Subgroups — #32973129

Order 18688.

402 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22609960	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988350	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₇₃	—	2	1	no	—
5	C₂×#14556525	C₄	1	2	yes	complemented
6	#17988447	—	2	1	no	—
7	#17988350	—	2	1	no	—
8	#17988350	—	2	1	no	—
9	#17988350	C₄	1	2	yes	complemented
10	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₇₃	—	2	1	no	—
12	C₂²⋊C₁₆×C₇₃	—	2	1	no	—
13	C₂²⋊C₁₆×C₇₃	C₄	1	2	yes	complemented
14	C₂³×Dic₇₃	—	4	1	no	—
15	C₂×C₄⋊Dic₇₃	—	2	4	no	—
16	C₂×C₄⋊Dic₇₃	—	2	2	no	—
17	C₂×C₄⋊Dic₇₃	—	2	2	no	—
18	C₂×C₄⋊Dic₇₃	—	4	2	no	—
19	C₂×C₄⋊Dic₇₃	—	4	1	no	—
20	C₂×C₂²⋊Dic₇₃	—	4	1	no	—
21	#14556525	—	4	2	no	—
22	#14556498	—	4	1	no	—
23	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	—	4	2	no	—
25	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	—	2	2	no	—
32	C₂×C₁₆×C₇₃	—	4	2	no	—
33	C₂²×Dic₇₃	—	4	4	no	—
34	C₂²×Dic₇₃	—	4	2	no	—
35	C₂²×Dic₇₃	—	4	2	no	—
36	C₂²×Dic₇₃	—	8	2	no	—
37	C₂²×Dic₇₃	—	8	1	no	—
38	C₂²×Dic₇₃	—	8	1	no	—
39	C₄⋊Dic₇₃	—	2	4	no	—
40	C₄⋊Dic₇₃	—	4	4	no	—
41	C₄⋊Dic₇₃	—	4	2	no	—
42	C₄⋊Dic₇₃	—	4	2	no	—
43	C₄⋊Dic₇₃	—	4	2	no	—
44	C₂²⋊Dic₇₃	—	8	1	no	—
45	C₂²⋊Dic₇₃	—	8	1	no	—
46	C₂⁴×C₇₃	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	—	2	2	no	—
48	C₂²×C₄×C₇₃	—	2	2	no	—
49	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
55	C₂×C₈×C₇₃	—	2	2	no	—
56	C₂×C₈×C₇₃	—	2	2	no	—
57	C₂×C₈×C₇₃	—	4	1	no	—
58	C₂×C₈×C₇₃	—	4	1	no	—
59	C₂×C₈×C₇₃	—	4	1	no	—
60	C₂×C₈×C₇₃	—	4	1	no	—
61	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
65	C₁₆×C₇₃	—	4	2	no	—
66	C₂³×C₇₃	—	2	2	no	—
67	C₂³×C₇₃	—	2	2	no	—
68	C₂³×C₇₃	#102	1	1	yes	characteristic
69	C₂³×C₇₃	#102	1	1	yes	characteristic
70	C₂³×C₇₃	#107	1	1	yes	characteristic
71	C₂³×C₇₃	D₁₆	1	1	yes	characteristic
72	C₂³×C₇₃	SD₃₂	1	1	yes	characteristic
73	C₂³×C₇₃	SD₃₂	1	1	yes	characteristic
74	C₂³×C₇₃	Q₃₂	1	1	yes	characteristic
75	C₂×C₄×C₇₃	—	2	2	no	—
76	C₂×C₄×C₇₃	—	2	2	no	—
77	C₂×C₄×C₇₃	—	2	2	no	—
78	C₂×C₄×C₇₃	—	2	1	no	—
79	C₂×C₄×C₇₃	—	2	1	no	—
80	C₂×C₄×C₇₃	—	2	1	no	—
81	C₂×C₄×C₇₃	—	2	1	no	—
82	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
83	C₂×C₄×C₇₃	#99	1	2	yes	—
84	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
85	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#104	1	2	yes	—
87	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
88	C₂×Dic₇₃	—	4	4	no	—
89	C₂×Dic₇₃	—	8	4	no	—
90	C₂×Dic₇₃	—	8	4	no	—
91	C₂×Dic₇₃	—	8	2	no	—
92	C₂×Dic₇₃	—	8	2	no	—
