Subgroups — #34029824

Order 18688.

716 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626900	C₂	1	2	yes	maximal
2	#22626154	C₂	1	2	yes	maximal, complemented
3	#22626153	C₂	1	4	yes	maximal, complemented
4	#22626148	C₂	1	4	yes	maximal, complemented
5	C₂×#17989479	C₂	1	2	yes	maximal, complemented
6	#22618360	C₂	1	1	yes	maximal, characteristic
7	#17989646	C₄	1	16	yes	complemented
8	#17989643	C₂²	1	8	yes	—
9	C₂×#14556665	C₂²	1	4	yes	—
10	#17989480	C₂²	1	8	yes	—
11	#17989479	C₂²	1	4	yes	—
12	#17989479	C₂²	1	4	yes	—
13	C₂×#14556599	C₄	1	1	yes	characteristic
14	C₂×#14556599	C₂²	1	1	yes	characteristic
15	C₄×#11924028	C₄	1	1	yes	characteristic
16	#17988545	C₄	1	1	yes	characteristic
17	#17988545	C₂²	1	1	yes	characteristic
18	#17988541	C₄	1	2	yes	—
19	#17988541	C₂²	1	2	yes	—
20	#17988539	C₄	1	2	yes	—
21	C₂×#14556502	C₂²	1	1	yes	characteristic
22	#97×D₇₃	C₂²	1	2	yes	—
23	C₄×M₁₆×C₇₃	C₄	1	1	yes	characteristic
24	#14556666	C₂×C₄	1	16	yes	complemented
25	#14556665	C₂×C₄	1	8	yes	complemented
26	#14556665	C₂³	1	8	yes	—
27	#14556664	C₂×C₄	1	16	yes	complemented
28	C₂×#11924031	C₂×C₄	1	8	yes	complemented
29	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
30	#14556599	—	2	4	no	—
31	#14556599	C₂×C₄	1	8	yes	—
32	C₂×#11924011	C₂×C₄	1	2	yes	—
33	C₂×#11924011	C₂×C₄	1	2	yes	—
34	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
35	C₂²×#10596831	C₂³	1	1	yes	characteristic
36	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₇₃	C₂³	1	2	yes	—
38	C₄×Dic₁₄₆	—	2	1	no	—
39	C₄×D₂₉₂	—	2	1	no	—
40	C₄×C₂²⋊D₇₃	—	2	2	no	—
41	#14556550	C₂×C₄	1	1	yes	characteristic
42	#14556548	C₂×C₄	1	2	yes	—
43	C₄²×D₇₃	C₂×C₄	1	2	yes	—
44	#14556502	C₂×C₄	1	1	yes	characteristic
45	#14556502	C₂×C₄	1	1	yes	characteristic
46	#14556502	C₂³	1	1	yes	characteristic
47	#14556502	C₂³	1	1	yes	characteristic
48	#14556501	C₂×C₄	1	2	yes	—
49	#14556501	C₂³	1	2	yes	—
50	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
51	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
55	#97×C₇₃	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
57	#11924031	C₂²×C₄	1	16	yes	—
58	#11924028	C₄²	1	4	yes	—
59	#11924028	M₁₆	1	4	yes	—
60	C₂×Dic₁₄₆	M₁₆	1	1	yes	characteristic
61	C₂×D₂₉₂	M₁₆	1	1	yes	characteristic
62	C₂×C₂²⋊D₇₃	M₁₆	1	2	yes	—
63	C₂×C₄×D₇₃	—	2	2	no	—
64	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
65	#11924011	—	2	4	no	—
66	#11924011	C₄²	1	8	yes	—
67	C₂×#10596831	—	2	2	no	—
68	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
69	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
70	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
71	C₂×#10596831	C₂²×C₄	1	4	yes	—
72	C₂×#10596831	C₂⁴	1	1	yes	characteristic
73	C₈×D₇₃	—	2	4	no	—
74	C₈×D₇₃	C₂²×C₄	1	8	yes	—
75	C₄⋊Dic₇₃	—	2	1	no	—
76	#11924007	—	2	2	no	—
77	C₂²⋊Dic₇₃	—	2	1	no	—
78	#11924005	—	2	2	no	—
