Subgroups — #347×#64408 #34012086

Order 18688.

418 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626148	C₂	1	1	yes	maximal, characteristic
2	#22626146	C₂	1	2	yes	maximal
3	M₃₂×#64408	C₂	1	2	yes	maximal
4	#97×#64408	C₂	1	1	yes	maximal, characteristic
5	#347×D₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989643	C₂²	1	2	yes	—
7	#17989546	C₄	1	1	yes	characteristic
8	C₂×#14556645	C₄	1	1	yes	characteristic
9	C₂×#14556645	C₂²	1	1	yes	characteristic
10	C₂×C₈×#64408	C₄	1	1	yes	characteristic
11	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
12	C₄²×#64408	C₄	1	1	yes	characteristic
13	#17989480	C₄	1	2	yes	complemented
14	#17989479	C₄	1	2	yes	complemented
15	#17989478	C₄	1	4	yes	complemented
16	#17989477	C₄	1	4	yes	complemented
17	#17989474	C₄	1	4	yes	complemented
18	C₁₆×#64408	C₄	1	4	yes	complemented
19	M₃₂×D₇₃	C₂²	1	2	yes	—
20	#97×D₇₃	C₂²	1	1	yes	characteristic
21	#17988365	C₄	1	1	yes	characteristic, complemented
22	#17988364	C₄	1	2	yes	complemented
23	#347×C₇₃	C₄	1	1	yes	characteristic, complemented
24	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
25	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×#64408	—	2	2	no	—
27	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
29	#14556666	C₂×C₄	1	4	yes	—
30	#14556665	C₂×C₄	1	4	yes	—
31	#14556664	C₂×C₄	1	4	yes	—
32	#14556646	—	2	2	no	—
33	#14556645	—	2	2	no	—
34	#14556645	C₂×C₄	1	2	yes	—
35	#14556645	C₂×C₄	1	2	yes	—
36	C₈×#64408	—	2	2	no	—
37	C₈×#64408	C₂×C₄	1	2	yes	—
38	C₈×#64408	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
41	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
42	#14556531	C₂×C₄	1	2	yes	—
43	#14556530	C₂×C₄	1	4	yes	—
44	C₁₆×D₇₃	C₂×C₄	1	4	yes	—
45	#14556502	C₂×C₄	1	1	yes	characteristic
46	#14556501	C₂×C₄	1	1	yes	characteristic
47	#14556501	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₇₃	C₂×C₄	1	2	yes	—
49	#97×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×#64408	—	2	1	no	—
51	C₂²×#64408	M₁₆	1	2	yes	—
52	#11924036	—	2	1	no	—
53	#11924036	C₄²	1	1	yes	characteristic
54	#11924036	C₄²	1	1	yes	characteristic
55	#11924036	C₄²	1	2	yes	—
56	#11924036	M₁₆	1	2	yes	—
57	C₂×#10596833	—	2	1	no	—
58	C₂×#10596833	M₁₆	1	2	yes	—
59	C₄×#64408	—	2	1	no	—
60	C₄×#64408	—	4	4	no	—
61	C₄×#64408	C₄²	1	2	yes	—
62	C₄×#64408	C₄²	1	1	yes	characteristic
63	C₄×#64408	C₄²	1	1	yes	characteristic
64	C₄×#64408	M₁₆	1	2	yes	—
65	#11924031	C₄²	1	8	yes	complemented
66	C₂×C₄×D₇₃	—	2	1	no	—
67	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×#10596831	C₄²	1	1	yes	characteristic
69	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
70	C₈×D₇₃	—	2	1	no	—
71	C₈×D₇₃	—	2	1	no	—
72	C₈×D₇₃	C₂²×C₄	1	2	yes	—
73	C₈×D₇₃	C₂²×C₄	1	2	yes	—
74	C₄×Dic₇₃	—	2	1	no	—
75	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
