Subgroups — #34007640

Order 18688.

502 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626100	C₂	1	2	yes	maximal
2	C₂×#17989450	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#10596833	C₂	1	2	yes	maximal
4	C₂×C₈×#10596833	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
7	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
9	#17989453	—	2	4	no	—
10	#17989450	—	2	2	no	—
11	Q₈⋊#10596833	—	2	4	no	—
12	C₈×#10596833	—	2	2	no	—
13	C₂×Q₈×Dic₇₃	C₂²	1	2	yes	—
14	Q₁₆×Dic₇₃	—	2	2	no	—
15	Q₁₆×Dic₇₃	—	2	2	no	—
16	C₂×#14556525	C₄	1	1	yes	characteristic
17	C₂×#14556515	C₄	1	2	yes	—
18	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
19	C₂²×Q₁₆×C₇₃	C₄	1	1	yes	characteristic
20	C₂²×#10596833	—	2	2	no	—
21	C₂²×#10596833	—	2	1	no	—
22	C₂²×#10596833	D₄	1	2	yes	—
23	C₄⋊#10596833	—	2	2	no	—
24	C₄⋊#10596833	—	2	2	no	—
25	C₄×#10596833	—	2	1	no	—
26	C₄×#10596833	—	2	1	no	—
27	C₄×#10596833	—	2	1	no	—
28	C₄×#10596833	D₄	1	2	yes	—
29	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
30	Q₈×Dic₇₃	—	2	2	no	—
31	Q₈×Dic₇₃	—	2	2	no	—
32	Q₈×Dic₇₃	—	2	2	no	—
33	Q₈×Dic₇₃	—	2	2	no	—
34	C₂×C₄⋊Dic₇₃	—	2	2	no	—
35	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₇₃	—	2	2	no	—
37	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
38	#14556525	—	2	2	no	—
39	#14556515	—	2	4	no	—
40	C₈×Dic₇₃	D₄	1	2	yes	—
41	C₈×Dic₇₃	D₄	1	2	yes	—
42	C₂²×Q₈×C₇₃	C₂×C₄	1	2	yes	—
43	C₂×Q₁₆×C₇₃	—	2	2	no	—
44	C₂×Q₁₆×C₇₃	—	2	2	no	—
45	C₂×Q₁₆×C₇₃	C₈	1	4	yes	complemented
46	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×#10596833	—	2	4	no	—
48	C₂×#10596833	—	2	2	no	—
49	C₂×#10596833	—	2	2	no	—
50	C₂×#10596833	—	2	2	no	—
51	C₂×#10596833	—	4	2	no	—
52	C₂×#10596833	—	4	4	no	—
53	C₂×#10596833	Q₁₆	1	4	yes	complemented
54	C₂²×Dic₇₃	—	2	2	no	—
55	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₇₃	#41	1	1	yes	characteristic
57	C₂×#10596831	—	2	1	no	—
58	C₂×#10596831	—	2	1	no	—
59	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
60	C₄⋊Dic₇₃	—	2	2	no	—
61	C₄⋊Dic₇₃	—	2	2	no	—
62	C₄⋊Dic₇₃	—	4	2	no	—
63	C₄⋊Dic₇₃	—	4	2	no	—
64	C₄×Dic₇₃	—	4	2	no	—
65	C₄×Dic₇₃	—	4	2	no	—
66	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
67	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
68	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
69	C₄×Dic₇₃	#41	1	1	yes	characteristic
70	C₂×Q₈×C₇₃	—	2	2	no	—
71	C₂×Q₈×C₇₃	—	2	2	no	—
72	C₂×Q₈×C₇₃	—	2	2	no	—
73	C₂×Q₈×C₇₃	—	2	2	no	—
74	C₂×Q₈×C₇₃	—	2	2	no	—
75	C₂×Q₈×C₇₃	C₂×C₈	1	4	yes	—
76	C₂²×C₄×C₇₃	—	2	2	no	—
77	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
78	Q₁₆×C₇₃	—	2	8	no	—
79	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
81	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
