Subgroups — #34020780

Order 18688.

914 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#10596833	C₂	1	2	yes	maximal
2	#22625727	C₂	1	6	yes	maximal
3	#22625717	C₂	1	3	yes	maximal
4	#22625646	C₂	1	3	yes	maximal
5	C₄×Q₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596833	C₄	1	16	yes	complemented
7	C₂×C₄⋊#10596833	C₂²	1	6	yes	—
8	C₂×#14556646	C₂²	1	6	yes	—
9	C₂×C₄×#10596833	C₂²	1	6	yes	—
10	#17989453	C₂²	1	1	yes	characteristic
11	#17989453	C₂²	1	3	yes	—
12	#17989453	C₂²	1	6	yes	—
13	C₂×Q₈×Dic₇₃	C₄	1	1	yes	characteristic
14	C₂×Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₇₃	C₄	1	3	yes	—
16	C₄⋊C₄×Dic₇₃	C₂²	1	3	yes	—
17	C₄×C₄⋊Dic₇₃	C₄	1	3	yes	—
18	C₄²×Dic₇₃	C₂²	1	3	yes	—
19	C₂×C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
20	C₂²×#10596833	C₂³	1	2	yes	—
21	C₂²×#10596833	C₂³	1	6	yes	—
22	C₄⋊#10596833	C₂×C₄	1	24	yes	—
23	#14556646	—	4	6	no	—
24	C₄×#10596833	C₂×C₄	1	24	yes	—
25	Q₈×Dic₇₃	—	2	4	no	—
26	Q₈×Dic₇₃	C₂×C₄	1	8	yes	—
27	C₂×C₄⋊Dic₇₃	C₂×C₄	1	3	yes	—
28	C₂×C₄⋊Dic₇₃	C₂×C₄	1	3	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
31	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
33	C₂×C₄×Dic₇₃	C₂³	1	3	yes	—
34	C₂×C₄×Dic₇₃	C₂³	1	3	yes	—
35	#14556498	C₂×C₄	1	6	yes	—
36	C₂²×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄×Q₈×C₇₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	3	yes	—
39	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
40	C₂×#10596833	—	2	6	no	—
41	C₂×#10596833	—	2	2	no	—
42	C₂×#10596833	—	2	2	no	—
43	C₂×#10596833	—	2	6	no	—
44	C₂×#10596833	C₂²×C₄	1	4	yes	—
45	C₂×#10596833	C₂²×C₄	1	12	yes	—
46	C₂²×Dic₇₃	C₂²×C₄	1	3	yes	—
47	C₂²×Dic₇₃	C₂²×C₄	1	3	yes	—
48	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₇₃	C₂⁴	1	1	yes	characteristic
50	C₄⋊Dic₇₃	—	2	6	no	—
51	C₄⋊Dic₇₃	C₄²	1	12	yes	—
52	C₄×Dic₇₃	—	2	6	no	—
53	C₄×Dic₇₃	—	2	6	no	—
54	C₄×Dic₇₃	—	2	6	no	—
55	C₄×Dic₇₃	—	2	3	no	—
56	C₄×Dic₇₃	—	2	2	no	—
57	C₄×Dic₇₃	—	2	3	no	—
58	C₄×Dic₇₃	C₂²×C₄	1	12	yes	—
59	C₂×Q₈×C₇₃	C₄²	1	4	yes	—
60	C₂×Q₈×C₇₃	M₁₆	1	4	yes	—
61	C₂×Q₈×C₇₃	M₁₆	1	4	yes	—
62	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
65	C₄⋊C₄×C₇₃	—	2	6	no	—
66	C₄²×C₇₃	—	2	6	no	—
67	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
68	Q₈×C₇₃	#97	1	16	yes	complemented
69	C₂×C₄×C₇₃	—	2	1	no	—
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	—	2	3	no	—
72	C₂×C₄×C₇₃	—	2	3	no	—
73	C₂×C₄×C₇₃	—	2	3	no	—
74	C₂×C₄×C₇₃	C₂×C₄²	1	6	yes	—
75	C₂×C₄×C₇₃	C₂×M₁₆	1	6	yes	—
76	C₂×C₄×C₇₃	C₂×M₁₆	1	6	yes	—
