Subgroups — #34004011

Order 18688.

414 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625727	C₂	1	2	yes	maximal
2	C₂×#17989451	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#10596833	C₂	1	1	yes	maximal, characteristic
4	#22624767	C₂	1	2	yes	maximal
5	#22609823	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
7	C₂×#14556646	—	2	2	no	—
8	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
10	#17989453	—	2	2	no	—
11	#17989451	C₄	1	4	yes	complemented
12	C₈×#10596833	C₄	1	4	yes	complemented
13	C₂×#14556524	C₄	1	2	yes	—
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₇₃	C₂²	1	2	yes	—
16	#17988350	C₄	1	1	yes	characteristic
17	#340×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	C₂×C₄	1	4	yes	—
24	#14556646	—	4	4	no	—
25	C₄×#10596833	—	2	1	no	—
26	C₄×#10596833	C₂×C₄	1	2	yes	—
27	C₄×#10596833	C₂×C₄	1	2	yes	—
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	#14556524	—	2	4	no	—
34	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
35	C₈×Dic₇₃	C₂×C₄	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	C₂×#10596833	—	2	4	no	—
40	C₂×#10596833	—	2	2	no	—
41	C₂×#10596833	—	2	1	no	—
42	C₂×#10596833	—	2	1	no	—
43	C₂×#10596833	—	2	4	no	—
44	C₂×#10596833	—	4	2	no	—
45	C₂×#10596833	—	4	2	no	—
46	C₂×#10596833	C₄⋊C₄	1	2	yes	—
47	C₂×#10596833	C₄⋊C₄	1	2	yes	—
48	C₂×#10596833	D₈	1	2	yes	—
49	C₂×#10596833	Q₁₆	1	2	yes	—
50	C₂²×Dic₇₃	—	2	2	no	—
51	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₇₃	#41	1	1	yes	characteristic
53	C₂×#10596831	C₄²	1	2	yes	—
54	C₂×#10596831	M₁₆	1	4	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	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₇₃	C₂²×C₄	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	—	2	2	no	—
66	C₂×C₈×C₇₃	C₄²	1	2	yes	—
67	C₄⋊C₄×C₇₃	—	2	2	no	—
68	C₄⋊C₄×C₇₃	—	2	2	no	—
69	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	4	2	no	—
71	C₂×C₄×C₇₃	—	4	2	no	—
72	C₂×C₄×C₇₃	—	4	2	no	—
73	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₇₃	#131	1	2	yes	—
77	C₂×Dic₇₃	—	2	2	no	—
78	C₂×Dic₇₃	—	4	2	no	—
79	C₂×Dic₇₃	—	4	2	no	—
80	C₂×Dic₇₃	—	4	2	no	—
81	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×Dic₇₃	#117	1	1	yes	characteristic
83	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
84	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
85	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
86	C₂×Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×Dic₇₃	#135	1	1	yes	characteristic
88	C₂×Dic₇₃	#135	1	1	yes	characteristic
89	#10596833	—	2	4	no	—
90	#10596833	—	2	2	no	—
91	#10596833	—	4	4	no	—
92	#10596833	#107	1	4	yes	complemented
93	C₈×C₇₃	—	2	2	no	—
94	#10596831	#97	1	4	yes	complemented
95	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₇₃	#436	1	2	yes	—
97	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
98	C₂²×C₇₃	C₄×Q₁₆	1	1	yes	characteristic
99	C₂²×C₇₃	#443	1	2	yes	—
100	C₄×C₇₃	—	8	2	no	—
101	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
102	C₄×C₇₃	#405	1	1	yes	characteristic
103	C₄×C₇₃	#446	1	2	yes	—
104	Dic₇₃	—	2	1	no	—
105	Dic₇₃	—	2	1	no	—
106	Dic₇₃	—	8	2	no	—
107	Dic₇₃	C₂×#107	1	1	yes	characteristic
108	Dic₇₃	#427	1	1	yes	characteristic
109	Dic₇₃	C₄×D₈	1	1	yes	characteristic
110	Dic₇₃	C₄×Q₁₆	1	1	yes	characteristic
111	#8282	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	#1589	1	2	yes	—
113	C₂×C₇₃	#1591	1	2	yes	—
114	C₂×C₇₃	#1770	1	1	yes	characteristic
115	C₂×C₇₃	#1772	1	1	yes	characteristic
