Subgroups — #34003921

Order 18688.

550 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556658	C₂	1	1	yes	maximal, characteristic
2	C₄²×#10596833	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#10596833	C₂	1	2	yes	maximal
4	#22624753	C₂	1	2	yes	maximal
5	#22609779	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556658	C₄	1	2	yes	—
7	C₂×C₄×#10596833	C₄	1	2	yes	—
8	C₂×C₄×#10596833	C₄	1	2	yes	—
9	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
10	C₂×C₄×#10596833	C₂²	1	2	yes	—
11	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
12	#17989453	C₄	1	2	yes	—
13	C₈×#10596833	—	2	4	no	—
14	C₂×#14556520	C₄	1	2	yes	—
15	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
16	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
17	#17988342	C₄	1	1	yes	characteristic
18	#336×C₇₃	C₄	1	1	yes	characteristic
19	C₂²×#10596833	C₂×C₄	1	4	yes	—
20	C₂²×#10596833	C₂×C₄	1	2	yes	—
21	C₂²×#10596833	C₂×C₄	1	2	yes	—
22	#14556658	C₈	1	8	yes	complemented
23	C₄×#10596833	—	2	4	no	—
24	C₄×#10596833	—	2	2	no	—
25	C₄×#10596833	—	2	1	no	—
26	C₄×#10596833	—	2	2	no	—
27	C₄×#10596833	—	2	1	no	—
28	C₄×#10596833	C₈	1	8	yes	complemented
29	C₄×#10596833	D₄	1	2	yes	—
30	C₄×#10596833	Q₈	1	2	yes	—
31	C₂²×#10596831	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
38	#14556520	C₈	1	8	yes	complemented
39	C₈×Dic₇₃	—	2	2	no	—
40	C₈×Dic₇₃	—	2	2	no	—
41	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×#10596833	—	2	2	no	—
44	C₂×#10596833	—	2	4	no	—
45	C₂×#10596833	—	2	2	no	—
46	C₂×#10596833	—	2	4	no	—
47	C₂×#10596833	C₄⋊C₄	1	4	yes	—
48	C₂×#10596833	C₂×C₈	1	4	yes	—
49	C₂×#10596833	C₂×C₈	1	8	yes	—
50	C₂×#10596833	C₂×C₈	1	4	yes	—
51	C₂×#10596833	M₁₆	1	4	yes	—
52	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
53	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
54	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×#10596831	—	2	2	no	—
57	C₂×#10596831	—	2	2	no	—
58	C₂×#10596831	C₂×C₈	1	4	yes	—
59	C₄×Dic₇₃	—	2	2	no	—
60	C₄×Dic₇₃	—	2	2	no	—
61	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
62	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
63	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
64	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
65	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
66	C₄×Dic₇₃	#41	1	1	yes	characteristic
67	C₄×Dic₇₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₃	—	2	2	no	—
72	C₂×C₈×C₇₃	—	2	2	no	—
73	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
74	C₄²×C₇₃	—	2	2	no	—
75	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₇₃	—	2	1	no	—
77	C₂×C₄×C₇₃	—	2	1	no	—
78	C₂×C₄×C₇₃	—	2	1	no	—
79	C₂×C₄×C₇₃	—	2	1	no	—
80	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
81	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
82	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₇₃	#131	1	2	yes	—
