Subgroups — #33999908

Order 18688.

320 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625846	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#10596833	C₂	1	1	yes	maximal, characteristic
3	#22609786	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10596833	—	2	1	no	—
5	C₂×C₄⋊#10596833	—	2	1	no	—
6	C₂×C₄×#10596833	C₄	1	2	yes	—
7	#17989453	—	2	1	no	—
8	#17989453	—	2	1	no	—
9	#17989453	C₄	1	2	yes	—
10	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
11	C₂×#14556501	—	2	1	no	—
12	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
13	#17988341	—	2	1	no	—
14	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
15	C₂²×#10596833	—	2	2	no	—
16	C₂²×#10596833	—	2	2	no	—
17	C₄⋊#10596833	—	2	2	no	—
18	C₄⋊#10596833	—	2	2	no	—
19	C₄⋊#10596833	—	4	1	no	—
20	C₄⋊#10596833	—	4	1	no	—
21	C₄×#10596833	—	4	2	no	—
22	C₂²×#10596831	—	2	1	no	—
23	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
26	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
28	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
29	C₄⋊#10596831	—	2	1	no	—
30	C₄⋊#10596831	C₈	1	2	yes	complemented
31	#14556501	—	4	1	no	—
32	#14556501	—	4	1	no	—
33	C₂²×C₈×C₇₃	—	2	1	no	—
34	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₈×C₇₃	—	2	1	no	—
36	C₄⋊C₈×C₇₃	C₈	1	2	yes	complemented
37	C₂×#10596833	—	2	4	no	—
38	C₂×#10596833	—	4	2	no	—
39	C₂×#10596833	—	4	2	no	—
40	C₂×#10596833	—	4	2	no	—
41	C₂×#10596833	—	4	2	no	—
42	C₂×#10596833	—	4	2	no	—
43	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×#10596831	—	2	2	no	—
48	C₂×#10596831	—	4	1	no	—
49	C₂×#10596831	—	4	1	no	—
50	C₄×Dic₇₃	—	2	1	no	—
51	C₄×Dic₇₃	—	2	1	no	—
52	C₄×Dic₇₃	—	2	1	no	—
53	C₄×Dic₇₃	—	2	1	no	—
54	C₄×Dic₇₃	—	4	1	no	—
55	C₄×Dic₇₃	—	4	1	no	—
56	C₄×Dic₇₃	—	4	1	no	—
57	C₄×Dic₇₃	D₈	1	1	yes	characteristic
58	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
59	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
60	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	—	2	2	no	—
65	C₂×C₈×C₇₃	—	4	1	no	—
66	C₂×C₈×C₇₃	—	4	1	no	—
67	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
69	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
70	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
71	C₂³×C₇₃	#95	1	1	yes	characteristic
72	C₂×C₄×C₇₃	—	2	1	no	—
73	C₂×C₄×C₇₃	—	2	1	no	—
74	C₂×C₄×C₇₃	—	2	1	no	—
75	C₂×C₄×C₇₃	—	2	1	no	—
76	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#104	1	1	yes	characteristic
83	C₂×C₄×C₇₃	#104	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
86	C₂×Dic₇₃	—	2	1	no	—
87	C₂×Dic₇₃	—	2	1	no	—
88	C₂×Dic₇₃	—	2	1	no	—
89	C₂×Dic₇₃	—	2	1	no	—
90	C₂×Dic₇₃	—	2	1	no	—
91	C₂×Dic₇₃	—	2	1	no	—
92	C₂×Dic₇₃	—	2	1	no	—
93	C₂×Dic₇₃	—	2	1	no	—
94	C₂×Dic₇₃	—	2	1	no	—
95	C₂×Dic₇₃	#100	1	1	yes	characteristic
