Subgroups — C₂×C₇₃×#1694 #32928552

Order 18688.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#462×C₇₃	C₂	1	2	yes	maximal
3	C₂×#446×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1694	C₂	1	8	yes	maximal, direct product factor
5	C₂×#334×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#332×C₇₃	C₂	1	2	yes	maximal
7	C₂²×Q₁₆×C₇₃	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
10	#500×C₇₃	—	2	2	no	—
11	#500×C₇₃	C₂²	1	4	yes	—
12	#462×C₇₃	—	2	4	no	—
13	#462×C₇₃	C₂²	1	8	yes	—
14	#446×C₇₃	—	2	2	no	—
15	#446×C₇₃	C₂²	1	4	yes	—
16	C₂×#107×C₇₃	—	2	1	no	—
17	C₂×#107×C₇₃	—	2	1	no	—
18	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
19	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
21	C₂×#103×C₇₃	—	2	2	no	—
22	C₂×#97×C₇₃	—	2	1	no	—
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	#334×C₇₃	C₂²	1	4	yes	—
25	#332×C₇₃	C₂²	1	8	yes	—
26	C₂²×Q₈×C₇₃	—	2	2	no	—
27	C₂²×Q₈×C₇₃	D₄	1	1	yes	characteristic
28	C₂×Q₁₆×C₇₃	—	2	8	no	—
29	C₂×Q₁₆×C₇₃	—	4	8	no	—
30	C₂²×C₈×C₇₃	—	2	1	no	—
31	C₂²×C₈×C₇₃	—	2	2	no	—
32	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₇₃	—	2	2	no	—
35	C₄⋊Q₈×C₇₃	—	2	2	no	—
36	C₄⋊Q₈×C₇₃	—	2	2	no	—
37	C₄⋊Q₈×C₇₃	C₂³	1	4	yes	—
38	C₄×Q₈×C₇₃	—	2	1	no	—
39	C₄×Q₈×C₇₃	—	2	1	no	—
40	C₄×Q₈×C₇₃	D₄	1	2	yes	—
41	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₃	—	2	1	no	—
47	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	#107×C₇₃	—	2	4	no	—
49	#107×C₇₃	—	2	4	no	—
50	C₄⋊C₈×C₇₃	—	2	4	no	—
51	C₄⋊C₈×C₇₃	—	2	1	no	—
52	C₄⋊C₈×C₇₃	—	2	2	no	—
53	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
54	C₄⋊C₈×C₇₃	C₂³	1	4	yes	—
55	#103×C₇₃	—	2	8	no	—
56	#97×C₇₃	—	2	4	no	—
57	C₄×C₈×C₇₃	D₄	1	2	yes	—
58	C₄×C₈×C₇₃	C₂³	1	2	yes	—
59	C₂×Q₈×C₇₃	—	2	1	no	—
60	C₂×Q₈×C₇₃	—	2	4	no	—
61	C₂×Q₈×C₇₃	—	2	4	no	—
62	C₂×Q₈×C₇₃	—	4	2	no	—
63	C₂×Q₈×C₇₃	—	4	4	no	—
64	C₂×Q₈×C₇₃	—	4	4	no	—
65	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₇₃	—	2	1	no	—
67	C₂²×C₄×C₇₃	—	2	1	no	—
68	C₂²×C₄×C₇₃	—	4	2	no	—
69	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
72	Q₁₆×C₇₃	—	4	16	no	—
73	C₂×C₈×C₇₃	—	2	2	no	—
74	C₂×C₈×C₇₃	—	2	2	no	—
75	C₂×C₈×C₇₃	—	2	4	no	—
76	C₂×C₈×C₇₃	—	4	4	no	—
77	C₂×C₈×C₇₃	—	4	2	no	—
78	C₂×C₈×C₇₃	Q₁₆	1	4	yes	complemented
79	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
81	C₄⋊C₄×C₇₃	—	2	2	no	—
82	C₄⋊C₄×C₇₃	—	2	1	no	—
83	C₄⋊C₄×C₇₃	—	2	2	no	—
84	C₄⋊C₄×C₇₃	—	2	4	no	—
85	C₄⋊C₄×C₇₃	—	2	4	no	—
86	C₄⋊C₄×C₇₃	—	4	2	no	—
87	C₄⋊C₄×C₇₃	—	4	2	no	—
88	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
89	C₄²×C₇₃	—	2	2	no	—
90	C₄²×C₇₃	—	2	2	no	—
91	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₇₃	#41	1	1	yes	characteristic
94	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
95	C₂³×C₇₃	#121	1	1	yes	characteristic