116	C₂×C₇₃	C₄×#107	1	1	yes	characteristic
117	C₄×#107	—	73	1	no	—
118	#1770	—	73	2	no	—
119	C₂×#335	—	73	1	no	—
120	C₂×C₈²	—	73	1	no	—
121	#1279	—	73	2	no	—
122	C₇₃	#8282	1	1	yes	characteristic, complemented, Sylow
123	C₂×#107	—	73	2	no	—
124	C₂×C₄⋊C₈	—	73	2	no	—
125	C₂×#97	—	146	2	no	—
126	C₂×C₄×C₈	—	73	1	no	—
127	C₂×C₄×C₈	—	73	1	no	—
128	C₂×C₄×C₈	—	73	1	no	—
129	C₄×C₄⋊C₄	—	73	2	no	—
130	#340	—	73	1	no	—
131	#340	#64408	1	1	yes	characteristic
132	#336	—	146	2	no	—
133	#335	—	73	4	no	—
134	C₈²	—	73	4	no	—
135	C₂²×C₈	—	73	1	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂²×C₈	—	146	2	no	—
139	C₂²×C₈	—	146	1	no	—
140	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	73	2	no	—
142	C₂×C₄⋊C₄	C₂×#64408	1	2	yes	—
143	C₂×C₄²	—	73	1	no	—
144	C₂×C₄²	—	146	2	no	—
145	#107	—	146	4	no	—
146	C₄⋊C₈	—	73	4	no	—
147	C₄⋊C₈	—	146	2	no	—
148	#97	—	292	4	no	—
149	C₄×C₈	—	73	2	no	—
150	C₄×C₈	—	73	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	#95	—	146	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	73	1	no	—
158	C₂²×C₄	—	73	1	no	—
159	C₂²×C₄	—	146	2	no	—
160	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	292	2	no	—
163	C₂×C₈	—	292	2	no	—
164	C₂×C₈	—	73	2	no	—
165	C₂×C₈	—	73	2	no	—
166	C₂×C₈	—	73	2	no	—
167	C₂×C₈	—	73	4	no	—
168	C₂×C₈	—	73	2	no	—
169	C₂×C₈	—	73	2	no	—
170	C₂×C₈	—	146	4	no	—
171	C₂×C₈	—	146	2	no	—
172	C₂×C₈	—	146	1	no	—
173	C₂×C₈	—	146	1	no	—
174	C₂×C₈	—	146	4	no	—
175	C₂×C₈	C₄×#64408	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	146	2	no	—
179	C₄⋊C₄	—	146	2	no	—
180	C₄²	—	292	2	no	—
181	C₄²	—	292	2	no	—
182	C₄²	—	73	1	no	—
183	C₄²	—	73	1	no	—
184	C₄²	—	73	1	no	—
185	C₄²	—	73	1	no	—
186	C₂³	D₄×#64408	1	1	yes	characteristic
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	292	2	no	—
191	C₂×C₄	—	292	2	no	—
192	C₂×C₄	—	292	2	no	—
193	C₂×C₄	—	73	1	no	—
194	C₂×C₄	—	73	1	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	73	1	no	—
197	C₂×C₄	—	73	1	no	—
198	C₂×C₄	—	73	1	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	D₇₃⋊M₁₆	1	2	yes	—
205	C₂×C₄	#14556678	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	292	4	no	—
208	C₈	—	73	4	no	—
209	C₈	—	73	4	no	—
210	C₈	—	146	2	no	—
211	C₈	—	146	4	no	—
212	C₂²	C₄⋊C₄×#64408	1	1	yes	characteristic
213	C₂²	#17989620	1	2	yes	—
214	C₂²	D₈×#64408	1	1	yes	characteristic
215	C₂²	Q₁₆×#64408	1	1	yes	characteristic
216	C₂²	#17989630	1	2	yes	—
217	C₄	—	8	2	no	—
218	C₄	—	584	2	no	—
219	C₄	—	73	1	no	—
220	C₄	—	73	1	no	—
221	C₄	—	73	1	no	—
222	C₄	—	73	1	no	—
223	C₄	—	146	1	no	—
224	C₄	—	146	1	no	—
225	C₄	#17989546	1	1	yes	characteristic
226	C₄	#17989551	1	1	yes	characteristic
227	C₄	#17989639	1	2	yes	—
228	C₂	#22625562	1	2	yes	—
229	C₂	#22625569	1	2	yes	—
230	C₂	#22625726	1	1	yes	characteristic
231	C₂	#22625734	1	1	yes	characteristic
232	C₂	#107×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#340×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	#340	#64408
hypocenter	C₇₃	#8282
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

Complements

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

N	H ≅ G/N	type
11. #17989451	C₄	semidirect
12. C₈×#10596833	C₄	semidirect
92. #10596833	#107	semidirect
94. #10596831	#97	semidirect
122. C₇₃	#8282	semidirect

Maximal subgroups

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

subgroup	index
#22625727	2
C₂×#17989451	2
C₂×C₈×#10596833	2
#22624767	2
#22609823	2
#8282	73