96	C₂×Dic₇₃	#101	1	1	yes	characteristic
97	C₂×Dic₇₃	#102	1	1	yes	characteristic
98	C₂×Dic₇₃	#103	1	1	yes	characteristic
99	C₂×Dic₇₃	#106	1	1	yes	characteristic
100	C₂×Dic₇₃	#107	1	1	yes	characteristic
101	#10596833	—	4	4	no	—
102	#10596833	—	8	2	no	—
103	C₈×C₇₃	—	4	2	no	—
104	#10596831	—	4	2	no	—
105	C₂²×C₇₃	#336	1	1	yes	characteristic
106	C₂²×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
107	C₂²×C₇₃	#338	1	1	yes	characteristic
108	C₂²×C₇₃	#339	1	1	yes	characteristic
109	C₂²×C₇₃	#340	1	1	yes	characteristic
110	C₂²×C₇₃	#343	1	1	yes	characteristic
111	C₂²×C₇₃	#344	1	1	yes	characteristic
112	C₄×C₇₃	—	2	1	no	—
113	C₄×C₇₃	—	2	1	no	—
114	C₄×C₇₃	—	2	1	no	—
115	C₄×C₇₃	—	2	1	no	—
116	C₄×C₇₃	D₄⋊C₈	1	1	yes	characteristic
117	C₄×C₇₃	D₄⋊C₈	1	1	yes	characteristic
118	C₄×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
119	C₄×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
120	Dic₇₃	—	2	1	no	—
121	Dic₇₃	—	2	1	no	—
122	Dic₇₃	—	2	1	no	—
123	Dic₇₃	—	2	1	no	—
124	Dic₇₃	—	2	1	no	—
125	Dic₇₃	—	2	1	no	—
126	Dic₇₃	—	4	1	no	—
127	#7022	—	73	1	no	maximal, Sylow
128	C₂×C₇₃	#1275	1	1	yes	characteristic
129	C₂×C₇₃	#1279	1	1	yes	characteristic
130	C₂×C₇₃	#1283	1	1	yes	characteristic
131	C₂×C₇₃	#1285	1	1	yes	characteristic
132	C₂×C₇₃	#1286	1	1	yes	characteristic
133	C₂×C₇₃	#1291	1	1	yes	characteristic
134	C₂×C₇₃	#1292	1	1	yes	characteristic
135	#1831	—	73	1	no	—
136	#1767	—	73	1	no	—
137	C₄×C₄⋊C₈	—	73	1	no	—
138	C₇₃	#7022	1	1	yes	characteristic, complemented, Sylow
139	C₂×C₄⋊C₈	—	73	1	no	—
140	C₂×C₄⋊C₈	—	146	1	no	—
141	C₂×C₄⋊C₈	—	146	1	no	—
142	C₂×C₄⋊C₈	#64408	1	1	yes	characteristic
143	C₂×#97	—	146	1	no	—
144	C₂×C₄×C₈	—	73	2	no	—
145	C₄³	—	73	1	no	—
146	#336	—	73	2	no	—
147	#336	—	146	1	no	—
148	#336	—	146	1	no	—
149	#336	—	146	1	no	—
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	146	2	no	—
152	C₂²×C₈	—	146	2	no	—
153	C₂²×C₈	—	146	1	no	—
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	—	2	1	no	—
162	C₄⋊C₈	—	292	1	no	—
163	C₄⋊C₈	—	292	1	no	—
164	C₄⋊C₈	—	73	2	no	—
165	C₄⋊C₈	—	146	2	no	—
166	C₄⋊C₈	—	146	2	no	—
167	C₄⋊C₈	—	146	1	no	—
168	C₄⋊C₈	#10596833	1	2	yes	complemented
169	#97	—	292	1	no	—
170	#97	—	292	1	no	—
171	C₄×C₈	—	292	2	no	—
172	C₂²×C₄	—	73	1	no	—
173	C₂²×C₄	—	73	1	no	—
174	C₂²×C₄	—	73	1	no	—
175	C₂²×C₄	—	73	1	no	—
176	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
177	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	292	2	no	—
183	C₂×C₈	—	292	2	no	—
184	C₂×C₈	—	292	1	no	—
185	C₂×C₈	—	292	1	no	—
186	C₂×C₈	—	292	2	no	—
187	C₂×C₈	—	292	2	no	—
188	C₂×C₈	—	292	2	no	—
189	C₂×C₈	—	146	4	no	—
190	C₂×C₈	—	146	2	no	—
191	C₄²	—	292	1	no	—
192	C₄²	—	292	1	no	—
193	C₄²	—	292	1	no	—
194	C₄²	—	73	1	no	—
195	C₄²	—	73	1	no	—
196	C₄²	—	73	1	no	—
