Subgroups — #32979118

Order 18688.

466 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22610069	C₂	1	1	yes	maximal, characteristic, complemented
2	#22608679	C₂	1	1	yes	maximal, characteristic, complemented
3	#22608356	C₂	1	2	yes	maximal
4	#22607828	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	#17988611	C₂²	1	2	yes	—
8	#17988352	—	2	1	no	—
9	#17988352	—	2	1	no	—
10	#17988340	—	2	1	no	—
11	#17988338	—	2	1	no	—
12	D₄⋊#10596831	—	2	4	no	—
13	C₂³×C₈×C₇₃	—	2	1	no	—
14	C₂×C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₇₃	—	2	1	no	—
16	C₈×D₄×C₇₃	—	2	1	no	—
17	C₈×D₄×C₇₃	C₄	1	2	yes	—
18	C₈×D₄×C₇₃	C₄	1	2	yes	—
19	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₇₃	—	2	1	no	—
21	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×#11924012	—	2	2	no	—
23	C₄⋊#10596831	—	2	2	no	—
24	C₄⋊#10596831	—	2	2	no	—
25	C₄⋊#10596831	—	2	2	no	—
26	C₂²⋊#10596831	—	2	2	no	—
27	C₄×#10596831	—	2	2	no	—
28	C₂²×D₄×C₇₃	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₇₃	—	2	1	no	—
30	C₂²×C₈×C₇₃	—	2	1	no	—
31	C₂²×C₈×C₇₃	—	2	2	no	—
32	C₂²×C₈×C₇₃	—	2	2	no	—
33	C₂²×C₈×C₇₃	—	4	4	no	—
34	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
36	C₄×D₄×C₇₃	—	2	1	no	—
37	C₄×D₄×C₇₃	—	2	1	no	—
38	C₄×D₄×C₇₃	C₂×C₄	1	2	yes	—
39	C₄×D₄×C₇₃	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₇₃	Q₈	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
42	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₇₃	—	2	1	no	—
44	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
45	C₂²⋊C₈×C₇₃	—	2	2	no	—
46	C₂²⋊C₈×C₇₃	—	2	2	no	—
47	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
51	#11924012	—	4	2	no	—
52	#11924012	—	4	2	no	—
53	C₂×#10596831	—	4	2	no	—
54	C₂×#10596831	—	4	2	no	—
55	C₂⁴×C₇₃	—	2	1	no	—
56	C₂×D₄×C₇₃	—	2	4	no	—
57	C₂×D₄×C₇₃	—	2	1	no	—
58	C₂×D₄×C₇₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇₃	—	2	1	no	—
60	C₂²×C₄×C₇₃	—	2	2	no	—
61	C₂²×C₄×C₇₃	—	2	2	no	—
62	C₂²×C₄×C₇₃	—	4	2	no	—
63	C₂²×C₄×C₇₃	—	4	2	no	—
64	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
67	C₂×C₈×C₇₃	—	2	2	no	—
68	C₂×C₈×C₇₃	—	2	2	no	—
69	C₂×C₈×C₇₃	—	2	2	no	—
70	C₂×C₈×C₇₃	—	4	4	no	—
71	C₂×C₈×C₇₃	—	4	4	no	—
72	C₂×C₈×C₇₃	—	4	2	no	—
73	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄⋊C₄×C₇₃	—	2	1	no	—
78	C₄⋊C₄×C₇₃	C₄⋊C₄	1	2	yes	—
79	C₂²⋊C₄×C₇₃	—	2	2	no	—
80	C₂²⋊C₄×C₇₃	—	2	2	no	—
81	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₇₃	#41	1	1	yes	characteristic
85	C₂³×C₇₃	—	2	4	no	—
86	C₂³×C₇₃	—	2	2	no	—
87	C₂³×C₇₃	—	4	2	no	—
88	C₂³×C₇₃	—	4	2	no	—
89	C₂³×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
90	D₄×C₇₃	—	2	4	no	—
91	D₄×C₇₃	—	2	2	no	—
92	D₄×C₇₃	#108	1	4	yes	—
93	C₂×C₄×C₇₃	—	2	2	no	—
94	C₂×C₄×C₇₃	—	2	2	no	—
95	C₂×C₄×C₇₃	—	4	1	no	—
96	C₂×C₄×C₇₃	—	4	2	no	—
97	C₂×C₄×C₇₃	—	4	2	no	—
98	C₂×C₄×C₇₃	—	4	2	no	—
99	C₂×C₄×C₇₃	—	4	4	no	—
100	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
103	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
104	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₇₃	#135	1	2	yes	—
107	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
108	C₈×C₇₃	—	2	1	no	—
109	C₈×C₇₃	—	2	1	no	—
110	C₈×C₇₃	—	2	1	no	—
111	C₈×C₇₃	—	2	1	no	—
112	C₈×C₇₃	—	4	2	no	—
113	#10596831	—	8	2	no	—
114	#10596831	—	8	2	no	—
115	C₂²×C₇₃	—	2	1	no	—
116	C₂²×C₇₃	—	2	1	no	—
117	C₂²×C₇₃	—	2	4	no	—
118	C₂²×C₇₃	—	4	8	no	—
119	C₂²×C₇₃	—	4	2	no	—
120	C₂²×C₇₃	—	4	2	no	—
121	C₂²×C₇₃	#380	1	1	yes	characteristic
122	C₂²×C₇₃	#478	1	2	yes	—
123	C₄×C₇₃	—	2	1	no	—
124	C₄×C₇₃	—	2	1	no	—
125	C₄×C₇₃	—	2	1	no	—
126	C₄×C₇₃	—	4	2	no	—
127	C₄×C₇₃	#416	1	1	yes	characteristic
128	C₄×C₇₃	#418	1	1	yes	characteristic
129	C₄×C₇₃	C₂×#108	1	2	yes	—
130	C₄×C₇₃	#463	1	2	yes	—
131	#8660	—	73	1	no	maximal, Sylow
132	C₂×C₇₃	—	2	2	no	—
133	C₂×C₇₃	—	4	4	no	—
