Subgroups — #34012106

Order 18688.

726 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556664	C₂	1	2	yes	maximal
2	#22624970	C₂	1	2	yes	maximal, complemented
3	#22624969	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556664	—	2	4	no	—
6	C₂×#14556664	—	2	4	no	—
7	C₂×#14556664	C₄	1	8	yes	complemented
8	C₂²×#11924031	C₂²	1	4	yes	—
9	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×D₇₃	C₄	1	1	yes	characteristic
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₇₃	—	2	4	no	—
16	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
17	#14556664	—	2	16	no	—
18	C₂×#11924031	—	2	4	no	—
19	C₂×#11924031	—	2	4	no	—
20	C₂×#11924031	C₂×C₄	1	8	yes	complemented
21	C₂²×C₄×D₇₃	C₈	1	2	yes	—
22	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
24	C₂²×#10596831	C₂³	1	1	yes	characteristic
25	C₂×C₈×D₇₃	—	2	2	no	—
26	C₂×C₈×D₇₃	—	2	2	no	—
27	C₂×C₈×D₇₃	C₈	1	4	yes	—
28	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
29	C₂×C₈×D₇₃	D₄	1	4	yes	—
30	C₂×C₈×D₇₃	D₄	1	2	yes	—
31	C₂×C₈×D₇₃	Q₈	1	2	yes	—
32	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
33	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₇₃	—	2	2	no	—
35	C₄²×D₇₃	—	2	2	no	—
36	C₄×#10596831	—	2	2	no	—
37	C₈×Dic₇₃	—	2	2	no	—
38	C₈×Dic₇₃	—	2	2	no	—
39	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₇₃	—	2	2	no	—
43	#11924031	—	2	8	no	—
44	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
46	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×D₇₃	—	2	4	no	—
48	C₂×C₄×D₇₃	—	2	4	no	—
49	C₂×C₄×D₇₃	C₁₆	1	8	yes	complemented
50	C₂×C₄×D₇₃	C₄²	1	2	yes	—
51	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
54	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
55	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
56	C₂×#10596831	—	2	1	no	—
57	C₂×#10596831	—	2	1	no	—
58	C₂×#10596831	C₂×C₈	1	2	yes	—
59	C₂×#10596831	C₂²×C₄	1	2	yes	—
60	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
61	C₂×#10596831	C₂×D₄	1	2	yes	—
62	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
63	C₈×D₇₃	—	2	8	no	—
64	C₈×D₇₃	C₂²⋊C₄	1	8	yes	—
65	C₈×D₇₃	C₄⋊C₄	1	8	yes	—
66	C₄×Dic₇₃	—	2	1	no	—
67	C₄×Dic₇₃	—	2	1	no	—
68	C₄×Dic₇₃	—	2	4	no	—
69	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₃	—	2	1	no	—
72	C₂×C₈×C₇₃	—	2	1	no	—
73	C₂×C₈×C₇₃	C₄²	1	2	yes	—
74	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
77	C₄²×C₇₃	—	2	1	no	—
78	C₄²×C₇₃	—	2	1	no	—
79	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
80	C₂²×D₇₃	C₄×C₈	1	2	yes	—
81	C₂²×D₇₃	C₂²⋊C₈	1	2	yes	—
82	C₂²×D₇₃	C₄⋊C₈	1	2	yes	—
83	C₂²×D₇₃	C₂×C₁₆	1	4	yes	—
84	C₂²×D₇₃	M₃₂	1	4	yes	—
85	C₂×C₄×C₇₃	—	2	2	no	—
86	C₂×C₄×C₇₃	—	2	2	no	—
87	C₂×C₄×C₇₃	C₂×C₁₆	1	4	yes	complemented
88	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
94	C₂×Dic₇₃	—	2	2	no	—
95	C₂×Dic₇₃	—	2	2	no	—
96	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
