Subgroups — #34003927

Order 18688.

502 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#10596833	C₂	1	1	yes	maximal, characteristic
2	C₄²×#10596833	C₂	1	1	yes	maximal, characteristic
3	C₂×#17989428	C₂	1	2	yes	maximal
4	#22624753	C₂	1	2	yes	maximal
5	C₄⋊C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596833	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	#17989428	—	2	4	no	—
14	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
16	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
17	#17988341	C₄	1	2	yes	—
18	C₂×C₄⋊C₈×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	C₄⋊#10596833	—	2	4	no	—
23	C₄×#10596833	—	2	4	no	—
24	C₄×#10596833	—	2	4	no	—
25	C₄×#10596833	—	2	2	no	—
26	C₄×#10596833	—	2	1	no	—
27	C₄×#10596833	—	2	2	no	—
28	C₄×#10596833	—	2	1	no	—
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	C₄⋊#10596831	C₈	1	4	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	C₄⋊C₈×C₇₃	C₈	1	4	yes	complemented
44	C₂×#10596833	—	2	2	no	—
45	C₂×#10596833	—	2	2	no	—
46	C₂×#10596833	—	2	2	no	—
47	C₂×#10596833	—	2	2	no	—
48	C₂×#10596833	—	2	4	no	—
49	C₂×#10596833	—	2	4	no	—
50	C₂×#10596833	—	2	4	no	—
51	C₂×#10596833	C₄⋊C₄	1	4	yes	—
52	C₂×#10596833	M₁₆	1	4	yes	—
53	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
54	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
55	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×#10596831	—	2	2	no	—
58	C₂×#10596831	—	2	2	no	—
59	C₂×#10596831	C₂×C₈	1	4	yes	—
60	C₄×Dic₇₃	—	2	1	no	—
61	C₄×Dic₇₃	—	2	1	no	—
62	C₄×Dic₇₃	—	2	2	no	—
63	C₄×Dic₇₃	—	2	2	no	—
64	C₄×Dic₇₃	M₁₆	1	4	yes	—
65	C₄×Dic₇₃	M₁₆	1	4	yes	—
66	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
67	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
68	C₄×Dic₇₃	#41	1	1	yes	characteristic
69	C₄×Dic₇₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₃	—	2	2	no	—
74	C₂×C₈×C₇₃	—	2	2	no	—
75	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
76	C₄²×C₇₃	C₂×C₈	1	2	yes	—
77	C₄²×C₇₃	C₂×C₈	1	2	yes	—
78	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₇₃	—	2	1	no	—
80	C₂×C₄×C₇₃	—	2	1	no	—
81	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
82	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
83	C₂×C₄×C₇₃	#97	1	2	yes	—
84	C₂×C₄×C₇₃	#97	1	2	yes	—
85	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
90	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
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	#97	1	2	yes	—
96	C₂×Dic₇₃	#97	1	2	yes	—
97	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×Dic₇₃	#117	1	1	yes	characteristic
99	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
100	C₂×Dic₇₃	C₄×Q₈	1	1	yes	characteristic
101	C₂×Dic₇₃	C₂×M₁₆	1	2	yes	—
102	C₂×Dic₇₃	C₂×M₁₆	1	2	yes	—
103	C₂×Dic₇₃	C₂×M₁₆	1	2	yes	—
104	C₂×Dic₇₃	#131	1	2	yes	—
105	#10596833	—	2	4	no	—