93	C₂×Dic₇₃	—	8	2	no	—
94	C₂×Dic₇₃	—	8	2	no	—
95	C₂×Dic₇₃	—	8	2	no	—
96	C₈×C₇₃	—	2	1	no	—
97	C₈×C₇₃	—	2	1	no	—
98	C₈×C₇₃	—	2	1	no	—
99	C₈×C₇₃	—	2	1	no	—
100	C₂²×C₇₃	—	2	1	no	—
101	C₂²×C₇₃	—	2	1	no	—
102	C₂²×C₇₃	—	2	2	no	—
103	C₂²×C₇₃	—	2	2	no	—
104	C₂²×C₇₃	—	2	2	no	—
105	C₂²×C₇₃	—	2	2	no	—
106	C₂²×C₇₃	—	2	1	no	—
107	C₂²×C₇₃	—	2	1	no	—
108	C₂²×C₇₃	#327	1	2	yes	—
109	C₂²×C₇₃	#340	1	1	yes	characteristic
110	C₂²×C₇₃	#357	1	1	yes	characteristic
111	C₂²×C₇₃	#357	1	1	yes	characteristic
112	C₂²×C₇₃	#358	1	1	yes	characteristic
113	C₂²×C₇₃	#358	1	1	yes	characteristic
114	C₂²×C₇₃	#360	1	2	yes	—
115	C₂²×C₇₃	#366	1	1	yes	characteristic
116	C₂²×C₇₃	#367	1	1	yes	characteristic
117	C₄×C₇₃	—	2	1	no	—
118	C₄×C₇₃	—	2	1	no	—
119	C₄×C₇₃	#328	1	2	yes	—
120	C₄×C₇₃	#339	1	1	yes	characteristic
121	C₄×C₇₃	#342	1	1	yes	characteristic
122	Dic₇₃	—	8	4	no	—
123	Dic₇₃	—	16	2	no	—
124	#7258	—	73	1	no	maximal, Sylow
125	C₂×C₇₃	—	2	1	no	—
126	C₂×C₇₃	—	2	1	no	—
127	C₂×C₇₃	—	2	1	no	—
128	C₂×C₇₃	—	2	1	no	—
129	C₂×C₇₃	#1294	1	1	yes	characteristic
130	C₂×C₇₃	#1347	1	2	yes	—
131	C₂×C₇₃	#1348	1	2	yes	—
132	C₂×C₇₃	#1380	1	1	yes	characteristic
133	C₂×C₇₃	#1386	1	1	yes	characteristic
134	C₂×C₂²⋊C₁₆	D₇₃	1	1	yes	characteristic
135	#1808	—	73	1	no	—
136	C₂×#340	—	73	1	no	—
137	C₇₃	#7258	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
139	C₂²×C₄⋊C₄	—	146	1	no	—
140	C₂²×C₁₆	—	2	1	no	—
141	C₂×#107	—	73	2	no	—
142	#380	—	146	1	no	—
143	C₂²⋊C₁₆	—	2	1	no	—
144	C₂²⋊C₁₆	Dic₇₃	1	2	yes	complemented
145	#340	—	73	2	no	—
146	#340	—	146	1	no	—
147	#340	—	146	1	no	—
148	C₂³×C₄	—	292	1	no	—
149	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
150	C₂²×C₈	—	4	2	no	—
151	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
152	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
153	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
154	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	292	2	no	—
158	C₂×C₄⋊C₄	—	292	1	no	—
159	C₂×C₄⋊C₄	—	146	2	no	—
160	C₂×C₄⋊C₄	—	146	4	no	—
161	C₂×C₄⋊C₄	—	146	2	no	—
162	C₂×C₂²⋊C₄	—	292	1	no	—
163	C₂×C₁₆	—	2	2	no	—
164	C₂×C₁₆	—	4	2	no	—
165	#107	—	292	2	no	—
166	#95	—	292	1	no	—
167	C₂⁴	D₅₈₄	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	292	4	no	—
171	C₂²×C₄	—	292	2	no	—
172	C₂²×C₄	—	292	2	no	—
173	C₂²×C₄	—	584	2	no	—
174	C₂²×C₄	—	584	1	no	—
175	C₂²×C₄	—	584	1	no	—
176	C₂²×C₄	#11924005	1	1	yes	characteristic
177	C₂²×C₄	#11924005	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
179	C₂²×C₄	#11924017	1	1	yes	characteristic