76	#11924003	C₄²	1	4	yes	complemented
77	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
78	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₇₃	C₄²	1	1	yes	characteristic
80	C₁₆×C₇₃	C₄²	1	4	yes	complemented
81	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×#64408	—	2	4	no	—
83	C₂×#64408	—	2	2	no	—
84	C₂×#64408	—	4	2	no	—
85	C₂×#64408	#97	1	2	yes	—
86	C₂×#64408	#97	1	2	yes	—
87	C₂×C₄×C₇₃	—	2	1	no	—
88	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
89	C₂×Dic₇₃	—	2	1	no	—
90	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
91	C₄×D₇₃	—	4	2	no	—
92	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
93	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
94	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
95	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
96	#10596833	—	2	1	no	—
97	#10596833	—	2	1	no	—
98	#10596833	#97	1	4	yes	—
99	C₈×C₇₃	—	2	1	no	—
100	C₈×C₇₃	C₂×C₄²	1	2	yes	—
101	#10596831	—	2	1	no	—
102	#10596831	C₂×C₄²	1	2	yes	—
103	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
104	C₂×D₇₃	—	2	2	no	—
105	C₂×D₇₃	C₂×#97	1	1	yes	characteristic
106	C₂×D₇₃	C₄×M₁₆	1	1	yes	characteristic
107	#64408	—	2	2	no	—
108	#64408	—	4	2	no	—
109	#64408	#347	1	4	yes	direct product factor
110	C₄×C₇₃	—	4	1	no	—
111	C₄×C₇₃	C₄³	1	1	yes	characteristic
112	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
113	Dic₇₃	—	4	1	no	—
114	Dic₇₃	C₂×#97	1	1	yes	characteristic
115	Dic₇₃	C₄×M₁₆	1	1	yes	characteristic
116	C₄×#347	—	73	1	no	maximal, Sylow
117	C₂×C₇₃	—	2	1	no	—
118	C₂×C₇₃	C₄×#97	1	1	yes	characteristic
119	D₇₃	—	2	1	no	—
120	D₇₃	C₂×#347	1	2	yes	—
121	C₂×#347	—	73	2	no	—
122	C₂×#347	—	73	1	no	—
123	C₂×#347	—	73	1	no	—
124	C₄×M₃₂	—	73	2	no	—
125	C₄×#97	—	73	1	no	—
126	C₇₃	C₄×#347	1	1	yes	characteristic, complemented, Sylow
127	C₂×M₃₂	—	73	2	no	—
128	C₂×M₃₂	—	73	2	no	—
129	C₂×#97	—	73	1	no	—
130	C₂×#97	—	73	1	no	—
131	C₂×#97	—	73	1	no	—
132	C₂×#97	—	73	1	no	—
133	C₂×C₄×C₈	—	73	1	no	—
134	C₂×C₄×C₈	—	73	1	no	—
135	C₄³	—	73	1	no	—
136	#347	—	73	4	no	—
137	#347	—	73	4	no	—
138	#347	—	73	2	no	—
139	#347	—	73	2	no	—
140	#347	—	73	2	no	—
141	#347	—	73	1	no	—
142	#347	#64408	1	1	yes	characteristic, direct product factor
143	#346	—	73	4	no	—
144	C₄×C₁₆	—	73	4	no	—
145	C₂²×C₈	—	73	1	no	—
146	C₂²×C₈	—	73	1	no	—
147	C₂²×C₈	—	73	1	no	—
148	C₂²×C₈	—	73	1	no	—
149	C₂×C₄²	—	73	1	no	—
150	C₂×C₄²	—	73	1	no	—
151	C₂×C₄²	—	73	1	no	—
152	C₂×C₄²	—	146	2	no	—
153	M₃₂	—	73	4	no	—
154	M₃₂	—	73	4	no	—
155	M₃₂	—	73	2	no	—
156	M₃₂	—	73	4	no	—
157	M₃₂	C₂×#64408	1	2	yes	—
158	C₂×C₁₆	—	73	4	no	—
159	C₂×C₁₆	—	73	4	no	—
160	#97	—	73	2	no	—
161	#97	—	73	2	no	—
162	#97	—	73	1	no	—
163	#97	—	73	1	no	—
164	#97	—	73	1	no	—
165	#97	—	146	2	no	—
166	#97	—	146	2	no	—