86	C₂×Dic₇₃	—	2	1	no	—
87	C₂×Dic₇₃	—	2	1	no	—
88	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
89	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
90	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
91	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
92	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×Dic₇₃	#117	1	1	yes	characteristic
94	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
95	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
96	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
97	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
98	C₂×Dic₇₃	C₂×M₁₆	1	2	yes	—
99	C₂×Dic₇₃	#131	1	2	yes	—
100	#10596833	—	2	4	no	—
101	#10596833	—	2	8	no	—
102	#10596833	C₄⋊C₈	1	8	yes	complemented
103	C₈×C₇₃	—	2	4	no	—
104	#10596831	—	2	4	no	—
105	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
107	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
108	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
109	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
110	C₂²×C₇₃	#405	1	1	yes	characteristic
111	C₂²×C₇₃	#405	1	1	yes	characteristic
112	C₄×C₇₃	—	2	2	no	—
113	C₄×C₇₃	—	2	2	no	—
114	C₄×C₇₃	C₈×D₄	1	2	yes	—
115	C₄×C₇₃	C₈×Q₈	1	2	yes	—
116	Dic₇₃	—	2	2	no	—
117	Dic₇₃	C₈²	1	4	yes	complemented
118	Dic₇₃	C₂×C₄⋊C₈	1	2	yes	—
119	Dic₇₃	#431	1	2	yes	—
120	Dic₇₃	C₈×D₄	1	2	yes	—
121	Dic₇₃	C₈×Q₈	1	2	yes	—
122	C₈×C₄⋊C₈	—	73	1	no	maximal, Sylow
123	C₂×C₇₃	C₂×C₈²	1	1	yes	characteristic
124	C₂×C₇₃	C₈×M₁₆	1	1	yes	characteristic
125	C₂×C₇₃	C₈×M₁₆	1	1	yes	characteristic
126	C₂×C₇₃	#1450	1	1	yes	characteristic
127	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
128	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₄⋊C₈	—	73	1	no	—
131	C₄×C₄⋊C₈	—	73	1	no	—
132	C₄²×C₈	—	73	1	no	—
133	C₂×C₈²	—	73	2	no	—
134	#1273	—	73	2	no	—
135	C₇₃	C₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
136	C₂×C₄⋊C₈	—	73	2	no	—
137	C₂×C₄⋊C₈	—	73	2	no	—
138	C₂×C₄×C₈	—	73	2	no	—
139	C₂×C₄×C₈	—	73	1	no	—
140	C₂×C₄×C₈	—	73	2	no	—
141	C₂×C₄×C₈	—	73	2	no	—
142	C₂×C₄×C₈	—	73	1	no	—
143	C₂×C₄×C₈	—	73	2	no	—
144	C₄³	—	73	1	no	—
145	#336	—	73	2	no	—
146	#336	—	73	1	no	—
147	#336	#64408	1	1	yes	characteristic
148	C₈²	—	146	4	no	—
149	C₂²×C₈	—	73	2	no	—
150	C₂²×C₈	—	73	4	no	—
151	C₂²×C₈	—	73	2	no	—
152	C₂²×C₈	—	73	2	no	—
153	C₂²×C₈	C₂×#64408	1	2	yes	—
154	C₂×C₄²	—	73	1	no	—
155	C₂×C₄²	—	73	1	no	—
156	C₂×C₄²	—	73	1	no	—
157	C₂×C₄²	—	73	1	no	—
158	C₂×C₄²	—	73	1	no	—
159	C₂×C₄²	—	73	1	no	—
160	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
161	C₄⋊C₈	—	73	8	no	—
162	C₄⋊C₈	—	73	8	no	—
163	C₄×C₈	—	73	2	no	—
164	C₄×C₈	—	73	8	no	—
165	C₄×C₈	—	73	2	no	—
166	C₄×C₈	—	146	2	no	—
167	C₄×C₈	—	146	2	no	—
168	C₄×C₈	—	146	2	no	—
169	C₄×C₈	—	146	2	no	—
170	C₄×C₈	—	146	1	no	—
171	C₄×C₈	—	146	1	no	—
172	C₄×C₈	—	146	4	no	—
