Subgroups — #34007481

Order 18688.

382 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626028	C₂	1	2	yes	maximal
2	C₂×#17989450	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989428	C₂	1	1	yes	maximal, characteristic
4	#22624767	C₂	1	2	yes	maximal
5	#106×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596833	—	2	2	no	—
7	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
8	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
10	#17989453	—	2	2	no	—
11	#17989450	—	2	2	no	—
12	#17989428	—	2	2	no	—
13	C₂×#14556523	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₇₃	C₂²	1	2	yes	—
16	#17988349	C₄	1	2	yes	—
17	C₂×#106×C₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596833	—	2	2	no	—
19	C₂²×#10596833	—	2	1	no	—
20	C₂²×#10596833	D₄	1	1	yes	characteristic
21	C₂²×#10596833	D₄	1	1	yes	characteristic
22	C₄⋊#10596833	—	2	2	no	—
23	C₄⋊#10596833	—	2	2	no	—
24	C₄⋊#10596833	—	4	4	no	—
25	C₄×#10596833	—	2	1	no	—
26	C₄×#10596833	—	2	1	no	—
27	C₄×#10596833	—	2	1	no	—
28	C₄×#10596833	Q₈	1	2	yes	—
29	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₇₃	—	2	2	no	—
32	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
33	#14556523	—	2	2	no	—
34	C₈×Dic₇₃	D₄	1	2	yes	—
35	C₈×Dic₇₃	Q₈	1	2	yes	—
36	#14556498	—	2	2	no	—
37	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
39	#106×C₇₃	C₈	1	4	yes	complemented
40	C₂×#10596833	—	2	1	no	—
41	C₂×#10596833	—	2	1	no	—
42	C₂×#10596833	—	2	2	no	—
43	C₂×#10596833	—	2	2	no	—
44	C₂×#10596833	—	2	2	no	—
45	C₂×#10596833	—	4	2	no	—
46	C₂×#10596833	—	4	2	no	—
47	C₂×#10596833	—	4	2	no	—
48	C₂×#10596833	—	4	2	no	—
49	C₂²×Dic₇₃	—	2	2	no	—
50	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₇₃	#41	1	1	yes	characteristic
52	C₂×#10596831	—	2	1	no	—
53	C₂×#10596831	—	2	1	no	—
54	C₂×#10596831	C₄⋊C₄	1	2	yes	—
55	C₄⋊Dic₇₃	—	2	2	no	—
56	C₄⋊Dic₇₃	—	2	2	no	—
57	C₄×Dic₇₃	—	4	2	no	—
58	C₄×Dic₇₃	—	4	2	no	—
59	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
60	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
61	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
62	C₄×Dic₇₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	—	2	2	no	—
64	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
68	C₄⋊C₄×C₇₃	—	2	2	no	—
69	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
70	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	—	2	4	no	—
72	C₂×C₄×C₇₃	—	4	2	no	—
73	C₂×C₄×C₇₃	—	4	2	no	—
74	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
79	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
80	C₂×Dic₇₃	—	2	2	no	—
81	C₂×Dic₇₃	—	4	2	no	—
82	C₂×Dic₇₃	—	4	2	no	—
83	C₂×Dic₇₃	—	4	2	no	—
84	C₂×Dic₇₃	#121	1	1	yes	characteristic
85	C₂×Dic₇₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×Dic₇₃	#125	1	1	yes	characteristic
87	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
89	C₂×Dic₇₃	#137	1	2	yes	—
90	#10596833	—	4	2	no	—
91	#10596833	—	4	1	no	—
92	#10596833	—	4	1	no	—
93	#10596833	—	8	2	no	—
94	C₈×C₇₃	C₄⋊C₈	1	4	yes	complemented
95	#10596831	—	2	2	no	—
96	C₂²×C₇₃	#389	1	1	yes	characteristic
97	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
98	C₂²×C₇₃	#436	1	1	yes	characteristic
99	C₂²×C₇₃	#440	1	2	yes	—
100	C₂²×C₇₃	#444	1	2	yes	—
101	C₄×C₇₃	—	4	4	no	—
102	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₄×C₇₃	#424	1	1	yes	characteristic
104	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
105	C₄×C₇₃	#446	1	1	yes	characteristic
106	Dic₇₃	—	2	1	no	—
107	Dic₇₃	—	2	1	no	—
108	Dic₇₃	—	8	2	no	—
109	Dic₇₃	#469	1	1	yes	characteristic
110	Dic₇₃	#470	1	1	yes	characteristic
111	Dic₇₃	#501	1	2	yes	—
112	#9596	—	73	1	no	maximal, Sylow
113	C₂×C₇₃	SD₁₆⋊C₈	1	2	yes	—
114	C₂×C₇₃	#1592	1	2	yes	—
115	C₂×C₇₃	#1939	1	1	yes	characteristic
116	C₂×C₇₃	#1944	1	1	yes	characteristic
117	C₂×C₇₃	#1946	1	1	yes	characteristic