106	#10596833	—	2	4	no	—
107	#10596833	—	4	4	no	—
108	C₈×C₇₃	—	2	4	no	—
109	#10596831	—	2	4	no	—
110	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
112	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
113	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
114	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
115	C₂²×C₇₃	#405	1	1	yes	characteristic
116	C₂²×C₇₃	#405	1	1	yes	characteristic
117	C₄×C₇₃	—	2	2	no	—
118	C₄×C₇₃	#322	1	4	yes	complemented
119	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
120	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
121	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
122	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
123	Dic₇₃	—	2	2	no	—
124	Dic₇₃	—	2	2	no	—
125	Dic₇₃	#423	1	2	yes	—
126	Dic₇₃	#432	1	2	yes	—
127	Dic₇₃	#436	1	2	yes	—
128	Dic₇₃	#446	1	2	yes	—
129	#8261	—	73	1	no	maximal, Sylow
130	C₂×C₇₃	C₂×#322	1	1	yes	characteristic
131	C₂×C₇₃	C₈×M₁₆	1	1	yes	characteristic
132	C₂×C₇₃	#1451	1	1	yes	characteristic
133	C₂×C₇₃	#1452	1	1	yes	characteristic
134	C₂×C₇₃	#1767	1	1	yes	characteristic
135	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₇₃	#1770	1	1	yes	characteristic
137	C₄×C₄⋊C₈	—	73	1	no	—
138	C₄×C₄⋊C₈	—	73	1	no	—
139	C₄²×C₈	—	73	1	no	—
140	C₂×#322	—	73	2	no	—
141	#1273	—	73	2	no	—
142	C₇₃	#8261	1	1	yes	characteristic, complemented, Sylow
143	C₂×C₄⋊C₈	—	73	1	no	—
144	C₂×C₄⋊C₈	—	73	2	no	—
145	C₂×C₄⋊C₈	#64408	1	1	yes	characteristic
146	C₂×C₄×C₈	—	73	1	no	—
147	C₂×C₄×C₈	—	73	2	no	—
148	C₂×C₄×C₈	—	73	2	no	—
149	C₂×C₄×C₈	—	73	2	no	—
150	C₂×C₄×C₈	—	73	1	no	—
151	C₂×C₄×C₈	—	73	2	no	—
152	C₄³	—	73	1	no	—
153	#336	—	73	2	no	—
154	#336	—	73	2	no	—
155	#322	—	146	4	no	—
156	C₂²×C₈	—	73	2	no	—
157	C₂²×C₈	—	73	2	no	—
158	C₂²×C₈	—	73	2	no	—
159	C₂²×C₈	—	73	4	no	—
160	C₂²×C₈	C₂×#64408	1	2	yes	—
161	C₂×C₄²	—	73	1	no	—
162	C₂×C₄²	—	73	1	no	—
163	C₂×C₄²	—	73	1	no	—
164	C₂×C₄²	—	73	1	no	—
165	C₂×C₄²	—	73	1	no	—
166	C₂×C₄²	—	73	1	no	—
167	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
168	C₄⋊C₈	—	73	4	no	—
169	C₄⋊C₈	—	146	4	no	—
170	C₄⋊C₈	#10596833	1	4	yes	complemented
171	C₄×C₈	—	73	2	no	—
172	C₄×C₈	—	73	2	no	—
173	C₄×C₈	—	146	2	no	—
174	C₄×C₈	—	146	2	no	—
175	C₄×C₈	—	146	2	no	—
176	C₄×C₈	—	146	2	no	—
177	C₄×C₈	—	146	1	no	—
178	C₄×C₈	—	146	4	no	—
179	C₄×C₈	—	146	1	no	—
180	C₄×C₈	—	146	4	no	—
181	C₂²×C₄	—	73	1	no	—
182	C₂²×C₄	—	73	1	no	—
183	C₂²×C₄	—	73	1	no	—
184	C₂²×C₄	—	73	1	no	—
185	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
186	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
187	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	73	4	no	—
191	C₂×C₈	—	73	4	no	—
192	C₂×C₈	—	73	4	no	—
193	C₂×C₈	—	146	4	no	—
194	C₂×C₈	—	146	4	no	—
195	C₂×C₈	—	146	4	no	—
196	C₂×C₈	—	146	2	no	—
197	C₂×C₈	—	146	2	no	—