134	C₂×C₇₃	#1804	1	1	yes	characteristic
135	C₂×C₇₃	#1810	1	1	yes	characteristic
136	C₂×C₇₃	#1814	1	1	yes	characteristic
137	C₂×C₈×D₄	D₇₃	1	1	yes	characteristic
138	#1837	—	73	1	no	—
139	#1564	—	73	1	no	—
140	#1486	—	73	2	no	—
141	#1301	—	73	2	no	—
142	C₇₃	#8660	1	1	yes	characteristic, complemented, Sylow
143	C₂³×C₈	—	2	1	no	—
144	C₂×C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
145	C₈×D₄	—	2	1	no	—
146	C₈×D₄	—	2	1	no	—
147	C₈×D₄	Dic₇₃	1	2	yes	—
148	C₈×D₄	Dic₇₃	1	2	yes	—
149	#431	—	73	2	no	—
150	#423	—	73	2	no	—
151	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
152	C₂×C₂²⋊C₈	—	2	1	no	—
153	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
154	#341	—	146	1	no	—
155	#341	—	146	1	no	—
156	#335	—	146	1	no	—
157	#334	—	146	1	no	—
158	D₄⋊C₈	—	146	4	no	—
159	C₂²×D₄	Dic₁₄₆	1	1	yes	characteristic
160	C₂³×C₄	—	2	1	no	—
161	C₂×M₁₆	—	146	2	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	4	4	no	—
166	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	D₂₉₂	1	2	yes	—
171	C₄×D₄	C₂×Dic₇₃	1	2	yes	—
172	C₂×C₄⋊C₄	Dic₁₄₆	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	—	2	1	no	—
174	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	—	146	2	no	—
177	C₄⋊C₈	—	146	2	no	—
178	C₄⋊C₈	—	146	2	no	—
179	C₄⋊C₈	C₂×Dic₇₃	1	2	yes	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₂²⋊C₈	—	146	2	no	—
183	C₄×C₈	—	146	2	no	—
184	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
185	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
188	C₂⁴	—	2	1	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	C₄⋊Dic₇₃	1	2	yes	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
198	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
199	C₂²×C₄	#11924028	1	1	yes	characteristic
200	M₁₆	—	292	2	no	—
201	M₁₆	—	292	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	292	2	no	—
209	C₂×C₈	—	292	2	no	—
210	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	C₄⋊Dic₇₃	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
219	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
220	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
221	C₄²	#11924028	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	#14556570	1	1	yes	characteristic
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	#14556527	1	4	yes	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
239	C₂×C₄	#14556550	1	1	yes	characteristic
240	C₂×C₄	#14556565	1	1	yes	characteristic
241	C₂×C₄	#14556565	1	1	yes	characteristic
242	C₂×C₄	#14556570	1	1	yes	characteristic
243	C₂×C₄	#14556611	1	2	yes	—
244	C₂×C₄	#14556617	1	2	yes	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	4	2	no	—
250	C₈	—	584	2	no	—
251	C₈	—	584	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	8	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	#17988447	1	1	yes	characteristic
259	C₂²	#17988887	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	#17988583	1	1	yes	characteristic
265	C₄	#17988592	1	1	yes	characteristic
266	C₄	C₂×#14556527	1	2	yes	—
267	C₄	#17988802	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	4	no	—
270	C₂	#22609941	1	1	yes	characteristic
271	C₂	#22609969	1	1	yes	characteristic
272	C₂	#22609985	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556565
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×C₈×D₄×C₇₃	C₂
socle	C₂²×C₇₃	#380
prime radical	C₂²×D₄×C₇₃	Q₈
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×C₈×D₄	D₇₃
hypocenter	C₇₃	#8660
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #22610069	C₂	semidirect
2. #22608679	C₂	semidirect
4. #22607828	C₂	semidirect
142. C₇₃	#8660	semidirect

Maximal subgroups

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

subgroup	index
#22610069	2
#22608679	2
#22608356	2
#22607828	2
C₂×C₈×D₄×C₇₃	2
#8660	73