118	#1939	—	73	2	no	—
119	C₄×#106	—	73	1	no	—
120	C₂×#335	—	73	1	no	—
121	C₂×#322	—	73	1	no	—
122	#1279	—	73	2	no	—
123	C₇₃	#9596	1	1	yes	characteristic, complemented, Sylow
124	C₂×#106	—	73	1	no	—
125	C₂×#106	#64408	1	1	yes	characteristic
126	C₂×C₄⋊C₈	—	73	2	no	—
127	C₂×C₄⋊C₈	—	146	2	no	—
128	C₂×C₄×C₈	—	73	1	no	—
129	C₂×C₄×C₈	—	73	1	no	—
130	C₂×C₄×C₈	—	73	1	no	—
131	C₄×C₄⋊C₄	—	73	2	no	—
132	#340	—	73	2	no	—
133	#336	—	146	2	no	—
134	#335	—	146	2	no	—
135	#322	—	146	2	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂²×C₈	—	73	1	no	—
139	C₂²×C₈	—	146	2	no	—
140	C₂²×C₈	—	146	1	no	—
141	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	73	2	no	—
143	C₂×C₄⋊C₄	C₂×#64408	1	2	yes	—
144	C₂×C₄²	—	73	1	no	—
145	C₂×C₄²	—	146	2	no	—
146	#106	—	146	2	no	—
147	#106	#10596833	1	4	yes	complemented
148	C₄⋊C₈	—	292	4	no	—
149	C₄⋊C₈	—	146	2	no	—
150	C₄⋊C₈	—	146	2	no	—
151	C₄×C₈	—	73	2	no	—
152	C₄×C₈	—	73	2	no	—
153	C₄×C₈	—	73	2	no	—
154	C₄×C₈	—	146	1	no	—
155	C₄×C₈	—	146	1	no	—
156	C₄×C₈	—	146	1	no	—
157	#95	—	146	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	73	1	no	—
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	—	146	2	no	—
162	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
163	C₂×C₈	—	292	2	no	—
164	C₂×C₈	—	292	2	no	—
165	C₂×C₈	—	292	2	no	—
166	C₂×C₈	—	292	2	no	—
167	C₂×C₈	—	73	2	no	—
168	C₂×C₈	—	146	2	no	—
169	C₂×C₈	—	146	1	no	—
170	C₂×C₈	—	146	1	no	—
171	C₂×C₈	—	146	2	no	—
172	C₂×C₈	—	146	2	no	—
173	C₂×C₈	—	146	1	no	—
174	C₂×C₈	—	146	1	no	—
175	C₂×C₈	C₄⋊#64408	1	2	yes	—
176	C₂×C₈	C₂×#10596833	1	2	yes	—
177	C₂×C₈	#11924038	1	2	yes	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	146	2	no	—
180	C₄⋊C₄	—	146	2	no	—
181	C₄⋊C₄	C₂×#10596833	1	4	yes	—
182	C₄²	—	292	2	no	—
183	C₄²	—	292	2	no	—
184	C₄²	—	73	1	no	—
185	C₄²	—	73	1	no	—
186	C₄²	—	73	1	no	—
187	C₄²	—	73	1	no	—
188	C₂³	D₄×#64408	1	1	yes	characteristic
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	292	2	no	—
193	C₂×C₄	—	292	2	no	—
194	C₂×C₄	—	292	2	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	73	1	no	—
197	C₂×C₄	—	73	2	no	—
198	C₂×C₄	—	73	2	no	—
199	C₂×C₄	—	73	1	no	—
200	C₂×C₄	—	73	1	no	—
201	C₂×C₄	—	146	2	no	—
202	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
203	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
204	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
205	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
206	C₂×C₄	#14556678	1	1	yes	characteristic
207	C₂×C₄	#14556679	1	1	yes	characteristic
208	C₈	—	292	1	no	—
209	C₈	—	292	1	no	—
210	C₈	—	292	2	no	—
211	C₈	—	584	2	no	—
212	C₈	—	146	2	no	—
213	C₈	C₄⋊#10596833	1	4	yes	complemented
214	C₂²	#17989538	1	1	yes	characteristic
215	C₂²	D₄×#10596833	1	1	yes	characteristic
216	C₂²	#17989621	1	1	yes	characteristic
217	C₂²	#17989626	1	2	yes	—
218	C₂²	#17989632	1	2	yes	—
219	C₄	—	4	4	no	—
220	C₄	—	584	2	no	—
221	C₄	—	73	1	no	—
222	C₄	—	73	1	no	—
223	C₄	—	73	2	no	—
224	C₄	—	146	1	no	—
225	C₄	—	146	1	no	—
226	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
227	C₄	#17989594	1	1	yes	characteristic
228	C₄	Q₈×#10596833	1	1	yes	characteristic
229	C₄	#17989637	1	1	yes	characteristic
230	C₂	SD₁₆⋊#10596833	1	2	yes	—
231	C₂	#22625573	1	2	yes	—
232	C₂	#22626028	1	1	yes	characteristic
233	C₂	#22626037	1	1	yes	characteristic
234	C₂	#22626041	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#106×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#106	#64408
hypocenter	C₇₃	#9596
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
39. #106×C₇₃	C₈	semidirect
94. C₈×C₇₃	C₄⋊C₈	semidirect
123. C₇₃	#9596	semidirect
147. #106	#10596833	semidirect
213. C₈	C₄⋊#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22626028	2
C₂×#17989450	2
C₂×#17989428	2
#22624767	2
#106×Dic₇₃	2
#9596	73