79	C₄×Dic₇₃	—	2	2	no	—
80	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇₃	—	2	1	no	—
82	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	M₁₆×C₇₃	—	2	2	no	—
84	M₁₆×C₇₃	C₄²	1	4	yes	—
85	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₇₃	—	2	1	no	—
88	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂²×D₇₃	C₂×M₁₆	1	2	yes	—
92	C₂×C₄×C₇₃	—	2	1	no	—
93	C₂×C₄×C₇₃	—	2	1	no	—
94	C₂×C₄×C₇₃	—	8	1	no	—
95	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
96	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
99	C₂²⋊D₇₃	#97	1	8	yes	—
100	C₂×Dic₇₃	—	2	2	no	—
101	C₂×Dic₇₃	C₂×M₁₆	1	2	yes	—
102	D₂₉₂	#97	1	4	yes	—
103	C₄×D₇₃	—	8	2	no	—
104	C₄×D₇₃	C₂×C₄²	1	4	yes	—
105	C₄×D₇₃	C₂×M₁₆	1	4	yes	—
106	Dic₁₄₆	#97	1	4	yes	—
107	C₈×C₇₃	—	2	2	no	—
108	C₈×C₇₃	C₂×C₄²	1	4	yes	—
109	#10596831	—	2	1	no	—
110	#10596831	—	2	1	no	—
111	#10596831	C₂×C₄²	1	2	yes	—
112	#10596831	C₂³×C₄	1	2	yes	—
113	C₂²×C₇₃	—	4	1	no	—
114	C₂²×C₇₃	C₂×#97	1	2	yes	—
115	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
116	C₂×D₇₃	—	4	2	no	—
117	C₂×D₇₃	C₂×#97	1	4	yes	—
118	C₄×C₇₃	—	4	1	no	—
119	C₄×C₇₃	—	4	1	no	—
120	C₄×C₇₃	C₂×#97	1	2	yes	—
121	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
122	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
123	Dic₇₃	—	4	2	no	—
124	Dic₇₃	C₂×#97	1	4	yes	—
125	#33329	—	73	1	no	maximal, Sylow
126	C₂×C₇₃	—	2	1	no	—
127	C₂×C₇₃	—	2	2	no	—
128	C₂×C₇₃	C₂²×#97	1	1	yes	characteristic
129	D₇₃	—	2	4	no	—
130	#3407	—	73	2	no	—
131	#2875	—	73	1	no	—
132	#2110	—	73	4	no	—
133	#2110	—	73	2	no	—
134	C₂×#347	—	73	2	no	—
135	C₂×#347	—	73	4	no	—
136	C₇₃	#33329	1	1	yes	characteristic, complemented, Sylow
137	C₂×#131	—	73	1	no	—
138	C₂×#131	—	73	1	no	—
139	C₄×#41	—	73	1	no	—
140	#504	—	73	16	no	—
141	C₂×M₃₂	—	73	4	no	—
142	C₂×M₃₂	—	73	8	no	—
143	#405	—	73	1	no	—
144	#405	—	73	1	no	—
145	#405	—	73	2	no	—
146	#405	—	73	2	no	—
147	C₄×M₁₆	—	73	2	no	—
148	C₄×M₁₆	#64408	1	1	yes	characteristic
149	C₂×#97	—	73	1	no	—
150	C₂×#97	—	73	2	no	—
151	#347	—	73	4	no	—
152	#347	—	73	4	no	—
153	#347	—	73	8	no	—
154	C₂×#41	—	73	1	no	—
155	#131	—	73	8	no	—
156	#131	—	146	4	no	—
157	C₂×M₁₆	—	73	2	no	—
158	C₂×M₁₆	—	73	2	no	—
159	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
160	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
161	C₂²×C₈	—	73	1	no	—
162	C₂²×C₈	—	73	2	no	—
163	C₂²×C₈	—	73	1	no	—
164	C₂²×C₈	—	73	2	no	—
165	C₄×Q₈	—	146	1	no	—
166	C₄×D₄	—	146	1	no	—
167	C₄×D₄	—	146	2	no	—
168	#117	—	73	1	no	—
169	#117	—	73	2	no	—
170	C₂×C₄²	—	73	2	no	—
171	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
172	M₃₂	—	73	8	no	—
173	M₃₂	—	73	8	no	—
174	M₃₂	—	73	16	no	—
175	C₂×C₁₆	—	73	8	no	—
176	C₂×C₁₆	—	73	16	no	—
177	#97	—	73	2	no	—
178	#97	—	73	2	no	—
179	#97	—	73	1	no	—
180	#97	—	73	1	no	—
181	#97	—	73	1	no	—