97	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
98	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
99	C₂×Dic₇₃	C₂×C₁₆	1	4	yes	complemented
100	C₄×D₇₃	—	2	16	no	—
101	C₄×D₇₃	#95	1	4	yes	—
102	C₄×D₇₃	#97	1	4	yes	—
103	C₄×D₇₃	C₂²⋊C₈	1	4	yes	—
104	C₄×D₇₃	C₄⋊C₈	1	4	yes	—
105	C₈×C₇₃	—	2	2	no	—
106	C₈×C₇₃	#95	1	4	yes	—
107	#10596831	—	2	2	no	—
108	#10596831	C₂×C₂²⋊C₄	1	2	yes	—
109	#10596831	C₂×C₄⋊C₄	1	2	yes	—
110	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
111	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
112	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₂²×C₇₃	C₂²×C₁₆	1	2	yes	—
114	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
115	C₂×D₇₃	#336	1	4	yes	—
116	C₂×D₇₃	C₄×C₁₆	1	4	yes	—
117	C₂×D₇₃	#346	1	4	yes	—
118	C₂×D₇₃	C₂²⋊C₁₆	1	8	yes	—
119	C₂×D₇₃	C₄⋊C₁₆	1	8	yes	—
120	C₄×C₇₃	—	2	4	no	—
121	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
122	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
123	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
124	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
125	Dic₇₃	—	2	4	no	—
126	Dic₇₃	#336	1	4	yes	—
127	C₂×#1374	—	73	1	no	maximal, Sylow
128	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
129	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
130	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
131	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	2	yes	—
132	C₂×C₇₃	C₂×C₄⋊C₁₆	1	2	yes	—
133	D₇₃	#1374	1	8	yes	complemented
134	C₂³×C₁₆	—	73	2	no	—
135	C₂²×C₄×C₈	—	73	1	no	—
136	#1374	—	73	2	no	—
137	#1374	—	73	2	no	—
138	C₇₃	C₂×#1374	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₈	—	73	1	no	—
140	C₂³×C₈	—	73	1	no	—
141	C₂²×C₄²	—	73	1	no	—
142	C₂²×C₁₆	—	73	4	no	—
143	C₂²×C₁₆	—	73	8	no	—
144	C₂²×C₁₆	—	146	4	no	—
145	C₂²×C₁₆	—	146	4	no	—
146	C₂×C₄×C₈	—	73	1	no	—
147	C₂×C₄×C₈	—	73	1	no	—
148	C₂×C₄×C₈	—	73	1	no	—
149	C₂×C₄×C₈	—	146	4	no	—
150	C₂×C₄×C₈	#64408	1	1	yes	characteristic
151	C₂³×C₄	—	73	1	no	—
152	C₂³×C₄	—	73	2	no	—
153	C₂²×C₈	—	73	1	no	—
154	C₂²×C₈	—	73	4	no	—
155	C₂²×C₈	—	73	4	no	—
156	C₂²×C₈	—	73	2	no	—
157	C₂²×C₈	—	73	2	no	—
158	C₂²×C₈	—	73	1	no	—
159	C₂²×C₈	—	73	4	no	—
160	C₂²×C₈	—	146	2	no	—
161	C₂²×C₈	—	146	2	no	—
162	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
163	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
164	C₂×C₄²	—	73	2	no	—
165	C₂×C₄²	—	73	1	no	—
166	C₂×C₄²	—	146	2	no	—
167	C₂×C₄²	—	146	2	no	—
168	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
169	C₂×C₁₆	—	73	8	no	—
170	C₂×C₁₆	—	146	4	no	—
171	C₂×C₁₆	—	146	4	no	—
172	C₂×C₁₆	—	146	16	no	—
173	C₄×C₈	—	2	2	no	—
174	C₄×C₈	—	146	2	no	—
175	C₄×C₈	—	146	2	no	—
176	C₄×C₈	—	146	2	no	—
177	C₂⁴	—	73	1	no	—
178	C₂²×C₄	—	73	1	no	—
179	C₂²×C₄	—	73	2	no	—
180	C₂²×C₄	—	73	2	no	—
181	C₂²×C₄	—	73	2	no	—