77	C₂×C₄×C₇₃	#131	1	2	yes	—
78	C₂×Dic₇₃	—	2	1	no	—
79	C₂×Dic₇₃	—	2	1	no	—
80	C₂×Dic₇₃	—	2	3	no	—
81	C₂×Dic₇₃	—	2	3	no	—
82	C₂×Dic₇₃	—	2	3	no	—
83	C₂×Dic₇₃	—	2	3	no	—
84	C₂×Dic₇₃	—	2	3	no	—
85	C₂×Dic₇₃	C₂×C₄²	1	6	yes	—
86	C₂×Dic₇₃	#131	1	2	yes	—
87	C₂×Dic₇₃	C₂³×C₄	1	2	yes	—
88	C₂×Dic₇₃	H(5,𝔽₂)	1	1	yes	characteristic
89	C₂×Dic₇₃	2₋¹⁺⁴	1	1	yes	characteristic
90	C₂×Dic₇₃	2₋¹⁺⁴	1	2	yes	—
91	#10596833	—	2	4	no	—
92	#10596833	—	2	12	no	—
93	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
94	C₂²×C₇₃	#519	1	1	yes	characteristic
95	C₂²×C₇₃	#520	1	1	yes	characteristic
96	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
97	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
98	C₂²×C₇₃	C₂×#131	1	1	yes	characteristic
99	C₂²×C₇₃	C₂×#131	1	1	yes	characteristic
100	C₄×C₇₃	—	2	2	no	—
101	C₄×C₇₃	—	4	3	no	—
102	C₄×C₇₃	C₂×#97	1	12	yes	—
103	Dic₇₃	—	2	6	no	—
104	Dic₇₃	—	2	2	no	—
105	Dic₇₃	—	4	3	no	—
106	Dic₇₃	#519	1	2	yes	—
107	Dic₇₃	#520	1	2	yes	—
108	#20572	—	73	1	no	maximal, Sylow
109	C₂×C₇₃	#2276	1	1	yes	characteristic
110	C₂×C₇₃	C₂²×#97	1	1	yes	characteristic
111	C₂×C₇₃	C₄×#131	1	1	yes	characteristic
112	C₂×C₇₃	#2984	1	1	yes	characteristic
113	C₂×C₇₃	#2984	1	1	yes	characteristic
114	C₂×C₇₃	#2991	1	1	yes	characteristic
115	C₂×C₇₃	#2991	1	1	yes	characteristic
116	C₂×C₈×Q₈	—	73	2	no	—
117	C₄²×Q₈	—	73	1	no	—
118	#1770	—	73	6	no	—
119	#1767	—	73	3	no	—
120	#1744	—	73	3	no	—
121	C₇₃	#20572	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄×Q₈	—	73	1	no	—
123	C₂×C₄×Q₈	—	73	1	no	—
124	C₂×C₄×Q₈	#64408	1	1	yes	characteristic
125	C₈×Q₈	—	73	16	no	—
126	C₂×C₄⋊C₈	—	73	6	no	—
127	C₂×#97	—	73	6	no	—
128	C₂×C₄×C₈	—	73	6	no	—
129	C₄×C₄⋊C₄	—	73	3	no	—
130	C₄×C₄⋊C₄	—	73	3	no	—
131	C₄×C₄⋊C₄	—	73	3	no	—
132	C₄³	—	73	3	no	—
133	#336	—	73	1	no	—
134	#336	—	73	3	no	—
135	#336	—	73	6	no	—
136	C₂²×Q₈	C₂×#64408	1	1	yes	characteristic
137	C₂²×C₈	—	73	2	no	—
138	C₂²×C₈	—	73	6	no	—
139	C₄×Q₈	—	2	4	no	—
140	C₄×Q₈	—	73	8	no	—
141	C₄×Q₈	—	146	4	no	—
142	C₂×C₄⋊C₄	—	73	3	no	—
143	C₂×C₄⋊C₄	—	73	3	no	—
144	C₂×C₄⋊C₄	C₂×#64408	1	3	yes	—
145	C₂×C₄²	—	73	3	no	—
146	C₂×C₄²	—	73	3	no	—
147	C₂×C₄²	—	73	1	no	—
148	C₂×C₄²	—	73	3	no	—
149	C₂×C₄²	—	73	3	no	—
150	C₂×C₄²	—	73	3	no	—
151	C₂×C₄²	C₂×#64408	1	3	yes	—
152	C₄⋊C₈	—	73	24	no	—
153	#97	—	292	6	no	—
154	C₄×C₈	—	73	24	no	—
155	#95	—	73	6	no	—
156	C₂×Q₈	C₄×#64408	1	4	yes	—
157	C₂×Q₈	#11924037	1	4	yes	—
158	C₂×Q₈	#11924038	1	4	yes	—
159	C₂²×C₄	—	73	1	no	—
160	C₂²×C₄	—	73	3	no	—