182	#97	—	73	1	no	—
183	#97	C₂×#64408	1	2	yes	—
184	C₄×C₈	—	73	2	no	—
185	C₄×C₈	—	73	2	no	—
186	C₄×C₈	C₂×#64408	1	2	yes	—
187	#41	—	73	4	no	—
188	#41	—	73	4	no	—
189	C₂×Q₈	—	73	1	no	—
190	C₂×D₄	—	73	2	no	—
191	C₂×D₄	—	73	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	73	2	no	—
194	C₂²×C₄	—	146	2	no	—
195	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
196	M₁₆	—	2	2	no	—
197	M₁₆	—	73	8	no	—
198	M₁₆	—	146	4	no	—
199	M₁₆	C₄×#64408	1	4	yes	—
200	C₂×C₈	—	73	1	no	—
201	C₂×C₈	—	73	1	no	—
202	C₂×C₈	—	73	4	no	—
203	C₂×C₈	—	73	8	no	—
204	C₂×C₈	—	73	1	no	—
205	C₂×C₈	—	73	1	no	—
206	C₂×C₈	—	146	4	no	—
207	C₂×C₈	—	146	2	no	—
208	C₂×C₈	C₂²×#64408	1	2	yes	—
209	C₂×C₈	C₂²×#64408	1	2	yes	—
210	C₄⋊C₄	—	146	2	no	—
211	C₄⋊C₄	—	146	1	no	—
212	C₂²⋊C₄	—	146	2	no	—
213	C₂²⋊C₄	—	146	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	73	2	no	—
216	C₄²	—	146	2	no	—
217	C₄²	C₂²×#64408	1	1	yes	characteristic
218	C₄²	C₂²×#64408	1	1	yes	characteristic
219	C₁₆	—	73	16	no	—
220	C₂³	—	73	2	no	—
221	C₂³	D₇₃⋊M₁₆	1	1	yes	characteristic
222	Q₈	—	73	4	no	—
223	D₄	—	73	8	no	—
224	D₄	—	73	4	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	8	1	no	—
228	C₂×C₄	—	584	2	no	—
229	C₂×C₄	—	73	2	no	—
230	C₂×C₄	—	73	4	no	—
231	C₂×C₄	—	73	4	no	—
232	C₂×C₄	—	146	2	no	—
233	C₂×C₄	C₂×C₄×#64408	1	2	yes	—
234	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
235	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
236	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	73	2	no	—
239	C₈	—	73	2	no	—
240	C₈	—	146	1	no	—
241	C₈	—	146	1	no	—
242	C₈	C₂×C₄×#64408	1	4	yes	—
243	C₂²	—	4	1	no	—
244	C₂²	—	292	2	no	—
245	C₂²	—	73	4	no	—
246	C₂²	#17989546	1	2	yes	—
247	C₂²	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
248	C₄	—	4	1	no	—
249	C₄	—	4	1	no	—
250	C₄	—	292	2	no	—
251	C₄	—	73	4	no	—
252	C₄	#17989546	1	2	yes	—
253	C₄	C₂²×C₄×#64408	1	1	yes	characteristic
254	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	2	no	—
257	C₂	—	146	4	no	—
258	C₂	C₂×#17989546	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×#64408
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₄×M₁₆×C₇₃	C₄
socle	C₂×C₇₃	C₂²×#97
prime radical	C₂×#11924028	C₂×C₄
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₄×M₁₆	#64408
hypocenter	C₇₃	#33329
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #22626154	C₂	semidirect
3. #22626153	C₂	semidirect
4. #22626148	C₂	semidirect
5. C₂×#17989479	C₂	semidirect
7. #17989646	C₄	semidirect
24. #14556666	C₂×C₄	semidirect
25. #14556665	C₂×C₄	semidirect
27. #14556664	C₂×C₄	semidirect
28. C₂×#11924031	C₂×C₄	semidirect
136. C₇₃	#33329	semidirect

Maximal subgroups

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

subgroup	index
#22626900	2
#22626154	2
#22626153	2
#22626148	2
C₂×#17989479	2
#22618360	2
#33329	73