180	C₂²×C₄	#11924017	1	1	yes	characteristic
181	C₂²×C₄	Dic₂₉₂	1	1	yes	characteristic
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
190	C₂×C₈	#11924007	1	1	yes	characteristic
191	C₂×C₈	#11924007	1	1	yes	characteristic
192	C₄⋊C₄	—	292	2	no	—
193	C₄⋊C₄	—	292	4	no	—
194	C₄⋊C₄	—	292	2	no	—
195	C₄⋊C₄	—	292	2	no	—
196	C₄⋊C₄	—	146	4	no	—
197	C₂²⋊C₄	—	584	1	no	—
198	C₂²⋊C₄	—	584	1	no	—
199	C₁₆	—	4	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	#14556513	1	1	yes	characteristic
203	C₂³	#14556513	1	1	yes	characteristic
204	C₂³	#14556525	1	1	yes	characteristic
205	C₂³	D₁₁₆₈	1	1	yes	characteristic
206	C₂³	#14556536	1	1	yes	characteristic
207	C₂³	#14556536	1	1	yes	characteristic
208	C₂³	Dic₅₈₄	1	1	yes	characteristic
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	292	4	no	—
217	C₂×C₄	—	584	4	no	—
218	C₂×C₄	—	584	4	no	—
219	C₂×C₄	—	584	2	no	—
220	C₂×C₄	—	584	2	no	—
221	C₂×C₄	—	584	2	no	—
222	C₂×C₄	—	584	2	no	—
223	C₂×C₄	—	584	2	no	—
224	C₂×C₄	#14556498	1	1	yes	characteristic
225	C₂×C₄	#14556505	1	2	yes	—
226	C₂×C₄	#14556516	1	1	yes	characteristic
227	C₂×C₄	#14556516	1	1	yes	characteristic
228	C₂×C₄	#14556517	1	2	yes	—
229	C₂×C₄	#14556523	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	#17988322	1	2	yes	—
243	C₂²	#17988350	1	1	yes	characteristic
244	C₂²	#17988389	1	1	yes	characteristic
245	C₂²	#17988389	1	1	yes	characteristic
246	C₂²	#17988392	1	1	yes	characteristic
247	C₂²	#17988392	1	1	yes	characteristic
248	C₂²	#17988398	1	2	yes	—
249	C₂²	#17988413	1	1	yes	characteristic
250	C₂²	#17988415	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	584	4	no	—
254	C₄	—	1168	2	no	—
255	C₄	#17988325	1	2	yes	—
256	C₄	#17988348	1	1	yes	characteristic
257	C₄	#17988353	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#22607811	1	1	yes	characteristic
263	C₂	#22607941	1	2	yes	—
264	C₂	#22607944	1	2	yes	—
265	C₂	#22608023	1	1	yes	characteristic
266	C₂	#22608037	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556513
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂³×C₈	C₂×D₇₃
Fitting	C₂×C₂²⋊C₁₆×C₇₃	C₂
socle	C₂³×C₇₃	#102
prime radical	C₂⁴×C₇₃	D₈
cosocle	C₂³×C₈×C₇₃	C₂²
hypercenter	C₂×C₂²⋊C₁₆	D₇₃
hypocenter	C₇₃	#7258
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
5. C₂×#14556525	C₄	semidirect
9. #17988350	C₄	semidirect
13. C₂²⋊C₁₆×C₇₃	C₄	semidirect
137. C₇₃	#7258	semidirect
144. C₂²⋊C₁₆	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22609960	2
C₂×#17988350	2
C₂×C₂²⋊C₁₆×C₇₃	2
#7258	73