167	#97	C₂×#64408	1	1	yes	characteristic
168	C₄×C₈	—	73	2	no	—
169	C₄×C₈	—	73	2	no	—
170	C₄×C₈	—	146	2	no	—
171	C₂²×C₄	—	73	1	no	—
172	C₂²×C₄	—	73	2	no	—
173	C₂²×C₄	—	146	1	no	—
174	C₂²×C₄	—	146	1	no	—
175	C₂×C₈	—	73	2	no	—
176	C₂×C₈	—	73	2	no	—
177	C₂×C₈	—	73	2	no	—
178	C₂×C₈	—	73	1	no	—
179	C₂×C₈	—	73	2	no	—
180	C₂×C₈	—	73	2	no	—
181	C₂×C₈	—	73	1	no	—
182	C₂×C₈	—	73	1	no	—
183	C₂×C₈	—	73	1	no	—
184	C₂×C₈	—	146	1	no	—
185	C₂×C₈	—	146	1	no	—
186	C₂×C₈	—	146	1	no	—
187	C₂×C₈	—	146	1	no	—
188	C₂×C₈	C₄×#64408	1	1	yes	characteristic
189	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
190	C₄²	—	292	4	no	—
191	C₄²	—	73	1	no	—
192	C₄²	—	73	1	no	—
193	C₄²	—	73	2	no	—
194	C₄²	—	73	2	no	—
195	C₄²	—	73	1	no	—
196	C₄²	—	146	1	no	—
197	C₄²	—	146	1	no	—
198	C₄²	C₄×#64408	1	1	yes	characteristic
199	C₁₆	—	73	8	no	—
200	C₁₆	—	73	4	no	—
201	C₁₆	C₄×#64408	1	4	yes	complemented
202	C₂³	—	73	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	292	2	no	—
205	C₂×C₄	—	292	2	no	—
206	C₂×C₄	—	73	2	no	—
207	C₂×C₄	—	73	1	no	—
208	C₂×C₄	—	73	1	no	—
209	C₂×C₄	—	73	1	no	—
210	C₂×C₄	—	73	1	no	—
211	C₂×C₄	—	73	1	no	—
212	C₂×C₄	—	73	2	no	—
213	C₂×C₄	—	146	2	no	—
214	C₂×C₄	—	146	1	no	—
215	C₂×C₄	—	146	4	no	—
216	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	73	4	no	—
219	C₈	—	73	2	no	—
220	C₈	—	146	1	no	—
221	C₈	—	146	1	no	—
222	C₈	—	146	1	no	—
223	C₈	C₂×C₄×#64408	1	2	yes	—
224	C₂²	—	73	1	no	—
225	C₂²	—	73	1	no	—
226	C₂²	—	146	2	no	—
227	C₂²	M₁₆×#64408	1	1	yes	characteristic
228	C₄	—	4	1	no	—
229	C₄	—	292	1	no	—
230	C₄	—	292	2	no	—
231	C₄	—	73	4	no	—
232	C₄	—	73	1	no	—
233	C₄	—	73	1	no	—
234	C₄	—	146	2	no	—
235	C₄	C₄²×#64408	1	1	yes	characteristic
236	C₄	M₁₆×#64408	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	73	2	no	—
239	C₂	—	146	1	no	—
240	C₂	#97×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×#64408
commutator	C₄×C₇₃	C₄³
Frattini	C₂×C₈	C₂²×#64408
Fitting	#347×C₇₃	C₄
socle	C₂×C₇₃	C₄×#97
prime radical	C₂²×D₇₃	C₂×M₁₆
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	#347	#64408
hypocenter	C₇₃	C₄×#347
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ #347

Complements

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

N	H ≅ G/N	type
13. #17989480	C₄	semidirect
14. #17989479	C₄	semidirect
15. #17989478	C₄	semidirect
16. #17989477	C₄	semidirect
17. #17989474	C₄	semidirect
18. C₁₆×#64408	C₄	semidirect
21. #17988365	C₄	semidirect
22. #17988364	C₄	semidirect
23. #347×C₇₃	C₄	semidirect
65. #11924031	C₄²	semidirect
76. #11924003	C₄²	semidirect
80. C₁₆×C₇₃	C₄²	semidirect
109. #64408	#347	direct (N × H)
126. C₇₃	C₄×#347	semidirect
142. #347	#64408	direct (N × H)
201. C₁₆	C₄×#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626148	2
#22626146	2
M₃₂×#64408	2
#97×#64408	2
#347×D₇₃	2
C₄×#347	73