Subgroups — #34003444

Order 18688.

506 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625162	C₂	1	2	yes	maximal
2	#22625134	C₂	1	2	yes	maximal, complemented
3	#22624771	C₂	1	2	yes	maximal, complemented
4	C₂×#17988542	C₂	1	1	yes	maximal, characteristic
5	#17989603	C₂²	1	4	yes	—
6	#17989432	C₄	1	4	yes	—
7	#17989430	C₄	1	4	yes	—
8	#17989428	—	2	2	no	—
9	C₈×#10596833	—	2	2	no	—
10	C₂²×#11924012	C₄	1	1	yes	characteristic
11	C₂×#14556550	C₂²	1	1	yes	characteristic
12	#17988542	—	2	2	no	—
13	#17988542	—	2	2	no	—
14	C₂×#14556501	C₄	1	1	yes	characteristic
15	C₂×#14556501	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
19	C₄⋊#10596833	—	2	4	no	—
20	C₂²⋊#10596833	C₂×C₄	1	8	yes	—
21	C₄×#10596833	—	2	4	no	—
22	C₂×#11924012	—	2	2	no	—
23	C₂×#11924012	—	2	2	no	—
24	C₂×#11924012	C₈	1	4	yes	complemented
25	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	#14556550	—	2	2	no	—
28	#14556550	—	2	2	no	—
29	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₂²⋊Dic₇₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
33	#14556501	—	2	2	no	—
34	#14556501	—	2	2	no	—
35	C₈×Dic₇₃	—	2	1	no	—
36	C₈×Dic₇₃	—	2	1	no	—
37	C₈×Dic₇₃	D₄	1	2	yes	—
38	C₈×Dic₇₃	D₄	1	2	yes	—
39	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₃	—	2	1	no	—
41	C₂²×C₈×C₇₃	—	2	1	no	—
42	C₂²×C₈×C₇₃	—	2	2	no	—
43	C₂²×C₈×C₇₃	C₈	1	4	yes	complemented
44	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×#10596833	—	2	8	no	—
47	C₂²×Dic₇₃	C₄²	1	2	yes	—
48	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
49	#11924012	—	2	8	no	—
50	C₂×#10596831	—	2	1	no	—
51	C₂×#10596831	—	2	1	no	—
52	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
53	C₂×#10596831	C₂×C₈	1	2	yes	—
54	C₂×#10596831	C₂×C₈	1	2	yes	—
55	C₂×#10596831	M₁₆	1	2	yes	—
56	C₄⋊Dic₇₃	—	2	4	no	—
57	C₂²⋊Dic₇₃	—	2	4	no	—
58	C₄×Dic₇₃	—	2	1	no	—
59	C₄×Dic₇₃	—	2	1	no	—
60	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
61	C₄×Dic₇₃	#41	1	2	yes	—
62	C₂⁴×C₇₃	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	—	2	1	no	—
64	C₂²×C₄×C₇₃	—	2	1	no	—
65	C₂²×C₄×C₇₃	—	2	1	no	—
66	C₂²×C₄×C₇₃	—	2	1	no	—
67	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₃	—	2	1	no	—
72	C₂×C₈×C₇₃	—	2	1	no	—
73	C₂×C₈×C₇₃	—	2	4	no	—
74	C₂×C₈×C₇₃	—	2	4	no	—
75	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
80	C₂³×C₇₃	—	2	1	no	—
81	C₂³×C₇₃	—	2	1	no	—
82	C₂³×C₇₃	—	4	2	no	—
83	C₂³×C₇₃	C₄×C₈	1	2	yes	—
84	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₇₃	—	2	1	no	—
86	C₂×C₄×C₇₃	—	2	1	no	—
87	C₂×C₄×C₇₃	—	2	2	no	—
88	C₂×C₄×C₇₃	—	2	2	no	—
89	C₂×C₄×C₇₃	—	2	2	no	—
90	C₂×C₄×C₇₃	—	2	2	no	—
91	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
92	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×Dic₇₃	—	2	2	no	—
99	C₂×Dic₇₃	—	4	2	no	—
100	C₂×Dic₇₃	—	4	2	no	—
101	C₂×Dic₇₃	C₄×D₄	1	4	yes	—
102	#10596833	—	4	8	no	—
103	C₈×C₇₃	—	2	2	no	—
104	C₈×C₇₃	C₂²⋊C₈	1	2	yes	—
105	C₈×C₇₃	C₂²⋊C₈	1	2	yes	—
106	#10596831	—	2	2	no	—
107	#10596831	C₂²⋊C₈	1	4	yes	—
108	C₂²×C₇₃	—	2	2	no	—
109	C₂²×C₇₃	—	2	2	no	—
110	C₂²×C₇₃	—	4	2	no	—
111	C₂²×C₇₃	—	4	2	no	—
112	C₂²×C₇₃	—	4	2	no	—
113	C₂²×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
115	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
116	C₄×C₇₃	—	2	2	no	—
117	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₄×C₇₃	#431	1	1	yes	characteristic
120	C₄×C₇₃	#431	1	1	yes	characteristic
121	Dic₇₃	—	2	4	no	—
122	Dic₇₃	—	4	2	no	—
123	#8156	—	73	1	no	maximal, Sylow
124	C₂×C₇₃	—	2	1	no	—
125	C₂×C₇₃	—	2	1	no	—
126	C₂×C₇₃	—	4	2	no	—
127	C₂×C₇₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
128	C₂×C₇₃	#1757	1	2	yes	—
129	C₂×#405	—	73	1	no	—
130	#1458	—	73	2	no	—
131	#1450	—	73	2	no	—
132	#1281	—	73	2	no	—
133	C₇₃	#8156	1	1	yes	characteristic, complemented, Sylow
134	C₂²×M₁₆	—	73	1	no	—
