C₄⋊C₈	—	73	2	no	—
178	C₄⋊C₈	—	73	2	no	—
179	C₄⋊C₈	—	73	4	no	—
180	C₄⋊C₈	—	146	2	no	—
181	C₄⋊C₈	—	146	2	no	—
182	C₄⋊C₈	—	146	1	no	—
183	C₄⋊C₈	—	146	1	no	—
184	C₄×C₈	—	292	2	no	—
185	C₄×C₈	—	146	2	no	—
186	C₄×C₈	—	146	2	no	—
187	C₂²×C₄	—	73	1	no	—
188	C₂²×C₄	—	73	1	no	—
189	C₂²×C₄	—	73	1	no	—
190	C₂²×C₄	—	73	1	no	—
191	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
192	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
193	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	292	1	no	—
198	C₂×C₈	—	292	1	no	—
199	C₂×C₈	—	292	1	no	—
200	C₂×C₈	—	292	1	no	—
201	C₂×C₈	—	292	2	no	—
202	C₂×C₈	—	292	2	no	—
203	C₂×C₈	—	292	2	no	—
204	C₂×C₈	—	73	4	no	—
205	C₂×C₈	—	146	2	no	—
206	C₂×C₈	—	146	2	no	—
207	C₂×C₈	—	146	2	no	—
208	C₂×C₈	—	146	2	no	—
209	C₄²	—	292	1	no	—
210	C₄²	—	73	1	no	—
211	C₄²	—	73	1	no	—
212	C₄²	—	73	2	no	—
213	C₄²	—	73	2	no	—
214	C₄²	—	73	1	no	—
215	C₄²	—	73	1	no	—
216	C₄²	—	146	1	no	—
217	C₄²	—	146	1	no	—
218	C₄²	—	146	1	no	—
219	C₄²	—	146	1	no	—
220	C₄²	—	146	2	no	—
221	C₄²	#11924036	1	2	yes	—
222	C₄²	#11924037	1	2	yes	—
223	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	73	1	no	—
227	C₂×C₄	—	73	1	no	—
228	C₂×C₄	—	73	1	no	—
229	C₂×C₄	—	73	1	no	—
230	C₂×C₄	—	73	1	no	—
231	C₂×C₄	—	73	1	no	—
232	C₂×C₄	—	73	1	no	—
233	C₂×C₄	—	73	1	no	—
234	C₂×C₄	—	73	2	no	—
235	C₂×C₄	—	146	1	no	—
236	C₂×C₄	—	146	1	no	—
237	C₂×C₄	—	146	1	no	—
238	C₂×C₄	—	146	1	no	—
239	C₂×C₄	—	146	1	no	—
240	C₂×C₄	—	146	1	no	—
241	C₂×C₄	—	146	1	no	—
242	C₂×C₄	#14556658	1	2	yes	—
243	C₂×C₄	#14556658	1	2	yes	—
244	C₂×C₄	#14556660	1	2	yes	—
245	C₂×C₄	#14556661	1	2	yes	—
246	C₂×C₄	D₄×#64408	1	1	yes	characteristic
247	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
248	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
249	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
250	C₂×C₄	#14556678	1	1	yes	characteristic
251	C₂×C₄	#14556679	1	1	yes	characteristic
252	C₈	—	4	2	no	—
253	C₈	—	292	2	no	—
254	C₈	—	292	2	no	—
255	C₈	—	584	2	no	—
256	C₈	—	146	4	no	—
257	C₂²	#17989538	1	1	yes	characteristic
258	C₂²	C₂×#14556658	1	1	yes	characteristic
259	C₂²	#17989594	1	1	yes	characteristic
260	C₂²	C₂×#14556660	1	1	yes	characteristic
261	C₂²	#17989598	1	1	yes	characteristic
262	C₂²	C₂×#14556661	1	1	yes	characteristic
263	C₂²	#17989602	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	292	1	no	—
266	C₄	—	73	1	no	—
267	C₄	—	73	1	no	—
268	C₄	—	73	1	no	—
269	C₄	—	73	1	no	—
270	C₄	—	146	2	no	—
271	C₄	—	146	1	no	—
272	C₄	—	146	1	no	—
273	C₄	#17989451	1	4	yes	complemented
274	C₄	#17989615	1	1	yes	characteristic
275	C₄	#17989622	1	1	yes	characteristic
276	C₄	#17989636	1	1	yes	characteristic
277	C₄	#17989638	1	1	yes	characteristic
278	C₂	C₂×#17989451	1	1	yes	characteristic
279	C₂	#22625484	1	1	yes	characteristic
280	C₂	#22625501	1	1	yes	characteristic
281	C₂	#22625508	1	1	yes	characteristic
282	C₂	#22626029	1	1	yes	characteristic
283	C₂	#22626042	1	1	yes	characteristic
284	C₂	#22626046	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄⋊C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×C₄⋊C₈	#64408
hypocenter	C₇₃	#9601
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
38. C₄⋊#10596831	C₈	semidirect
122. C₄×C₇₃	#335	semidirect
148. C₇₃	#9601	semidirect
273. C₄	#17989451	semidirect

Maximal subgroups

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

subgroup	index
#22625863	2
C₄×C₄⋊#10596833	2
C₂×#17989451	2
#22624761	2
C₄⋊C₈×Dic₇₃	2
#9601	73