198	C₂×C₈	—	146	2	no	—
199	C₂×C₈	—	146	2	no	—
200	C₂×C₈	—	146	2	no	—
201	C₂×C₈	—	146	2	no	—
202	C₂×C₈	C₂×#10596833	1	4	yes	—
203	C₄²	—	73	1	no	—
204	C₄²	—	73	1	no	—
205	C₄²	—	73	4	no	—
206	C₄²	—	73	4	no	—
207	C₄²	—	73	1	no	—
208	C₄²	—	73	1	no	—
209	C₄²	—	146	2	no	—
210	C₄²	—	146	2	no	—
211	C₄²	—	146	1	no	—
212	C₄²	—	146	1	no	—
213	C₄²	C₂×#10596833	1	2	yes	—
214	C₄²	#11924036	1	2	yes	—
215	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	73	1	no	—
219	C₂×C₄	—	73	1	no	—
220	C₂×C₄	—	73	1	no	—
221	C₂×C₄	—	73	1	no	—
222	C₂×C₄	—	73	2	no	—
223	C₂×C₄	—	73	2	no	—
224	C₂×C₄	—	73	2	no	—
225	C₂×C₄	—	73	2	no	—
226	C₂×C₄	—	73	2	no	—
227	C₂×C₄	—	73	2	no	—
228	C₂×C₄	—	146	1	no	—
229	C₂×C₄	—	146	1	no	—
230	C₂×C₄	—	146	1	no	—
231	C₂×C₄	—	146	1	no	—
232	C₂×C₄	C₄×#10596833	1	2	yes	—
233	C₂×C₄	C₄×#10596833	1	2	yes	—
234	C₂×C₄	#14556645	1	2	yes	—
235	C₂×C₄	#14556645	1	2	yes	—
236	C₂×C₄	D₄×#64408	1	1	yes	characteristic
237	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
238	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
239	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
240	C₂×C₄	#14556678	1	1	yes	characteristic
241	C₂×C₄	#14556679	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	—	292	4	no	—
244	C₈	—	146	4	no	—
245	C₈	—	146	4	no	—
246	C₈	—	146	4	no	—
247	C₂²	C₄⋊C₄×#64408	1	1	yes	characteristic
248	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
249	C₂²	C₂×#14556645	1	1	yes	characteristic
250	C₂²	C₂×#14556645	1	1	yes	characteristic
251	C₂²	M₁₆×#64408	1	1	yes	characteristic
252	C₂²	#17989552	1	1	yes	characteristic
253	C₂²	#17989553	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	73	2	no	—
256	C₄	—	73	2	no	—
257	C₄	—	73	2	no	—
258	C₄	—	73	2	no	—
259	C₄	—	146	2	no	—
260	C₄	—	146	2	no	—
261	C₄	#17989428	1	4	yes	complemented
262	C₄	D₄×#10596833	1	1	yes	characteristic
263	C₄	#17989615	1	1	yes	characteristic
264	C₄	Q₈×#10596833	1	1	yes	characteristic
265	C₄	#17989636	1	1	yes	characteristic
266	C₂	C₂×#17989428	1	1	yes	characteristic
267	C₂	M₁₆×#10596833	1	1	yes	characteristic
268	C₂	#22625143	1	1	yes	characteristic
269	C₂	#22625147	1	1	yes	characteristic
270	C₂	#22625718	1	1	yes	characteristic
271	C₂	C₄⋊C₄×#10596833	1	1	yes	characteristic
272	C₂	#22625728	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₂×C₄⋊C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄²
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×C₄⋊C₈	#64408
hypocenter	C₇₃	#8261
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
38. C₄⋊#10596831	C₈	semidirect
43. C₄⋊C₈×C₇₃	C₈	semidirect
118. C₄×C₇₃	#322	semidirect
142. C₇₃	#8261	semidirect
170. C₄⋊C₈	#10596833	semidirect
261. C₄	#17989428	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10596833	2
C₄²×#10596833	2
C₂×#17989428	2
#22624753	2
C₄⋊C₈×Dic₇₃	2
#8261	73