82	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
83	Q₈×C₇₃	—	2	4	no	—
84	Q₈×C₇₃	—	2	4	no	—
85	Q₈×C₇₃	—	4	2	no	—
86	Q₈×C₇₃	—	4	2	no	—
87	C₂×C₄×C₇₃	—	2	4	no	—
88	C₂×C₄×C₇₃	—	4	2	no	—
89	C₂×C₄×C₇₃	—	4	2	no	—
90	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
95	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
96	C₂×Dic₇₃	—	2	2	no	—
97	C₂×Dic₇₃	—	4	2	no	—
98	C₂×Dic₇₃	—	4	4	no	—
99	C₂×Dic₇₃	#121	1	1	yes	characteristic
100	C₂×Dic₇₃	#121	1	1	yes	characteristic
101	C₂×Dic₇₃	#124	1	1	yes	characteristic
102	C₂×Dic₇₃	#127	1	1	yes	characteristic
103	C₂×Dic₇₃	C₂×Q₁₆	1	2	yes	—
104	C₂×Dic₇₃	#135	1	2	yes	—
105	#10596833	—	2	4	no	—
106	#10596833	—	2	4	no	—
107	#10596833	—	8	2	no	—
108	C₈×C₇₃	C₂²⋊C₈	1	4	yes	—
109	#10596831	—	2	2	no	—
110	C₂²×C₇₃	#390	1	1	yes	characteristic
111	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
112	C₂²×C₇₃	#436	1	1	yes	characteristic
113	C₂²×C₇₃	C₄×Q₁₆	1	2	yes	—
114	C₂²×C₇₃	#443	1	2	yes	—
115	C₄×C₇₃	—	4	4	no	—
116	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₄×C₇₃	#408	1	1	yes	characteristic
118	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
119	C₄×C₇₃	#436	1	1	yes	characteristic
120	Dic₇₃	—	2	1	no	—
121	Dic₇₃	—	2	1	no	—
122	Dic₇₃	—	8	2	no	—
123	Dic₇₃	#467	1	2	yes	—
124	Dic₇₃	#494	1	1	yes	characteristic
125	Dic₇₃	#495	1	1	yes	characteristic
126	#9656	—	73	1	no	maximal, Sylow
127	C₂×C₇₃	C₈×Q₁₆	1	2	yes	—
128	C₂×C₇₃	#1591	1	2	yes	—
129	C₂×C₇₃	#1954	1	1	yes	characteristic
130	C₂×C₇₃	#1969	1	1	yes	characteristic
131	C₂×C₇₃	#1974	1	1	yes	characteristic
132	C₂×C₄×Q₁₆	—	73	1	no	—
133	#1984	—	73	2	no	—
134	C₂×#335	—	73	1	no	—
135	C₂×Q₈⋊C₈	—	73	2	no	—
136	C₂×C₈²	—	73	1	no	—
137	C₇₃	#9656	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₁₆	#64408	1	1	yes	characteristic
139	C₂×C₄×Q₈	—	73	2	no	—
140	C₄×Q₁₆	—	146	2	no	—
141	C₄×Q₁₆	—	146	2	no	—
142	C₂×#107	—	73	1	no	—
143	C₂×C₄⋊C₈	—	73	2	no	—
144	C₂×#103	—	73	2	no	—
145	C₂×C₄×C₈	—	73	1	no	—
146	C₂×C₄×C₈	—	73	1	no	—
147	C₂×C₄×C₈	—	73	1	no	—
148	#336	—	146	4	no	—
149	#335	—	146	2	no	—
150	Q₈⋊C₈	—	146	4	no	—
151	C₈²	—	146	2	no	—
152	C₂²×Q₈	C₂×#64408	1	2	yes	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	2	2	no	—
155	C₂×Q₁₆	#10596833	1	4	yes	complemented
156	C₂²×C₈	—	73	2	no	—
157	C₂²×C₈	—	73	1	no	—
158	C₂²×C₈	—	146	2	no	—
159	C₂²×C₈	—	146	1	no	—
160	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
161	C₄×Q₈	—	146	2	no	—
162	C₄×Q₈	—	146	2	no	—
163	C₄×Q₈	—	146	2	no	—
164	C₄×Q₈	—	146	2	no	—
165	C₂×C₄⋊C₄	—	73	2	no	—
166	C₂×C₄⋊C₄	—	146	2	no	—
167	C₂×C₄²	—	73	1	no	—
168	C₂×C₄²	—	146	2	no	—
169	#107	—	146	2	no	—
170	C₄⋊C₈	—	146	2	no	—
171	C₄⋊C₈	—	146	2	no	—
172	#103	—	146	4	no	—
173	C₄×C₈	—	73	2	no	—
174	C₄×C₈	—	73	2	no	—
175	C₄×C₈	—	73	2	no	—
