Subgroups — #34019736

Order 18688.

572 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626842	C₂	1	2	yes	maximal
2	#22625571	C₂	1	2	yes	maximal, complemented
3	#22625556	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#10596833	C₂	1	2	yes	maximal, complemented
5	#22625481	C₂	1	1	yes	maximal, characteristic
6	#22625254	C₂	1	2	yes	maximal
7	C₈×#11924038	C₂	1	1	yes	maximal, characteristic
8	#22620515	C₂	1	1	yes	maximal, characteristic
9	#17989638	C₂²	1	4	yes	—
10	#17989637	—	2	2	no	—
11	#17989636	—	2	2	no	—
12	Q₈×#10596833	C₂²	1	4	yes	—
13	#17989612	—	2	4	no	—
14	#17989606	C₂²	1	2	yes	—
15	#17989606	C₂²	1	1	yes	characteristic
16	C₄×#11924038	C₂²	1	1	yes	characteristic
17	#17989450	C₂²	1	1	yes	characteristic
18	#17989450	C₂²	1	1	yes	characteristic
19	#17989449	C₂²	1	2	yes	—
20	Q₈⋊#10596833	C₂²	1	4	yes	—
21	Q₈⋊#10596833	C₂²	1	4	yes	—
22	#17989429	C₂²	1	2	yes	—
23	C₈×#10596833	C₂²	1	2	yes	—
24	#17989142	C₂²	1	2	yes	—
25	#17988992	C₄	1	1	yes	characteristic
26	#17988937	C₄	1	2	yes	—
27	#17988825	C₄	1	2	yes	—
28	#17988825	C₄	1	2	yes	—
29	Q₁₆×Dic₇₃	C₂²	1	2	yes	—
30	Q₁₆×Dic₇₃	C₂²	1	2	yes	—
31	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
32	#494×C₇₃	C₄	1	1	yes	characteristic
33	C₂×#11924038	—	2	1	no	—
34	#14556658	—	2	4	no	—
35	C₄⋊#10596833	—	2	4	no	—
36	C₄⋊#10596833	D₄	1	1	yes	characteristic
37	C₄⋊#10596833	D₄	1	1	yes	characteristic
38	C₄⋊#10596833	C₂³	1	2	yes	—
39	C₄⋊#10596833	C₂³	1	2	yes	—
40	#14556648	—	2	2	no	—
41	#14556648	D₄	1	2	yes	—
42	#14556646	—	2	4	no	—
43	#14556646	—	2	1	no	—
44	C₄×#10596833	—	2	4	no	—
45	C₄×#10596833	—	2	2	no	—
46	C₄×#10596833	C₂³	1	2	yes	—
47	C₄×#10596833	C₂³	1	2	yes	—
48	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
50	#14556591	C₂×C₄	1	2	yes	—
51	#14556582	—	2	2	no	—
52	#14556571	—	2	2	no	—
53	#14556567	C₂×C₄	1	4	yes	—
54	Q₈×Dic₇₃	C₂³	1	4	yes	—
55	#14556556	—	2	2	no	—
56	#14556549	—	2	2	no	—
57	C₄²×D₇₃	C₂³	1	1	yes	characteristic
58	#14556525	C₂×C₄	1	2	yes	—
59	#14556525	C₂×C₄	1	2	yes	—
60	#14556515	C₂×C₄	1	4	yes	—
61	#14556515	C₂×C₄	1	4	yes	—
62	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
63	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
64	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
65	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
66	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
67	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
69	#11924038	—	2	4	no	—
70	C₂×#10596833	—	2	2	no	—
71	C₂×#10596833	—	2	2	no	—
72	C₂×#10596833	—	2	2	no	—
73	C₂×#10596833	C₂×D₄	1	2	yes	—
74	C₂×#10596833	C₂×D₄	1	2	yes	—
75	C₂×Dic₁₄₆	—	2	2	no	—
76	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
79	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
80	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
81	C₈×D₇₃	—	2	2	no	—
82	C₈×D₇₃	—	4	1	no	—
83	C₄⋊Dic₇₃	—	2	2	no	—
84	C₄⋊Dic₇₃	—	2	4	no	—