197	C₄²	—	73	1	no	—
198	C₄²	—	146	1	no	—
199	C₄²	—	146	1	no	—
200	C₄²	—	146	1	no	—
201	C₄²	—	146	1	no	—
202	C₄²	C₂×#10596833	1	1	yes	characteristic
203	C₄²	#11924036	1	1	yes	characteristic
204	C₄²	#11924037	1	1	yes	characteristic
205	C₄²	#11924038	1	1	yes	characteristic
206	C₂³	(C₂×#64408)⋊C₄	1	1	yes	characteristic
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	73	1	no	—
212	C₂×C₄	—	73	1	no	—
213	C₂×C₄	—	73	1	no	—
214	C₂×C₄	—	73	1	no	—
215	C₂×C₄	—	73	1	no	—
216	C₂×C₄	—	73	1	no	—
217	C₂×C₄	—	146	1	no	—
218	C₂×C₄	—	146	1	no	—
219	C₂×C₄	—	146	1	no	—
220	C₂×C₄	—	146	1	no	—
221	C₂×C₄	—	146	1	no	—
222	C₂×C₄	—	146	1	no	—
223	C₂×C₄	—	146	1	no	—
224	C₂×C₄	—	146	1	no	—
225	C₂×C₄	—	146	1	no	—
226	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
227	C₂×C₄	#14556646	1	1	yes	characteristic
228	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
229	C₂×C₄	#14556648	1	1	yes	characteristic
230	C₂×C₄	#14556653	1	1	yes	characteristic
231	C₂×C₄	#14556654	1	1	yes	characteristic
232	C₂×C₄	#14556655	1	1	yes	characteristic
233	C₂×C₄	#14556656	1	1	yes	characteristic
234	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
235	C₂×C₄	#14556658	1	1	yes	characteristic
236	C₈	—	4	2	no	—
237	C₈	—	292	4	no	—
238	C₈	—	292	2	no	—
239	C₈	—	584	2	no	—
240	C₂²	#17989453	1	1	yes	characteristic
241	C₂²	#17989455	1	1	yes	characteristic
242	C₂²	#17989456	1	1	yes	characteristic
243	C₂²	#17989457	1	1	yes	characteristic
244	C₂²	#17989460	1	1	yes	characteristic
245	C₂²	#17989466	1	1	yes	characteristic
246	C₂²	#17989470	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	292	1	no	—
252	C₄	—	146	1	no	—
253	C₄	—	146	1	no	—
254	C₄	—	146	1	no	—
255	C₄	—	146	1	no	—
256	C₄	—	146	1	no	—
257	C₄	—	146	1	no	—
258	C₄	D₄⋊#10596833	1	1	yes	characteristic
259	C₄	#17989435	1	1	yes	characteristic
260	C₄	Q₈⋊#10596833	1	1	yes	characteristic
261	C₄	#17989437	1	1	yes	characteristic
262	C₂	#22624757	1	1	yes	characteristic
263	C₂	#22624767	1	1	yes	characteristic
264	C₂	#22624777	1	1	yes	characteristic
265	C₂	#22624783	1	1	yes	characteristic
266	C₂	#22624786	1	1	yes	characteristic
267	C₂	#22624797	1	1	yes	characteristic
268	C₂	#22624800	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	(C₂×#64408)⋊C₄
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₂×C₄²	C₂×#64408
Fitting	C₂×C₄⋊C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	#95
cosocle	C₄²×Dic₇₃	C₂²
hypercenter	C₂×C₄⋊C₈	#64408
hypocenter	C₇₃	#7022
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
30. C₄⋊#10596831	C₈	semidirect
36. C₄⋊C₈×C₇₃	C₈	semidirect
138. C₇₃	#7022	semidirect
168. C₄⋊C₈	#10596833	semidirect

Maximal subgroups

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

subgroup	index
#22625846	2
C₄×C₄⋊#10596833	2
#22609786	2
#7022	73