135	C₂³×C₈	#64408	1	1	yes	characteristic
136	C₂×#117	—	73	1	no	—
137	#431	—	73	4	no	—
138	#405	—	146	2	no	—
139	#405	—	146	2	no	—
140	C₂×#97	—	73	1	no	—
141	C₂×#97	—	73	1	no	—
142	C₂×C₄×C₈	—	73	1	no	—
143	C₂×C₄×C₈	—	73	1	no	—
144	#324	—	73	4	no	—
145	#323	—	73	4	no	—
146	#322	—	146	2	no	—
147	C₈²	—	146	2	no	—
148	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
149	C₂×M₁₆	—	73	4	no	—
150	C₂×M₁₆	—	146	2	no	—
151	C₂×M₁₆	—	146	2	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	73	1	no	—
156	C₂²×C₈	—	73	1	no	—
157	C₂²×C₈	#10596833	1	4	yes	complemented
158	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
159	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
160	#117	—	146	2	no	—
161	#117	—	146	2	no	—
162	C₂×C₄⋊C₄	—	73	2	no	—
163	C₂×C₂²⋊C₄	—	73	2	no	—
164	C₂×C₄²	—	73	1	no	—
165	C₂×C₄²	—	73	1	no	—
166	C₄⋊C₈	—	146	4	no	—
167	C₂²⋊C₈	—	73	8	no	—
168	#97	—	146	2	no	—
169	#97	—	146	2	no	—
170	C₄×C₈	—	73	2	no	—
171	C₄×C₈	—	73	2	no	—
172	C₄×C₈	—	146	4	no	—
173	C₄×C₈	—	146	1	no	—
174	C₄×C₈	—	146	1	no	—
175	C₂⁴	C₄×#64408	1	1	yes	characteristic
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	73	2	no	—
181	C₂²×C₄	—	73	2	no	—
182	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
183	C₂²×C₄	C₂×#10596833	1	2	yes	—
184	C₂²×C₄	#11924036	1	2	yes	—
185	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
186	M₁₆	—	146	8	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	73	2	no	—
192	C₂×C₈	—	73	2	no	—
193	C₂×C₈	—	73	2	no	—
194	C₂×C₈	—	73	2	no	—
195	C₂×C₈	—	146	8	no	—
196	C₂×C₈	—	146	1	no	—
197	C₂×C₈	—	146	1	no	—
198	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
199	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
200	C₂×C₈	C₂×#10596833	1	2	yes	—
201	C₂×C₈	C₂×#10596833	1	2	yes	—
202	C₂×C₈	#11924037	1	2	yes	—
203	C₄⋊C₄	—	146	4	no	—
204	C₂²⋊C₄	—	146	4	no	—
205	C₄²	—	73	2	no	—
206	C₄²	—	73	2	no	—
207	C₄²	—	146	1	no	—
208	C₄²	—	146	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	4	2	no	—
212	C₂³	C₄×#10596833	1	2	yes	—
213	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	292	2	no	—
221	C₂×C₄	—	292	2	no	—
222	C₂×C₄	—	73	4	no	—
223	C₂×C₄	—	146	2	no	—
224	C₂×C₄	C₄×#10596833	1	2	yes	—
225	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
226	C₂×C₄	#14556670	1	1	yes	characteristic
227	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
228	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
229	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
230	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	292	8	no	—
233	C₈	—	73	4	no	—
234	C₈	—	146	2	no	—
235	C₈	C₂²⋊#10596833	1	2	yes	—
236	C₈	C₂²⋊#10596833	1	2	yes	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	C₄×C₂²⋊#64408	1	1	yes	characteristic
243	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
244	C₂²	C₄×#11924037	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	292	2	no	—
247	C₄	—	146	4	no	—
248	C₄	C₂×C₂²⋊#10596833	1	1	yes	characteristic
249	C₄	#17989555	1	1	yes	characteristic
250	C₄	#17989603	1	1	yes	characteristic
251	C₄	#17989604	1	1	yes	characteristic
252	C₂	—	2	1	no	—
253	C₂	—	2	1	no	—
254	C₂	—	4	2	no	—
255	C₂	C₄×C₂²⋊#10596833	1	1	yes	characteristic
256	C₂	#22625686	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊#64408
commutator	C₂²×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂³×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₇₃	C₄²
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂³×C₈	#64408
hypocenter	C₇₃	#8156
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #22625134	C₂	semidirect
3. #22624771	C₂	semidirect
24. C₂×#11924012	C₈	semidirect
43. C₂²×C₈×C₇₃	C₈	semidirect
133. C₇₃	#8156	semidirect
157. C₂²×C₈	#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22625162	2
#22625134	2
#22624771	2
C₂×#17988542	2
#8156	73