85	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
86	#11924007	—	2	4	no	—
87	#11924005	—	2	4	no	—
88	C₄×Dic₇₃	—	2	4	no	—
89	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
90	C₄×Dic₇₃	#41	1	1	yes	characteristic
91	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
92	C₂×Q₈×C₇₃	C₂²×C₄	1	4	yes	—
93	Q₁₆×C₇₃	—	2	4	no	—
94	Q₁₆×C₇₃	—	4	2	no	—
95	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₇₃	—	2	2	no	—
98	C₄⋊C₄×C₇₃	—	2	2	no	—
99	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
101	Q₈×C₇₃	—	2	4	no	—
102	Q₈×C₇₃	—	2	4	no	—
103	C₂×C₄×C₇₃	—	2	4	no	—
104	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×Dic₇₃	—	2	4	no	—
108	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
109	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
111	C₄×D₇₃	—	2	1	no	—
112	C₄×D₇₃	—	2	1	no	—
113	C₄×D₇₃	—	2	2	no	—
114	C₄×D₇₃	—	2	1	no	—
115	C₄×D₇₃	—	2	1	no	—
116	Dic₁₄₆	—	4	4	no	—
117	#10596833	—	2	4	no	—
118	#10596833	—	2	2	no	—
119	#10596833	—	2	2	no	—
120	#10596833	—	4	2	no	—
121	#10596833	—	4	2	no	—
122	C₈×C₇₃	—	2	1	no	—
123	C₈×C₇₃	#131	1	2	yes	—
124	#10596831	—	2	1	no	—
125	#10596831	—	2	1	no	—
126	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂×D₇₃	—	2	1	no	—
128	C₂×D₇₃	—	2	1	no	—
129	C₂×D₇₃	—	2	1	no	—
130	C₄×C₇₃	—	2	1	no	—
131	C₄×C₇₃	—	4	4	no	—
132	C₄×C₇₃	#443	1	2	yes	—
133	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
134	C₄×C₇₃	#568	1	1	yes	characteristic
135	Dic₇₃	—	2	1	no	—
136	Dic₇₃	—	2	1	no	—
137	Dic₇₃	—	4	4	no	—
138	Dic₇₃	#578	1	1	yes	characteristic
139	Dic₇₃	#578	1	1	yes	characteristic
140	#18086	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	#2935	1	1	yes	characteristic
142	C₂×C₇₃	#2950	1	1	yes	characteristic
143	C₂×C₇₃	C₂×#443	1	1	yes	characteristic
144	D₇₃	—	4	1	no	—
145	#3170	—	73	1	no	—
146	#2986	—	73	2	no	—
147	#1591	—	73	2	no	—
148	#1587	—	73	4	no	—
149	C₈×Q₁₆	—	73	2	no	—
150	#1564	—	73	1	no	—
151	#1485	—	73	2	no	—
152	C₈×M₁₆	—	73	1	no	—
153	C₇₃	#18086	1	1	yes	characteristic, complemented, Sylow
154	#533	—	73	2	no	—
155	#500	—	73	1	no	—
156	#494	#64408	1	1	yes	characteristic
157	#487	—	73	2	no	—
158	#467	—	73	2	no	—
159	#467	—	73	2	no	—
160	#446	—	73	4	no	—
161	#446	—	146	2	no	—
162	C₈×Q₈	—	73	4	no	—
163	C₈×Q₈	—	146	2	no	—
164	C₄×Q₁₆	—	73	2	no	—
165	C₄×Q₁₆	—	73	2	no	—
166	#433	—	146	4	no	—
167	#431	—	73	1	no	—
168	#431	—	73	2	no	—
169	C₄×M₁₆	—	73	1	no	—
170	C₂×C₄×C₈	—	73	1	no	—
171	#335	—	73	1	no	—
172	#335	—	73	1	no	—
173	#334	—	73	2	no	—
174	Q₈⋊C₈	—	73	4	no	—
175	Q₈⋊C₈	—	73	4	no	—
176	#322	—	73	2	no	—
177	C₈²	—	73	2	no	—
178	C₂×Q₁₆	C₂×#64408	1	2	yes	—
179	C₂×Q₁₆	C₂×#64408	1	2	yes	—
180	C₂×M₁₆	—	146	1	no	—
181	C₂²×C₈	—	73	1	no	—
182	C₂²×C₈	—	73	1	no	—
183	C₄⋊Q₈	—	73	2	no	—
184	C₄⋊Q₈	C₂×#64408	1	2	yes	—
185	#125	—	146	2	no	—
186	C₂²⋊Q₈	—	73	4	no	—
187	C₂²⋊Q₈	—	146	2	no	—
188	C₄×Q₈	—	73	4	no	—
189	C₄×Q₈	—	146	2	no	—
190	#117	—	146	2	no	—