176	C₄×C₈	—	146	1	no	—
177	C₄×C₈	—	146	1	no	—
178	C₄×C₈	—	146	1	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	C₂×#10596833	1	4	yes	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	73	1	no	—
187	C₂²×C₄	—	73	1	no	—
188	C₂²×C₄	—	146	2	no	—
189	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
190	Q₁₆	—	2	8	no	—
191	C₂×C₈	—	292	2	no	—
192	C₂×C₈	—	292	4	no	—
193	C₂×C₈	—	73	4	no	—
194	C₂×C₈	—	73	2	no	—
195	C₂×C₈	—	146	2	no	—
196	C₂×C₈	—	146	2	no	—
197	C₂×C₈	—	146	2	no	—
198	C₂×C₈	—	146	4	no	—
199	C₂×C₈	—	146	1	no	—
200	C₂×C₈	—	146	1	no	—
201	C₂×C₈	C₂²⋊#64408	1	2	yes	—
202	C₂×C₈	C₂×#10596833	1	2	yes	—
203	C₂×C₈	#11924037	1	2	yes	—
204	C₄⋊C₄	—	292	2	no	—
205	C₄⋊C₄	—	292	2	no	—
206	C₄⋊C₄	—	146	2	no	—
207	C₄⋊C₄	—	146	2	no	—
208	C₄²	—	292	2	no	—
209	C₄²	—	292	2	no	—
210	C₄²	—	73	1	no	—
211	C₄²	—	73	1	no	—
212	C₄²	—	73	1	no	—
213	C₄²	—	73	1	no	—
214	C₂³	D₄×#64408	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	2	4	no	—
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	292	2	no	—
223	C₂×C₄	—	292	4	no	—
224	C₂×C₄	—	73	1	no	—
225	C₂×C₄	—	73	1	no	—
226	C₂×C₄	—	73	2	no	—
227	C₂×C₄	—	73	2	no	—
228	C₂×C₄	—	73	1	no	—
229	C₂×C₄	—	73	1	no	—
230	C₂×C₄	—	146	2	no	—
231	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
232	C₂×C₄	D₄×#64408	1	1	yes	characteristic
233	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
234	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
235	C₂×C₄	#14556678	1	1	yes	characteristic
236	C₂×C₄	#14556678	1	1	yes	characteristic
237	C₈	—	584	2	no	—
238	C₈	—	146	4	no	—
239	C₈	—	146	4	no	—
240	C₈	—	146	2	no	—
241	C₈	C₂²⋊#10596833	1	4	yes	—
242	C₂²	#17989539	1	1	yes	characteristic
243	C₂²	D₄×#10596833	1	1	yes	characteristic
244	C₂²	#17989622	1	1	yes	characteristic
245	C₂²	Q₁₆×#64408	1	2	yes	—
246	C₂²	#17989631	1	2	yes	—
247	C₄	—	4	4	no	—
248	C₄	—	584	2	no	—
249	C₄	—	73	1	no	—
250	C₄	—	73	2	no	—
251	C₄	—	73	1	no	—
252	C₄	—	146	1	no	—
253	C₄	—	146	1	no	—
254	C₄	C₂×C₂²⋊#10596833	1	1	yes	characteristic
255	C₄	#17989561	1	1	yes	characteristic
256	C₄	D₄×#10596833	1	1	yes	characteristic
257	C₄	#17989622	1	1	yes	characteristic
258	C₂	Q₁₆×#10596833	1	2	yes	—
259	C₂	#22625571	1	2	yes	—
260	C₂	#22626052	1	1	yes	characteristic
261	C₂	#22626077	1	1	yes	characteristic
262	C₂	#22626082	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂²×Q₁₆×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂²×Q₁₆	#64408
hypocenter	C₇₃	#9656
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
45. C₂×Q₁₆×C₇₃	C₈	semidirect
53. C₂×#10596833	Q₁₆	semidirect
137. C₇₃	#9656	semidirect
155. C₂×Q₁₆	#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22626100	2
C₂×#17989450	2
C₂×Q₈⋊#10596833	2
C₂×C₈×#10596833	2
C₂×Q₁₆×Dic₇₃	2
#9656	73