173	C₂²×C₄	—	73	1	no	—
174	C₂²×C₄	—	73	1	no	—
175	C₂²×C₄	—	73	1	no	—
176	C₂²×C₄	—	73	1	no	—
177	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
178	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
179	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	73	4	no	—
183	C₂×C₈	—	73	4	no	—
184	C₂×C₈	—	73	4	no	—
185	C₂×C₈	—	73	4	no	—
186	C₂×C₈	—	73	8	no	—
187	C₂×C₈	—	73	4	no	—
188	C₂×C₈	—	146	4	no	—
189	C₂×C₈	—	146	2	no	—
190	C₂×C₈	—	146	2	no	—
191	C₂×C₈	—	146	4	no	—
192	C₂×C₈	—	146	2	no	—
193	C₂×C₈	—	146	2	no	—
194	C₂×C₈	#11924036	1	4	yes	—
195	C₄²	—	2	2	no	—
196	C₄²	—	73	1	no	—
197	C₄²	—	73	1	no	—
198	C₄²	—	73	4	no	—
199	C₄²	—	73	4	no	—
200	C₄²	—	73	4	no	—
201	C₄²	—	73	1	no	—
202	C₄²	—	73	1	no	—
203	C₄²	—	146	2	no	—
204	C₄²	—	146	2	no	—
205	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	73	1	no	—
211	C₂×C₄	—	73	1	no	—
212	C₂×C₄	—	73	1	no	—
213	C₂×C₄	—	73	1	no	—
214	C₂×C₄	—	73	2	no	—
215	C₂×C₄	—	73	2	no	—
216	C₂×C₄	—	73	2	no	—
217	C₂×C₄	—	73	2	no	—
218	C₂×C₄	—	73	2	no	—
219	C₂×C₄	—	73	2	no	—
220	C₂×C₄	—	73	2	no	—
221	C₂×C₄	—	73	2	no	—
222	C₂×C₄	—	146	1	no	—
223	C₂×C₄	—	146	1	no	—
224	C₂×C₄	C₈×#64408	1	2	yes	—
225	C₂×C₄	C₈×#64408	1	2	yes	—
226	C₂×C₄	D₄×#64408	1	1	yes	characteristic
227	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
228	C₂×C₄	C₂×#11924036	1	2	yes	—
229	C₂×C₄	#14556678	1	2	yes	—
230	C₈	—	2	4	no	—
231	C₈	—	73	8	no	—
232	C₈	—	146	4	no	—
233	C₈	—	146	8	no	—
234	C₈	—	146	4	no	—
235	C₂²	C₄⋊C₄×#64408	1	1	yes	characteristic
236	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
237	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
238	C₂²	C₄×#11924036	1	1	yes	characteristic
239	C₂²	M₁₆×#64408	1	1	yes	characteristic
240	C₂²	#17989551	1	1	yes	characteristic
241	C₂²	#17989553	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	73	4	no	—
245	C₄	—	73	2	no	—
246	C₄	—	73	2	no	—
247	C₄	—	73	2	no	—
248	C₄	—	73	2	no	—
249	C₄	—	146	2	no	—
250	C₄	#17989614	1	2	yes	—
251	C₄	#17989636	1	2	yes	—
252	C₂	C₈×#11924036	1	1	yes	characteristic
253	C₂	#22625129	1	1	yes	characteristic
254	C₂	#22625132	1	1	yes	characteristic
255	C₂	#22625136	1	1	yes	characteristic
256	C₂	C₄⋊C₈×#64408	1	1	yes	characteristic
257	C₂	#22625723	1	1	yes	characteristic
258	C₂	#22625724	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂×C₇₃	C₂×C₈²
Fitting	#336×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄²
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	#336	#64408
hypocenter	C₇₃	C₈×C₄⋊C₈
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
22. #14556658	C₈	semidirect
28. C₄×#10596833	C₈	semidirect
38. #14556520	C₈	semidirect
102. #10596833	C₄⋊C₈	semidirect
117. Dic₇₃	C₈²	semidirect
135. C₇₃	C₈×C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556658	2
C₄²×#10596833	2
C₂×C₈×#10596833	2
#22624753	2
#22609779	2
C₈×C₄⋊C₈	73