191	C₂×C₄²	—	73	1	no	—
192	#107	—	73	2	no	—
193	#107	—	73	2	no	—
194	C₄⋊C₈	—	73	1	no	—
195	C₄⋊C₈	—	73	1	no	—
196	C₄⋊C₈	—	73	2	no	—
197	C₄⋊C₈	—	73	2	no	—
198	C₄⋊C₈	—	146	4	no	—
199	C₄⋊C₈	—	146	4	no	—
200	#103	—	73	4	no	—
201	#103	—	73	4	no	—
202	C₂²⋊C₈	—	73	2	no	—
203	C₂²⋊C₈	—	146	2	no	—
204	#97	—	146	1	no	—
205	#97	—	146	4	no	—
206	C₄×C₈	—	73	2	no	—
207	C₄×C₈	—	73	1	no	—
208	C₄×C₈	—	73	1	no	—
209	C₄×C₈	—	73	2	no	—
210	C₄×C₈	—	73	1	no	—
211	C₄×C₈	—	146	2	no	—
212	C₄×C₈	—	146	4	no	—
213	C₄×C₈	C₂×#64408	1	1	yes	characteristic
214	C₂×Q₈	—	146	2	no	—
215	C₂×Q₈	C₂²×#64408	1	4	yes	—
216	C₂²×C₄	—	73	1	no	—
217	C₂²×C₄	—	73	1	no	—
218	C₂²×C₄	—	73	1	no	—
219	Q₁₆	—	2	4	no	—
220	Q₁₆	—	4	2	no	—
221	M₁₆	—	146	4	no	—
222	C₂×C₈	—	292	1	no	—
223	C₂×C₈	—	73	2	no	—
224	C₂×C₈	—	73	2	no	—
225	C₂×C₈	—	73	1	no	—
226	C₂×C₈	—	73	1	no	—
227	C₂×C₈	—	146	2	no	—
228	C₂×C₈	—	146	2	no	—
229	C₂×C₈	—	146	2	no	—
230	C₂×C₈	—	146	2	no	—
231	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
232	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	73	4	no	—
236	C₄⋊C₄	—	146	4	no	—
237	C₄⋊C₄	—	146	4	no	—
238	C₄⋊C₄	—	146	2	no	—
239	C₂²⋊C₄	—	146	4	no	—
240	C₄²	—	73	1	no	—
241	C₄²	—	73	1	no	—
242	C₄²	—	73	1	no	—
243	C₄²	—	146	4	no	—
244	C₄²	C₂²×#64408	1	1	yes	characteristic
245	C₂³	—	73	1	no	—
246	Q₈	—	2	4	no	—
247	Q₈	—	2	4	no	—
248	Q₈	—	292	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	73	1	no	—
251	C₂×C₄	—	73	1	no	—
252	C₂×C₄	—	73	1	no	—
253	C₂×C₄	—	146	1	no	—
254	C₂×C₄	—	146	1	no	—
255	C₂×C₄	—	146	4	no	—
256	C₂×C₄	—	146	1	no	—
257	C₂×C₄	—	146	1	no	—
258	C₂×C₄	—	146	2	no	—
259	C₂×C₄	D₄×#64408	1	1	yes	characteristic
260	C₂×C₄	D₄×#64408	1	1	yes	characteristic
261	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	292	2	no	—
264	C₈	—	292	2	no	—
265	C₈	—	146	2	no	—
266	C₈	—	146	2	no	—
267	C₈	—	146	4	no	—
268	C₈	—	146	1	no	—
269	C₈	—	146	1	no	—
270	C₈	#14556678	1	2	yes	—
271	C₂²	—	146	1	no	—
272	C₂²	—	146	1	no	—
273	C₂²	—	146	1	no	—
274	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	—	292	4	no	—
278	C₄	—	73	1	no	—
279	C₄	—	73	1	no	—
280	C₄	—	146	1	no	—
281	C₄	—	146	1	no	—
282	C₄	#17989631	1	2	yes	—
283	C₄	C₂×#14556678	1	1	yes	characteristic
284	C₄	#17989665	1	1	yes	characteristic
285	C₂	—	292	1	no	—
286	C₂	#22626674	1	1	yes	characteristic
287	C₂	#22626692	1	1	yes	characteristic
288	C₂	C₂×#17989631	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	#494×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	#494	#64408
hypocenter	C₇₃	#18086
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #22625571	C₂	semidirect
3. #22625556	C₂	semidirect
4. Q₁₆×#10596833	C₂	semidirect
153. C₇₃	#18086	semidirect

Maximal subgroups

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

subgroup	index
#22626842	2
#22625571	2
#22625556	2
Q₁₆×#10596833	2
#22625481	2
#22625254	2
C₈×#11924038	2
#22620515	2
#18086	73