161	C₂²×C₄	—	73	1	no	—
162	C₂²×C₄	—	73	3	no	—
163	C₂²×C₄	C₂²×#64408	1	3	yes	—
164	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
165	C₂²×C₄	C₂²×#64408	1	3	yes	—
166	C₂×C₈	—	73	4	no	—
167	C₂×C₈	—	73	12	no	—
168	C₂×C₈	—	146	6	no	—
169	C₂×C₈	—	146	2	no	—
170	C₂×C₈	—	146	6	no	—
171	C₂×C₈	—	146	2	no	—
172	C₄⋊C₄	—	2	6	no	—
173	C₄⋊C₄	—	73	12	no	—
174	C₄⋊C₄	—	146	6	no	—
175	C₄²	—	2	6	no	—
176	C₄²	—	73	12	no	—
177	C₄²	—	146	3	no	—
178	C₄²	—	146	2	no	—
179	C₄²	—	146	3	no	—
180	C₄²	—	146	6	no	—
181	C₄²	—	146	6	no	—
182	C₄²	—	146	6	no	—
183	C₂³	C₂³×#64408	1	1	yes	characteristic
184	Q₈	#14556646	1	16	yes	complemented
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	3	no	—
188	C₂×C₄	—	2	3	no	—
189	C₂×C₄	—	2	3	no	—
190	C₂×C₄	—	73	2	no	—
191	C₂×C₄	—	73	2	no	—
192	C₂×C₄	—	73	1	no	—
193	C₂×C₄	—	73	1	no	—
194	C₂×C₄	—	73	6	no	—
195	C₂×C₄	—	73	2	no	—
196	C₂×C₄	—	146	3	no	—
197	C₂×C₄	—	146	3	no	—
198	C₂×C₄	—	146	3	no	—
199	C₂×C₄	—	146	1	no	—
200	C₂×C₄	—	146	1	no	—
201	C₂×C₄	—	146	3	no	—
202	C₂×C₄	—	146	3	no	—
203	C₂×C₄	C₂×C₄×#64408	1	6	yes	—
204	C₂×C₄	C₂×#11924037	1	6	yes	—
205	C₂×C₄	C₂×#11924038	1	6	yes	—
206	C₂×C₄	#14556679	1	2	yes	—
207	C₈	—	146	12	no	—
208	C₈	—	146	4	no	—
209	C₂²	C₂²×C₄×#64408	1	1	yes	characteristic
210	C₂²	#17989655	1	1	yes	characteristic
211	C₂²	#17989656	1	1	yes	characteristic
212	C₂²	C₂²×#11924037	1	1	yes	characteristic
213	C₂²	C₂²×#11924038	1	1	yes	characteristic
214	C₂²	D₇₃⋊#131	1	1	yes	characteristic
215	C₂²	C₂×#14556679	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	4	3	no	—
218	C₄	—	292	3	no	—
219	C₄	—	73	2	no	—
220	C₄	—	73	2	no	—
221	C₄	—	146	2	no	—
222	C₄	—	146	6	no	—
223	C₄	C₂×#14556646	1	12	yes	—
224	C₂	#22626371	1	1	yes	characteristic
225	C₂	C₂²×#14556646	1	1	yes	characteristic
226	C₂	C₄×#14556679	1	1	yes	characteristic
227	C₂	#22626833	1	1	yes	characteristic
228	C₂	#22626834	1	1	yes	characteristic
229	C₂	#22626863	1	1	yes	characteristic
230	C₂	#22626865	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×#64408
commutator	C₂²×C₇₃	C₂²×C₄²
Fitting	C₂×C₄×Q₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂³×C₄
cosocle	C₂²×Dic₇₃	C₂⁴
hypercenter	C₂×C₄×Q₈	#64408
hypocenter	C₇₃	#20572
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
6. Q₈×#10596833	C₄	semidirect
68. Q₈×C₇₃	#97	semidirect
121. C₇₃	#20572	semidirect
184. Q₈	#14556646	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#10596833	2
#22625727	2
#22625717	2
#22625646	2
C₄×Q₈×Dic₇₃	2
#20572	73