182	C₂²×C₄	—	73	2	no	—
183	C₂²×C₄	—	73	2	no	—
184	C₂²×C₄	—	73	2	no	—
185	C₂²×C₄	—	73	2	no	—
186	C₂²×C₄	—	73	8	no	—
187	C₂²×C₄	—	146	4	no	—
188	C₂²×C₄	—	146	4	no	—
189	C₂²×C₄	#11924036	1	2	yes	—
190	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	73	2	no	—
194	C₂×C₈	—	73	2	no	—
195	C₂×C₈	—	73	1	no	—
196	C₂×C₈	—	73	1	no	—
197	C₂×C₈	—	73	8	no	—
198	C₂×C₈	—	73	8	no	—
199	C₂×C₈	—	73	2	no	—
200	C₂×C₈	—	146	8	no	—
201	C₂×C₈	—	146	1	no	—
202	C₂×C₈	—	146	1	no	—
203	C₂×C₈	C₄×#64408	1	2	yes	—
204	C₂×C₈	C₂²⋊#64408	1	2	yes	—
205	C₂×C₈	C₄⋊#64408	1	2	yes	—
206	C₂×C₈	#11924036	1	2	yes	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	146	4	no	—
210	C₄²	—	146	1	no	—
211	C₄²	—	146	1	no	—
212	C₁₆	—	146	8	no	—
213	C₂³	—	73	4	no	—
214	C₂³	—	73	4	no	—
215	C₂³	—	73	2	no	—
216	C₂³	—	73	2	no	—
217	C₂³	—	73	2	no	—
218	C₂³	C₂×#11924036	1	1	yes	characteristic
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	73	2	no	—
222	C₂×C₄	—	73	2	no	—
223	C₂×C₄	—	73	2	no	—
224	C₂×C₄	—	73	4	no	—
225	C₂×C₄	—	73	4	no	—
226	C₂×C₄	—	73	4	no	—
227	C₂×C₄	—	73	4	no	—
228	C₂×C₄	—	73	4	no	—
229	C₂×C₄	—	146	16	no	—
230	C₂×C₄	—	146	2	no	—
231	C₂×C₄	—	146	2	no	—
232	C₂×C₄	#14556664	1	4	yes	complemented
233	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
236	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
237	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
238	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	73	2	no	—
241	C₈	—	73	2	no	—
242	C₈	—	146	2	no	—
243	C₈	(C₂×#64408)⋊C₄	1	4	yes	—
244	C₂²	—	73	4	no	—
245	C₂²	—	73	4	no	—
246	C₂²	—	73	4	no	—
247	C₂²	—	73	8	no	—
248	C₂²	—	73	8	no	—
249	C₂²	C₄×#11924036	1	1	yes	characteristic
250	C₂²	#17989555	1	1	yes	characteristic
251	C₂²	#17989588	1	1	yes	characteristic
252	C₂²	C₂×#14556664	1	2	yes	—
253	C₂²	#17989643	1	2	yes	—
254	C₄	—	2	4	no	—
255	C₄	—	73	4	no	—
256	C₄	—	146	4	no	—
257	C₄	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic
258	C₄	#17989546	1	1	yes	characteristic
259	C₄	#17989555	1	1	yes	characteristic
260	C₄	#17989588	1	1	yes	characteristic
261	C₂	—	73	8	no	—
262	C₂	#22625592	1	1	yes	characteristic
263	C₂	C₄×#14556664	1	1	yes	characteristic
264	C₂	#22626132	1	1	yes	characteristic
265	C₂	#22626156	1	2	yes	—
266	C₂	#22626261	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#64408
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₂×#1374
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22624970	C₂	semidirect
3. #22624969	C₂	semidirect
7. C₂×#14556664	C₄	semidirect
20. C₂×#11924031	C₂×C₄	semidirect
49. C₂×C₄×D₇₃	C₁₆	semidirect
87. C₂×C₄×C₇₃	C₂×C₁₆	semidirect
99. C₂×Dic₇₃	C₂×C₁₆	semidirect
133. D₇₃	#1374	semidirect
138. C₇₃	C₂×#1374	semidirect
232. C₂×C₄	#14556664	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556664	2
#22624970	2
#22624969	2
C₂×C₄×C₈×D₇₃	2
C₂×#1374	73