96	Q₈×C₇₃	—	2	4	no	—
97	Q₈×C₇₃	—	4	2	no	—
98	Q₈×C₇₃	—	4	4	no	—
99	Q₈×C₇₃	—	4	4	no	—
100	Q₈×C₇₃	—	8	4	no	—
101	C₂×C₄×C₇₃	—	2	2	no	—
102	C₂×C₄×C₇₃	—	2	2	no	—
103	C₂×C₄×C₇₃	—	2	2	no	—
104	C₂×C₄×C₇₃	—	4	2	no	—
105	C₂×C₄×C₇₃	—	4	2	no	—
106	C₂×C₄×C₇₃	—	4	4	no	—
107	C₂×C₄×C₇₃	—	8	4	no	—
108	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
112	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
113	C₂×C₄×C₇₃	#137	1	2	yes	—
114	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
117	C₈×C₇₃	—	2	2	no	—
118	C₈×C₇₃	—	4	4	no	—
119	C₈×C₇₃	—	4	2	no	—
120	C₈×C₇₃	C₂×Q₁₆	1	4	yes	complemented
121	C₂²×C₇₃	#462	1	2	yes	—
122	C₂²×C₇₃	#469	1	2	yes	—
123	C₂²×C₇₃	#473	1	2	yes	—
124	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
125	C₄×C₇₃	—	2	1	no	—
126	C₄×C₇₃	—	2	1	no	—
127	C₄×C₇₃	—	4	2	no	—
128	C₄×C₇₃	—	4	2	no	—
129	C₄×C₇₃	—	8	4	no	—
130	C₄×C₇₃	#462	1	2	yes	—
131	C₄×C₇₃	#469	1	2	yes	—
132	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
133	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
134	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
135	C₂×#1694	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₇₃	#1694	1	4	yes	direct product factor
137	C₂×C₇₃	C₂×#462	1	1	yes	characteristic
138	C₂×C₇₃	C₂×#469	1	1	yes	characteristic
139	C₂×C₇₃	C₂×#473	1	1	yes	characteristic
140	C₂×#500	C₂×C₇₃	1	1	yes	characteristic
141	C₂×#462	C₂×C₇₃	1	2	yes	—
142	C₂×#446	C₂×C₇₃	1	1	yes	characteristic
143	#1694	C₂×C₇₃	1	8	yes	direct product factor
144	C₂×#334	C₂×C₇₃	1	1	yes	characteristic
145	C₂×#332	C₂×C₇₃	1	2	yes	—
146	C₇₃	C₂×#1694	1	1	yes	characteristic, direct product factor, Sylow
147	C₂²×Q₁₆	—	2	2	no	—
148	C₂×C₄⋊Q₈	C₂²×C₇₃	1	2	yes	—
149	C₂×C₄×Q₈	C₂²×C₇₃	1	1	yes	characteristic
150	#500	—	2	2	no	—
151	#500	C₂²×C₇₃	1	4	yes	—
152	#462	—	2	4	no	—
153	#462	C₂²×C₇₃	1	8	yes	—
154	#446	—	2	2	no	—
155	#446	C₂²×C₇₃	1	4	yes	—
156	C₂×#107	—	2	1	no	—
157	C₂×#107	—	2	1	no	—
158	C₂×C₄⋊C₈	—	2	1	no	—
159	C₂×C₄⋊C₈	C₂²×C₇₃	1	1	yes	characteristic
160	C₂×C₄⋊C₈	C₂²×C₇₃	1	2	yes	—
161	C₂×#103	—	2	2	no	—
162	C₂×#97	—	2	1	no	—
163	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
164	#334	C₂²×C₇₃	1	4	yes	—
165	#332	C₂²×C₇₃	1	8	yes	—
166	C₂²×Q₈	—	2	2	no	—
167	C₂²×Q₈	D₄×C₇₃	1	1	yes	characteristic
168	C₂×Q₁₆	—	2	8	no	—
169	C₂×Q₁₆	—	4	8	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
173	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	—	2	2	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	C₂³×C₇₃	1	4	yes	—
178	C₄×Q₈	—	2	1	no	—
179	C₄×Q₈	—	2	1	no	—
180	C₄×Q₈	D₄×C₇₃	1	2	yes	—
181	C₄×Q₈	C₂³×C₇₃	1	2	yes	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	D₄×C₇₃	1	1	yes	characteristic
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
188	#107	—	2	4	no	—
189	#107	—	2	4	no	—
190	C₄⋊C₈	—	2	4	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	C₂³×C₇₃	1	2	yes	—
194	C₄⋊C₈	C₂³×C₇₃	1	4	yes	—
195	#103	—	2	8	no	—
196	#97	—	2	4	no	—
197	C₄×C₈	D₄×C₇₃	1	2	yes	—
198	C₄×C₈	C₂³×C₇₃	1	2	yes	—
199	C₂×Q₈	—	2	1	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×Q₈	—	4	4	no	—
205	C₂×Q₈	C₂×D₄×C₇₃	1	2	yes	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
211	C₂²×C₄	#41×C₇₃	1	1	yes	characteristic
212	Q₁₆	—	4	16	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	4	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	Q₁₆×C₇₃	1	4	yes	complemented
219	C₂×C₈	C₂×D₄×C₇₃	1	2	yes	—
220	C₂×C₈	C₂×D₄×C₇₃	1	2	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	C₂×D₄×C₇₃	1	2	yes	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
232	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
233	C₄²	#41×C₇₃	1	1	yes	characteristic
234	C₄²	C₂⁴×C₇₃	1	1	yes	characteristic
235	C₂³	#121×C₇₃	1	1	yes	characteristic
236	Q₈	—	2	4	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	4	4	no	—
239	Q₈	—	4	4	no	—
240	Q₈	—	8	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	#121×C₇₃	1	1	yes	characteristic
249	C₂×C₄	#121×C₇₃	1	1	yes	characteristic
250	C₂×C₄	#121×C₇₃	1	1	yes	characteristic
251	C₂×C₄	C₂×Q₁₆×C₇₃	1	2	yes	—
252	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	4	yes	—
253	C₂×C₄	#137×C₇₃	1	2	yes	—
254	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
256	C₂×C₄	C₂×#41×C₇₃	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	4	4	no	—
259	C₈	—	4	2	no	—
260	C₈	C₂×Q₁₆×C₇₃	1	4	yes	complemented
261	C₂²	#462×C₇₃	1	2	yes	—
262	C₂²	#469×C₇₃	1	2	yes	—
263	C₂²	#473×C₇₃	1	2	yes	—
264	C₂²	C₂×#121×C₇₃	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	8	4	no	—
270	C₄	#462×C₇₃	1	2	yes	—
271	C₄	#469×C₇₃	1	2	yes	—
272	C₄	C₂²×Q₁₆×C₇₃	1	1	yes	characteristic
273	C₄	C₂×AGL(1,ℤ/8)×C₇₃	1	2	yes	—
274	C₄	C₂×#137×C₇₃	1	1	yes	characteristic
275	C₂	C₇₃×#1694	1	4	yes	direct product factor
276	C₂	C₂×#462×C₇₃	1	1	yes	characteristic
277	C₂	C₂×#469×C₇₃	1	1	yes	characteristic
278	C₂	C₂×#473×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₃	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇₃×#1694	C₂	direct (N × H)
78. C₂×C₈×C₇₃	Q₁₆	semidirect
120. C₈×C₇₃	C₂×Q₁₆	semidirect
135. C₂×#1694	C₇₃	direct (N × H)
136. C₂×C₇₃	#1694	direct (N × H)
143. #1694	C₂×C₇₃	direct (N × H)
146. C₇₃	C₂×#1694	direct (N × H)
218. C₂×C₈	Q₁₆×C₇₃	semidirect
260. C₈	C₂×Q₁₆×C₇₃	semidirect
275. C₂	C₇₃×#1694	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#500×C₇₃	2
C₂×#462×C₇₃	2
C₂×#446×C₇₃	2
C₇₃×#1694	2
C₂×#334×C₇₃	2
C₂×#332×C₇₃	2
C₂×#1694	73

Subgroups — C2×C73×#1694 #32928552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₃×#1694 #32928552